MATHEMATICS, CIVILIZATION AND PROGRESS

 

Sal Restivo, (2012), MATHEMATICS, CIVILIZATION, AND PROGRESS, in

History of Mathematics, [Eds.UNESCO-EOLSS Joint Commitee], in Encyclopedia of Life Support Systems (EOLSS), Developed under the Auspices of the UNESCO, Eolss Publishers, Paris, France, [http://www.eolss.net]

 

MATHEMATICS, CIVILIZATION, AND PROGRESS

 

Sal Restivo

Department of Intercultural Communication and Interaction, University of Ghent, Ghent Belgium

 

Keywords: abstraction, arithmetic, Chinese mathematics, civilization, ethnomathematics, European mathematics, generalization, geometry, Greek mathematics, Indian mathematics, mathematical workers, mathematicians, mathematics, non-Euclidean geometry, Platonism, proofs, progress, puzzle solving, social constructionism, sociology, transcendental

 

Contents

 

1.  Prologue: Interrogating the terms of our discourse

2.  Introduction: mathematics and civilization

3.  Mathematics and Civilization: Case Studies

4.  The Social Roots of Mathematics

5.  Puzzles and Proofs

6.  Generalization (“Abstraction”) and Self-consciousness

7.  Conclusion Acknowledgement Glossary Bibliography Biographical Sketch

 

Summary

 

This chapter provides an overview of the comparative history of mathematics in the context of the ideas of civilization and progress. The narrative is guided by the idea that mathematics is a social practice and not a set of ideas revealed through some sort of heavenly or Platonic discourse. This implies that mathematics has a normative dimension in terms of its cultural and professional settings. The chapter begins with a critical review of the terms of the discourse, namely mathematics, civilization, and progress. Following some general remarks on mathematics and civilization, the reader is guided through case studies of mathematics in its civilizational and cultural contexts including mathematics in China, India, and Greece, as well as modern Europe. A more analytical section follows, summarizing the idea of the social roots of mathematics, the development and functions of puzzle solving and proofs, and generalization as a way to think about “abstraction” in concrete terms. The narrative follows an historical social science perspective that draws on the ideas of classical and contemporary social theorists from Durkheim and Spengler to Randall Collins and Sal Restivo.

 

1.  Prologue: Interrogating the Terms of Our Discourse

 

Let us begin by briefly interrogating the three terms that make up the title of this essay. First, we should ask, borrowing the title of the book by Reuben Hersh (1999): What is mathematics, really? Mathematics has been shrouded in mystery and halos for most of

its history. The reason for this is that it has seemed impossible to account for the nature and successes of mathematics without granting it some sort of transcendental status. Classically, this is most dramatically expressed in the Platonic notion of mathematics.

 

Briefly, what we call Platonism in mathematics refers to Plato’s theory of Forms. Skirting the complexities of scholarly discourse, Plato is associated with the idea that there are “Forms” or “ideals” that are transcendent and pure. These immaterial Forms exist in a realm outside of our everyday space and time. They are the pure types of the ideas and concepts we manage in our everyday world.

 

This over-simplifies Plato but is consistent with a long tradition in the history and philosophy of mathematics. Consider, for example, the way some scholars have viewed the development of non-Euclidean geometries (NEGs). The mathematician Dirk Struik (1967: 167), for example, described that development as “remarkable” in two respects. First, he claimed, the ideas emerged independently in Gȍttingen, Budapest, and Kazan; second, they emerged on the periphery of the world mathematical community (most notably in the case of Kazan and to a lesser extent Budapest). And the distinguished historian of mathematics, Carl Boyer (1968: 585) characterized the case as one of “startling…simultaneity.”

 

These reflect classical Platonic, transcendental views of mathematics. One even finds such views in the forms of the sociology of knowledge and science developed from the 1920s on in the works of Karl Mannheim and Robert K. Merton and their followers. Mannheim, for example, wrote in 1936 that 2+2 = 4 exists outside of history; and Merton championed a sociology of science that focused on the social system of science and not on scientific knowledge which he claimed lay outside of the influences of society and culture.

 

His ambivalence about this is reflected in his critical reply to G.N. Clark’s criticism of Boris Hessen’s historical materialism of Newton’s Principia (Merton, 1967: 661-663). Clark opposed Hessen’s (Marxist) political economy of the Principia with a defense of Newton’s “purely” scientific motives. Merton argued that individual motivations do not change the structural facts of the matter and in this case they support Hessen’s argument. This doesn’t reach to the core of the social construction of scientific knowledge but it does demonstrate at least an appreciation for the contextual foundations of that knowledge.

 

There are a couple of curiosities in the case of non-Euclidean geometry (NEGs). Even a cursory review of the facts reveals that NEGs have a history that begins already with Euclid’s earliest commentators, runs over the centuries through names like Saccheri, Lambert, Klügel, and Legendre, and culminates in the works of Lobachevsky (1793- 1856), Reimann (1826-1866), and J. Bolyai (1802-1860). The concerns over Euclid’s parallels postulate moved geometers eventually to the systematic development of NEGs. The issue was that the parallels postulate, the fifth postulate in Euclid’s system, did not possess the axiomatic self-evidence of the first four postulates, and it could not be derived from the first four. The three creators of NEGs were by no means isolated and working independently. All were connected to Gauss (1777-1855) who had been working on NEGs since the late 1700s.

J. Bolyai was the son of one of Gauss’ friends, W. Bolyai. Gauss and his friend Bolyai were at the University of Göttingen where the parallels postulate was the subject of lectures by Kastner and a number of dissertations. Reimann was Gauss’ dissertation student. And as for Lobachevsky, he did indeed work at a university on the periphery of the European mathematical community, the University of Kazan. However, the university was staffed by distinguished German professors, including Gauss’ teacher,

J.M. Bartels. J. Bolyai developed ideas on non-Euclidean geometries (NEGs) as early as 1823. His “The Science of Absolute Space” was published ten years later in a book written by his father. Lobachevsky published on the foundations of geometry from 1825 on. Reimann’s Habilitationschrift was on the foundations of geometry. Gauss, we know, wrote about NEGs in letters to W. Bolyai (December 17, 1799), Taurinus (November 8, 1824), and to Besel (January 27, 1829). He also wrote about NEGs in published notes from 1831 on. There are two short reviews on NEGs in Göttingische Anziegen in 1816 and 1822. One has to wonder why in the face of the facts of the case Struik and Boyer chose to view things as “remarkable” and “startling.”

 

Classically, the story of the development of NEGs was told in the context of “pure” mathematics. Thus, to take the case of Riemann as an example, the story was that he constructed the generalization of elliptic geometry as a purely mathematical exercise. The idea that there was a concrete possibility of practical applications for this exercise was not a consideration. In the light of a more realistic sociological and network analysis, Riemann’s work along with that of Gauss, Lobachevsky, Bolyai, Helmoltz, and Clifford, the story of NEGs takes on a different shape.

 

To some extent, they all agreed that Euclidean geometry was an unimpeachable system of ideal space and logic. It could be read as a game played in accordance with a set of formal rules. In fact, however, they interrogated Euclidean geometry in terms of whether it was a valid representation of “actual space.” This should be tested not by mathematics, not what is within the confines of the social world of mathematics per se, but should be tested scientifically – by observation and some mode of experimentation.

 

The sociological generalization this leads to is that if you are given a “genius” or a startling event, search for a social network –cherchez le réseau. No one has made the case for social networks as the roots of ideas more powerfully than Randall Collins (1998). The rationale here should become clearer over the course of this chapter.

 

Even more curious in the case of the sociology of knowledge is the fact that already in his The Elementary Forms of Religious Life published in French in 1912, Emile Durkheim had linked the social construction of religion and the gods to the social construction of logical concepts. Durkheim’s program in the rejection of transcendence languished until the emergence of the science studies movement in the late 1960s and the works of David Bloor, Donald MacKenzie, and Sal Restivo in the sociology of mathematics.

 

It is interesting that a focus on practice as opposed to cognition was already adumbrated in Courant’s and Robbins’ classic “What is Mathematics?” (1906/1995). We must turn to active experience, not philosophy, they wrote, to answer the question “What is mathematics”? They challenged the idea of mathematics as nothing more than a set of

consistent conclusions and postulates produced by the “free will” of mathematicians. Forty years later, Davis and Hersh (1981) wrote an introduction to “the mathematical experience” for a general readership that already reflected the influence of the emergent sociology of mathematics. They eschewed Platonism in favor of grounding the meaning of mathematics in “the shared understanding of human beings…” Their ideas reflect a kind of weak sociology of mathematics that still privileges the mind and the individual as the creative founts of a real objective mathematics.

 

Almost twenty years later, Hersh, now clearly well-read in the sociology of mathematics, wrote “What is Mathematics, Really?” (1997). The allusion to Courant and Robbins is not an accident. Hersh does not find their definition of mathematics satisfactory. In spite of his emphasis on the social nature of mathematics, Hersh views this anti-Platonic anti-foundationalist perspective as a philosophical humanism. While he makes some significant progress by comparison to his work with Davis, by conflating and confusing philosophical and sociological discourses, he ends up once again defending a weak sociology of mathematics. The modern sociology of mathematics associated with the science and technology studies movement that emerged in the late 1960s has established mathematics as a human construction, a social construction. Mathematics is manufactured by humans on the earth out of the material and symbolic resources at their disposal in their local environments. While traditional philosophical and sociological discourses have become estranged, especially in the arena of science studies, there are efforts abroad to reconcile the two disciplines consistent with the interdisciplinary turn in contemporary research and theory. In this sense, one can consider Hersh’s philosophical humanism a step in that direction.

 

The second term in my title, “civilization” is not without its controversial features. In the ancient world, “civilized” peoples contrasted themselves with “barbarians;” in modern times, “civilized” has been opposed to “primitive” or “savage.” The relevance of the concept of civilization to the topic of mathematics lies in its association with the idea of progress. More to the point is the fact that different civilizations (variously “nations,” “societies,” and “cultures”) are associated with different mathematical traditions (v. Restivo, 1992, 23-88). To the extent that humans have developed in ways that can be captured in the ideas of “evolution” and “progress,” mathematics, in conjunction with science and technology more generally, is assumed to have contributed positively to and benefited from those developments. The Scottish philosopher Adam Ferguson (1723-1816) is credited by Benveniste (1954) with introducing the term “civilization” in its modern sense into the English language in his Essay on the History of Civil Society (1767), and perhaps as early as 1759. The term also appears in the works of Boswell (1772), Adam Smith (1776), and John Millar (1771). Mirabeau (1757) introduces the term in French in his L'Ami des hommes ou traité de la population. Just as the individual grows from infancy to adulthood, Ferguson wrote, the species advances from “rudeness to civilization.” Set at the pinnacle of forms of society, civilizations are characterized by complexity, hierarchies, dynamism, advanced levels of rationality, social progress, and a statist form of government. Rousseau, by contrast, viewed civilization as opposed to human nature.

 

If we adopt Ferguson’s view of civilization, then clearly mathematics has been both a result of the emergence and development of civilization and a contributor to that

development. If on the other hand we adopt Rousseau’s viewpoint, the virtues of science and mathematics and the very idea of “civilization” are made severely problematic.

 

What about progress, the very idea? Arguably, the idea comes into Western and world culture in the Old Testament with its conception of linear time and a God that moves through time with humans (e.g., Sedlacek, 2011: 47). The idea of scientific and technological progress was fueled by the seventeenth century advances in science and literature by such cultural giants as Galileo, Newton, Descartes, Molière, and Racine. The idea of social progress was added later. Early in the eighteenth century, the Abbé de Saint Pierre advocated establishing political and ethical academies to promote social progress. Saint Pierre and Turgot influenced the Encyclopedists. The great Encyclopédie was produced by a group of eighteenth century philosophers under the direction of Denis Diderot.

 

It defines the Enlightenment program of promoting reason and unified knowledge. It was at this point that social progress became mated to the values of industrialization and incorporated into the ideology of the bourgeoisie. Scientific, technological, and social progress were all aspects of the ideology of industrial civilization. Veblen, for example, argued that the various sciences could be distinguished in terms of their proximity to the domain of technology. Thus, the physical sciences were closest to that domain, even integral with it, whereas such areas as political theory and economics were farther afield. We have entered an era of machine discipline unlike any in human history. And now we stand on the threshold of machines that will discipline us with conscious awareness and values, including social and sociable robots (the so-called robosapiens), and cyborgs.

 

There have been attempts to identify a type of progress that is independent of material or technological criteria (see, for example, the discussion in Almond, Chodorow, and Pearce, 1985, and the classic criticisms in Roszak, 1969/1995). For many ancient as well as modern thinkers, the idea of progress has always been problematic. We are right to be concerned about the actual and potential impacts of our new bio- and nano- technologies. But one finds similar concerns in Plato’s Phaedrus. There, in the dialogue between Theuth and king Thamus concerning the new technology of writing, Theuth makes promising predictions about the impact of writing.

 

The king claims to be in a better position to do what in effect is a “technology assessment,” and concludes that writing will have the opposite of the effects predicted by Theuth. The cultural meaning of science has fared no better. Where the Rousseaus and the Roszaks saw danger and alienation in science, the Francis Bacons and Bronowskis saw civilization and progress. When the biochemist J.B.S. Haldane wrote about a future of human happiness built on the application of science, Bertrand Russell replied with a vision of science used to promote power and privilege rather than to improve the human condition. St. Augustine worried about the invention of machines of destruction; Spengler predicted that humans would be annihilated by Faustian man. Fontenelle, in the first modern secular treatise on progress published in 1688 argued that science was the clearest and most reliable path to progress. Rousseau, by contrast,

argued that science and the arts have corrupted our minds. The author will draw attention to some additional examples in his conclusion in this chapter.

 

By its intimate association with the very foundations of science, mathematics does not escape this ambivalence. But it stands apart from science in terms of its stronger association with human progress. In the seventeenth and eighteenth centuries, a wave of positivism fueled by Newton’s achievements evoked nothing but the promise of progress among mathematicians of that period. The historian Florian Cajori (1894: 4) had no question about the connection between mathematics and human progress. For Alex Bellos (2010: ix), mathematics is (“arguably”) the foundation of all human progress.

 

Progress, then, can be viewed in terms of “amelioration” or “improvement” in a social or ethical sense. Are we more advanced than cultures that are less dominated by machines and machine ideology? How do we measure the primacy of humans and ecologies and how do we sustain them in any given culture? Can we bring them to fruition and nourish them in any culture, or are some more friendly to the primacy of humans and ecologies than others? These issues are really matters of degree associated with the degree to which individuation of the self (and then the myth of individualism, selfishness, and greed) has progressed in any given society. Furthermore, the degree of awareness of and attention to ethics, values, and social justice has to come into consideration here. It is impossible to even discuss the idea of progress without engaging ideas about and the value of the person, freedom, and democracy.

 

It may be possible to define progress in a way that takes it out of the realm of hopes, wishes, and dreams and plants it more firmly on a meaningful (and even perhaps measurable) foundation. Following Gerhard Lenski (1974: 59), progress can be defined as the process by which human beings raise the upper limit of their capacity for perceiving, conceptualizing, accumulating, processing, mobilizing, distributing, and utilizing information, resources, and energy in the adaptive-evolutionary process. The relationship between adaptation and evolution is a paradoxical one. On the one hand, survival depends on the capacity to adapt to surroundings; on the other hand, adaptation involves increasing specialization and decreasing evolutionary potential. Adaptation is a dead end. As a given entity adapts to a given set of conditions, it specializes to the point that it begins to lose any capacity for adapting to significant changes in those conditions. The anthropologists Sahlins and Service (1960: 95-97) summarize these ideas as follows:

 

Principle of Stabilization: specific evolution (the increase in adaptive specialization by a given system) is ultimately self-limiting.

 

General evolution (progressive advance measured in absolute terms rather than in terms of degrees of adaptation in particular environments) occurs because of the emergence of new, relatively unspecialized forms.

 

Law of Evolutionary Potential: increasing specialization narrows adaptive potential. The more specialized and adaptive a mechanism or form is at any given point in

evolutionary history, the smaller is its potential for adapting to new situations and passing on to a new stage of development.

 

We can add here the Law of Adaptive Levels: adaptation occurs at different levels across various life orders and systems and occurs at different speeds in different spatial arenas. This law draws attention to the complexity of adaptation and the general processes of variation and selection. Adaptation suggests an active agent in a stable environment. But active agents can and do change their environments in ways that make different demands on the adapting agents. Looked at another way, environments have agential like dynamics. Law of Agent-Environment Entanglement.

 

Perhaps the most important aspect of the ideology of science is that it is (in its mythical pure form) completely independent of technology. This serves among other things to deflect social criticism from science onto technology and to justify the separation of science from concerns about ethics and values. Interestingly, this idea seems to be more readily appreciated in general by third world intellectuals than by the Brahmin scholars of the West and their emulators. Careful study of the history of contemporary Western science has demonstrated the intimate connection between what we often distinguish as science and technology. It has also revealed the intimate connection between technoscience research and development and the production, maintenance, and use of the means (and the most advanced means) of violence in society. Not only that, but this is true in general for the most advanced systems of knowledge in at least every society that has reached a level of complexity that gives rise to a system of social stratification.

 

Contradictions and ambivalence about science, technology, and progress may be built into the very core of our cultural machinery. Agricultural activities in the ancient Near East reduced vast forests to open plains, and wind erosion and over-grazing turned those areas into deserts. Deforestation in ancient China led to the development of the loess plateau. Loess sediment gives the Yellow River (nicknamed “China’s Sorrow”) its signature color and flooding pattern. Was deforestation necessary for building China into the greatest civilizational area on earth between the first and sixteenth centuries of the common era? Or were there conservation principles that the ancient Chinese could have relied on without detracting from their cultural development? There is some evidence that at least some of the deforestation they caused could have been avoided. The deforestation experiences of China, Rome, and other civilizational areas of the ancient world are being repeated today and offer cautionary tales for an era characterized by many hard to monitor emerging and converging technologies, that is, technocultural systems.

 

At the end of the day, it should be clear that progress is not easy to define, and that it is even harder to point to examples of progress that resist critical interrogation. How can we sustain the idea of progress in the face of the widespread ecological, environmental, and human destruction that has characterized the industrial age? The fact is that the destruction and danger we see all around us is integrally connected to the very things we use to mark the progress of humanity. For these reasons, we must be cautious when considering whether any of the sciences, engineering disciplines, or mathematics have contributed to or served as signposts of progress. Mathematics, like all systems of knowledge, does not exist in a vacuum. It is always connected to social institutions and

under the control of the most powerful institutions in any given society. All of this may put too much of the onus on the sciences and technology when what we are dealing with is culture in general. Is it possible that cultures by their very natures inevitably destroy planets?

 

It should be clear from this brief introduction that the terms of our title, “mathematics,” “civilization,” and “progress” are all imbued with some level of ambivalence and uncertainty. It remains to be seen whether in the rest of this chapter we can find our way to greater certainty about the meaning and implications of these terms.

 

2.  Introduction: Mathematics and Civilization

 

Karl Mannheim (1893-1947) and Oswald Spengler (1880-1936) defend diametrically opposed positions on the possibility of a sociology of mathematics. Mannheim argues that mathematics is exempt from sociocultural and historical explanations; mathematics is not an ideology, and mathematical truths are not culturally relative. This view has been reinforced by Pythagoreans and Platonists who believe that mathematical truths are eternal objects that exist independently of the flux of historical experience and outside of time and space. Most historians, philosophers, and sociologists of science have traditionally adopted a basically Mannheimian view of mathematics.

 

Spengler, on the other hand, holds that each culture has its own conception of number. Spengler's notion of the "soul” of a civilization cannot provide the basis for an adequate sociological analysis. However, Spengler's goal of explaining mathematics in terms of the particular social and historical forms in which it is produced is sociologically viable. Spengler’s argument is summarized in two statements: (1) “There is not, and cannot be, number as such." There are several number-worlds as there are several cultures; and (2) "There is no Mathematik but only mathematics." Spengler's objective in his analysis of "number" is to show how a crystallized culture demonstrates its idea of the human condition, of what it means to be human. The "peculiar position" of mathematics rests on the fact that it is at once science, art, and metaphysics. It is safe to assume, taking some liberties, that Spengler’s use of “Culture” is more or less commensurate with what we commonly mean by “civilization.”

 

The author takes some additional liberties here with Spengler’s spiritualized materialism in order to avoid some of the idiosyncrasies of his vocabulary. Number, like God, represents the ultimate meaning of the natural world. And like myth, number originated in naming, an act that gives humans power over features of their experience and environments. Nature, the numerable, is contrasted with history, the aggregate of all things that have no relationship to number. Note that in this moment it appears that Mannheim and Spengler might be at one on the nature of mathematics. While Spengler clearly sees a closer connection between mathematics and culture than Mannheim does, he does as the following paragraph illustrates harbor some ambivalence.

 

Spengler argues against treating earlier mathematical events as stages in the development of "mathematics”. This is consistent with his thesis on the incommensurability of Cultures and with his cyclical view of historical change. His general schema of Classical and Western styles and stages in "Culture," "number," and

"mind" is essentially an analysis of world views. This is reflected in Spengler's attempt to correlate mathematical and other sociocultural "styles.” For example, he argues that Gothic cathedrals and Doric temples are “mathematics in stone." Spengler is aware of the problem of the limits of a "naturalistic" approach to number and pessimistic about a solution. It is impossible, he writes, to distinguish between cultural features that are independent of time and space, and those that follow from the forms of culture manufactured by humans.

 

Finally, Spengler claims that a deep religious intuition is behind the greatest creative acts of mathematicians. Number thought is not merely a matter of knowledge and experience, it is a "view of the universe," that is, a world view. The second claim Spengler makes is that a "high mathematical endowment" may exist without any "mathematical science.” For example, the discovery of the boomerang can only be attributed to people having a sense of mathematics that we must recognize as a reflection of the higher geometry.

 

Sociologists of mathematics have been bold enough about challenging the Platonic conception of number, but they have hesitated to follow Spengler. His ideas must seem mad to scholars and laypersons, and specialists and non-specialists alike, to whom the truth of number relations appears to be self-evident. And yet, the "necessary truth" of numbers has been challenged by mathematical insiders and outsiders. One of the outsiders is Dostoevsky. In his Notes from Underground (1864/1918), Dostoevsky argues that 2+2 = 4 is not life but death, impudent, a farce. It’s nonetheless “excellent” and we must give it its due, but then we must recognize that 2+2=5 is also sometimes “a most charming little thing”. Contrast Dostoevsky’s perspective with Orwell’s (1949) use of these two equalities in 1984. For Orwell, 2+2=4 stands for freedom and liberty; 2+2=5 stands for Big Brother totalitarianism. Dostoevsky uses 2+2=4 to stand for everyday routines and tradition; 2+2=5 represents creativity.

 

Dostoevsky's remarks are not merely a matter of literary privilege. Mathematicians and historians and philosophers of mathematics have also challenged the conventional wisdom on number. We should not expect ordinary arithmetic to apply in every physical situation; whether it does or does not has to be based on our experience in different physical situations. In other words, whether or not 2+2=4 is always an empirical question. Where we have long term experiences with situations in which 2+2=4 we are justified in considering those situations closed to further interrogation, that is, we are justified in taking the equality for granted. This is not a warrant for universalizing mathematics uncritically.

 

Studies in ethnomathematics have helped to reinforce and ground the notion that mathematics and logics are culturally situated (cf. Benesch, 1992). African mathematics has posed problems for European intellectuals since at least the publication of Robin Horton’s (1997) studies on patterns of thought in Africa and the West beginning in the 1960s. The earliest findings and interpretations suggested a different way of reasoning and a different logic in African cultures by comparison with the Europeans. Ethnomathematics has helped to sort out the early discourses and ground differences in cultural patterns rather than in mental proclivities. Malagasy divination rituals, for example, rely on complex algebraic algorithms. Some peoples use calendars far more

abstract and elegant than those used in European cultures, notably the Chinese and the Maya. Certain concepts about time and equality that Westerners assumed to be universal in fact vary across cultures. The Basque notion of equivalence, for example, is a dynamic and temporal one not adequately captured by the familiar equal sign. Other ideas taken to be the exclusive province of professionally trained Western mathematicians are, in fact, shared by people in many societies. (see D’Ambrosio, 2006; Mesquita, Restivo, and D’Ambrosio, 2011; Eglash, 1999; Ascher, 2004; Verran, 1992).

 

There is, in brief, a rationale for pursuing the Spenglerian program for a sociology of mathematics based on the views of at least some mathematicians, historians of mathematics, and observers of numbers. The author refers here to authors such as Dostoevsky and Orwell, mathematicians such as Dirk Struik and Chandler Davis, philosophers of mathematics such as Paul Ernest and Leone Burton, and historians of mathematics such as Morris Kline.

 

Sociologists are obliged to interrogate mathematics in a comparative perspective: how does mathematics develop at different times and places; who are the noted mathematicians, what are the social positions they held and how were they related to one another; and what are the social conditions within and outside of mathematical communities as they go through phases of growth, stagnation, and decline. The degree of "community" among mathematicians, the level of specialization, the extent of institutionalization and the relative autonomy of the social activity of mathematics, it should be stressed, are variable across time and space.

 

It is possible to narrate the history of mathematics as a more or less linear unfolding that gives the appearance of an inevitable “logical evolution.” However, the evidence is that there are a number of variations among the types of mathematics produced in different cultures. The latter "horizontal" variations are prima .facie evidence for the Spengler thesis. But what about the long-term trends? These too are socially determined, and in two different senses. First, and in a weaker sense, the "longitudinal" development of mathematics does not occur without interruptions, nor does it unfold in a single cultural context. Mathematicians move along a certain path at some times and not others. This implies among other things that the concepts of truth and what counts as a proof in mathematics vary over time. A sequence in mathematical development or in the network of mathematicians will stop, start, stop, and start again over a period of a hundred or a thousand years. What drives these processes? And why do particular mathematicians at particular times and places make the major advances and not others? More to the point, how is it that some mathematicians and not others who might be better candidates get credit for discoveries and inventions? This question opens up an opportunity to consider eponymy and mis-eponymy in mathematics.

 

In the sciences, eponymy refers to the process of naming an invention or discovery after the person who made the invention or discovery. Some well-known examples are Ohm’s law, the Pythagorean theorem, and Pascal’s triangle. In 1980, the statistics professor Stephen Stigler (1980) formulated "Stigler’s law of eponymy:” stated in general terms it states that no discovery or invention is named after its original discoverer or inventor. Stigler then attributed “Stigler’s Law” to the sociologist of science Robert K. Merton (1961, 1963; and on the related Matthew effect, see Merton,

1968), thus making Stigler’s Law self-exemplifying (further exemplifying Merton’s fondness for the self-exemplifying hypothesis). Merton had already formulated the idea that all discoveries are in principle multiples. Consider the following examples: Pythagoras’ theorem was already known in Mesopotamia at least a thousand years earlier (crediting Pythagoras with a proof is entirely speculative). The Chinese had Pascal’s triangle at least 320 years before he was born. L’Hôpital’s Rule is probably due to Johann Bernoulli; Cardano’s formula for solving cubic equations comes from Tartaglia; Pell’s equation is due to Fermat; Benford’s Law is due to Simon Newcomb; and the Möbius strip could just as easily be named for Listing (Polster and Moss, 2011).

 

Following Spengler and more recent developments in the sociology of mathematics, it is possible to identify social factors that affect the variations, interruptions, progressions, and retrogressions characteristic of the longitudinal development of mathematics. There is a stronger sense in which the longitudinal development of mathematics towards more "advanced" forms is socially determined. The longitudinal development of mathematics reveals the social aspects of mathematical work. Much of professionalized mathematics is created in response to stimuli from within the mathematical community, especially as mathematicians go about playing competitive games with one another. The development of higher and higher levels of “abstraction” (see immediately below), for example, reflects the increasing self-consciousness of mathematicians about their own operations. This in turn reflects higher levels of specialization and institutional autonomy among mathematicians. The Spengler thesis is true in a very strong form: “number”, and all that it stands for metaphorically, is a socially created activity, or more technically, a social construction (see the detailed explanation of this much abused term in Restivo and Croissant (2008).

 

We should consider replacing the term “abstraction” with the term “generalization.” The reason is that what we commonly understand as abstractions are simply concrete forms constructed under conditions of professionalization and disciplinary closure. As mathematics becomes more organized and disciplined, mathematicians build new levels of mathematics on the grounds of earlier mathematical forms. This removes mathematics further and further ceteris paribus from the everyday world and gives rise to the idea of abstraction. But it is important to understand that we are dealing with new levels of concreteness. The term “abstraction” makes us vulnerable to the myths and ideologies of purity and even to ideas about heavens and gods. It is in my view better to eliminate it from our vocabulary in the sense that it is used in the sciences and mathematics.

 

3.  Mathematics and Civilization: Case Studies

 

The world history of mathematics has not unfolded in a unilinear, unidirectional manner. The Greeks, for example, took a step backward from the Babylonian achievements in notation. Different types of mathematical systems have developed in different parts of the world; and rival forms of mathematics have sometimes developed within societies and professional networks. Hindu mathematics, especially in the period before the influx of Greek astronomy (ca. 400CE) placed unique emphasis upon large numbers. Geometry, arithmetic, number theory, and algebra were ignored in favor of the use of numbers in "sociological" schemes. The Upanishads (ca. 700 to 500BCE) are full

of numerical (or more accurately, numerological) descriptions: 72,000 arteries; 36,360, or 36,000 syllables; the 33, 303, or 3306 gods; the 5, 6, or 12 basic elements out of which the world is composed. The wisdom of the Buddha is illustrated by the gigantic numbers he can count out (on the order of 8 times 23 series of 107), and his magnificence is shown by the huge number of Bodhisattvas and other celestial beings who gather to set the scenes for his various sutras. The Hindu cosmology includes a cyclical view of time that enumerates great blocks of years called yugas. There are four yugas ranging from 432,000 to 1,728,000 years, all of which together make up one thousandth of a kalpa or 4,320,000,000 years.

 

This emphasis upon immense, cosmological numbers, gives a distinctively Hindu view of the near-infinite stretches of being that surround the empirical world. It seems almost inevitable that the Hindus should have invented zero (sunya, emptiness, in Sanskrit). The concept sunya, developed about 100CE, was the central concept in Madhyamika Buddhist mysticism, and preceded the invention of the mathematical zero about 600CE. Brahmagupta published a number of rules governing the use of zero and negative numbers in his Brahmasputha Siddhanta (ca. 630CE). Classical Indian world views are permeated with a “mathematics of transcendence.” Numbers were used as a technology for transcending experience not as a mathematics that was directed toward rationalistic generalization. Numbers were used rhetorically to mystify, impress, and awe. In general, numbers were used numerologically rather than mathematically. The social roots of this distinctive mathematical system lie in the particularly exalted status of Indian religious specialists. A sociologist of mathematics would search for the roots of the Hindu emphasis on large numbers in the great variety of ethnic groups making up Indian society, institutionalized in the ramifications of the caste system.

 

The cosmological significance of Chinese mathematics has an ideographic bias. Numbers, and higher mathematical expressions, are written as concrete pictures. The system of hexagrams that make up the I Ching, the ancient book of divination, was continuously reinterpreted in successive Chinese cosmologies as the basic form of the changing universe. Chinese arithmetic and algebra were always worked out in positional notation. Different algebraic unknowns, for example, could be represented by counting sticks laid out in different directions from a central point. Chinese algebra, at its height around 1300CE, could be used to represent fairly complex equations, and included some notion of determinants (i.e., the pattern of coefficients). But it could not be developed in the direction of increasingly general rules. The ideographs (and the social conditions of their use) helped preserve the everyday roots of mathematics.

 

Why did Chinese mathematics take this form? Probably for some of the same reasons that account for the maintenance of ideographic writing among Chinese intellectuals. Both gave a concrete aesthetic emphasis to Chinese culture. The ideographic form had technical limitations that a more generalized form - an alphabet, a more mechanical mathematical symbolism - would have overcome. Ideographs are hard to learn; they require a great deal of memorization. But these limitations may in fact have been the reason why Chinese intellectuals preferred them. A difficult notation is a social advantage to a group attempting to monopolize intellectual positions. This may be contrasted with the algorithmic imperative characteristic of periods of rapid commercial expansion.

Writing and mathematics were highly esoteric skills in the ancient civilizations when they were first developed. Those who possessed these skills were almost exclusively state or religious dignitaries. Writing and mathematical notation tended therefore to be retained in forms that were very difficult to read and interpret, except by those who could spend a long time in acquiring familiarity with them. Sanskrit, for example, was written without vowels and without spaces between the words. Egyptian writing was similarly conservative. Chinese writing and mathematics are notable because archaic styles lasted much longer than anywhere else. The over-riding cultural issue might be that China was, as Leon Stover argued (1974: 24-25), a “once and always Bronze Age culture,” the only primary civilization to develop its Bronze Age to the fullest.

 

The development of ideographs and mathematical notation in China was in the direction of greater aggregative complexity and aesthetic elaboration, not of simplification and generalization. The Chinese literati thus managed to make their tools progressively more difficult to acquire. This is in keeping with the unusually high social position of Chinese intellectuals. They maintained their status through an examination system that was used to select officials in many dynasties. Many students of mathematics have contended that a “good” notation is a condition for progress in mathematics. The question for China or for any given society or mathematical community is: why wasn’t a more appropriate symbolism invented at some particular point in the history of mathematics? To answer that question we should envision a struggle between monopolizing and democratizing forces over access to writing and mathematics.

 

Monopolistic groups were strong in highly centralized administrations such as ancient Egypt, the Mesopotamian states, and China. Democratizing forces won the upper hand in decentralized situations, and/or under social conditions where there was a great deal of private business activity – as in ancient (especially Ionian) Greece, and certain periods in ancient and medieval India. The predominance of these forces was to varying degrees opposed by counter-forces. Greek mathematics also had some conservative elements, especially in the Alexandrian period when difficult rhetorical forms of exposition limited the development of algebra. The specific character of mathematics in given world cultures is due to the differential incidence of such conditions.

 

Greek mathematics is distinguished by its emphasis on geometry, generalized puzzles, and formal logical proofs. This is the intellectual lineage of modern Western mathematics. But the history of Greek and European mathematics also shows a divergent type that rose to prominence following the establishment of the classical form. During the Alexandrian period, another form of arithmetic was developed that was used neither for practical calculations nor for puzzle-contests. This was a type of numerology that used the real relations among numbers to reveal a mystical cosmology. The system was connected with verbal symbolism through a set of correspondences between numbers and letters of the Hebrew or Greek alphabets. Any word could be transformed into a related number that in turn would reveal mathematical relations to other words.

 

The social conditions involved in the creation and development of this alternative mathematics are connected with religious movements. Numerology is related to Hebrew Cabbalism, Christian Gnosticism, and the Neo-Pythagorean revival associated with Philo of Alexandria, ca. 20BCE-50CE). The most prominent expositor of this new

mathematics was Nichomachus (ca.100CE). Like Philo, he was a Hellenistic Jew (living in Syria). It was in this Jewish-Greek intellectual milieu of the Levant that the major religious movements of the time were organized.

 

There are variants even in modern European mathematics. There are conflicts between alternative notational systems in the 1500s and 1600s; and a century-long battle between the followers of Newton and those of Leibniz over the calculus. In the nineteenth century, a major dispute arose between Riemann, Dedekind, Cantor, Klein, and Hilbert and critics such as Kronecker and Brouwer. This split continued and widened in the twentieth century. The result was the emergence of schools in conflict over the foundations of mathematics. The main competitors in this arena were the logicists, the formalists, and the intuitionists

 

4.  The Social Roots of Mathematics

 

The social activities of everyday life in all the ancient civilizations gave rise to arithmetic and geometry, the two major modes of mathematical work. Each of these modes is associated with specific types of social activity. The development of arithmetic is stimulated by problems in accounting, taxation, stock-piling, and commerce; and by religious, magical, and artistic concerns in astronomy, in the construction of altars and temples, in the design of musical instruments, and in divination. Geometry is the product of problems that arise in measurement, land surveying, construction and engineering in general. Arithmetic and geometrical systems appear in conjunction with the emergence of literacy in all the earliest civilizations – China, India, Mesopotamia, Egypt, and Greece. These mathematical systems are, to varying degrees in the different civilizations, products of independent invention and diffusion. Note that while it is analytically “simple” to distinguish geometrical and arithmetical systems and methods it is not always so simple to do this in practice.

 

While we have found arithmetic, geometry, number work, and general mathematics in cultures throughout recorded history, special conditions were required for the emergence and crystallization of the discipline of mathematics. The general human ecological conditions for the emergence of modern science are discussed in Restivo and Karp (1974; and see Restivo, 1979, and 1994: 29-48). A combination of organizational and institutional factors rooted in a human ecology was required to foster and sustain the development of mathematical communities with generational continuity. As those conditions crystallized in Western Europe beginning in the 1500s and earlier, the discipline of mathematics emerged when sets of arithmetic and geometrical problems were assembled for purposes of codification and teaching, and to facilitate mathematical studies. Assembling problems was an important step toward unifying mathematics and stimulating generalization.

 

One of the most important steps in unifying and disciplining mathematics arose from efforts to state general rules for solving all problems of a given type. A further step could be taken once problems were arranged so that they could be treated in more general terms. Problems that had arisen in practical settings could now be transformed into hypothetical puzzles, and problems could be invented without explicit reference to practical issues. The three famous puzzles proposed by Greek geometers of the 5th and

4th centuries BCE are among the earliest examples of such puzzles: to double the volume of a cube (duplication of the cube), to construct a square with the same area as a given circle (quadrature of the circle), and to divide a given angle into three equal parts (trisection of the angle). Such problems were related to the non-mathematical riddles religious oracles commonly posed for one another. One account of the origin of the problem of duplicating the cube, for example, is that the oracle at Delos, in reply to an appeal from the Athenians concerning the plague of 430 BCE, recommended doubling the size of the altar of Apollo. The altar was a cube. The early Hindu literature already refers to problems about the size and shape of altars, and these may have been transmitted to Greece by the Pythagoreans, a secret religio-political society. The problem is also a translation into spatial geometric algebra of the Babylonian cubic

equation X ^{3 =}V .

The duplication, quadrature, and trisection problems were popular with the Sophists, who made a specialty out of debates of all kinds. A generation or two later, Plato introduced the constraint that the only valid solutions to these problems were those in which only an unmarked straightedge and a compass were used. This meant that special mechanical devices for geometrical forms could not be used in mathematical competitions. The result was stiffer competitive conditions and an emphasis on intellectual means and “gentlemanly” norms.        Plato's Academy was organized to help an elite group of intellectuals gain political power; and it represented the opposition of an aristocracy to democratization and commercialization. It is not surprising that this elite group of intellectuals developed an ideology of extreme intellectual purity, glorifying the extreme separation of hand and brain in the slave economy of classical Greece.

 

The three famous Greek puzzles and other problems became the basis for mathematical games (i.e., competitions) of challenge-and-response. Various forms of these games are important throughout most of the subsequent history of Western mathematics. Prior to the nineteenth and twentieth centuries, however, the challenge-and-response competitions were often initiated, endorsed, or rewarded by patrons, scientific academies, and governments. Prizes were sometimes offered for solutions to practical problems. Economic concerns as well as governmental prestige were often mixed in with the struggles for intellectual preeminence. At about the same time that they initiated mathematical contests, the Greek mathematicians took two further steps that led to new mathematical forms. They stipulated that a formal, logical mode of argument must be used in solving problems; this represented a further development of earlier methods of proof. And by extending this idea they created systems of interrelated proofs. This culminated in the Elements of Euclid shortly after 300 BCE. In addition to a collection of problems, Euclid presented an explicit body of generalizations in the form of definitions, postulates, and axioms. Euclid, like Aristotle, did not use the term "axiom" but something closer to "common notion." They both self-consciously worked at codifying past human experiences. The process of "systematization-and- generalization” is one of the two major paths to new mathematical forms. The other major path is an "empirical" one.

 

The empirical path to new mathematical forms involves applying existing mathematical concepts and methods to new areas of experience. Most of the early Greek geometrical

puzzles, for example, concerned flat figures. But the methods of plane geometry could be easily extended to solid geometry, and then to the properties of spheres or of conic sections; the work on conic sections eventually led to work on curves of various shapes. The intermittent periods of creativity in Alexandrian mathematics (especially from 300 to 200 BCE and 150-200 CE) were largely devoted to these extensions. No new level of generalization (with the exception of trigonometry, considered below) was achieved, but a number of new specialties appeared.

 

The history of arithmetic shows some of the same processes that occur in the history of geometry. The effort to find general rules for solving numerical problems led gradually to what we now call algebra. Here again we see mathematicians developing the practice of posing problems primarily to challenge other mathematicians. For example, there is this famous problem, attributed to Archimedes (287 to 212 BCE): find the number of bulls and cows of various colors in a herd, if the number of white cows is one third plus one quarter of the total number of black cattle; the number of black bulls is one quarter plus one fifth the number of the spotted bulls in excess of the number of brown bulls, etc. Such problems, involving unknown quantities, led over a very long period to the introduction of various kinds of notations and symbolisms.

 

These took quite different directions in ancient and medieval China and India, the Arab world and later in medieval and Renaissance Europe. The creation of a highly generalized symbolism which could be mechanically manipulated to find solutions did not appear until the late1500s and 1600s in Europe.

 

Over this period, and to different degrees in different parts of the world, algebra underwent an empirical extension. Problems were deliberately created to increase the number of unknowns, and to raise them to successively higher powers. Equations of the form ax + b = c gave way to those on the order of ax⁴ + by³ + cz² = g . The complexity of these, of course, could be extended indefinitely (Vieta in the1580s, for example, was

challenged to solve an equation involving x⁴⁵ ); but the extensions also gave rise to

efforts to find general rules for solving higher order equations. In other words, empirical extensions tended to promote generalized extensions. At the same time, arithmetic was developing in other directions.

 

What is generally called elementary arithmetic (solving numerical problems in, for example, addition, subtraction, multiplication, and division) continued to stimulate efforts to find general rules for solving particular problems. There was tremendous variation from one system of numerical symbols and calculating rules to another in terms of the ease or difficulty with which they could be applied to solving practical problems. Most of the ancient forms of notation made working with large numbers, fractions, or complex operations like division or the extraction of roots difficult; the exposition of problems was usually carried out in words. A great deal of mathematical creativity went into the development of notational systems that could be readily manipulated. Among the most important of these innovations were the invention of decimal place notation and the zero sign in India; the standardization of positional methods for writing multiplication and division (in Europe ca. 1600); and the invention of logarithms by the Scotsman Napier in 1614, for use in astronomy, navigation, and commerce.

A different development in arithmetic led to what we now call “number theory.” This focused on the properties of numbers themselves. As early as Eratosthenes (ca. 230 BCE), efforts were made to find a general method for identifying prime numbers. There were also various propositions about how numbers are composed of other numbers. The Pythagorean work on “triangular’ and “square” numbers anticipated Fermat’s theorem that every prime number of the form 4n +1 is a sum of two squares. Number theory was particularly popular in the Alexandrian period in an occultist, cabalistic form. In its more standard puzzle-solving form, it has remained popular among mathematicians from the Renaissance through the modern period.

 

One more branch of mathematics, based on a combination of arithmetic and geometry, developed in the Alexandrian period. Measurements of angles and lines, and the calculation of their ratios, led to the creation of trigonometry, notably by Hipparchus (ca. 140 BCE) and Menelaus (ca, 100 BCE). Trigonometry spread to medieval India and the Arab world, and in Renaissance Europe provided the basis for Napier's development of logarithms.

 

The overall picture so far, then, shows mathematics arising from practical geometry and arithmetic. The development of general mathematical puzzles and the extension of mathematics to new areas led to the emergence of new fields. Geometry became increasingly systematic, and progressively applied to plane and solid figures, to conics, and eventually to trigonometry. Arithmetic gave rise to algebra in successfully more complex forms (based on practical calculating systems), and to number theory. The creation of new fields continued in modern Europe. They grew out of the processes of generalizing (commonly understood in terms of increasing levels of abstraction). New fields were furthermore the result of extending results to new empirical areas, and the combination of existing mathematical fields into hybrid fields. The combination of algebra with a new coordinate representation in geometry by Descartes and Fermat produced analytic geometry.

 

Consideration of the problems of motion and the study of curves gave rise to the calculus in the 1600s. Calculus was then applied to successively more complex functions (empirical extension); and eventually (in the 1800s) it was generalized into a theory concerning such things as the rules for solving equations, and the properties of all functions (generalized extension). It should be noted that the drive towards creating new fields by generalization and extension seems to be characteristic of highly competitive periods. Geometry itself experienced a rapid series of branching around 1800 and thereafter, the best known being the non-Euclidean geometries. But there was also the creation of descriptive geometry by Monge, projective geometry by Poncelet, higher analytical geometry by Plucker, modern synthetic geometry by Steiner and Von Staudt, and topology by Mobius, Klein, and Poincare. In the late nineteenth and early twentieth centuries, Klein, Hilbert, and Cartan unified these different geometries. This unification occurs prominently in Klein’s Erlangen Program and its generalization in Cartan’s program which was designed to place the unification into the framework of Riemannian geometry.

 

In algebra, there was a parallel set of developments after 1800.The effort to find a general solution for the quintic and other higher-order equations led to the creation of

the theory of groups by Abel, Galois, Cauchy, and others. This theory focused on an abstract pattern among the coefficients of equations, and opened up a new area of inquiry in higher mathematics. “Abstract” algebras were created by Boole, Cayley, Sylvester, Hamilton, and Grassman. All of these new tools were applied to other branches of mathematics. Dedekind applied set theory to the calculus, Cantor applied it to the concept of infinity, and others applied it to topology, number theory, and geometry. These developments led to the creation of yet another even more general field toward the end of the nineteenth century, “foundations.” "Foundations" focused on the nature of mathematical objects themselves and with the rules by which mathematics should be carried out. Foundations research has been the focus of a number of opposing schools, and has led to what are probably the most intense controversies in the history of mathematics.

 

The basic forms of mathematics, arithmetic and geometry arise from practical problems in construction, taxation, administration, astronomy, and commerce. Moreover, the stimulus of practical concerns does not simply disappear once mathematics is launched. For example, the basic forms of arithmetic, including the number system, developed over a very long period, during which virtually the sole interest in improvement was to facilitate practical calculations. The same can be said for the invention of logarithms, and much of the development of trigonometry. Other advanced forms of mathematics were also stimulated by efforts to solve practical problems. The development of the calculus was linked to problems in ballistics and navigational astronomy. Newton’s mechanics makes an interesting case study in this regard. In a hallmark paper in the history and sociology of science presented at the Second International Congress of the History of Science in London in 1931, Boris Hessen (1893-1936) situated Newton’s work in the technological problems of his time, his social class position, and Newton’s lack of familiarity with the steam engine and the conservation of energy principle. Hessen’s views contradicted the view that Newton was divinely placed at a particular historical juncture and that his genius was the fountain out of which his physics sprung. The present author noted earlier R.K. Merton’s defense of Hessen’s sociology of science against the idealistic and motivational explanations offered by G.N. Clark. Merton’s argument implicitly opposes Carlyles’s Great Man theory, Alfred North Whitehead’s mystical explanation of Newton’s successes (he was born in the same year that Galileo died), and Alexander Pope’s divine providence conjecture: “Nature and nature’s laws lay hid in night; God said ‘Let Newton be!’ and all was light.”

 

Descriptive geometry and Fourier's analysis answered problems in the production of new machinery in the industrial revolution. Practical concerns do not tell the whole story of mathematics, but they are one component that continuously shapes its history. This suggests a general principle: an increase in the amount, type, intensity, or scope of practical concerns in a society will stimulate mathematical activity. The relationship between economic concerns and mathematics is especially strong; commercial growth tends to be very stimulating for mathematics. Mathematical innovations will also tend to occur when there is a shift to new productive technologies (and when there are shifts to new technologies of warfare and transportation, and shifts to more intensive administrative modes of organization). This implies a link between the development of modern European mathematics and the development of modern industrial technological societies (loosely, “capitalism” because the term “capitalism” does not refer to an actual

economic system but rather expresses an economic ideology). Since this is one factor among several, it does not imply that mathematics must come to an end in non-capitalist societies. It does, however, suggest that the form and content of mathematics (within the constraints noted by Spengler) as we know it today is a product of specific lines of cultural development. One could say that modern mathematics, like modern science, is part of the knowledge system generated by and supportive of capitalism.

The roots of mathematics in practical concerns are more apparent in some cases than in others. For example, the history of Chinese mathematics is primarily a history of an inductive “mathematics of survival." Its origins can be traced from the myth of Yü the Great Engineer's discovery of a magic square on the back of a Lo River tortoise (ca. 500 BCE). China’s mathematical evolution reaches its high point in the late Sung and early Yuan dynasties with the publication of Chu Shih-Chieh's "Precious Mirror of the Four Elements," written in 1313. Chinese mathematics never ventured far from problems of everyday life such as taxation, barter, canal and dike construction, surveying, warfare, and property matters. Chinese mathematical workers could not organize an autonomous mathematical community, and consequently failed to establish the level of generational continuity that is a necessary condition for long-term mathematical development. This helps to explain why the Chinese did not develop the more general forms of higher mathematics.

 

Conditions in ancient Greece were more favorable for generalizing mathematics. The commercial expansion in Greece in the 600s BCE stimulated mathematical growth. Learned merchants practiced and taught mathematical arts, and master-student relationships across generations fostered mathematical progress. Political and economic changes in Greek civilization led to the development of an increasingly elitist and self- perpetuating intellectual community, culminating in the oligarchic conditions and intellectual elitism of Plato's time. The achievements of the "thinking Greeks" depended on a division of labor that divorced hand and brain. The "thinkers" had the "leisure" to reflect on and elaborate mathematics. The class structure of the slave-based society that developed in the post-Ionian period conditioned the development of classical mathematics. Arithmetic was left to the slaves who carried out most commercial transactions, and householders for whom simple calculations were a part of everyday life. The elite intellectual class, by contrast, courted geometry which was considered democratic and more readily adapted to the interests of the ruling classes than arithmetic. What we know as "Greek mathematics" is a product of the classical period.

 

The development of specialties within the division of labor, left unchecked, tends to foster virtuosity. Such specialization tends to increase the specialists' distance from the order and distribution of everyday phenomena and to increase the importance of human- created phenomena, especially symbols. The result is an increase in the level of generalization and the development of ideologies of purity. This is essentially what occurred in classical Greece. Hand and brain slowly reunited following Plato's death; there is already evidence of an increased interest in linking mathematical and practical concerns in Aristotle. In the Alexandrian period, hand and brain were more or less united, but the ideology of purity retained some vitality. This is notably illustrated by Archimedes, whose work clearly exhibited a unity of hand and brain but whose philosophy echoed Platonist purity.

The decline of Greek commercial culture was accompanied by the decline of Greek mathematical culture. The achievements of Archimedes, which brought Greek mathematics to the threshold of the calculus, mark the high point of Greek mathematics. When mathematics was revived in the European commercial revolution (beginning haltingly as early as the twelfth century, the Gothic cathedrals signaling the coming economic revolution by serving as accumulating devices for labor, resources, and machinery), many aspects of the Greek case were recapitulated. European mathematics moved on in the direction of the calculus, rooted in problems of motion. It picked up, in other words, essentially where Archimedes had left off, and under the influence of the Archimedean (and more generally, Greek) writings as they were recovered and translated (and here, of course, the Arabic-Islamic transmissions were critical). By 1676, Newton was writing about mathematical quantities "described by continual motion."

 

The concept of function, central to practically all seventeenth and eighteenth century mathematics, was derived from studies of motion. Newton and Leibniz helped to reduce the basic problems addressed in the development of the calculus – rates of change, tangents, maxima and minima, and summations – to differentiation and anti- differentiation. Infinitesimals, nurtured earlier in the debates of theologians and the scholastics, entered into the process of production. Highly general intellectual ideas in the Euclidean realm of the straight, the flat, and the uniform gave way to the ideas of a dynamic world of guns and machinery, and global navigation and commerce characterized by skews, curves, and accelerations. The search for algorithms, time- saving rules for solving problems, is evident in the writings of the inventors of the calculus (e.g. in Leibniz's "De geometria recondita et analysi indivisibilium atque infinitorium" of 1686).

 

As the industrial “machine" of capitalist society was fashioned, so was the “machine of the calculus." Descartes' analytic geometry, the other great contribution to the development of pre-modern European mathematics, was also characterized by an algorithmic imperative.

 

It was, in spite of the conflicts between Cartesians and Newtonians, from the very beginning in constant association with the development of the Newtonian-Leibnizian calculus. Let us note parenthetically that Newton and Leibniz invented two different calculuses. Newton’s “method of fluxions” was much more indebted to classical geometry (and especially to Archimedes) than is commonly supposed, and Leibniz’s “differential calculus” used a much better notation (Restivo, 1992: 134).

 

The historian of mathematics Boutroux (1919) characterized Descartes’ analytic geometry as an industrial process; it transformed mathematical research into “manufacturing.” The idea that the calculus is linked to the emergence of capitalism (that is, early industrialization) is further suggested by the Japanese case. When the Japanese established a monetary economy and experienced a commercial revolution in the seventeenth century, they also worked out a "native calculus." This was not entirely indigenous since they had probably come into contact with Europe’s calculus from European contacts at their ports of entry.

5.  Puzzles and Proofs

 

Mathematical workers and mathematicians, from the earliest times onward, and especially in the West, have posed puzzles for one another. This practice tends to make mathematics a competitive game. Some periods have been dominated by public challenges such as those that the Emperor Frederick's court mathematician posed to Leonardo Fibonacci (ca. 1200), those that Tartaglia and Cardano posed for one another in sixteenth century Italy, or those that led to such high acclaim for Vieta at the French court in the 1570s. Such puzzle-contests have been important for several reasons. They often involved pushing mathematics into more general realms. Mathematicians would try to invent problems which were unknown in practical life in order to stump their opponents. The search for general solutions to equations, such as those that Tartaglia found for cubic equations and Vieta found for the reduction of equations from one form to another, was directly motivated by these contests.

 

The emphasis on proofs which has characterized various periods in the development of mathematics was partly due to a heightening of the competitiveness in these contests. Greek mathematicians rationalized the concept and method of proof at a time when mathematics was popular among the elite class of philosophers and there was a lot of competition for power and attention in the intellectual arena. This was the same period during which the wandering Sophists challenged one another to debating contests and in doing so began to develop canons of logic. This is completely analogous to the development in mathematics, in terms of both cause and effect. The analogy turns into a virtual identity when we realize that many of the mathematicians of the time were Sophists, and that many of the formal schools that were organized in the classical period (e.g., the Academy) used prowess in mathematics as a grounds for claiming superiority over competing institutions. Stressing proofs was a way of clarifying the rules of the game and escalating the intensity of competition. In general, competitive puzzle- contests are probably responsible for much of the inventiveness characteristic of Western mathematics. This analysis should not obscure the economic stimulus to the initial development of proofs.

 

Thales, the philosopher-merchant (who might have been a composite or imaginary construction), is credited with carrying the idea of a proof to a more general level than the Babylonians and Egyptians. Thales symbolically and iconically personifies the need among the Ionians of his era to develop a comprehensive and organized understanding of physical reality and successful computational methods in the context of the increasingly well-organized economy that they were products of and helped to fashion. Thales' proofs were probably crude extensions of Babylonian or Egyptian "rules" for checking results. In any case, the process of constructing proofs was rationalized over the next three hundred years and eventually led to Euclidean-type proofs.

 

Concern for proof has varied a great deal in the history of mathematics. The Chinese and Hindu mathematicians ignored proofs almost entirely; indeed, they would often present problems without solutions, or with incorrect solutions. These practices were the result of a relatively uncompetitive situation in mathematics in these societies. The social density of mathematicians in these societies was rather low; we rarely hear of more than a few mathematicians working at the same time, whereas in Greece and

Europe the numbers in creative periods are quite high. Most of the Oriental mathematicians were government officials, and thus were insulated from outside competition, while most of the ancient Greek and modern mathematicians were private individuals or teachers in competitive itinerant or formal educational systems. In the Islamic-Arabic world, there was a flurry of mathematical activity in the period 800- 1000 (and later to some extent).There was some concern for proofs (in the works of Tahbit Ibn Qurra, for example), but this was much more limited than in classical Greece. The Greek works they translated stimulated an awareness of and interest in proofs among the Islamic-Arabic mathematicians.

 

The limited emphasis on proofs reflects the fact that their community (or network) was not as densely populated as the Greek mathematical community, competition was not as intense, master-student chains and schools were not as well organized, and generational continuity was limited. Similarly, the episodic history of mathematics in India constantly interrupted generational continuity, the Bronze Age dynastic history of mathematics in China undermined specialization free of the centripetal force of the Emperor’s bureaucracy, and the delimited mathematical renaissance of Japan in the seventeenth century ended abruptly with the consolidation of Tokogawa power (Restivo, 1992: 22-60). Only Europe from the 1500s on was able to sustain generational continuity on a level no other civilization had achieved.

 

In modern Europe, the emphasis on proofs has grown steadily. In the 1600s, Fermat presented his theorems without proofs, and in the 1700s, Euler offered proofs that were not very rigorous. The early 1800s saw a shift towards more rigorous standards of proof; earlier solutions were rejected, not because they were incorrect, but because the reasoning behind them was not sufficiently universal and comprehensive. This went along with a massive increase in the number of people engaged in mathematics (which in turn was the result of the expansion of educational systems, especially in Germany and France and other social changes related to the rationalization of social and economic life). Both this shift towards rigor, and the earlier invention of proofs, had important effects on the nature of mathematics. Both pressed mathematics toward new levels of generalization: proofs had to invoke more general elements than particular numerical examples, and rigorous proofs stimulated the systematic consideration of the nature of mathematics in the nineteenth century.

 

6.  Generalization (“Abstraction”) and Self-Consciousness

 

The "main line" development of Western puzzle-solving mathematics is characterized by an increasing awareness that levels of generalization have been created by the mathematicians themselves. Mathematicians moved beyond a naïve realism about mathematical objects (sometimes real in the material sense, sometimes real in the Platonic sense) when they gradually began to use negative numbers instead of dropping negative roots of equations (as Hindu, Arab, and medieval European mathematicians had done). Later they came to recognize that imaginary numbers (an unfortunate and distracting nomenclature) could be used despite their apparent absurdity. Gauss established a new basis for modern algebra by creating a representational system for complex numbers. Nineteenth century higher mathematics took off from this point.

Mathematicians finally realized that they were not tied to common-sense representations of the world, but that mathematical concepts and systems could be deliberately created. These creations were not ab novo but situated in the cultural objects of the increasingly well-organized mathematical community. The new, more general geometries (including projective and non-Euclidean geometries) popularized the point, and stimulated the creation of new algebras and more generalized forms of analysis. The objects with which modern mathematics deals, however, are real in a particular sense. They are not simply things, as was once naively believed; they are, rather, operations, activities that mathematicians carry out. The imaginary number i is a shorthand for an activity, the operation of extracting a square root from a negative number. This operation, of course, cannot be carried out. But mathematicians had long been used to working backwards from solutions-not-yet-found, to the premises, by symbolizing the solution using an arbitrary designation (e.g., x ). This symbol represented the result of an imaginary operation. The imaginary number i, then, could be used as the basis for other mathematical operations, even though the operation of producing it could never actually be performed. The ordinary arithmetic operations, the concept of a function, the concept of a group - all of these are operations of different degrees of complexity. A natural whole number itself is not a thing but an operation - the operation of counting (and perhaps also other operations whose nature modern mathematicians are untangling). Modern mathematics has proceeded by taking its operations as its units. These are crystallized into new symbols which can then be manipulated as if they were things. In fact, within the social world of mathematics these symbols are things, the matters of fact, and the material resources of the human ecology of the mathematics community (see Alexksandrov, Kolmogorov, and Lavrent’ev, 1963 for a thoroughly materialistic and realistic history of mathematics).

 

A process of reification has gone on in conjunction with the emergence of the notion that generalized operations (what are commonly referred to as “abstractions”) are socially created by mathematicians. Thus mathematics has built upon itself hierarchically by treating operations as entities upon which other operations can be performed. The Western trend in symbolism, then, is not an "accidental" feature of Western mathematical uniqueness; the symbolism was created precisely because the mathematical community was pushing towards this degree of self-consciousness and reflexivity. What I am pointing to here is an emerging awareness among the more reflective mathematicians and philosophers of mathematics that crystallized in the late nineteenth and early twentieth century. That awareness that the creative construction of mathematics came from the hands and brains of situated human beings was enough to fuel the more recent efforts to ground mathematics in the material world of activities and experience. It has not, however, been enough to eliminate all vestiges of Platonic thinking among mathematicians.

 

Mathematics, like other modern activities, has been affected by specialization on a level unknown in earlier historical periods. As a result, the "causal power" of mathematics itself in the relationships between mathematical and other social activities has steadily increased. Mathematical ideas have increasingly become the generative basis for new mathematical ideas. The work setting and institutional context of mathematical activity has become a social foundation of a higher order than the social foundation of subsistence productive activity. Mathematics continues to be socially rooted within the

mathematical community; it is especially important to recognize the social nature of the symbols mathematicians create for communication within their own ranks.

 

This perspective throws new light on an old problem reflected in the title of a famous paper by Eugene Wigner (1960) on “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” The utility of “pure” mathematics for the physical and natural science is not just a coincidence; it must reflect some larger and deeper truth about both mathematics and physics. In fact, the “coincidence” reflects a constant interplay between mathematics and physics. As soon as this is interrupted by the professional and bureaucratic closure of the mathematics community, we can expect the cycles of effective applications to become increasingly rare. This is not just a sociological conclusion based on the theory of closure and autonomy in social systems (Restivo, 1992: 171-175; Restivo, 1993: 263-267) but something that has been recognized by professional mathematicians (e.g., Boos and Niss, eds., 1979).

 

The development of Western higher mathematics, then, is a social and a cultural development. The objects with which mathematicians deal are activities of mathematicians. In building upon the operations already in existence, and making them symbolic entities on which further operations can be performed, mathematicians are self-consciously building upon previous activities in their intellectual community. Mathematics thus embodies its own social and cultural history, and uses it as the base upon which its current community activities are constructed. Western mathematics thus depends upon a particular kind of long-term organization of the intellectual community. This is an organization in which strong links are maintained across generations, and in a highly self-conscious and competitive form. The new attempts to competitively consume the old. The important linkages of teachers and pupils typically found among European mathematicians, together with strong external competition among different mathematical "lineages," have been the social basis for this pattern (on the general theory of lineages and networks as the progenitors of ideas, see Collins, 1998). Once the pattern of competitive self-consciousness was established, subsequent rounds of competition could only escalate the degree of self-reflection and inventiveness among mathematicians. Out of this situation arose the hyper-reflexive concerns of twentieth- century foundations research. It is important to keep in the mind the self-conscious creativity of mathematicians is inseparable from the crystallization of an autonomous mathematical community.

 

7.  Conclusion

 

All thought, in its early stages, begins as action. The actions which you [King Arthur] have been wading through have been ideas, clumsy ones of

course, but they had to be established as a foundation before we could begin to think in earnest.

T.H. White’s Merlyn the Magician What, “In the beginning was the Word?” Absurd.

Then maybe it should say “In the beginning was the Mind?” Or better “…there was Force?”

Yet something warns me as I grasp the pen,

 

 

Act.”

That my translation must be changed again.

The spirit helps me. Now it is exact. I write: “In the beginning was the

 

Goethe’s Faust

 

The history of mathematics can be situated socially and culturally. This task requires a sociology both of the external institutional and organizational conditions of the societies within which mathematical activities are situated, and a sociology of the internal organization of the mathematical activities within communities and social networks of mathematicians. The notion of "internal and external factors" is an analytic device. The Spenglerian idea of mathematics as a world view is not, in the end, compatible with a strict adherence to internal-external analysis. The mathematics of any particular time embodies its own social history. This process becomes increasingly intense as and to the extent that mathematical activity becomes and remains more clearly differentiated from other social activities and more autonomous. But "autonomy" simply means that mathematicians communicate more intensively with each other than with outsiders. It does not mean that mathematicians are more removed from social determinants or that they have unimpeded access to "objective reality." Their activities remain at all times coupled to the social activities of insiders and outsiders, and thus unfold in an environment of multiple social, cultural and historical determinants. This is the rationale for a Spenglerian approach to the sociology of mathematics. That this is becoming more accepted is indicated by a clear turn to practice, experience, and shared meaning in the philosophy of mathematics, the philosophy of mathematics education, and among reflective mathematicians.

 

We are no longer entranced by the idea that the power of mathematics lies in formal relations among meaningless symbols, nor are we as ready as in the past to take seriously Platonic and foundationalist perspectives on mathematics. We do, however, need to be more radical in our sociological imagination if we are going to release ourselves from the strong hold that philosophy has on our intellectual lives. Philosophy, indeed, can be viewed as a general Platonism and equally detrimental in its classical forms and agendas to our efforts to ground mathematics (as well as science and logic) in social life. It is to philosophy in its more recent turn to practice that we must look if we are going to salvage philosophy as a credible intellectual activity. The strengths of philosophy as a guide to reasoned speculation, thought experiments, and ethical stands will be heightened by the turn to practice and the empirical arena. This is how we can save philosophy as an analytical tool for understanding mathematics. And here is where philosophy is drawn into the interdisciplinary boundary breaking modalities of late twentieth and twenty first century intellectual life. We are witnessing the demise of the traditional disciplines as we have known them for more than one hundred years. As the new inter-disciplines come to the fore, it begins to appear that we are witnessing the emergence of a second generation natural philosophy. For the moment, let us bracket this development and focus on mathematics and sociology as distinct, viable disciplinary projects. Both of these disciplines are undergoing interdisciplinary developments that are contributing to the emergence of a neo-natural philosophy.

 

Once again, then, what is mathematics? Technical talk about mathematics – trying to understand mathematics in terms of mathematics or mathematical philosophy – has the

effect of isolating mathematics from practice, experience, and shared meaning; it tends to “spiritualize” the technical. It is important to understand technical talk as social talk, to recognize that mathematics and mathematical objects are not (to borrow terms from the anthropologist Clifford Geertz' (1983: 94-120) “concatenations of pure form,” “parades of syntactic variations,” or sets of “structural transformations.” To address the question “What is mathematics?” is to reveal a sensibility, a collective formation, a worldview, a form of life. This implies that we can understand mathematics and mathematical objects in terms of a natural history, or an ethnography of a cultural system. We can only answer this question by immersing ourselves in the social worlds in which mathematicians work, in their networks of cooperating and conflicting human beings. It is these “math worlds” that produce mathematics, not individual mathematicians or mathematicians’ minds or brains. It is easy to interpret this perspective as somehow “mystical,” or as implying a “super-organic” entity. A realistic interpretation requires understanding human individuals as social things and their ideas as the “voice” of their social networks.

 

Mathematics, mathematical objects, and mathematicians themselves are manufactured out of the social ecology of everyday interactions, the locally available social, material, and symbolic interpersonally meaningful resources. All of what has been written in the last two paragraphs is captured by the short hand phrase, “the social construction of mathematics.” This phrase and the concept it conveys are widely misunderstood. It is not a philosophical statement or claim but rather a statement of the fundamental theorem of a sociology broadly conceived, a sociology being transformed by the interdisciplinary movements of our era. Everything we do and think is a product of our social ecologies. Our thoughts and actions are not simple products of revelation, genetics, biology, or mind or brain. To put it the simplest terms, all of our cultural productions come out of our social interactions in the context of sets of locally available material and symbolic resources. The idea of the social seems to be transparent, but in fact it is one of the most profound discoveries about the natural world, a discovery that still eludes the majority of our intellectuals and scholars. The interdisciplinary imperative can sustain this idea of the social even while it brings it into closer and closer association with our biology (cf. Clark, 2010, Noë, 2010, and Reyna, 2007).

 

Mathematics is a human, and thus a social, creation rooted in the materials and symbols of our everyday lives. It is earthbound and rooted in human labor. We can account for the Platonic angels and devils that accompany mathematics everywhere in two ways. First, there are certain human universals and environmental overlaps across the variety of our material environments, the physics, biology, and chemistry of life, culture, space, and time that can account for certain “universalistic” features of mathematics. Everywhere in everyday life, putting two apples together with two apples gives us phenomenologically four apples. But the generalization that 2+2 = 4 is culturally glossed and means something very different in Plato, Leibniz, Peano, and Russell and Whitehead. The earthbound everyday world of apples is commensurable for Plato and Russell and Whitehead; that world has not changed in the millennia that separate their lives. However, the discipline and then the profession of mathematics has changed dramatically across the centuries and created new incommensurable experiences within the social world of mathematics. The professionalization of mathematics gives rise to the phenomenon of mathematics giving rise to mathematics, an outcome that reinforces

the idea of a mathematics independent of work, space-time, and culture. Mathematics is always and everywhere culturally, historically, and locally embedded. There is, to recall Spengler, only mathematics and not Mathematik. There is, however, number work that carries across all cultures. Culture always intrudes in our mathematics; but while there is no Mathematik, there are ways for us to translate and communicate commensurable number and math work across cultures.

 

The concept-phrase “mathematics is a social construction” must be unpacked in order to give us what we see when we look at working mathematicians and the products of their work. We need to describe how mathematicians come to be mathematicians, the conditions under which mathematicians work, their work sites, the materials they work with, and the things they produce. This comes down to describing their culture – their material culture (tools, techniques, and products), their social culture (patterns of organization – social networks and structures, patterns of social interaction, rituals, norms, values, ideas, concepts, theories, and beliefs), and their symbolic culture (the reservoir of past and present symbolic resources that they manipulate in order to manufacture equations, theorems, proofs, and so on). This implies that in order to understand mathematics at all, we must carry out ethnographies – studies of mathematicians in action. To say, furthermore, that “mathematics is a social construction” is to say that the products of mathematics – mathematical objects – embody the social relations of mathematics. They are not free standing, culturally or historically independent, Platonic objects. To view a mathematical object is to view a social history of mathematicians at work. It is in this sense that mathematical objects are real. Before there is mathematics there is number work; before there are professional mathematicians there are number workers and then mathematics workers.

 

Arithmetic, geometry, and the higher mathematics are produced originally by number or mathematical workers and later on by variously disciplined and ultimately professional mathematicians. Ethnographies and historical sociologies of mathematics must, to be complete, situate mathematics cultures in their wider social, cultural, historical, and global settings. They must also attend to issues of power, class, gender, ethnicity, and status inside and outside more or less well-defined mathematical communities.

 

There is a hidden interrogation in the interrogation of mathematics that undermines its claims to transcendence and purity. That hidden interrogation is the interrogation of the very idea of the transcendent, of a Platonic realm of ideas. It is not too much of a stretch to see that this sort of interrogation will sooner or later have us interrogating religion and the gods. This is not the place to follow this line of inquiry. However, it is important to note that any deep understanding of the nature of mathematics as a this- worldly phenomenon is necessarily linked to the possibility of bringing religion and the gods down to earth. One only has to consider that the sociologist Emile Durkheim (1912/1995) concludes his remarkable study of the social construction of religion and the gods by arguing that logical concepts are, like religion and the gods, collective representations, this-worldly social constructs. Durkheim manifests the unfolding of the sociological enterprise as an exercise in the rejection of the transcendental.

 

Finally, let us revisit the ideas of civilization and progress. As we saw earlier, we could try to place ourselves on a continuum of civilization and progress between the polar

positions of Rousseau and Ferguson; everything about human society and culture as we have known it (especially since the coming of modern science and technology and the industrial revolution) denies our humanity, or everything trumpets its triumphs. This inevitably enmeshes us in a conflictful conversation without stop signs.

 

We could try to adopt Lenski’s notion of progress introduced earlier and bring the idea of civilization into his framework as a form of social and cultural organization. This would give us a more “scientific” foundation to rest on, but not one without its own ethical and value biases and implications. In the end, we are probably safest at this point if we recognize that mathematics is one of the many tools humans have fashioned to help them wind their way through the complex tapestries of life, and then one of humanity’s most useful as well as awe inspiring tools, even if more often than not actualized in the service of state power and ruling elites as “weapons of math destruction.” Moreover, it is one of the achievements that Nietzsche would have recognized as part of the fragile reason and sense of freedom humans have purchased at great cost.

 

Every entity recognized in history as a “civilization” from ancient Sumer and Babylonia to contemporary China has fed mathematical traditions into the ocean of a mathematical heritage that is in principle a part of humanity’s world culture. We can value the mathematics that has been developed out of and applied to the best of our activities as humans, activities that have been liberating, supported social justice, and in general made our planet a better and safer place on which to pursue the betterment of our species and its social and natural ecologies. We are by many measures and accounts at a cross-roads in human and planetary history, perhaps the cross-roads that opens a road just a couple of hundred years into the industrial-technological global society to annihilation on various levels and scales. It may be that our fate as a species and a part of a global and planetary ecology doomed to annihilation on a scale of millennia is now sealed on a scale of centuries or even years. Our survival, and especially our survival with some decent quality of life distributed across the planet and its life forms, will depend on long-term thinking that is at once broad, deep, and wise. All of our civilizational tools will be needed in perhaps our last chance in the short run to “save” ourselves and our planet.

 

Ours is an era when the very idea of progress, under attack throughout the twentieth century as wars, holocausts, ecological disasters, and radioactive fallout crushed romantic and idealistic dreams of a better world, has given way to concerns about unintended consequences, precautionary principles, and technology assessments. How dangerous it sounds today to hear the echoes of the British Labor Party’s celebration in the mid-1960s of “the white heat of technology revolution.” And yet we must ask if we are still subject to the hypnotic seductions of the technological lottery. Imagine a group of distinguished scientists brought together to speculate on “the next hundred years,” and to speculate optimistically because the idea (read “ideology’) of science was by definition inclined to optimism. And consider that when just such an event was organized to celebrate the centennial of Joseph E. Seagram & Sons (purveyors of whiskey and related spirited drinks) in 1957, they all choked on their optimism as they offered their speculations. Distinguished speakers such as Nobelist geneticist Herman J. Muller, geochemist Harrison Brown, economist and college president John Weir, and

infamous rocket scientist Wernher Von Braun found it necessary to qualify their optimism. Muller said the future would be rosy if we could avoid war, dictatorship, overpopulation, or fanaticism; Brown began by saying “If we survive the next century; Weir began with “If man survives…” Braun, in the most bizarre opening sentence, said “I believe the intercontinental ballistic missile is actually merely a humble beginning of much greater things to come.”

 

No matter. It is beyond doubt that mathematics will play a key role in our efforts, successful or not, realistic or delusional, to realize civilization and progress in terms that even Rousseau might embrace. This is not meant to lock into mathematics as simply a means to quantifying problem formulation and problem solving across all arenas of human experience. It is meant mathematics as a humanistic mode of knowing. It is not only or at all to Plato’s view of the role of mathematics for his guardian-rulers we must turn to (keeping in mind that for all his resistance to poetry and metaphor, Plato was a master of both). The author wants rather to endorse something like Nietzsche’s notion of mathematics. That is, the rigor and refinement of mathematics must be brought into all of our endeavors but not because this is necessarily the best way to apprehend things (recall Nietzsche’s views on the limits of science). We should want to do this in order to become more aware of our relationships as human beings to the things in our world. Mathematics does not give us the most general or the ultimate form of knowledge but opens a path (only asymptotically, at best) to such a goal. This view of mathematics has been promoted for almost half a century by the humanistic mathematics movement. If civilization has a future, if progress can become at least a Lenskian possibility, a humanistic mathematics will have to take hold in our efforts to identify, specify, and solve the problems of planetary survival we face today.

 

Acknowledgments

 

The author wants to acknowledge my collaboration with and indebtedness to Randall Collins of the University of Pennsylvania with whom he first tackled the problem of mathematics and civilization in the early 1980s. Randy is an important inspiration for the perspective developed here. He also wants to thank his collaborators and colleagues in the sociology of mathematics over the many years in which he worked in this field: Wenda Bauchspies, Deborah Sloan, Monica Mesquita, Ubiratan D’Ambrosio, the late Dirk Struik, the late Joseph Needham, the late Leone Burton, Paul Ernest, David Bloor, Donald MacKenzie, Jean Paul Bendegem, and Jens Hoyrup,

 

Glossary

 

Abstraction                : Classically, a process by which higher level concepts are derived from the usage and classification of literal ("real" or "concrete") concepts, first principles, or other methods; used as a noun, a concept that acts as a super-categorical noun for all subordinate concepts, and connects any related concepts as a group, field, or category. Abstractions may be formed by reducing the information content of a concept or an observable phenomenon, typically to retain only information which is relevant for a particular purpose. For example, abstracting a

leather soccer ball to the more general idea of a ball retains only the information on general ball attributes and behavior, eliminating the other characteristics of that particular ball. In this chapter, abstractions are replaced by the term “generalization” and given a concrete sense. In this sense, what we normally refer to as abstractions are the concrete terms of an autonomous professional community of practice.

Arithmetic                  : From the Greek word ἀριθμός, arithmos “number”; historically, the oldest and most elementary branch of mathematics, used for everything from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of operations that combine numbers. In common usage, it refers to the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers. The term “higher arithmetic” as distinguished from elementary arithmetic, is used in professional mathematics to refer to more advanced practices related to number theory.

Civilization                 : A relatively high level of cultural and technological development; specifically associated with the development of writing and written records in the ancient world. Classically, “civilized” peoples contrasted themselves with “barbarians;” in modern times, “civilized” has been opposed to “primitive” or “savage.” Viewed in positive and optimistic terms, civilizations are characterized by complexity, hierarchies, dynamism, advanced levels of rationality, social progress, and a statist form of government. Viewed in negative pessimistic terms, notably by the philosopher Rousseau, civilization is conceived to be opposed to human nature.

Ethnomathematics : The study of the relationship between mathematics and culture. This research brings to light the mathematical and logical traditions of non-literate, but it in general the study of the mathematical and logical practices of specific cultures. It refers to a broad cluster of ideas ranging from distinct numerical and mathematical systems to multicultural mathematics education.

Formalists                  : Formalists, following the mathematical philosophy of one of the most influential leaders of this school, David Hilbert (1862- 1943), treat mathematics as a “game.” In its most extreme version, formalism claims that mathematics is not about anything but rather sets of rules of inference that can be applied to given “strings” (axioms) to generate new strings. You can, for example, use the “game” Euclidean geometry (which is viewed as some strings) to generate a new string such as the Pythagorean theorem. This is roughly the equivalent of proving the theorem in classical mathematics. See also intuitionists and logicists.

Foundationalism  : In epistemology (theories of knowledge) the idea that there are basic (foundational) beliefs, assumptions, etc. that are the grounds for beliefs in general. Basic beliefs justify other beliefs. Basic beliefs are said to be self-evident or self-justifying. Basic beliefs

can also derive their warrant from sensory experience. Anti- foundationalists have a problem with giving an uncontroversial or principled account of which beliefs are self-evident or indubitable and see foundationalism as a form of an unexamined a priori or even as a God surrogate. from the Ancient Greek

Geometry                   : From the Ancient Greek γεωμετρία; geo- "earth", -metria "measurement;” the branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, Euclid (3rd century BCE) put it into an axiomatic form,

Intuitionists                : The intuitionists approached mathematics as the result of constructive cognition. Humans make mathematics and logic as mental constructs by applying internally consistent methods.

L.E.J. Brouwer (1881-1996) is considered the father of modern intuitionism. Intuitionists view the truth of mathematics as a subjective claim. Brouwer rejected realist/Platonist ideas about the truth or reality of mathematical objects. Intuitionism substitutes constructability for abstract truth and is the provocation for the transition from the proof theory of truth to the model theory of truth in modern mathematics. See also logicists and formalists.

Logicists                     : The logicists believed that mathematics was an extension of logic and therefore that all or at least some of mathematics was reducible to logic. Richard Dedekind (1831-1916) and Gottlob Frege (1848-1925) are considered the founders of this school. The logicist culminated in the monumental Principia Mathematica (published in three volumes in 1910, 1912, and 1913, and in a second edition in 1927 by Cambridge University Press )by Bertrand Russell (1872-1970) and Alfred North Whitehead (1861-1947) . Logicism survives today in Zermelo- Frankael set theory or one of its variations (such as Zermelo- Frankael set theory with the axiom of choice, or ZFC). Most of mathematics is believed to be reducible to the logical foundations provided by the axioms of ZF, ZFC, and derivatives. See also formalism and intuitionism.

Mathematical worker

: (Also number worker). To be distinguished from “mathematician,” someone who works with numbers and “does” mathematics as a member of a specialized occupational or professional class. The mathematical worker works with numbers as a member of a relatively informal and unorganized specialty outside of a formal credentializing system. The two terms are sometimes used interchangeably in the text for convenience but readers alerted to the distinction should be able to identify which term is most appropriate from the context.

Mathematics              : In its standard dictionary sense, the science of numbers and their operations,  interrelations,  combinations,  generalizations,  and

 

 

 

 

 

 

 

 

 

Non-Euclidean Geometry

abstractions and of space configurations and their structure, measurement, transformations, and generalizations . Classically, “mathematics” refers to the two fundamental ways in which humans work with numbers, arithmetic and geometry. In the Platonic view, this number work reveals a transcendental realm of ideal numbers revealed to humans as they develop over time. More recently the Platonic view has been opposed by the idea that mathematics is a social practice; it is manufactured by humans on the earth out of the material and symbolic resources at their disposal in their local environments.

: Refers to the geometries generated by denying Euclid’s fifth postulate, the parallels postulate. Consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line. In Euclidean geometry the lines remain at a constant distance from each other even if extended to infinity, and are known as parallels. In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels. In elliptic geometry the lines "curve toward" each other and eventually intersect. Spherical geometry is the geometry of the two- dimensional surface of a sphere.

Objectivity                 : This term is traditionally associated with ideas about value- neutrality, observations and interpretations uncolored by personal biases, and methods that are systematically developed and applied and not based on the idiosyncrasies of individual scientists. In this context, an objective statement is based on replicable research processes, and beyond fundamental dispute. In the sociology of science, objectivity has been shown to be an achievement of social groups and thus situated in “cultures of objectivity” or “objectivity communities.

Progress                     : In general terms, the forward development and betterment of humankind over time. The idea of scientific and technological progress was fueled by the seventeenth century advances in science and literature by Galileo, Newton, Descartes, Molière, and Racine. The idea of social progress was added later. The wars, holocausts, and human and environmental degradations characteristic of the twentieth century contributed to the development of critiques of the very idea of progress.

Proof                           : As a noun, evidence or an argument establishing or helping to establish a fact or the truth of a statement; the cogency of evidence that compels rational acceptance of a truth or a fact; the process or an instance of establishing the validity of a statement especially by derivation from other statements in accordance with principles of reasoning. As in the case other traditionally taken for granted ideas in the sciences, proof has become the subject of empirically grounded interrogations by sociologists of science who now refer proof to proof communities, and demonstrate that

 

Social Constructionism

 

 

 

 

Sociology of knowledge

what counts as a proof varies across time, space, and cultures.

: The fundamental theorem or central dogma of sociology, it refers to the fact that the only means humans have available for invention and discovery are their interactions with each other using the material resources of their environments and the symbolic resources of their cultures. It is fully compatible with a critical realism that assumes a “reality out there” that can only be known through the lenses of society and culture.

: The study of the relationship between human thought and the social, cultural, and historical contexts within which it arises, and of the effects prevailing ideas have on societies; deals with broad fundamental questions about the extent and limits of social influences on individual's lives and the social-cultural basics of our knowledge about the world When applied to the study of science in particular, an important specialty in sociology known as the sociology of science.

Spengler thesis   : Oswald Spengler argued that that there is no “Mathematik” but only “mathematics.” Mathematical forms are related to particular cultures. The weak form of the thesis is that there are as many mathematics as there are cultures. The strong form of the thesis is that mathematics is socially constructed.

Technoscience            : Variously, the idea the science and technology are intricately interrelated; the separation of “science” and “technology” may be analytically useful under some conditions but the idea that the two ideas are conceptually distinct is more ideological than substantive. This idea gives rise to such hybrid concepts as technocultural and technosocial.

Transcendental (transcendent; transcendence)

: Transcendence, transcendent, and transcendental are words that refer to an object (or a property of an object) as being comparatively beyond that of other objects. Such objects (or properties) transcend other objects (or properties) in some way. In philosophy, transcendence refers to climbing or going beyond some philosophical concept or limit In nineteenth century American, transcendentalism was developed within a religio- philosophical movement that claimed there is an ideal spiritual state that 'transcends' the physical and empirical. Following Restivo (2011 170) The Transcendental Fallacy (also known as the theologian’s fallacy) is that there is a world or that there are worlds beyond our own – transcendental worlds, supernatural worlds, worlds of souls, spirits and ghosts, gods, devils, and angels, heavens and hells. There are no such worlds. They are symbolic of social categories and heavens and hells. There are no such worlds. They are symbolic of social categories and classifications in our earthly societies and cultures. There is nothing beyond our material, organic, and social world. Death is final; there is no soul, there is no life after death. It is also possible that the so-called “many worlds interpretation” in quantum mechanics is contaminated by this fallacy as the result

of mathegrammatical illusions. The world, the universe, may be more complex than we can know or imagine, but that complexity does not include transcendental or supernatural features. Stated positively, this is Durkheim’s Law

 

Bibliography

 

Aleksandrov, A.D., A.N. Kolmogorov and M.A. Lavrent’ev (eds.) (1969), Mathematics: It’s Content, Methods, and Meanings, Cambridge, MA: MIT Press (orig. publ. in Russian in 1956). [A product of some the greatest mathematicians of the twentieth century, this is a readable review of the basic fields of mathematics. The authors are aware of the fact that mathematics is a human and a social creation and they do not mystify it or present it as a revelation of Platonic discourses.]

 

Almond, G., M. Chodorow, and R.H. Pearce (1985), Progress and its Discontents, Berkeley: University of California Press. [The twentieth century has not been kind to the idea of progress. Intellectuals across the spectrum of the disciplines are no longer that there are any viable grounds for the claim that science and technology have improved our lives morally or materially. The term “progress” is no longer meaningful empirically but is now about our aspirations and a “compelling obligation.”]

 

Ascher, M. (2004), Mathematics Elsewhere: An Exploration of Ideas Across Cultures, Princeton: Princeton University Press. [An introduction to mathematical ideas of peoples from a variety of small- scale and traditional cultures that challenges our conception of what mathematics is. Traditional cultures have mathematical ideas that are far more substantial and sophisticated than has been traditionally acknowledged.]

 

Baber, Z. (1996), The Science of Empire: Scientific Knowledges, Civilization, and Colonial Rule in India. Albany NY: SUNY Press. [Baber analyzes the reciprocal interactions between science technology and society in India from antiquity to modern times. The author analyzes institutional factors, including pre- colonial trading circuits, in the transfer of science and technology from India to other civilizational centers. He also explains the role of modern science and technology as factors in consolidating British rule in India.]

 

Ball, W. W. R. (1960), A Short Account of the History of Mathematics. New York: Dover. Unabridged and altered version of the fourth edition published in 1908. [A classic contribution to the history of mathematics.]

 

Bell, E. T. (1937/1986) Men of Mathematics, New York: [Simon and Schuster Touchstone Books. Bell is a mathematician and lacks the rigorous standards of the professional historians of science. But this book, for all of its idealism and focus on the dramatic, has been very inspiring for young people interested in mathematics and offers some insights into what it’s like to be a mathematician.]

 

Bellos, A. (2010), Here’s Looking at Euclid. New York: The Free Press. [This is a book designed to inspire and surprise. Bellos, who studied mathematics and philosophy at Oxford, takes the reader on a tour of mathematics based on his skills as a writer, traveler, and interviewer.]

 

Benesch, W. (1992), “Comparative Logics and the Comparative Study of Civilizations,” Comparative Civilizations Review, No. 27, Fall 1992, 88-105. [Benesch views “civilization as a state of mind.” Drawing heavily on Indian civilizational features, and especially the culture of the Jain for comparative materials, Benesch identifies four primary logics: object logics, subject logics, situational logics, and aspect logics. Benesch’s typology can be expanded to encompass propositional and predicate logic, first and higher order logics, modal logics, temporal logics, probability logics, multi-valued logics, negation logics, and quantum logic. Using another classification scheme we get: relevant logic (defined by some mathematicians as classical logic practiced with “good taste;” constructive logic (known in some of the literature by the less descriptive term "intuitionistic logic;" fuzzy logic, and comparative logic. The way to understand the proliferation of logics is to recognize that what we understand in everyday life and in a good deal of everyday science as the one universal logic is that set of rules and rules of inference that

reflect the workings of the macro-level physical world. In the worlds of the very small and very large, and the very fast and very slow, and across different levels of reality, different logics play out.]

 

Benveniste, É. (1966), Civilisation. “Contribution à l'histoire du mot” (Civilisation. Contribution to the history of the word), 1954, published in Problèmes de linguistique générale, Paris: Editions Gallimard ,

pp. 336-345 (translated by Mary Elizabeth Meek as Problems in general linguistics, 2 vols. 1971). [The author, a semiotician, traces the development of the concept “civilization” from its first occurrence in Adam Ferguson’s 1767 treatise on the history of civil society.]

 

Boos, B. and M. Niss, eds. (1979), Mathematics and the Real World, Boston: Birkhauser. [The editors recognize that to the extent that “pure” mathematics has gone too far and increasingly fails to interact with “real world” sciences and engineering disciplines, to that extent has it diminished its practical effectiveness. Mathematics that increasingly turns in on itself increasingly negates Wigner’s (1960) “unreasonable effectiveness” claim.]

 

Boutroux, P. (1919), L’ideal scientifique mathematiciens dans l’antiquité et dans les temps modernes. , Paris: Presses Universitaire. [Boutroux (1880-1922) was the son of the famous French philosopher Émile Boutroux and Aline Catherine Eugénie Boutroux, He was related to the statesman Raymond Poincaré¸the physicist Lucien Poincaré, and noted mathematician Henri Poincaré. Boutroux identifies three stages in the history of mathematics: the aesthetic, contemplate mathematics of the Greeks, the synthetic conception of Cartesian algebra, and what he saw as the incoherence of the mathematics of his time. There is a progressive theme in this history, an improving understanding of a higher reality. His didactic goals were to assert that progress in the history of science was a function of interactions between all the sciences, and to argue that problems should dictate approaches to solutions.]

 

Boyer, C. (1968), A History of Mathematics, Wiley, New York. [The novelist David F. Wallace called Boyer “the Gibbon of math history.” This book is to the history of mathematics what Paul Samuelson’s book on economics is to the study of economics. The reference of choice in the history of mathematics is now in its 3^rd edition (2011, updated by Uta Merzbach).]

 

Cajori, F. (1894), A History of Mathematics. New York: Macmillan & Co.. [Cajori’s (1859-1930) book was first published in 1893 and has gone through several editions. It is a reliable, readable treatment of the history of mathematics from antiquity to the early 1900s.]

 

Cajori, F. (1929/1993), A History of Mathematical Notations. Dover, New York. [This book is still today considered with good reason to be unsurpassed. Demonstrates how notation changes with changes in the shape of civilization. Here we discover, for example, how the Greeks, Romans, and Hebrews counted.]

 

Cantor, M. (1907), Vorlesungen uber die Geschichte del' Mathematik,. [Anastatischer Neudruck, Leipzig. Moritz Cantor (1829-1920) was an historian of mathematics who studied under some the giants of mathematics, including notably Gauss, and this book in four volumes is not just comprehensive but is considered a founding document in the history of mathematics as a critical, methodologically sound field of study.]

 

Clark, A. (2010), Supersizing the Mind: Embodiment, Action, and Cognitive Extension, New York: Oxford University Press. [Clark is yet another of a small but increasing number of authors who are focusing in on the problems and paradoxes that arise when we make the brain the source of our thinking and consciousness, and the source of our morals and our beliefs. In fact it is becoming increasingly clear that cognition is a complex result of tangled networks that criss-cross the boundaries of brain, body, and world. Mind is not bound by the brain; consciousness, as Nietzsche already intuited, is a network of relationships.]

 

Collins, R. (1998), The Sociology of Philosophies. Cambridge, MA: Harvard University Press. [A comprehensive social history of world philosophy in the context of global intellectual life. Collins traces the development of philosophical thought in China, Japan, India, ancient Greece, the medieval Islamic and Jewish world, medieval Christendom, and modern Europe. The result is an empirically grounded theory of ideas as the product of social networks. One of the most important contributions to modern sociology by a leading theorist.]

Collins, R. and S. Restivo. (1983), "Robber Barons and Politicians in Mathematics," The Canadian Journal of Sociology 8, 2 (Spring 1983), pp. 199-227. [Two of the leading contributors to the sociology of science offer an alternative to Kuhn’s theory of scientific change. Major scandals in mathematics are shown to be associated with shifts in the organizational structure of mathematical work. The article demonstrates the power of conflict theory as an explanatory approach in the sociology of mathematics.] Courant, R. and H. Robbins. (1996), What is Mathematics? New York: Oxford University Press (rev. by Ian Stewart, orig. publ. 1906). [A classic contribution to our understanding of mathematics as an enterprise of practice rather than philosophy.]

 

D’Ambrosio, U. (2006), Ethnomathematics, Rotterdam: Sense Publishers. An introduction to the concept of ethnomathematics by the founding father of the field.

 

Davis, P.J. and R. Hersh. (1981), The Mathematical Experience. Boston: Birkhauser. [Two mathematicians bring sociological, psychological, and philosophical perspectives to this popular exposition of the nature and relevance of mathematics.]

 

Dostoevsky, F. (1864), "Notes from the Underground," pp. 107-2-10 in The Best Short Stories of Dostoevsky. New York, n.d. (orig. publ, in Russian). [This classic piece of literature can be read as a contribution to the sociology of mathematics; the author’s discussion of 2+2=4 and 2+2=5 shows how arithmetic can be used to symbolize ideologies. Should be compared with how these equalities are treated by George Orwell in 1984.]

 

Dorrie, H. (1965), One Hundred Great Problems of Elementary Mathematics,Their History and Solution. New York: Dover. [This book was originally published in 1932 under the title Triumph der Mathematik. Fascinating well selected problems but not really “elementary.” An intriguing 2000 year survey but not for those without a strong background in mathematics.]

 

Durkheim, E. (912/1995), The Elementary Forms of Religious Life. New York: The Free Press 1 (trans. Karen Fields). [One of the most important studies in the history of scholarship. Durkheim crystallizes the idea of who and what God is and explains the function of religion in society. Criticisms abound, but they do not change the world transforming potential of this book.]

 

Eglash, R. (1999), African Fractals, Piscataway, NJ: Rutgers University Press. [Patterns across cultures are characterized by specific design themes. In Europe and America, cities are often laid out in a grid pattern of straight streets and right-angle corners. In contrast, traditional African settlements tend to use fractal structures--circles of circles of circular dwellings, rectangular walls enclosing ever-smaller rectangles, and streets in which broad avenues branch down to tiny footpaths with striking geometric repetition. These indigenous fractals are not limited to architecture; their recursive patterns echo throughout many disparate African designs and knowledge systems.]

 

Gasking, D. (1956), "Mathematics and the World," pp. 1708-1722 in J.R. Newman (ed.), The World of Mathematics. New York: Simon & Schuster. [Originally published in 1940, this is an exemplar of conventionalist philosophy of mathematics. Gasking views mathematics as true by virtue of linguistic conventions. Mathematics on this view is not empirical in the way, for example, that Wittgenstein argues it is or in the way it is understood today in social studies of mathematics.]

 

Geertz, C. (1983), “Art as a Cultural System,” pp. 94-120 in C. Geertz, Local Knowledge, New York: Basic Books. [Art, like speech, myth and other cultural systems are situated, ideationally connected to society; in a word cultural systems are socially constructed.]

 

Hersh, R. (1999), What is Mathematics, Really? New York: Oxford University Press. [Mathematics for the modern reader by a mathematician who appreciates the new sociology of mathematics developed by Bloor on the one hand and Restivo on the other.]

 

Horton, R. (1997), Patterns of Thought in Africa and the West: Essays on Magic, Science and Religion, Cambridge; Cambridge University Press. [The distinguished anthropologist and philosopher was at the center of debates that crystallized in the 1950s and 1960s concerning the nature of and relationship between “primitive” (read, primarily, African) and “advanced” (read “the West”) systems of knowledge

and belief. In this collection of his essays, Horton addresses the debates by exploring African beliefs, rituals, and cosmologies in the context of open and closed systems thinking. His objective is to demonstrate the kinship between primitive and modern thought and between science and religion more generally. Among the most important provocations for this and related debates are studies of Zande (n. Azande) logic. Were they “pre-logical,” as some claimed (e.g., in my view arguably Levy-Bruhl) or perfectly coherent in context (as Evans-Pritchard argued)? In the early years of the science studies movement David Bloor took up this question and argued for a cultural relativism that provoked new interest in this topic. This literature is relevant to my topic but requires more direct attention than I can give it here.]

 

Kavolis, V. (1985), “Civilizational Analysis as a Sociology of Culture,” Sociological Theory 85, 31, Spring 1985, 29-38. [The comparative civilizational analysis of any phenomenon, Kavolis argues, requires that we spell out the structure of each level of sociocultural organization and the connections and resources linking all the levels that bear on the particular phenomenon in a particular civilization. We don’t require an elaborate analytical apparatus but it helps to carry an awareness of such an apparatus in the study of a particular problem in terms of the relations between general civilizational structures, processes, and issues. Civilizational analysis is about the distinctive varieties of and changes in human experience, and the universe of symbolic designs in social behavior.]

 

Kline, M. (1962), Mathematics: A Cultural Approach, Reading, Massachusetts: Addison-Wesley. [A good introduction, even where dated factually and ideologically, to mathematics in relation to the humanities and the search for truth.]

 

Kramer, E. E. (1970), The Nature and Growth of Modern Mathematics. New York: Hawthorne Books. [Kramer discusses the lives and contributions of prominent mathematicians from Pythagoras and Newton to the modern period. Mathematical concepts such as binary operations, point-set topology, post-relativity geometries, optimization and decision processes ergodic theorems epsilon-delta arithmetization, and integral equations are discussed with admirable clarity.]

 

Lenski, G. (1974), Human Societies. New York: McGraw-Hill. [A cultural ecology of the evolution of societies. This is a major textbook contribution to macrosociology, now in its 11^th edition from Paradigm Publishers in Boulder, Colorado, 2008 co-authored by P. Nolan.]

 

Mannheim, K. (1936), Ideology and Utopia. Eugene, Oregon: Harvest Publishers. [This is a complex treatise in the context of Mannheim’s legacy in relationship to European and America sociology. It is a founding document in the emergence of sociology but especially of the sociology of knowledge. In terms of its relationship to the substance of this essay, the significance of this book is that while situating knowledge in its social, cultural, and historical contexts, Mannheim exempts the formal science from his analysis. There cannot be, he claims in these pages, a sociology of 2+2=4. This idea carried into the emergence of the sociology of science in the 1930s and was not seriously challenged until the birth of the science studies movement in the late 1960s.]

 

McClain, E.G. (1976), The Myth of Invariance: The Origins of the Gods, Mathematics and Music from the Rg Veda to Plato. York Beach, ME: Nicolas-Hays, Inc. [McClain’s thesis is that in the ancient civilizations music was a science that bridged the gap between the everyday world and the divine. The invariance of music contrasted sharply with the variability of the everyday world. Music expresses and motivates mathematics. Music as a science is revealed by studying the mathematical relationships between musical notes. In this context music is properly understood as an expression of and the motive for mathematical study. The "key" to unlocking this science comes from a study of the mathematical relationships between various musical notes. The Pythagorean notion of number was in fact more general and more widespread and an essential feature of the very idea of culture and civilization. The presentation is controversial in some details but we are learning more and more about the centrality of musicality in humans and this book speaks to that idea.]

 

Merton, R.K. (1968), Social Theory and Social Structure, enlarged ed., New York: The Free Press. [Merton founded and dominated the sociology of science with his students from the late 1930s to the late 1960s. The Mertonian paradigm, consonant with Mannheim’s sociology of knowledge, focused on the

social system of science – for example, norms, values, the reward system, stratification in science, age- grading – but exempted scientific knowledge per se from sociological scrutiny.]

 

Merton, R. K. (1961), , "Singletons and Multiples in Scientific Discovery: a Chapter in the Sociology of Science," Proceedings of the American Philosophical Society, 105: 470–86. Reprinted in R. K. Merton, The Sociology of Science: Theoretical and Empirical Investigations, Chicago, University of Chicago Press, 1973, pp. 343–70.

 

Merton, R. K. (1973), "Resistance to the Systematic Study of Multiple Discoveries in Science," European Journal of Sociology, 4:237–82, 1963. Reprinted in R.K. Merton, The Sociology of Science: Theoretical and Empirical Investigations, Chicago, University of Chicago Press , pp. 371–82. [This and the following paper introduce Merton’s idea that all discoveries are in principle multiples. These papers are milestones in the Merton corpus.]

 

Merton, R.K. (1958), “The Matthew Effect in Science,” Science, 159(3810): 56-63, January 5, 1968. [Widely cited in the sociology of science, the Matthew effect, or the principle of accumulated advantage, refers to situations in which the rich get richer and the poor get poor. Merton named the effect after Matthew 25: 29 (NRSV Bible). In science as in other activities, power and economic or social capital can be leveraged to gain additional power and capital. This is one the causes of mis-eponymy along with historical amnesia and random and deliberate acts of misattribution.]

 

Mesquita,M., S. Restivo and U. D'Ambrosio (2011), Asphalt Children and City Streets: A Life, a City, and a Case Study of History, Culture, and Ethnomathematics in São Paulo, Rotterdam: Sense Publishers. [An innovative plural voiced ethnography and history of street children that attends to how mathematics plays into their survival strategies.]

 

Morris-Suzuki, T. (1994), The Technological Transformation of Japan: From the Seventeenth to the Twenty-first Century, Cambridge: Cambridge University Press. [Japan did not miraculously leap into the technological forefront of twentieth century societies. Its rise to superpower status, as this book makes clear, is rooted in its history. This is the first general English language history of technology in modern Japan. One of the significant features of this book is its consideration of the social costs of rapid changes in technology.]

 

Nasr, S. H. (2007), Science and Civilization in Islam. Chicago: Kazi Publications. [The first one volume English language book on Islamic science from the Muslim perspective. Hossein explains the place of science in Muslim culture as he unfolds its content and spirit.]

 

Needham, J. (1959), Science and Civilization in China. Volume III: Mathematics and the Sciences of the Heavens and the Earth. Cambridge: Cambridge University Press. [One of several volumes in the monumental study that uncovered the hidden history of science and technology in China. Needham documented that not only did China have a history in science and technology, it was the leading civilizational center of science and technology in the world between the early Christian period in Palestine and 1500 CE. The explanatory framework is Marxist cultural ecology.]

 

Neugebauer, O. (1952), The Exact Sciences in Antiquity. Princeton: Princeton University Press. [It took a long time for historians of science to overcome the ideology of the once and always Greek miracle. Neugebauer contributes to the demise of this myth in this non-technical discussion of the influence of Egyptian and Babylonian mathematics and astronomy on the Hellenistic world. An early look into the sophistication of ancient Babylonian mathematics.]

 

Noë, A. (2010), Out of Our Heads: Why You Are Not Your Brain, and Other Lessons from the Biology of Consciousness, New York: Hill and Wang. [Makes an important contribution to getting away from classical ideas about the primacy of the brain in consciousness. His approach is radically social but in a strange way that makes biology, rather than sociology, the science of the social. But just because of this twisted logic, he furthers the interdisciplinary agenda of figuring out a non-reductionist way to link biology and society.]

Polster, B. and M. Ross, (2011), “Pythagoras’s Theorem ain’t Pythagoras’s,” h t t p : / / education . theage . co m . au / cmspage . php ? I ntid = 147 & intversion = 79, March 7, 2011.

 

Resnikoff, H.L. and R.O. Wells, Jr. (2011), Mathematics and Civilization. New York: Dover (paperback). [This book was originally published by Holt, Rinehart, and Winston in 1973 and later published by Dover with supplemental materials. Still a good general introduction to the reciprocal relations between mathematics and human culture with an emphasis on the technical mathematics. No great demands are made on the mathematical aptitudes of readers, and the more sophisticated reader will find some of the treatment, especially on the calculus, technically deficient.]

 

Restivo, S. (1979), "Joseph Needham and the Comparative Sociology of Chinese and Modern Science: A Critical Perspective," pp. 25-51 in R.A. Jones and H. Kuklick (eds.), Research in the Sociology of Knowledge, Sciences, and Art, Vol. II. JAI Press, Greenwich. [The most extensive journal length critical survey of Needham’s monumental history. Supports and extends his sociological materialism based on a human ecological perspective.]

 

Restivo, S. (1981), "Mathematics and the Limits of Sociology of Knowledge," Social Science Information, V. 20, 4/5: 679-701. [The new sociology of science associated with the science studies movement that emerged in the late 1960s challenged the status of mathematics as the arbiter of the limits of the sociology of science and knowledge. This is one of the early examples of the challenge by one of the founders of the modern sociology of mathematics.]

 

Restivo, S. (2001/1992), Mathematics in Society and History. New York. [This is the first book devoted completely to the sociology of mathematics as a subfield of sociology. The author reviews precursors, Spengler’s thesis on numbers and culture, mathematical traditions in different civilizations, and develops an original conception of the sociology of pure mathematics.]

 

Restivo, S. (1993), “The Social Life of Mathematics,” pp. 247-278 in S. Restivo, J.P. van Bendegem, and

R. Fischer (eds.), Math Worlds: Philosophical and Social Studies of Mathematics and Mathematics Education, Albany, NY: SUNY Press. [A collection of essays, including a set of papers by German scholars previously unavailable in English, focused on recent developments in the study and teaching of mathematics. The authors are all guided by the idea that mathematical knowledge must be grounded in and reflect the realities of mathematical practice.]

 

Restivo, S. (1994), Science, Society, and Values: Toward a Sociology of Objectivity. Bethlehem PA: Lehigh University Press. [This book introduces Restivo’s main contributions to the sociology of science between 1966 and the early 1990s. Based on his work in the ethnography of science, the history of science in China and the West, his social problems approach to understanding modern science, and other contributions, he develops a sociological perspective on objectivity.]

 

Restivo, S. (2007), “Mathematics,” Monza: Polimetrica. June, 2007 The Language of Science (ISSN 1971-1352). [An overview of the sociology of mathematics written for an online encyclopedia.]

 

Restivo, S. (2011), Red, Black, and Objective: Science, Sociology, and Anarchism, Surrey: Ashgate Publishers. [This book explores the implications of the science studies movement for science and society in the context of an anarchist tradition. The particular tradition the author has in mind here makes anarchism one of the sociological sciences. Here he follows Peter Kropotkin. The book is grounded in the empirical studies carried out over the last forty years by researchers in science studies (and more broadly science and technology studies). The author’s perspective is at once empirical, normative, and policy- oriented.]

 

Restivo, S. and H. Karp (1974), "Ecological Factors in the Emergence of Modern Science," pp. 123-142 in S. Restivo and C.K. Vanderpool (eds.), Comparative Studies in Science and Society. Columbus, OH: C. Merrill. [The authors theoretically link organizational and institutional structures to their ecological contexts and apply this theory to the problem of why modern science emerged in the West and not in China.]

Restivo, S. and R. Collins (1982), “Mathematics and Civilization,” The Centennial Review Vol. XXVI, No. 3 (Summer 1982), pp. 277-301. [An introduction to the comparative historical sociology of mathematics that modifies Spengler’s ideas on numbers and culture in the context of sociological theory.]

 

Restivo, S. and J. Croissant (2008), “Social Constructionism in Science and Technology Studies,” pp. 213-229 in J.A. Holstein & J.F. Gubrium, eds., Handbook of Constructionist Research, New York: Guilford. [The authors aim is to clarify the widespread misconceptions, misapplications, and misconstruals of this term which they identify as the fundamental theorem of sociology. They are at pains to argue that the term does not imply or entail any form of relativism; it is compatible with a realistic sociology that recognizes objectivity and truth as real. They are real however in an institutional sense. This view, as Durkheim pointed out more than one hundred years ago, is consistent with the idea that there is a reality “outside of us,” but we do not have access to a ding an sich.]

 

Reyna, S.P. (2007), Connections: Brain, Mind and Culture in Social Anthropology, New York: Taylor and Francis. [Reyna returns to the Boasian (Franz Boas) roots of anthropology to creatively construct a new paradigm for connecting the biological and cultural domains without submitting to the Sirens of reductionism.]

 

Rosental, C. (2008), Weaving Self-Evidence: A Sociology of Logic, Princeton: Princeton University Press. [Rosental traces the history of a theorem in the foundations of fuzzy logic to demonstrate the inherently social nature of logic. He describes the process by which logical propositions are produced, disseminated, and established as truths.]

 

Roszak, T. (1995), The Making of a Counter-Culture: Reflections on the Technocratic Society and its Youthful Opposition, with a new introduction, Berkeley; University of California Press (orig. publ. 1969). [Published in the middle of the 1960s sociocultural revolution, this book spoke directly to those who would become “the children of the 60s” while simultaneously bewildering their parents. Roszak coined the term “counter-culture” and damned the technocracy that was at the heart of the problems the protesters were angry about. A literate effort to explain the disaffection of young people and the young at heart during this tumultuous period.]

 

Sahlins, M. and E. Service, eds. (1960), Evolution and Culture, Ann Arbor: University of Michigan Press (co-authored by T.G. Harding, D. Kaplan, M.D. Sahlins, and E.R. Service). [In the hands of these authors, evolutionary anthropology becomes a predictive tool that can be applied to theorizing the future of human societies. Current events on the world scene, including the political and economic rise of China and the troubled status of the United States would not have surprised these anthropologists. This is a classic and still relevant contribution to our understanding of culture and cultural change writ large.]

 

Schechter, Eric (2005), Classical and Nonclassical Logics: An Introduction to the Mathematics of Propositions, Princeton: Princeton University Press. [Classical logic--the logic crystallized in the early twentieth century by Frege, Russell, and others--is computationally the simplest of the major logics, and adequate for the needs of most mathematicians. But it is just one of the many kinds of reasoning in everyday thought. Schechter introduces classical logic alongside constructive, relevant, comparative, and other nonclassical logics. This is the first textbook to make this subject accessible to beginners.]

 

Sedlacek, T. (2011), Economics of Good and Evil, New York, Oxford University Press. [Sedlacek understands economics as a social, cultural, and historical phenomenon. It is a product of our civilization not a pure, value free science. In this sense, Sedlacek lines up with contemporary students of the sociology of science and mathematics. Economics, at the end of the day, is about “good and evil.” In viewing economics as a moral enterprise he reminds us that the author of The Wealth of Nations, Adam Smith, is also the author of The Theory of Moral Sentiments.]

 

Selin, H. (ed.) (2001), Mathematics Across Cultures, New York: Springer. [A survey of Islamic, Chinese, Native American, Aboriginal Australian, Inca, Egyptian, and African mathematics, with essays on rationality, logic and mathematics. Shows how science and math practice is situated in its cultural context.]

Smith, D. E. (1958), History of Mathematics. New York: Dover. [A classical two volume introduction to the history of mathematics from ancient Egypt to modern times, very lucid survey including biographical notes and chronology.]

 

Sohn-Rethel, A. (1975), "Science as Alienated Consciousness." Radical Science Journal, Nos. 2/3: 65-

101. [Working from inside a sophisticated Marxist paradigm, Sohn-Rethel discusses science as an alienated and alienating form of knowledge and knowledge building. He is known for linking Kant and Marx in his work on epistemology.]

 

Spengler, O. (1926), The Decline of the West. New York: A. Knopf. [Spengler’s readers append adjectives like “audacious,” “profound,” “magnificent,” “exciting,” and “dazzlingly” to this book which flaws and all is one of the most amazing efforts in human history to capture humanity’s march through space and time. It’s relevance for the topic at hand is that Spengler’s theses are grounded in a radically cultural understanding of the relationship between culture and mathematics.]

 

Stigler, Stephen S. (1980), ‘Stigler’s Law of Eponymy’, Trans. N. Y. Acad. Sci. (2) 39, 147–157.

 

Leon Stover (1974), The Cultural Ecology of Chinese Civilization. New York: Signet. [An innovative interpretation of peasants and elites in what Stover refers to as a “once and always Bronze Age culture.” A paradigm for understanding the nature of Chinese culture and the continuities between the age of the dynasties and the modern era from Sun Yat Sen and Chiang Kai Shek to Maoism and modernism.]

 

Struik, D. (1967), A Concise History of Mathematics,. New York: Dover Publications. [The fourth revised edition of this classic was published by Dover in 1987. Struik, a Dutch mathematician and Marxist theory, was a professor at MIT for most of his professional career and to my knowledge the first person to identify the sociology of mathematics as a field of study. This book is a very readable introduction to the history of mathematics, concise as advertised but with a lot of substance poured into the books roughly 230 pages. Struik covers the period from the ancient world to the early twentieth century.]

 

Sugimoto, Masayoshi and D.L. Swain, (1978), Science & Culture in Traditional Japan. Cambridge MA: MIT Press. [Between 600 and 1854 CE, Japan was impacted by a first and second Chinese cultural wave and the first Western Cultural Wave in the nineteenth century. The authors focus on how these cultural waves set the stage for the development of an indigenous science and technology.]

 

Verran, Helen. (1992), Science and an African Logic. Chicago: The University of Chicago Press. [An empirical study that supports the idea of mathematics and logics as culturally situated. Quantity is not always absolute (as in 2=2=4) but sometimes relational, as in Yoruba. Verran’s experience and research as a teacher in Nigeria is the basis for this important contribution to the sociology of mathematics and ethnomathematics.]

 

Wigner, E. (1960), “On the Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Communications on Pure and Applied Mathematics , 13: 1–14. [A classic paper in defense of the idea that the mathematical structure of a physics theory often points the way to further advances in that theory and even to empirical predictions. It is basically an argument in support of “pure” mathematics.]

 

Wright, Ronald (2004), A Short History of Progress, Philadelphia: Da Capo Press. [Looking over the long history of humanity, Wright sees not the unfolding a linear evolution of progress but rather a series of “progress traps.” He reveals a history of “progress and disasters” that should serve as a warning to humanity and especially to those people who assume that progress is an inevitable and positive manifestation of human exceptionalism.]

 

Zaslavsky, C. (1999), Africa Counts: Number and Pattern in African Cultures, 3^rd ed. Chicago: Lawrence Hill Books (orig. publ. 1973 by Prindle, Weber, and Schmidt). [This is one of the earliest efforts to document the experience of mathematics in a non-Western culture and to view it in a positive civilizational perspective.]

 

Zeleza, Paul Tiyambe and Ibulaimu Kakoma (2005), Science and Technology in Africa. Trenton NJ: Africa World Press. [The authors deal with scientific and technology literacy, production, and

consumption in modern Africa. The focus is on developments in information technology and biotechnology in the context of The Knowledge Society in a globalizing context.]

 

Biographical Sketch

 

Dr. Sal Restivo is widely recognized as one of the founders of the field of Science and Technology Studies (STS), a pioneer in ethnographic studies of science, a founder of the modern sociology of mathematics, a contributor to public sociology and a prominent figure in the radical science movement of the 1960s. Dr. Restivo was Professor of Sociology, Science Studies, and Information Technology in the Department of Science and Technology Studies, at Rensselaer Polytechnic Institute in Troy, New York until his retirement in June 2012. He is Special Lecture Professor in STS at Northeastern University in Shenyang, China; a former Special Professor of Mathematics, Education, and Society at Nottingham University in Great Britain; and a former Hixon/Riggs Professor of Science, Technology, and Society at Harvey Mudd College. In 2012, he was a Senior Postdoctoral Fellow at the University of Ghent in Belgium. He is a founding member (1975) of and a former president (1994/95) of the Society for Social Studies of Science.

 

 

 

 

 

 

 

 

 

 

 

 

 

Sal Restivo, (2012), MATHEMATICS, CIVILIZATION, AND PROGRESS, in

History of Mathematics, [Eds.UNESCO-EOLSS Joint Commitee], in Encyclopedia of Life Support Systems (EOLSS), Developed under the Auspices of the UNESCO, Eolss Publishers, Paris, France, [http://www.eolss.net]

 

MATHEMATICS, CIVILIZATION, AND PROGRESS

 

Sal Restivo

Department of Intercultural Communication and Interaction, University of Ghent, Ghent Belgium

 

Keywords: abstraction, arithmetic, Chinese mathematics, civilization, ethnomathematics, European mathematics, generalization, geometry, Greek mathematics, Indian mathematics, mathematical workers, mathematicians, mathematics, non-Euclidean geometry, Platonism, proofs, progress, puzzle solving, social constructionism, sociology, transcendental

 

Contents

 

1.  Prologue: Interrogating the terms of our discourse

2.  Introduction: mathematics and civilization

3.  Mathematics and Civilization: Case Studies

4.  The Social Roots of Mathematics

5.  Puzzles and Proofs

6.  Generalization (“Abstraction”) and Self-consciousness

7.  Conclusion Acknowledgement Glossary Bibliography Biographical Sketch

 

Summary

 

This chapter provides an overview of the comparative history of mathematics in the context of the ideas of civilization and progress. The narrative is guided by the idea that mathematics is a social practice and not a set of ideas revealed through some sort of heavenly or Platonic discourse. This implies that mathematics has a normative dimension in terms of its cultural and professional settings. The chapter begins with a critical review of the terms of the discourse, namely mathematics, civilization, and progress. Following some general remarks on mathematics and civilization, the reader is guided through case studies of mathematics in its civilizational and cultural contexts including mathematics in China, India, and Greece, as well as modern Europe. A more analytical section follows, summarizing the idea of the social roots of mathematics, the development and functions of puzzle solving and proofs, and generalization as a way to think about “abstraction” in concrete terms. The narrative follows an historical social science perspective that draws on the ideas of classical and contemporary social theorists from Durkheim and Spengler to Randall Collins and Sal Restivo.

 

1.  Prologue: Interrogating the Terms of Our Discourse

 

Let us begin by briefly interrogating the three terms that make up the title of this essay. First, we should ask, borrowing the title of the book by Reuben Hersh (1999): What is mathematics, really? Mathematics has been shrouded in mystery and halos for most of

its history. The reason for this is that it has seemed impossible to account for the nature and successes of mathematics without granting it some sort of transcendental status. Classically, this is most dramatically expressed in the Platonic notion of mathematics.

 

Briefly, what we call Platonism in mathematics refers to Plato’s theory of Forms. Skirting the complexities of scholarly discourse, Plato is associated with the idea that there are “Forms” or “ideals” that are transcendent and pure. These immaterial Forms exist in a realm outside of our everyday space and time. They are the pure types of the ideas and concepts we manage in our everyday world.

 

This over-simplifies Plato but is consistent with a long tradition in the history and philosophy of mathematics. Consider, for example, the way some scholars have viewed the development of non-Euclidean geometries (NEGs). The mathematician Dirk Struik (1967: 167), for example, described that development as “remarkable” in two respects. First, he claimed, the ideas emerged independently in Gȍttingen, Budapest, and Kazan; second, they emerged on the periphery of the world mathematical community (most notably in the case of Kazan and to a lesser extent Budapest). And the distinguished historian of mathematics, Carl Boyer (1968: 585) characterized the case as one of “startling…simultaneity.”

 

These reflect classical Platonic, transcendental views of mathematics. One even finds such views in the forms of the sociology of knowledge and science developed from the 1920s on in the works of Karl Mannheim and Robert K. Merton and their followers. Mannheim, for example, wrote in 1936 that 2+2 = 4 exists outside of history; and Merton championed a sociology of science that focused on the social system of science and not on scientific knowledge which he claimed lay outside of the influences of society and culture.

 

His ambivalence about this is reflected in his critical reply to G.N. Clark’s criticism of Boris Hessen’s historical materialism of Newton’s Principia (Merton, 1967: 661-663). Clark opposed Hessen’s (Marxist) political economy of the Principia with a defense of Newton’s “purely” scientific motives. Merton argued that individual motivations do not change the structural facts of the matter and in this case they support Hessen’s argument. This doesn’t reach to the core of the social construction of scientific knowledge but it does demonstrate at least an appreciation for the contextual foundations of that knowledge.

 

There are a couple of curiosities in the case of non-Euclidean geometry (NEGs). Even a cursory review of the facts reveals that NEGs have a history that begins already with Euclid’s earliest commentators, runs over the centuries through names like Saccheri, Lambert, Klügel, and Legendre, and culminates in the works of Lobachevsky (1793- 1856), Reimann (1826-1866), and J. Bolyai (1802-1860). The concerns over Euclid’s parallels postulate moved geometers eventually to the systematic development of NEGs. The issue was that the parallels postulate, the fifth postulate in Euclid’s system, did not possess the axiomatic self-evidence of the first four postulates, and it could not be derived from the first four. The three creators of NEGs were by no means isolated and working independently. All were connected to Gauss (1777-1855) who had been working on NEGs since the late 1700s.

J. Bolyai was the son of one of Gauss’ friends, W. Bolyai. Gauss and his friend Bolyai were at the University of Göttingen where the parallels postulate was the subject of lectures by Kastner and a number of dissertations. Reimann was Gauss’ dissertation student. And as for Lobachevsky, he did indeed work at a university on the periphery of the European mathematical community, the University of Kazan. However, the university was staffed by distinguished German professors, including Gauss’ teacher,

J.M. Bartels. J. Bolyai developed ideas on non-Euclidean geometries (NEGs) as early as 1823. His “The Science of Absolute Space” was published ten years later in a book written by his father. Lobachevsky published on the foundations of geometry from 1825 on. Reimann’s Habilitationschrift was on the foundations of geometry. Gauss, we know, wrote about NEGs in letters to W. Bolyai (December 17, 1799), Taurinus (November 8, 1824), and to Besel (January 27, 1829). He also wrote about NEGs in published notes from 1831 on. There are two short reviews on NEGs in Göttingische Anziegen in 1816 and 1822. One has to wonder why in the face of the facts of the case Struik and Boyer chose to view things as “remarkable” and “startling.”

 

Classically, the story of the development of NEGs was told in the context of “pure” mathematics. Thus, to take the case of Riemann as an example, the story was that he constructed the generalization of elliptic geometry as a purely mathematical exercise. The idea that there was a concrete possibility of practical applications for this exercise was not a consideration. In the light of a more realistic sociological and network analysis, Riemann’s work along with that of Gauss, Lobachevsky, Bolyai, Helmoltz, and Clifford, the story of NEGs takes on a different shape.

 

To some extent, they all agreed that Euclidean geometry was an unimpeachable system of ideal space and logic. It could be read as a game played in accordance with a set of formal rules. In fact, however, they interrogated Euclidean geometry in terms of whether it was a valid representation of “actual space.” This should be tested not by mathematics, not what is within the confines of the social world of mathematics per se, but should be tested scientifically – by observation and some mode of experimentation.

 

The sociological generalization this leads to is that if you are given a “genius” or a startling event, search for a social network –cherchez le réseau. No one has made the case for social networks as the roots of ideas more powerfully than Randall Collins (1998). The rationale here should become clearer over the course of this chapter.

 

Even more curious in the case of the sociology of knowledge is the fact that already in his The Elementary Forms of Religious Life published in French in 1912, Emile Durkheim had linked the social construction of religion and the gods to the social construction of logical concepts. Durkheim’s program in the rejection of transcendence languished until the emergence of the science studies movement in the late 1960s and the works of David Bloor, Donald MacKenzie, and Sal Restivo in the sociology of mathematics.

 

It is interesting that a focus on practice as opposed to cognition was already adumbrated in Courant’s and Robbins’ classic “What is Mathematics?” (1906/1995). We must turn to active experience, not philosophy, they wrote, to answer the question “What is mathematics”? They challenged the idea of mathematics as nothing more than a set of

consistent conclusions and postulates produced by the “free will” of mathematicians. Forty years later, Davis and Hersh (1981) wrote an introduction to “the mathematical experience” for a general readership that already reflected the influence of the emergent sociology of mathematics. They eschewed Platonism in favor of grounding the meaning of mathematics in “the shared understanding of human beings…” Their ideas reflect a kind of weak sociology of mathematics that still privileges the mind and the individual as the creative founts of a real objective mathematics.

 

Almost twenty years later, Hersh, now clearly well-read in the sociology of mathematics, wrote “What is Mathematics, Really?” (1997). The allusion to Courant and Robbins is not an accident. Hersh does not find their definition of mathematics satisfactory. In spite of his emphasis on the social nature of mathematics, Hersh views this anti-Platonic anti-foundationalist perspective as a philosophical humanism. While he makes some significant progress by comparison to his work with Davis, by conflating and confusing philosophical and sociological discourses, he ends up once again defending a weak sociology of mathematics. The modern sociology of mathematics associated with the science and technology studies movement that emerged in the late 1960s has established mathematics as a human construction, a social construction. Mathematics is manufactured by humans on the earth out of the material and symbolic resources at their disposal in their local environments. While traditional philosophical and sociological discourses have become estranged, especially in the arena of science studies, there are efforts abroad to reconcile the two disciplines consistent with the interdisciplinary turn in contemporary research and theory. In this sense, one can consider Hersh’s philosophical humanism a step in that direction.

 

The second term in my title, “civilization” is not without its controversial features. In the ancient world, “civilized” peoples contrasted themselves with “barbarians;” in modern times, “civilized” has been opposed to “primitive” or “savage.” The relevance of the concept of civilization to the topic of mathematics lies in its association with the idea of progress. More to the point is the fact that different civilizations (variously “nations,” “societies,” and “cultures”) are associated with different mathematical traditions (v. Restivo, 1992, 23-88). To the extent that humans have developed in ways that can be captured in the ideas of “evolution” and “progress,” mathematics, in conjunction with science and technology more generally, is assumed to have contributed positively to and benefited from those developments. The Scottish philosopher Adam Ferguson (1723-1816) is credited by Benveniste (1954) with introducing the term “civilization” in its modern sense into the English language in his Essay on the History of Civil Society (1767), and perhaps as early as 1759. The term also appears in the works of Boswell (1772), Adam Smith (1776), and John Millar (1771). Mirabeau (1757) introduces the term in French in his L'Ami des hommes ou traité de la population. Just as the individual grows from infancy to adulthood, Ferguson wrote, the species advances from “rudeness to civilization.” Set at the pinnacle of forms of society, civilizations are characterized by complexity, hierarchies, dynamism, advanced levels of rationality, social progress, and a statist form of government. Rousseau, by contrast, viewed civilization as opposed to human nature.

 

If we adopt Ferguson’s view of civilization, then clearly mathematics has been both a result of the emergence and development of civilization and a contributor to that

development. If on the other hand we adopt Rousseau’s viewpoint, the virtues of science and mathematics and the very idea of “civilization” are made severely problematic.

 

What about progress, the very idea? Arguably, the idea comes into Western and world culture in the Old Testament with its conception of linear time and a God that moves through time with humans (e.g., Sedlacek, 2011: 47). The idea of scientific and technological progress was fueled by the seventeenth century advances in science and literature by such cultural giants as Galileo, Newton, Descartes, Molière, and Racine. The idea of social progress was added later. Early in the eighteenth century, the Abbé de Saint Pierre advocated establishing political and ethical academies to promote social progress. Saint Pierre and Turgot influenced the Encyclopedists. The great Encyclopédie was produced by a group of eighteenth century philosophers under the direction of Denis Diderot.

 

It defines the Enlightenment program of promoting reason and unified knowledge. It was at this point that social progress became mated to the values of industrialization and incorporated into the ideology of the bourgeoisie. Scientific, technological, and social progress were all aspects of the ideology of industrial civilization. Veblen, for example, argued that the various sciences could be distinguished in terms of their proximity to the domain of technology. Thus, the physical sciences were closest to that domain, even integral with it, whereas such areas as political theory and economics were farther afield. We have entered an era of machine discipline unlike any in human history. And now we stand on the threshold of machines that will discipline us with conscious awareness and values, including social and sociable robots (the so-called robosapiens), and cyborgs.

 

There have been attempts to identify a type of progress that is independent of material or technological criteria (see, for example, the discussion in Almond, Chodorow, and Pearce, 1985, and the classic criticisms in Roszak, 1969/1995). For many ancient as well as modern thinkers, the idea of progress has always been problematic. We are right to be concerned about the actual and potential impacts of our new bio- and nano- technologies. But one finds similar concerns in Plato’s Phaedrus. There, in the dialogue between Theuth and king Thamus concerning the new technology of writing, Theuth makes promising predictions about the impact of writing.

 

The king claims to be in a better position to do what in effect is a “technology assessment,” and concludes that writing will have the opposite of the effects predicted by Theuth. The cultural meaning of science has fared no better. Where the Rousseaus and the Roszaks saw danger and alienation in science, the Francis Bacons and Bronowskis saw civilization and progress. When the biochemist J.B.S. Haldane wrote about a future of human happiness built on the application of science, Bertrand Russell replied with a vision of science used to promote power and privilege rather than to improve the human condition. St. Augustine worried about the invention of machines of destruction; Spengler predicted that humans would be annihilated by Faustian man. Fontenelle, in the first modern secular treatise on progress published in 1688 argued that science was the clearest and most reliable path to progress. Rousseau, by contrast,

argued that science and the arts have corrupted our minds. The author will draw attention to some additional examples in his conclusion in this chapter.

 

By its intimate association with the very foundations of science, mathematics does not escape this ambivalence. But it stands apart from science in terms of its stronger association with human progress. In the seventeenth and eighteenth centuries, a wave of positivism fueled by Newton’s achievements evoked nothing but the promise of progress among mathematicians of that period. The historian Florian Cajori (1894: 4) had no question about the connection between mathematics and human progress. For Alex Bellos (2010: ix), mathematics is (“arguably”) the foundation of all human progress.

 

Progress, then, can be viewed in terms of “amelioration” or “improvement” in a social or ethical sense. Are we more advanced than cultures that are less dominated by machines and machine ideology? How do we measure the primacy of humans and ecologies and how do we sustain them in any given culture? Can we bring them to fruition and nourish them in any culture, or are some more friendly to the primacy of humans and ecologies than others? These issues are really matters of degree associated with the degree to which individuation of the self (and then the myth of individualism, selfishness, and greed) has progressed in any given society. Furthermore, the degree of awareness of and attention to ethics, values, and social justice has to come into consideration here. It is impossible to even discuss the idea of progress without engaging ideas about and the value of the person, freedom, and democracy.

 

It may be possible to define progress in a way that takes it out of the realm of hopes, wishes, and dreams and plants it more firmly on a meaningful (and even perhaps measurable) foundation. Following Gerhard Lenski (1974: 59), progress can be defined as the process by which human beings raise the upper limit of their capacity for perceiving, conceptualizing, accumulating, processing, mobilizing, distributing, and utilizing information, resources, and energy in the adaptive-evolutionary process. The relationship between adaptation and evolution is a paradoxical one. On the one hand, survival depends on the capacity to adapt to surroundings; on the other hand, adaptation involves increasing specialization and decreasing evolutionary potential. Adaptation is a dead end. As a given entity adapts to a given set of conditions, it specializes to the point that it begins to lose any capacity for adapting to significant changes in those conditions. The anthropologists Sahlins and Service (1960: 95-97) summarize these ideas as follows:

 

Principle of Stabilization: specific evolution (the increase in adaptive specialization by a given system) is ultimately self-limiting.

 

General evolution (progressive advance measured in absolute terms rather than in terms of degrees of adaptation in particular environments) occurs because of the emergence of new, relatively unspecialized forms.

 

Law of Evolutionary Potential: increasing specialization narrows adaptive potential. The more specialized and adaptive a mechanism or form is at any given point in

evolutionary history, the smaller is its potential for adapting to new situations and passing on to a new stage of development.

 

We can add here the Law of Adaptive Levels: adaptation occurs at different levels across various life orders and systems and occurs at different speeds in different spatial arenas. This law draws attention to the complexity of adaptation and the general processes of variation and selection. Adaptation suggests an active agent in a stable environment. But active agents can and do change their environments in ways that make different demands on the adapting agents. Looked at another way, environments have agential like dynamics. Law of Agent-Environment Entanglement.

 

Perhaps the most important aspect of the ideology of science is that it is (in its mythical pure form) completely independent of technology. This serves among other things to deflect social criticism from science onto technology and to justify the separation of science from concerns about ethics and values. Interestingly, this idea seems to be more readily appreciated in general by third world intellectuals than by the Brahmin scholars of the West and their emulators. Careful study of the history of contemporary Western science has demonstrated the intimate connection between what we often distinguish as science and technology. It has also revealed the intimate connection between technoscience research and development and the production, maintenance, and use of the means (and the most advanced means) of violence in society. Not only that, but this is true in general for the most advanced systems of knowledge in at least every society that has reached a level of complexity that gives rise to a system of social stratification.

 

Contradictions and ambivalence about science, technology, and progress may be built into the very core of our cultural machinery. Agricultural activities in the ancient Near East reduced vast forests to open plains, and wind erosion and over-grazing turned those areas into deserts. Deforestation in ancient China led to the development of the loess plateau. Loess sediment gives the Yellow River (nicknamed “China’s Sorrow”) its signature color and flooding pattern. Was deforestation necessary for building China into the greatest civilizational area on earth between the first and sixteenth centuries of the common era? Or were there conservation principles that the ancient Chinese could have relied on without detracting from their cultural development? There is some evidence that at least some of the deforestation they caused could have been avoided. The deforestation experiences of China, Rome, and other civilizational areas of the ancient world are being repeated today and offer cautionary tales for an era characterized by many hard to monitor emerging and converging technologies, that is, technocultural systems.

 

At the end of the day, it should be clear that progress is not easy to define, and that it is even harder to point to examples of progress that resist critical interrogation. How can we sustain the idea of progress in the face of the widespread ecological, environmental, and human destruction that has characterized the industrial age? The fact is that the destruction and danger we see all around us is integrally connected to the very things we use to mark the progress of humanity. For these reasons, we must be cautious when considering whether any of the sciences, engineering disciplines, or mathematics have contributed to or served as signposts of progress. Mathematics, like all systems of knowledge, does not exist in a vacuum. It is always connected to social institutions and

under the control of the most powerful institutions in any given society. All of this may put too much of the onus on the sciences and technology when what we are dealing with is culture in general. Is it possible that cultures by their very natures inevitably destroy planets?

 

It should be clear from this brief introduction that the terms of our title, “mathematics,” “civilization,” and “progress” are all imbued with some level of ambivalence and uncertainty. It remains to be seen whether in the rest of this chapter we can find our way to greater certainty about the meaning and implications of these terms.

 

2.  Introduction: Mathematics and Civilization

 

Karl Mannheim (1893-1947) and Oswald Spengler (1880-1936) defend diametrically opposed positions on the possibility of a sociology of mathematics. Mannheim argues that mathematics is exempt from sociocultural and historical explanations; mathematics is not an ideology, and mathematical truths are not culturally relative. This view has been reinforced by Pythagoreans and Platonists who believe that mathematical truths are eternal objects that exist independently of the flux of historical experience and outside of time and space. Most historians, philosophers, and sociologists of science have traditionally adopted a basically Mannheimian view of mathematics.

 

Spengler, on the other hand, holds that each culture has its own conception of number. Spengler's notion of the "soul” of a civilization cannot provide the basis for an adequate sociological analysis. However, Spengler's goal of explaining mathematics in terms of the particular social and historical forms in which it is produced is sociologically viable. Spengler’s argument is summarized in two statements: (1) “There is not, and cannot be, number as such." There are several number-worlds as there are several cultures; and (2) "There is no Mathematik but only mathematics." Spengler's objective in his analysis of "number" is to show how a crystallized culture demonstrates its idea of the human condition, of what it means to be human. The "peculiar position" of mathematics rests on the fact that it is at once science, art, and metaphysics. It is safe to assume, taking some liberties, that Spengler’s use of “Culture” is more or less commensurate with what we commonly mean by “civilization.”

 

The author takes some additional liberties here with Spengler’s spiritualized materialism in order to avoid some of the idiosyncrasies of his vocabulary. Number, like God, represents the ultimate meaning of the natural world. And like myth, number originated in naming, an act that gives humans power over features of their experience and environments. Nature, the numerable, is contrasted with history, the aggregate of all things that have no relationship to number. Note that in this moment it appears that Mannheim and Spengler might be at one on the nature of mathematics. While Spengler clearly sees a closer connection between mathematics and culture than Mannheim does, he does as the following paragraph illustrates harbor some ambivalence.

 

Spengler argues against treating earlier mathematical events as stages in the development of "mathematics”. This is consistent with his thesis on the incommensurability of Cultures and with his cyclical view of historical change. His general schema of Classical and Western styles and stages in "Culture," "number," and

"mind" is essentially an analysis of world views. This is reflected in Spengler's attempt to correlate mathematical and other sociocultural "styles.” For example, he argues that Gothic cathedrals and Doric temples are “mathematics in stone." Spengler is aware of the problem of the limits of a "naturalistic" approach to number and pessimistic about a solution. It is impossible, he writes, to distinguish between cultural features that are independent of time and space, and those that follow from the forms of culture manufactured by humans.

 

Finally, Spengler claims that a deep religious intuition is behind the greatest creative acts of mathematicians. Number thought is not merely a matter of knowledge and experience, it is a "view of the universe," that is, a world view. The second claim Spengler makes is that a "high mathematical endowment" may exist without any "mathematical science.” For example, the discovery of the boomerang can only be attributed to people having a sense of mathematics that we must recognize as a reflection of the higher geometry.

 

Sociologists of mathematics have been bold enough about challenging the Platonic conception of number, but they have hesitated to follow Spengler. His ideas must seem mad to scholars and laypersons, and specialists and non-specialists alike, to whom the truth of number relations appears to be self-evident. And yet, the "necessary truth" of numbers has been challenged by mathematical insiders and outsiders. One of the outsiders is Dostoevsky. In his Notes from Underground (1864/1918), Dostoevsky argues that 2+2 = 4 is not life but death, impudent, a farce. It’s nonetheless “excellent” and we must give it its due, but then we must recognize that 2+2=5 is also sometimes “a most charming little thing”. Contrast Dostoevsky’s perspective with Orwell’s (1949) use of these two equalities in 1984. For Orwell, 2+2=4 stands for freedom and liberty; 2+2=5 stands for Big Brother totalitarianism. Dostoevsky uses 2+2=4 to stand for everyday routines and tradition; 2+2=5 represents creativity.

 

Dostoevsky's remarks are not merely a matter of literary privilege. Mathematicians and historians and philosophers of mathematics have also challenged the conventional wisdom on number. We should not expect ordinary arithmetic to apply in every physical situation; whether it does or does not has to be based on our experience in different physical situations. In other words, whether or not 2+2=4 is always an empirical question. Where we have long term experiences with situations in which 2+2=4 we are justified in considering those situations closed to further interrogation, that is, we are justified in taking the equality for granted. This is not a warrant for universalizing mathematics uncritically.

 

Studies in ethnomathematics have helped to reinforce and ground the notion that mathematics and logics are culturally situated (cf. Benesch, 1992). African mathematics has posed problems for European intellectuals since at least the publication of Robin Horton’s (1997) studies on patterns of thought in Africa and the West beginning in the 1960s. The earliest findings and interpretations suggested a different way of reasoning and a different logic in African cultures by comparison with the Europeans. Ethnomathematics has helped to sort out the early discourses and ground differences in cultural patterns rather than in mental proclivities. Malagasy divination rituals, for example, rely on complex algebraic algorithms. Some peoples use calendars far more

abstract and elegant than those used in European cultures, notably the Chinese and the Maya. Certain concepts about time and equality that Westerners assumed to be universal in fact vary across cultures. The Basque notion of equivalence, for example, is a dynamic and temporal one not adequately captured by the familiar equal sign. Other ideas taken to be the exclusive province of professionally trained Western mathematicians are, in fact, shared by people in many societies. (see D’Ambrosio, 2006; Mesquita, Restivo, and D’Ambrosio, 2011; Eglash, 1999; Ascher, 2004; Verran, 1992).

 

There is, in brief, a rationale for pursuing the Spenglerian program for a sociology of mathematics based on the views of at least some mathematicians, historians of mathematics, and observers of numbers. The author refers here to authors such as Dostoevsky and Orwell, mathematicians such as Dirk Struik and Chandler Davis, philosophers of mathematics such as Paul Ernest and Leone Burton, and historians of mathematics such as Morris Kline.

 

Sociologists are obliged to interrogate mathematics in a comparative perspective: how does mathematics develop at different times and places; who are the noted mathematicians, what are the social positions they held and how were they related to one another; and what are the social conditions within and outside of mathematical communities as they go through phases of growth, stagnation, and decline. The degree of "community" among mathematicians, the level of specialization, the extent of institutionalization and the relative autonomy of the social activity of mathematics, it should be stressed, are variable across time and space.

 

It is possible to narrate the history of mathematics as a more or less linear unfolding that gives the appearance of an inevitable “logical evolution.” However, the evidence is that there are a number of variations among the types of mathematics produced in different cultures. The latter "horizontal" variations are prima .facie evidence for the Spengler thesis. But what about the long-term trends? These too are socially determined, and in two different senses. First, and in a weaker sense, the "longitudinal" development of mathematics does not occur without interruptions, nor does it unfold in a single cultural context. Mathematicians move along a certain path at some times and not others. This implies among other things that the concepts of truth and what counts as a proof in mathematics vary over time. A sequence in mathematical development or in the network of mathematicians will stop, start, stop, and start again over a period of a hundred or a thousand years. What drives these processes? And why do particular mathematicians at particular times and places make the major advances and not others? More to the point, how is it that some mathematicians and not others who might be better candidates get credit for discoveries and inventions? This question opens up an opportunity to consider eponymy and mis-eponymy in mathematics.

 

In the sciences, eponymy refers to the process of naming an invention or discovery after the person who made the invention or discovery. Some well-known examples are Ohm’s law, the Pythagorean theorem, and Pascal’s triangle. In 1980, the statistics professor Stephen Stigler (1980) formulated "Stigler’s law of eponymy:” stated in general terms it states that no discovery or invention is named after its original discoverer or inventor. Stigler then attributed “Stigler’s Law” to the sociologist of science Robert K. Merton (1961, 1963; and on the related Matthew effect, see Merton,

1968), thus making Stigler’s Law self-exemplifying (further exemplifying Merton’s fondness for the self-exemplifying hypothesis). Merton had already formulated the idea that all discoveries are in principle multiples. Consider the following examples: Pythagoras’ theorem was already known in Mesopotamia at least a thousand years earlier (crediting Pythagoras with a proof is entirely speculative). The Chinese had Pascal’s triangle at least 320 years before he was born. L’Hôpital’s Rule is probably due to Johann Bernoulli; Cardano’s formula for solving cubic equations comes from Tartaglia; Pell’s equation is due to Fermat; Benford’s Law is due to Simon Newcomb; and the Möbius strip could just as easily be named for Listing (Polster and Moss, 2011).

 

Following Spengler and more recent developments in the sociology of mathematics, it is possible to identify social factors that affect the variations, interruptions, progressions, and retrogressions characteristic of the longitudinal development of mathematics. There is a stronger sense in which the longitudinal development of mathematics towards more "advanced" forms is socially determined. The longitudinal development of mathematics reveals the social aspects of mathematical work. Much of professionalized mathematics is created in response to stimuli from within the mathematical community, especially as mathematicians go about playing competitive games with one another. The development of higher and higher levels of “abstraction” (see immediately below), for example, reflects the increasing self-consciousness of mathematicians about their own operations. This in turn reflects higher levels of specialization and institutional autonomy among mathematicians. The Spengler thesis is true in a very strong form: “number”, and all that it stands for metaphorically, is a socially created activity, or more technically, a social construction (see the detailed explanation of this much abused term in Restivo and Croissant (2008).

 

We should consider replacing the term “abstraction” with the term “generalization.” The reason is that what we commonly understand as abstractions are simply concrete forms constructed under conditions of professionalization and disciplinary closure. As mathematics becomes more organized and disciplined, mathematicians build new levels of mathematics on the grounds of earlier mathematical forms. This removes mathematics further and further ceteris paribus from the everyday world and gives rise to the idea of abstraction. But it is important to understand that we are dealing with new levels of concreteness. The term “abstraction” makes us vulnerable to the myths and ideologies of purity and even to ideas about heavens and gods. It is in my view better to eliminate it from our vocabulary in the sense that it is used in the sciences and mathematics.

 

3.  Mathematics and Civilization: Case Studies

 

The world history of mathematics has not unfolded in a unilinear, unidirectional manner. The Greeks, for example, took a step backward from the Babylonian achievements in notation. Different types of mathematical systems have developed in different parts of the world; and rival forms of mathematics have sometimes developed within societies and professional networks. Hindu mathematics, especially in the period before the influx of Greek astronomy (ca. 400CE) placed unique emphasis upon large numbers. Geometry, arithmetic, number theory, and algebra were ignored in favor of the use of numbers in "sociological" schemes. The Upanishads (ca. 700 to 500BCE) are full

of numerical (or more accurately, numerological) descriptions: 72,000 arteries; 36,360, or 36,000 syllables; the 33, 303, or 3306 gods; the 5, 6, or 12 basic elements out of which the world is composed. The wisdom of the Buddha is illustrated by the gigantic numbers he can count out (on the order of 8 times 23 series of 107), and his magnificence is shown by the huge number of Bodhisattvas and other celestial beings who gather to set the scenes for his various sutras. The Hindu cosmology includes a cyclical view of time that enumerates great blocks of years called yugas. There are four yugas ranging from 432,000 to 1,728,000 years, all of which together make up one thousandth of a kalpa or 4,320,000,000 years.

 

This emphasis upon immense, cosmological numbers, gives a distinctively Hindu view of the near-infinite stretches of being that surround the empirical world. It seems almost inevitable that the Hindus should have invented zero (sunya, emptiness, in Sanskrit). The concept sunya, developed about 100CE, was the central concept in Madhyamika Buddhist mysticism, and preceded the invention of the mathematical zero about 600CE. Brahmagupta published a number of rules governing the use of zero and negative numbers in his Brahmasputha Siddhanta (ca. 630CE). Classical Indian world views are permeated with a “mathematics of transcendence.” Numbers were used as a technology for transcending experience not as a mathematics that was directed toward rationalistic generalization. Numbers were used rhetorically to mystify, impress, and awe. In general, numbers were used numerologically rather than mathematically. The social roots of this distinctive mathematical system lie in the particularly exalted status of Indian religious specialists. A sociologist of mathematics would search for the roots of the Hindu emphasis on large numbers in the great variety of ethnic groups making up Indian society, institutionalized in the ramifications of the caste system.

 

The cosmological significance of Chinese mathematics has an ideographic bias. Numbers, and higher mathematical expressions, are written as concrete pictures. The system of hexagrams that make up the I Ching, the ancient book of divination, was continuously reinterpreted in successive Chinese cosmologies as the basic form of the changing universe. Chinese arithmetic and algebra were always worked out in positional notation. Different algebraic unknowns, for example, could be represented by counting sticks laid out in different directions from a central point. Chinese algebra, at its height around 1300CE, could be used to represent fairly complex equations, and included some notion of determinants (i.e., the pattern of coefficients). But it could not be developed in the direction of increasingly general rules. The ideographs (and the social conditions of their use) helped preserve the everyday roots of mathematics.

 

Why did Chinese mathematics take this form? Probably for some of the same reasons that account for the maintenance of ideographic writing among Chinese intellectuals. Both gave a concrete aesthetic emphasis to Chinese culture. The ideographic form had technical limitations that a more generalized form - an alphabet, a more mechanical mathematical symbolism - would have overcome. Ideographs are hard to learn; they require a great deal of memorization. But these limitations may in fact have been the reason why Chinese intellectuals preferred them. A difficult notation is a social advantage to a group attempting to monopolize intellectual positions. This may be contrasted with the algorithmic imperative characteristic of periods of rapid commercial expansion.

Writing and mathematics were highly esoteric skills in the ancient civilizations when they were first developed. Those who possessed these skills were almost exclusively state or religious dignitaries. Writing and mathematical notation tended therefore to be retained in forms that were very difficult to read and interpret, except by those who could spend a long time in acquiring familiarity with them. Sanskrit, for example, was written without vowels and without spaces between the words. Egyptian writing was similarly conservative. Chinese writing and mathematics are notable because archaic styles lasted much longer than anywhere else. The over-riding cultural issue might be that China was, as Leon Stover argued (1974: 24-25), a “once and always Bronze Age culture,” the only primary civilization to develop its Bronze Age to the fullest.

 

The development of ideographs and mathematical notation in China was in the direction of greater aggregative complexity and aesthetic elaboration, not of simplification and generalization. The Chinese literati thus managed to make their tools progressively more difficult to acquire. This is in keeping with the unusually high social position of Chinese intellectuals. They maintained their status through an examination system that was used to select officials in many dynasties. Many students of mathematics have contended that a “good” notation is a condition for progress in mathematics. The question for China or for any given society or mathematical community is: why wasn’t a more appropriate symbolism invented at some particular point in the history of mathematics? To answer that question we should envision a struggle between monopolizing and democratizing forces over access to writing and mathematics.

 

Monopolistic groups were strong in highly centralized administrations such as ancient Egypt, the Mesopotamian states, and China. Democratizing forces won the upper hand in decentralized situations, and/or under social conditions where there was a great deal of private business activity – as in ancient (especially Ionian) Greece, and certain periods in ancient and medieval India. The predominance of these forces was to varying degrees opposed by counter-forces. Greek mathematics also had some conservative elements, especially in the Alexandrian period when difficult rhetorical forms of exposition limited the development of algebra. The specific character of mathematics in given world cultures is due to the differential incidence of such conditions.

 

Greek mathematics is distinguished by its emphasis on geometry, generalized puzzles, and formal logical proofs. This is the intellectual lineage of modern Western mathematics. But the history of Greek and European mathematics also shows a divergent type that rose to prominence following the establishment of the classical form. During the Alexandrian period, another form of arithmetic was developed that was used neither for practical calculations nor for puzzle-contests. This was a type of numerology that used the real relations among numbers to reveal a mystical cosmology. The system was connected with verbal symbolism through a set of correspondences between numbers and letters of the Hebrew or Greek alphabets. Any word could be transformed into a related number that in turn would reveal mathematical relations to other words.

 

The social conditions involved in the creation and development of this alternative mathematics are connected with religious movements. Numerology is related to Hebrew Cabbalism, Christian Gnosticism, and the Neo-Pythagorean revival associated with Philo of Alexandria, ca. 20BCE-50CE). The most prominent expositor of this new

mathematics was Nichomachus (ca.100CE). Like Philo, he was a Hellenistic Jew (living in Syria). It was in this Jewish-Greek intellectual milieu of the Levant that the major religious movements of the time were organized.

 

There are variants even in modern European mathematics. There are conflicts between alternative notational systems in the 1500s and 1600s; and a century-long battle between the followers of Newton and those of Leibniz over the calculus. In the nineteenth century, a major dispute arose between Riemann, Dedekind, Cantor, Klein, and Hilbert and critics such as Kronecker and Brouwer. This split continued and widened in the twentieth century. The result was the emergence of schools in conflict over the foundations of mathematics. The main competitors in this arena were the logicists, the formalists, and the intuitionists

 

4.  The Social Roots of Mathematics

 

The social activities of everyday life in all the ancient civilizations gave rise to arithmetic and geometry, the two major modes of mathematical work. Each of these modes is associated with specific types of social activity. The development of arithmetic is stimulated by problems in accounting, taxation, stock-piling, and commerce; and by religious, magical, and artistic concerns in astronomy, in the construction of altars and temples, in the design of musical instruments, and in divination. Geometry is the product of problems that arise in measurement, land surveying, construction and engineering in general. Arithmetic and geometrical systems appear in conjunction with the emergence of literacy in all the earliest civilizations – China, India, Mesopotamia, Egypt, and Greece. These mathematical systems are, to varying degrees in the different civilizations, products of independent invention and diffusion. Note that while it is analytically “simple” to distinguish geometrical and arithmetical systems and methods it is not always so simple to do this in practice.

 

While we have found arithmetic, geometry, number work, and general mathematics in cultures throughout recorded history, special conditions were required for the emergence and crystallization of the discipline of mathematics. The general human ecological conditions for the emergence of modern science are discussed in Restivo and Karp (1974; and see Restivo, 1979, and 1994: 29-48). A combination of organizational and institutional factors rooted in a human ecology was required to foster and sustain the development of mathematical communities with generational continuity. As those conditions crystallized in Western Europe beginning in the 1500s and earlier, the discipline of mathematics emerged when sets of arithmetic and geometrical problems were assembled for purposes of codification and teaching, and to facilitate mathematical studies. Assembling problems was an important step toward unifying mathematics and stimulating generalization.

 

One of the most important steps in unifying and disciplining mathematics arose from efforts to state general rules for solving all problems of a given type. A further step could be taken once problems were arranged so that they could be treated in more general terms. Problems that had arisen in practical settings could now be transformed into hypothetical puzzles, and problems could be invented without explicit reference to practical issues. The three famous puzzles proposed by Greek geometers of the 5th and

4th centuries BCE are among the earliest examples of such puzzles: to double the volume of a cube (duplication of the cube), to construct a square with the same area as a given circle (quadrature of the circle), and to divide a given angle into three equal parts (trisection of the angle). Such problems were related to the non-mathematical riddles religious oracles commonly posed for one another. One account of the origin of the problem of duplicating the cube, for example, is that the oracle at Delos, in reply to an appeal from the Athenians concerning the plague of 430 BCE, recommended doubling the size of the altar of Apollo. The altar was a cube. The early Hindu literature already refers to problems about the size and shape of altars, and these may have been transmitted to Greece by the Pythagoreans, a secret religio-political society. The problem is also a translation into spatial geometric algebra of the Babylonian cubic

equation X ^{3 =}V .

The duplication, quadrature, and trisection problems were popular with the Sophists, who made a specialty out of debates of all kinds. A generation or two later, Plato introduced the constraint that the only valid solutions to these problems were those in which only an unmarked straightedge and a compass were used. This meant that special mechanical devices for geometrical forms could not be used in mathematical competitions. The result was stiffer competitive conditions and an emphasis on intellectual means and “gentlemanly” norms.        Plato's Academy was organized to help an elite group of intellectuals gain political power; and it represented the opposition of an aristocracy to democratization and commercialization. It is not surprising that this elite group of intellectuals developed an ideology of extreme intellectual purity, glorifying the extreme separation of hand and brain in the slave economy of classical Greece.

 

The three famous Greek puzzles and other problems became the basis for mathematical games (i.e., competitions) of challenge-and-response. Various forms of these games are important throughout most of the subsequent history of Western mathematics. Prior to the nineteenth and twentieth centuries, however, the challenge-and-response competitions were often initiated, endorsed, or rewarded by patrons, scientific academies, and governments. Prizes were sometimes offered for solutions to practical problems. Economic concerns as well as governmental prestige were often mixed in with the struggles for intellectual preeminence. At about the same time that they initiated mathematical contests, the Greek mathematicians took two further steps that led to new mathematical forms. They stipulated that a formal, logical mode of argument must be used in solving problems; this represented a further development of earlier methods of proof. And by extending this idea they created systems of interrelated proofs. This culminated in the Elements of Euclid shortly after 300 BCE. In addition to a collection of problems, Euclid presented an explicit body of generalizations in the form of definitions, postulates, and axioms. Euclid, like Aristotle, did not use the term "axiom" but something closer to "common notion." They both self-consciously worked at codifying past human experiences. The process of "systematization-and- generalization” is one of the two major paths to new mathematical forms. The other major path is an "empirical" one.

 

The empirical path to new mathematical forms involves applying existing mathematical concepts and methods to new areas of experience. Most of the early Greek geometrical

puzzles, for example, concerned flat figures. But the methods of plane geometry could be easily extended to solid geometry, and then to the properties of spheres or of conic sections; the work on conic sections eventually led to work on curves of various shapes. The intermittent periods of creativity in Alexandrian mathematics (especially from 300 to 200 BCE and 150-200 CE) were largely devoted to these extensions. No new level of generalization (with the exception of trigonometry, considered below) was achieved, but a number of new specialties appeared.

 

The history of arithmetic shows some of the same processes that occur in the history of geometry. The effort to find general rules for solving numerical problems led gradually to what we now call algebra. Here again we see mathematicians developing the practice of posing problems primarily to challenge other mathematicians. For example, there is this famous problem, attributed to Archimedes (287 to 212 BCE): find the number of bulls and cows of various colors in a herd, if the number of white cows is one third plus one quarter of the total number of black cattle; the number of black bulls is one quarter plus one fifth the number of the spotted bulls in excess of the number of brown bulls, etc. Such problems, involving unknown quantities, led over a very long period to the introduction of various kinds of notations and symbolisms.

 

These took quite different directions in ancient and medieval China and India, the Arab world and later in medieval and Renaissance Europe. The creation of a highly generalized symbolism which could be mechanically manipulated to find solutions did not appear until the late1500s and 1600s in Europe.

 

Over this period, and to different degrees in different parts of the world, algebra underwent an empirical extension. Problems were deliberately created to increase the number of unknowns, and to raise them to successively higher powers. Equations of the form ax + b = c gave way to those on the order of ax⁴ + by³ + cz² = g . The complexity of these, of course, could be extended indefinitely (Vieta in the1580s, for example, was

challenged to solve an equation involving x⁴⁵ ); but the extensions also gave rise to

efforts to find general rules for solving higher order equations. In other words, empirical extensions tended to promote generalized extensions. At the same time, arithmetic was developing in other directions.

 

What is generally called elementary arithmetic (solving numerical problems in, for example, addition, subtraction, multiplication, and division) continued to stimulate efforts to find general rules for solving particular problems. There was tremendous variation from one system of numerical symbols and calculating rules to another in terms of the ease or difficulty with which they could be applied to solving practical problems. Most of the ancient forms of notation made working with large numbers, fractions, or complex operations like division or the extraction of roots difficult; the exposition of problems was usually carried out in words. A great deal of mathematical creativity went into the development of notational systems that could be readily manipulated. Among the most important of these innovations were the invention of decimal place notation and the zero sign in India; the standardization of positional methods for writing multiplication and division (in Europe ca. 1600); and the invention of logarithms by the Scotsman Napier in 1614, for use in astronomy, navigation, and commerce.

A different development in arithmetic led to what we now call “number theory.” This focused on the properties of numbers themselves. As early as Eratosthenes (ca. 230 BCE), efforts were made to find a general method for identifying prime numbers. There were also various propositions about how numbers are composed of other numbers. The Pythagorean work on “triangular’ and “square” numbers anticipated Fermat’s theorem that every prime number of the form 4n +1 is a sum of two squares. Number theory was particularly popular in the Alexandrian period in an occultist, cabalistic form. In its more standard puzzle-solving form, it has remained popular among mathematicians from the Renaissance through the modern period.

 

One more branch of mathematics, based on a combination of arithmetic and geometry, developed in the Alexandrian period. Measurements of angles and lines, and the calculation of their ratios, led to the creation of trigonometry, notably by Hipparchus (ca. 140 BCE) and Menelaus (ca, 100 BCE). Trigonometry spread to medieval India and the Arab world, and in Renaissance Europe provided the basis for Napier's development of logarithms.

 

The overall picture so far, then, shows mathematics arising from practical geometry and arithmetic. The development of general mathematical puzzles and the extension of mathematics to new areas led to the emergence of new fields. Geometry became increasingly systematic, and progressively applied to plane and solid figures, to conics, and eventually to trigonometry. Arithmetic gave rise to algebra in successfully more complex forms (based on practical calculating systems), and to number theory. The creation of new fields continued in modern Europe. They grew out of the processes of generalizing (commonly understood in terms of increasing levels of abstraction). New fields were furthermore the result of extending results to new empirical areas, and the combination of existing mathematical fields into hybrid fields. The combination of algebra with a new coordinate representation in geometry by Descartes and Fermat produced analytic geometry.

 

Consideration of the problems of motion and the study of curves gave rise to the calculus in the 1600s. Calculus was then applied to successively more complex functions (empirical extension); and eventually (in the 1800s) it was generalized into a theory concerning such things as the rules for solving equations, and the properties of all functions (generalized extension). It should be noted that the drive towards creating new fields by generalization and extension seems to be characteristic of highly competitive periods. Geometry itself experienced a rapid series of branching around 1800 and thereafter, the best known being the non-Euclidean geometries. But there was also the creation of descriptive geometry by Monge, projective geometry by Poncelet, higher analytical geometry by Plucker, modern synthetic geometry by Steiner and Von Staudt, and topology by Mobius, Klein, and Poincare. In the late nineteenth and early twentieth centuries, Klein, Hilbert, and Cartan unified these different geometries. This unification occurs prominently in Klein’s Erlangen Program and its generalization in Cartan’s program which was designed to place the unification into the framework of Riemannian geometry.

 

In algebra, there was a parallel set of developments after 1800.The effort to find a general solution for the quintic and other higher-order equations led to the creation of

the theory of groups by Abel, Galois, Cauchy, and others. This theory focused on an abstract pattern among the coefficients of equations, and opened up a new area of inquiry in higher mathematics. “Abstract” algebras were created by Boole, Cayley, Sylvester, Hamilton, and Grassman. All of these new tools were applied to other branches of mathematics. Dedekind applied set theory to the calculus, Cantor applied it to the concept of infinity, and others applied it to topology, number theory, and geometry. These developments led to the creation of yet another even more general field toward the end of the nineteenth century, “foundations.” "Foundations" focused on the nature of mathematical objects themselves and with the rules by which mathematics should be carried out. Foundations research has been the focus of a number of opposing schools, and has led to what are probably the most intense controversies in the history of mathematics.

 

The basic forms of mathematics, arithmetic and geometry arise from practical problems in construction, taxation, administration, astronomy, and commerce. Moreover, the stimulus of practical concerns does not simply disappear once mathematics is launched. For example, the basic forms of arithmetic, including the number system, developed over a very long period, during which virtually the sole interest in improvement was to facilitate practical calculations. The same can be said for the invention of logarithms, and much of the development of trigonometry. Other advanced forms of mathematics were also stimulated by efforts to solve practical problems. The development of the calculus was linked to problems in ballistics and navigational astronomy. Newton’s mechanics makes an interesting case study in this regard. In a hallmark paper in the history and sociology of science presented at the Second International Congress of the History of Science in London in 1931, Boris Hessen (1893-1936) situated Newton’s work in the technological problems of his time, his social class position, and Newton’s lack of familiarity with the steam engine and the conservation of energy principle. Hessen’s views contradicted the view that Newton was divinely placed at a particular historical juncture and that his genius was the fountain out of which his physics sprung. The present author noted earlier R.K. Merton’s defense of Hessen’s sociology of science against the idealistic and motivational explanations offered by G.N. Clark. Merton’s argument implicitly opposes Carlyles’s Great Man theory, Alfred North Whitehead’s mystical explanation of Newton’s successes (he was born in the same year that Galileo died), and Alexander Pope’s divine providence conjecture: “Nature and nature’s laws lay hid in night; God said ‘Let Newton be!’ and all was light.”

 

Descriptive geometry and Fourier's analysis answered problems in the production of new machinery in the industrial revolution. Practical concerns do not tell the whole story of mathematics, but they are one component that continuously shapes its history. This suggests a general principle: an increase in the amount, type, intensity, or scope of practical concerns in a society will stimulate mathematical activity. The relationship between economic concerns and mathematics is especially strong; commercial growth tends to be very stimulating for mathematics. Mathematical innovations will also tend to occur when there is a shift to new productive technologies (and when there are shifts to new technologies of warfare and transportation, and shifts to more intensive administrative modes of organization). This implies a link between the development of modern European mathematics and the development of modern industrial technological societies (loosely, “capitalism” because the term “capitalism” does not refer to an actual

economic system but rather expresses an economic ideology). Since this is one factor among several, it does not imply that mathematics must come to an end in non-capitalist societies. It does, however, suggest that the form and content of mathematics (within the constraints noted by Spengler) as we know it today is a product of specific lines of cultural development. One could say that modern mathematics, like modern science, is part of the knowledge system generated by and supportive of capitalism.

The roots of mathematics in practical concerns are more apparent in some cases than in others. For example, the history of Chinese mathematics is primarily a history of an inductive “mathematics of survival." Its origins can be traced from the myth of Yü the Great Engineer's discovery of a magic square on the back of a Lo River tortoise (ca. 500 BCE). China’s mathematical evolution reaches its high point in the late Sung and early Yuan dynasties with the publication of Chu Shih-Chieh's "Precious Mirror of the Four Elements," written in 1313. Chinese mathematics never ventured far from problems of everyday life such as taxation, barter, canal and dike construction, surveying, warfare, and property matters. Chinese mathematical workers could not organize an autonomous mathematical community, and consequently failed to establish the level of generational continuity that is a necessary condition for long-term mathematical development. This helps to explain why the Chinese did not develop the more general forms of higher mathematics.

 

Conditions in ancient Greece were more favorable for generalizing mathematics. The commercial expansion in Greece in the 600s BCE stimulated mathematical growth. Learned merchants practiced and taught mathematical arts, and master-student relationships across generations fostered mathematical progress. Political and economic changes in Greek civilization led to the development of an increasingly elitist and self- perpetuating intellectual community, culminating in the oligarchic conditions and intellectual elitism of Plato's time. The achievements of the "thinking Greeks" depended on a division of labor that divorced hand and brain. The "thinkers" had the "leisure" to reflect on and elaborate mathematics. The class structure of the slave-based society that developed in the post-Ionian period conditioned the development of classical mathematics. Arithmetic was left to the slaves who carried out most commercial transactions, and householders for whom simple calculations were a part of everyday life. The elite intellectual class, by contrast, courted geometry which was considered democratic and more readily adapted to the interests of the ruling classes than arithmetic. What we know as "Greek mathematics" is a product of the classical period.

 

The development of specialties within the division of labor, left unchecked, tends to foster virtuosity. Such specialization tends to increase the specialists' distance from the order and distribution of everyday phenomena and to increase the importance of human- created phenomena, especially symbols. The result is an increase in the level of generalization and the development of ideologies of purity. This is essentially what occurred in classical Greece. Hand and brain slowly reunited following Plato's death; there is already evidence of an increased interest in linking mathematical and practical concerns in Aristotle. In the Alexandrian period, hand and brain were more or less united, but the ideology of purity retained some vitality. This is notably illustrated by Archimedes, whose work clearly exhibited a unity of hand and brain but whose philosophy echoed Platonist purity.

The decline of Greek commercial culture was accompanied by the decline of Greek mathematical culture. The achievements of Archimedes, which brought Greek mathematics to the threshold of the calculus, mark the high point of Greek mathematics. When mathematics was revived in the European commercial revolution (beginning haltingly as early as the twelfth century, the Gothic cathedrals signaling the coming economic revolution by serving as accumulating devices for labor, resources, and machinery), many aspects of the Greek case were recapitulated. European mathematics moved on in the direction of the calculus, rooted in problems of motion. It picked up, in other words, essentially where Archimedes had left off, and under the influence of the Archimedean (and more generally, Greek) writings as they were recovered and translated (and here, of course, the Arabic-Islamic transmissions were critical). By 1676, Newton was writing about mathematical quantities "described by continual motion."

 

The concept of function, central to practically all seventeenth and eighteenth century mathematics, was derived from studies of motion. Newton and Leibniz helped to reduce the basic problems addressed in the development of the calculus – rates of change, tangents, maxima and minima, and summations – to differentiation and anti- differentiation. Infinitesimals, nurtured earlier in the debates of theologians and the scholastics, entered into the process of production. Highly general intellectual ideas in the Euclidean realm of the straight, the flat, and the uniform gave way to the ideas of a dynamic world of guns and machinery, and global navigation and commerce characterized by skews, curves, and accelerations. The search for algorithms, time- saving rules for solving problems, is evident in the writings of the inventors of the calculus (e.g. in Leibniz's "De geometria recondita et analysi indivisibilium atque infinitorium" of 1686).

 

As the industrial “machine" of capitalist society was fashioned, so was the “machine of the calculus." Descartes' analytic geometry, the other great contribution to the development of pre-modern European mathematics, was also characterized by an algorithmic imperative.

 

It was, in spite of the conflicts between Cartesians and Newtonians, from the very beginning in constant association with the development of the Newtonian-Leibnizian calculus. Let us note parenthetically that Newton and Leibniz invented two different calculuses. Newton’s “method of fluxions” was much more indebted to classical geometry (and especially to Archimedes) than is commonly supposed, and Leibniz’s “differential calculus” used a much better notation (Restivo, 1992: 134).

 

The historian of mathematics Boutroux (1919) characterized Descartes’ analytic geometry as an industrial process; it transformed mathematical research into “manufacturing.” The idea that the calculus is linked to the emergence of capitalism (that is, early industrialization) is further suggested by the Japanese case. When the Japanese established a monetary economy and experienced a commercial revolution in the seventeenth century, they also worked out a "native calculus." This was not entirely indigenous since they had probably come into contact with Europe’s calculus from European contacts at their ports of entry.

5.  Puzzles and Proofs

 

Mathematical workers and mathematicians, from the earliest times onward, and especially in the West, have posed puzzles for one another. This practice tends to make mathematics a competitive game. Some periods have been dominated by public challenges such as those that the Emperor Frederick's court mathematician posed to Leonardo Fibonacci (ca. 1200), those that Tartaglia and Cardano posed for one another in sixteenth century Italy, or those that led to such high acclaim for Vieta at the French court in the 1570s. Such puzzle-contests have been important for several reasons. They often involved pushing mathematics into more general realms. Mathematicians would try to invent problems which were unknown in practical life in order to stump their opponents. The search for general solutions to equations, such as those that Tartaglia found for cubic equations and Vieta found for the reduction of equations from one form to another, was directly motivated by these contests.

 

The emphasis on proofs which has characterized various periods in the development of mathematics was partly due to a heightening of the competitiveness in these contests. Greek mathematicians rationalized the concept and method of proof at a time when mathematics was popular among the elite class of philosophers and there was a lot of competition for power and attention in the intellectual arena. This was the same period during which the wandering Sophists challenged one another to debating contests and in doing so began to develop canons of logic. This is completely analogous to the development in mathematics, in terms of both cause and effect. The analogy turns into a virtual identity when we realize that many of the mathematicians of the time were Sophists, and that many of the formal schools that were organized in the classical period (e.g., the Academy) used prowess in mathematics as a grounds for claiming superiority over competing institutions. Stressing proofs was a way of clarifying the rules of the game and escalating the intensity of competition. In general, competitive puzzle- contests are probably responsible for much of the inventiveness characteristic of Western mathematics. This analysis should not obscure the economic stimulus to the initial development of proofs.

 

Thales, the philosopher-merchant (who might have been a composite or imaginary construction), is credited with carrying the idea of a proof to a more general level than the Babylonians and Egyptians. Thales symbolically and iconically personifies the need among the Ionians of his era to develop a comprehensive and organized understanding of physical reality and successful computational methods in the context of the increasingly well-organized economy that they were products of and helped to fashion. Thales' proofs were probably crude extensions of Babylonian or Egyptian "rules" for checking results. In any case, the process of constructing proofs was rationalized over the next three hundred years and eventually led to Euclidean-type proofs.

 

Concern for proof has varied a great deal in the history of mathematics. The Chinese and Hindu mathematicians ignored proofs almost entirely; indeed, they would often present problems without solutions, or with incorrect solutions. These practices were the result of a relatively uncompetitive situation in mathematics in these societies. The social density of mathematicians in these societies was rather low; we rarely hear of more than a few mathematicians working at the same time, whereas in Greece and

Europe the numbers in creative periods are quite high. Most of the Oriental mathematicians were government officials, and thus were insulated from outside competition, while most of the ancient Greek and modern mathematicians were private individuals or teachers in competitive itinerant or formal educational systems. In the Islamic-Arabic world, there was a flurry of mathematical activity in the period 800- 1000 (and later to some extent).There was some concern for proofs (in the works of Tahbit Ibn Qurra, for example), but this was much more limited than in classical Greece. The Greek works they translated stimulated an awareness of and interest in proofs among the Islamic-Arabic mathematicians.

 

The limited emphasis on proofs reflects the fact that their community (or network) was not as densely populated as the Greek mathematical community, competition was not as intense, master-student chains and schools were not as well organized, and generational continuity was limited. Similarly, the episodic history of mathematics in India constantly interrupted generational continuity, the Bronze Age dynastic history of mathematics in China undermined specialization free of the centripetal force of the Emperor’s bureaucracy, and the delimited mathematical renaissance of Japan in the seventeenth century ended abruptly with the consolidation of Tokogawa power (Restivo, 1992: 22-60). Only Europe from the 1500s on was able to sustain generational continuity on a level no other civilization had achieved.

 

In modern Europe, the emphasis on proofs has grown steadily. In the 1600s, Fermat presented his theorems without proofs, and in the 1700s, Euler offered proofs that were not very rigorous. The early 1800s saw a shift towards more rigorous standards of proof; earlier solutions were rejected, not because they were incorrect, but because the reasoning behind them was not sufficiently universal and comprehensive. This went along with a massive increase in the number of people engaged in mathematics (which in turn was the result of the expansion of educational systems, especially in Germany and France and other social changes related to the rationalization of social and economic life). Both this shift towards rigor, and the earlier invention of proofs, had important effects on the nature of mathematics. Both pressed mathematics toward new levels of generalization: proofs had to invoke more general elements than particular numerical examples, and rigorous proofs stimulated the systematic consideration of the nature of mathematics in the nineteenth century.

 

6.  Generalization (“Abstraction”) and Self-Consciousness

 

The "main line" development of Western puzzle-solving mathematics is characterized by an increasing awareness that levels of generalization have been created by the mathematicians themselves. Mathematicians moved beyond a naïve realism about mathematical objects (sometimes real in the material sense, sometimes real in the Platonic sense) when they gradually began to use negative numbers instead of dropping negative roots of equations (as Hindu, Arab, and medieval European mathematicians had done). Later they came to recognize that imaginary numbers (an unfortunate and distracting nomenclature) could be used despite their apparent absurdity. Gauss established a new basis for modern algebra by creating a representational system for complex numbers. Nineteenth century higher mathematics took off from this point.

Mathematicians finally realized that they were not tied to common-sense representations of the world, but that mathematical concepts and systems could be deliberately created. These creations were not ab novo but situated in the cultural objects of the increasingly well-organized mathematical community. The new, more general geometries (including projective and non-Euclidean geometries) popularized the point, and stimulated the creation of new algebras and more generalized forms of analysis. The objects with which modern mathematics deals, however, are real in a particular sense. They are not simply things, as was once naively believed; they are, rather, operations, activities that mathematicians carry out. The imaginary number i is a shorthand for an activity, the operation of extracting a square root from a negative number. This operation, of course, cannot be carried out. But mathematicians had long been used to working backwards from solutions-not-yet-found, to the premises, by symbolizing the solution using an arbitrary designation (e.g., x ). This symbol represented the result of an imaginary operation. The imaginary number i, then, could be used as the basis for other mathematical operations, even though the operation of producing it could never actually be performed. The ordinary arithmetic operations, the concept of a function, the concept of a group - all of these are operations of different degrees of complexity. A natural whole number itself is not a thing but an operation - the operation of counting (and perhaps also other operations whose nature modern mathematicians are untangling). Modern mathematics has proceeded by taking its operations as its units. These are crystallized into new symbols which can then be manipulated as if they were things. In fact, within the social world of mathematics these symbols are things, the matters of fact, and the material resources of the human ecology of the mathematics community (see Alexksandrov, Kolmogorov, and Lavrent’ev, 1963 for a thoroughly materialistic and realistic history of mathematics).

 

A process of reification has gone on in conjunction with the emergence of the notion that generalized operations (what are commonly referred to as “abstractions”) are socially created by mathematicians. Thus mathematics has built upon itself hierarchically by treating operations as entities upon which other operations can be performed. The Western trend in symbolism, then, is not an "accidental" feature of Western mathematical uniqueness; the symbolism was created precisely because the mathematical community was pushing towards this degree of self-consciousness and reflexivity. What I am pointing to here is an emerging awareness among the more reflective mathematicians and philosophers of mathematics that crystallized in the late nineteenth and early twentieth century. That awareness that the creative construction of mathematics came from the hands and brains of situated human beings was enough to fuel the more recent efforts to ground mathematics in the material world of activities and experience. It has not, however, been enough to eliminate all vestiges of Platonic thinking among mathematicians.

 

Mathematics, like other modern activities, has been affected by specialization on a level unknown in earlier historical periods. As a result, the "causal power" of mathematics itself in the relationships between mathematical and other social activities has steadily increased. Mathematical ideas have increasingly become the generative basis for new mathematical ideas. The work setting and institutional context of mathematical activity has become a social foundation of a higher order than the social foundation of subsistence productive activity. Mathematics continues to be socially rooted within the

mathematical community; it is especially important to recognize the social nature of the symbols mathematicians create for communication within their own ranks.

 

This perspective throws new light on an old problem reflected in the title of a famous paper by Eugene Wigner (1960) on “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” The utility of “pure” mathematics for the physical and natural science is not just a coincidence; it must reflect some larger and deeper truth about both mathematics and physics. In fact, the “coincidence” reflects a constant interplay between mathematics and physics. As soon as this is interrupted by the professional and bureaucratic closure of the mathematics community, we can expect the cycles of effective applications to become increasingly rare. This is not just a sociological conclusion based on the theory of closure and autonomy in social systems (Restivo, 1992: 171-175; Restivo, 1993: 263-267) but something that has been recognized by professional mathematicians (e.g., Boos and Niss, eds., 1979).

 

The development of Western higher mathematics, then, is a social and a cultural development. The objects with which mathematicians deal are activities of mathematicians. In building upon the operations already in existence, and making them symbolic entities on which further operations can be performed, mathematicians are self-consciously building upon previous activities in their intellectual community. Mathematics thus embodies its own social and cultural history, and uses it as the base upon which its current community activities are constructed. Western mathematics thus depends upon a particular kind of long-term organization of the intellectual community. This is an organization in which strong links are maintained across generations, and in a highly self-conscious and competitive form. The new attempts to competitively consume the old. The important linkages of teachers and pupils typically found among European mathematicians, together with strong external competition among different mathematical "lineages," have been the social basis for this pattern (on the general theory of lineages and networks as the progenitors of ideas, see Collins, 1998). Once the pattern of competitive self-consciousness was established, subsequent rounds of competition could only escalate the degree of self-reflection and inventiveness among mathematicians. Out of this situation arose the hyper-reflexive concerns of twentieth- century foundations research. It is important to keep in the mind the self-conscious creativity of mathematicians is inseparable from the crystallization of an autonomous mathematical community.

 

7.  Conclusion

 

All thought, in its early stages, begins as action. The actions which you [King Arthur] have been wading through have been ideas, clumsy ones of

course, but they had to be established as a foundation before we could begin to think in earnest.

T.H. White’s Merlyn the Magician What, “In the beginning was the Word?” Absurd.

Then maybe it should say “In the beginning was the Mind?” Or better “…there was Force?”

Yet something warns me as I grasp the pen,

 

 

Act.”

That my translation must be changed again.

The spirit helps me. Now it is exact. I write: “In the beginning was the

 

Goethe’s Faust

 

The history of mathematics can be situated socially and culturally. This task requires a sociology both of the external institutional and organizational conditions of the societies within which mathematical activities are situated, and a sociology of the internal organization of the mathematical activities within communities and social networks of mathematicians. The notion of "internal and external factors" is an analytic device. The Spenglerian idea of mathematics as a world view is not, in the end, compatible with a strict adherence to internal-external analysis. The mathematics of any particular time embodies its own social history. This process becomes increasingly intense as and to the extent that mathematical activity becomes and remains more clearly differentiated from other social activities and more autonomous. But "autonomy" simply means that mathematicians communicate more intensively with each other than with outsiders. It does not mean that mathematicians are more removed from social determinants or that they have unimpeded access to "objective reality." Their activities remain at all times coupled to the social activities of insiders and outsiders, and thus unfold in an environment of multiple social, cultural and historical determinants. This is the rationale for a Spenglerian approach to the sociology of mathematics. That this is becoming more accepted is indicated by a clear turn to practice, experience, and shared meaning in the philosophy of mathematics, the philosophy of mathematics education, and among reflective mathematicians.

 

We are no longer entranced by the idea that the power of mathematics lies in formal relations among meaningless symbols, nor are we as ready as in the past to take seriously Platonic and foundationalist perspectives on mathematics. We do, however, need to be more radical in our sociological imagination if we are going to release ourselves from the strong hold that philosophy has on our intellectual lives. Philosophy, indeed, can be viewed as a general Platonism and equally detrimental in its classical forms and agendas to our efforts to ground mathematics (as well as science and logic) in social life. It is to philosophy in its more recent turn to practice that we must look if we are going to salvage philosophy as a credible intellectual activity. The strengths of philosophy as a guide to reasoned speculation, thought experiments, and ethical stands will be heightened by the turn to practice and the empirical arena. This is how we can save philosophy as an analytical tool for understanding mathematics. And here is where philosophy is drawn into the interdisciplinary boundary breaking modalities of late twentieth and twenty first century intellectual life. We are witnessing the demise of the traditional disciplines as we have known them for more than one hundred years. As the new inter-disciplines come to the fore, it begins to appear that we are witnessing the emergence of a second generation natural philosophy. For the moment, let us bracket this development and focus on mathematics and sociology as distinct, viable disciplinary projects. Both of these disciplines are undergoing interdisciplinary developments that are contributing to the emergence of a neo-natural philosophy.

 

Once again, then, what is mathematics? Technical talk about mathematics – trying to understand mathematics in terms of mathematics or mathematical philosophy – has the

effect of isolating mathematics from practice, experience, and shared meaning; it tends to “spiritualize” the technical. It is important to understand technical talk as social talk, to recognize that mathematics and mathematical objects are not (to borrow terms from the anthropologist Clifford Geertz' (1983: 94-120) “concatenations of pure form,” “parades of syntactic variations,” or sets of “structural transformations.” To address the question “What is mathematics?” is to reveal a sensibility, a collective formation, a worldview, a form of life. This implies that we can understand mathematics and mathematical objects in terms of a natural history, or an ethnography of a cultural system. We can only answer this question by immersing ourselves in the social worlds in which mathematicians work, in their networks of cooperating and conflicting human beings. It is these “math worlds” that produce mathematics, not individual mathematicians or mathematicians’ minds or brains. It is easy to interpret this perspective as somehow “mystical,” or as implying a “super-organic” entity. A realistic interpretation requires understanding human individuals as social things and their ideas as the “voice” of their social networks.

 

Mathematics, mathematical objects, and mathematicians themselves are manufactured out of the social ecology of everyday interactions, the locally available social, material, and symbolic interpersonally meaningful resources. All of what has been written in the last two paragraphs is captured by the short hand phrase, “the social construction of mathematics.” This phrase and the concept it conveys are widely misunderstood. It is not a philosophical statement or claim but rather a statement of the fundamental theorem of a sociology broadly conceived, a sociology being transformed by the interdisciplinary movements of our era. Everything we do and think is a product of our social ecologies. Our thoughts and actions are not simple products of revelation, genetics, biology, or mind or brain. To put it the simplest terms, all of our cultural productions come out of our social interactions in the context of sets of locally available material and symbolic resources. The idea of the social seems to be transparent, but in fact it is one of the most profound discoveries about the natural world, a discovery that still eludes the majority of our intellectuals and scholars. The interdisciplinary imperative can sustain this idea of the social even while it brings it into closer and closer association with our biology (cf. Clark, 2010, Noë, 2010, and Reyna, 2007).

 

Mathematics is a human, and thus a social, creation rooted in the materials and symbols of our everyday lives. It is earthbound and rooted in human labor. We can account for the Platonic angels and devils that accompany mathematics everywhere in two ways. First, there are certain human universals and environmental overlaps across the variety of our material environments, the physics, biology, and chemistry of life, culture, space, and time that can account for certain “universalistic” features of mathematics. Everywhere in everyday life, putting two apples together with two apples gives us phenomenologically four apples. But the generalization that 2+2 = 4 is culturally glossed and means something very different in Plato, Leibniz, Peano, and Russell and Whitehead. The earthbound everyday world of apples is commensurable for Plato and Russell and Whitehead; that world has not changed in the millennia that separate their lives. However, the discipline and then the profession of mathematics has changed dramatically across the centuries and created new incommensurable experiences within the social world of mathematics. The professionalization of mathematics gives rise to the phenomenon of mathematics giving rise to mathematics, an outcome that reinforces

the idea of a mathematics independent of work, space-time, and culture. Mathematics is always and everywhere culturally, historically, and locally embedded. There is, to recall Spengler, only mathematics and not Mathematik. There is, however, number work that carries across all cultures. Culture always intrudes in our mathematics; but while there is no Mathematik, there are ways for us to translate and communicate commensurable number and math work across cultures.

 

The concept-phrase “mathematics is a social construction” must be unpacked in order to give us what we see when we look at working mathematicians and the products of their work. We need to describe how mathematicians come to be mathematicians, the conditions under which mathematicians work, their work sites, the materials they work with, and the things they produce. This comes down to describing their culture – their material culture (tools, techniques, and products), their social culture (patterns of organization – social networks and structures, patterns of social interaction, rituals, norms, values, ideas, concepts, theories, and beliefs), and their symbolic culture (the reservoir of past and present symbolic resources that they manipulate in order to manufacture equations, theorems, proofs, and so on). This implies that in order to understand mathematics at all, we must carry out ethnographies – studies of mathematicians in action. To say, furthermore, that “mathematics is a social construction” is to say that the products of mathematics – mathematical objects – embody the social relations of mathematics. They are not free standing, culturally or historically independent, Platonic objects. To view a mathematical object is to view a social history of mathematicians at work. It is in this sense that mathematical objects are real. Before there is mathematics there is number work; before there are professional mathematicians there are number workers and then mathematics workers.

 

Arithmetic, geometry, and the higher mathematics are produced originally by number or mathematical workers and later on by variously disciplined and ultimately professional mathematicians. Ethnographies and historical sociologies of mathematics must, to be complete, situate mathematics cultures in their wider social, cultural, historical, and global settings. They must also attend to issues of power, class, gender, ethnicity, and status inside and outside more or less well-defined mathematical communities.

 

There is a hidden interrogation in the interrogation of mathematics that undermines its claims to transcendence and purity. That hidden interrogation is the interrogation of the very idea of the transcendent, of a Platonic realm of ideas. It is not too much of a stretch to see that this sort of interrogation will sooner or later have us interrogating religion and the gods. This is not the place to follow this line of inquiry. However, it is important to note that any deep understanding of the nature of mathematics as a this- worldly phenomenon is necessarily linked to the possibility of bringing religion and the gods down to earth. One only has to consider that the sociologist Emile Durkheim (1912/1995) concludes his remarkable study of the social construction of religion and the gods by arguing that logical concepts are, like religion and the gods, collective representations, this-worldly social constructs. Durkheim manifests the unfolding of the sociological enterprise as an exercise in the rejection of the transcendental.

 

Finally, let us revisit the ideas of civilization and progress. As we saw earlier, we could try to place ourselves on a continuum of civilization and progress between the polar

positions of Rousseau and Ferguson; everything about human society and culture as we have known it (especially since the coming of modern science and technology and the industrial revolution) denies our humanity, or everything trumpets its triumphs. This inevitably enmeshes us in a conflictful conversation without stop signs.

 

We could try to adopt Lenski’s notion of progress introduced earlier and bring the idea of civilization into his framework as a form of social and cultural organization. This would give us a more “scientific” foundation to rest on, but not one without its own ethical and value biases and implications. In the end, we are probably safest at this point if we recognize that mathematics is one of the many tools humans have fashioned to help them wind their way through the complex tapestries of life, and then one of humanity’s most useful as well as awe inspiring tools, even if more often than not actualized in the service of state power and ruling elites as “weapons of math destruction.” Moreover, it is one of the achievements that Nietzsche would have recognized as part of the fragile reason and sense of freedom humans have purchased at great cost.

 

Every entity recognized in history as a “civilization” from ancient Sumer and Babylonia to contemporary China has fed mathematical traditions into the ocean of a mathematical heritage that is in principle a part of humanity’s world culture. We can value the mathematics that has been developed out of and applied to the best of our activities as humans, activities that have been liberating, supported social justice, and in general made our planet a better and safer place on which to pursue the betterment of our species and its social and natural ecologies. We are by many measures and accounts at a cross-roads in human and planetary history, perhaps the cross-roads that opens a road just a couple of hundred years into the industrial-technological global society to annihilation on various levels and scales. It may be that our fate as a species and a part of a global and planetary ecology doomed to annihilation on a scale of millennia is now sealed on a scale of centuries or even years. Our survival, and especially our survival with some decent quality of life distributed across the planet and its life forms, will depend on long-term thinking that is at once broad, deep, and wise. All of our civilizational tools will be needed in perhaps our last chance in the short run to “save” ourselves and our planet.

 

Ours is an era when the very idea of progress, under attack throughout the twentieth century as wars, holocausts, ecological disasters, and radioactive fallout crushed romantic and idealistic dreams of a better world, has given way to concerns about unintended consequences, precautionary principles, and technology assessments. How dangerous it sounds today to hear the echoes of the British Labor Party’s celebration in the mid-1960s of “the white heat of technology revolution.” And yet we must ask if we are still subject to the hypnotic seductions of the technological lottery. Imagine a group of distinguished scientists brought together to speculate on “the next hundred years,” and to speculate optimistically because the idea (read “ideology’) of science was by definition inclined to optimism. And consider that when just such an event was organized to celebrate the centennial of Joseph E. Seagram & Sons (purveyors of whiskey and related spirited drinks) in 1957, they all choked on their optimism as they offered their speculations. Distinguished speakers such as Nobelist geneticist Herman J. Muller, geochemist Harrison Brown, economist and college president John Weir, and

infamous rocket scientist Wernher Von Braun found it necessary to qualify their optimism. Muller said the future would be rosy if we could avoid war, dictatorship, overpopulation, or fanaticism; Brown began by saying “If we survive the next century; Weir began with “If man survives…” Braun, in the most bizarre opening sentence, said “I believe the intercontinental ballistic missile is actually merely a humble beginning of much greater things to come.”

 

No matter. It is beyond doubt that mathematics will play a key role in our efforts, successful or not, realistic or delusional, to realize civilization and progress in terms that even Rousseau might embrace. This is not meant to lock into mathematics as simply a means to quantifying problem formulation and problem solving across all arenas of human experience. It is meant mathematics as a humanistic mode of knowing. It is not only or at all to Plato’s view of the role of mathematics for his guardian-rulers we must turn to (keeping in mind that for all his resistance to poetry and metaphor, Plato was a master of both). The author wants rather to endorse something like Nietzsche’s notion of mathematics. That is, the rigor and refinement of mathematics must be brought into all of our endeavors but not because this is necessarily the best way to apprehend things (recall Nietzsche’s views on the limits of science). We should want to do this in order to become more aware of our relationships as human beings to the things in our world. Mathematics does not give us the most general or the ultimate form of knowledge but opens a path (only asymptotically, at best) to such a goal. This view of mathematics has been promoted for almost half a century by the humanistic mathematics movement. If civilization has a future, if progress can become at least a Lenskian possibility, a humanistic mathematics will have to take hold in our efforts to identify, specify, and solve the problems of planetary survival we face today.

 

Acknowledgments

 

The author wants to acknowledge my collaboration with and indebtedness to Randall Collins of the University of Pennsylvania with whom he first tackled the problem of mathematics and civilization in the early 1980s. Randy is an important inspiration for the perspective developed here. He also wants to thank his collaborators and colleagues in the sociology of mathematics over the many years in which he worked in this field: Wenda Bauchspies, Deborah Sloan, Monica Mesquita, Ubiratan D’Ambrosio, the late Dirk Struik, the late Joseph Needham, the late Leone Burton, Paul Ernest, David Bloor, Donald MacKenzie, Jean Paul Bendegem, and Jens Hoyrup,

 

Glossary

 

Abstraction                : Classically, a process by which higher level concepts are derived from the usage and classification of literal ("real" or "concrete") concepts, first principles, or other methods; used as a noun, a concept that acts as a super-categorical noun for all subordinate concepts, and connects any related concepts as a group, field, or category. Abstractions may be formed by reducing the information content of a concept or an observable phenomenon, typically to retain only information which is relevant for a particular purpose. For example, abstracting a

leather soccer ball to the more general idea of a ball retains only the information on general ball attributes and behavior, eliminating the other characteristics of that particular ball. In this chapter, abstractions are replaced by the term “generalization” and given a concrete sense. In this sense, what we normally refer to as abstractions are the concrete terms of an autonomous professional community of practice.

Arithmetic                  : From the Greek word ἀριθμός, arithmos “number”; historically, the oldest and most elementary branch of mathematics, used for everything from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of operations that combine numbers. In common usage, it refers to the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers. The term “higher arithmetic” as distinguished from elementary arithmetic, is used in professional mathematics to refer to more advanced practices related to number theory.

Civilization                 : A relatively high level of cultural and technological development; specifically associated with the development of writing and written records in the ancient world. Classically, “civilized” peoples contrasted themselves with “barbarians;” in modern times, “civilized” has been opposed to “primitive” or “savage.” Viewed in positive and optimistic terms, civilizations are characterized by complexity, hierarchies, dynamism, advanced levels of rationality, social progress, and a statist form of government. Viewed in negative pessimistic terms, notably by the philosopher Rousseau, civilization is conceived to be opposed to human nature.

Ethnomathematics : The study of the relationship between mathematics and culture. This research brings to light the mathematical and logical traditions of non-literate, but it in general the study of the mathematical and logical practices of specific cultures. It refers to a broad cluster of ideas ranging from distinct numerical and mathematical systems to multicultural mathematics education.

Formalists                  : Formalists, following the mathematical philosophy of one of the most influential leaders of this school, David Hilbert (1862- 1943), treat mathematics as a “game.” In its most extreme version, formalism claims that mathematics is not about anything but rather sets of rules of inference that can be applied to given “strings” (axioms) to generate new strings. You can, for example, use the “game” Euclidean geometry (which is viewed as some strings) to generate a new string such as the Pythagorean theorem. This is roughly the equivalent of proving the theorem in classical mathematics. See also intuitionists and logicists.

Foundationalism  : In epistemology (theories of knowledge) the idea that there are basic (foundational) beliefs, assumptions, etc. that are the grounds for beliefs in general. Basic beliefs justify other beliefs. Basic beliefs are said to be self-evident or self-justifying. Basic beliefs

can also derive their warrant from sensory experience. Anti- foundationalists have a problem with giving an uncontroversial or principled account of which beliefs are self-evident or indubitable and see foundationalism as a form of an unexamined a priori or even as a God surrogate. from the Ancient Greek

Geometry                   : From the Ancient Greek γεωμετρία; geo- "earth", -metria "measurement;” the branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, Euclid (3rd century BCE) put it into an axiomatic form,

Intuitionists                : The intuitionists approached mathematics as the result of constructive cognition. Humans make mathematics and logic as mental constructs by applying internally consistent methods.

L.E.J. Brouwer (1881-1996) is considered the father of modern intuitionism. Intuitionists view the truth of mathematics as a subjective claim. Brouwer rejected realist/Platonist ideas about the truth or reality of mathematical objects. Intuitionism substitutes constructability for abstract truth and is the provocation for the transition from the proof theory of truth to the model theory of truth in modern mathematics. See also logicists and formalists.

Logicists                     : The logicists believed that mathematics was an extension of logic and therefore that all or at least some of mathematics was reducible to logic. Richard Dedekind (1831-1916) and Gottlob Frege (1848-1925) are considered the founders of this school. The logicist culminated in the monumental Principia Mathematica (published in three volumes in 1910, 1912, and 1913, and in a second edition in 1927 by Cambridge University Press )by Bertrand Russell (1872-1970) and Alfred North Whitehead (1861-1947) . Logicism survives today in Zermelo- Frankael set theory or one of its variations (such as Zermelo- Frankael set theory with the axiom of choice, or ZFC). Most of mathematics is believed to be reducible to the logical foundations provided by the axioms of ZF, ZFC, and derivatives. See also formalism and intuitionism.

Mathematical worker

: (Also number worker). To be distinguished from “mathematician,” someone who works with numbers and “does” mathematics as a member of a specialized occupational or professional class. The mathematical worker works with numbers as a member of a relatively informal and unorganized specialty outside of a formal credentializing system. The two terms are sometimes used interchangeably in the text for convenience but readers alerted to the distinction should be able to identify which term is most appropriate from the context.

Mathematics              : In its standard dictionary sense, the science of numbers and their operations,  interrelations,  combinations,  generalizations,  and

 

 

 

 

 

 

 

 

 

Non-Euclidean Geometry

abstractions and of space configurations and their structure, measurement, transformations, and generalizations . Classically, “mathematics” refers to the two fundamental ways in which humans work with numbers, arithmetic and geometry. In the Platonic view, this number work reveals a transcendental realm of ideal numbers revealed to humans as they develop over time. More recently the Platonic view has been opposed by the idea that mathematics is a social practice; it is manufactured by humans on the earth out of the material and symbolic resources at their disposal in their local environments.

: Refers to the geometries generated by denying Euclid’s fifth postulate, the parallels postulate. Consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line. In Euclidean geometry the lines remain at a constant distance from each other even if extended to infinity, and are known as parallels. In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels. In elliptic geometry the lines "curve toward" each other and eventually intersect. Spherical geometry is the geometry of the two- dimensional surface of a sphere.

Objectivity                 : This term is traditionally associated with ideas about value- neutrality, observations and interpretations uncolored by personal biases, and methods that are systematically developed and applied and not based on the idiosyncrasies of individual scientists. In this context, an objective statement is based on replicable research processes, and beyond fundamental dispute. In the sociology of science, objectivity has been shown to be an achievement of social groups and thus situated in “cultures of objectivity” or “objectivity communities.

Progress                     : In general terms, the forward development and betterment of humankind over time. The idea of scientific and technological progress was fueled by the seventeenth century advances in science and literature by Galileo, Newton, Descartes, Molière, and Racine. The idea of social progress was added later. The wars, holocausts, and human and environmental degradations characteristic of the twentieth century contributed to the development of critiques of the very idea of progress.

Proof                           : As a noun, evidence or an argument establishing or helping to establish a fact or the truth of a statement; the cogency of evidence that compels rational acceptance of a truth or a fact; the process or an instance of establishing the validity of a statement especially by derivation from other statements in accordance with principles of reasoning. As in the case other traditionally taken for granted ideas in the sciences, proof has become the subject of empirically grounded interrogations by sociologists of science who now refer proof to proof communities, and demonstrate that

 

Social Constructionism

 

 

 

 

Sociology of knowledge

what counts as a proof varies across time, space, and cultures.

: The fundamental theorem or central dogma of sociology, it refers to the fact that the only means humans have available for invention and discovery are their interactions with each other using the material resources of their environments and the symbolic resources of their cultures. It is fully compatible with a critical realism that assumes a “reality out there” that can only be known through the lenses of society and culture.

: The study of the relationship between human thought and the social, cultural, and historical contexts within which it arises, and of the effects prevailing ideas have on societies; deals with broad fundamental questions about the extent and limits of social influences on individual's lives and the social-cultural basics of our knowledge about the world When applied to the study of science in particular, an important specialty in sociology known as the sociology of science.

Spengler thesis   : Oswald Spengler argued that that there is no “Mathematik” but only “mathematics.” Mathematical forms are related to particular cultures. The weak form of the thesis is that there are as many mathematics as there are cultures. The strong form of the thesis is that mathematics is socially constructed.

Technoscience            : Variously, the idea the science and technology are intricately interrelated; the separation of “science” and “technology” may be analytically useful under some conditions but the idea that the two ideas are conceptually distinct is more ideological than substantive. This idea gives rise to such hybrid concepts as technocultural and technosocial.

Transcendental (transcendent; transcendence)

: Transcendence, transcendent, and transcendental are words that refer to an object (or a property of an object) as being comparatively beyond that of other objects. Such objects (or properties) transcend other objects (or properties) in some way. In philosophy, transcendence refers to climbing or going beyond some philosophical concept or limit In nineteenth century American, transcendentalism was developed within a religio- philosophical movement that claimed there is an ideal spiritual state that 'transcends' the physical and empirical. Following Restivo (2011 170) The Transcendental Fallacy (also known as the theologian’s fallacy) is that there is a world or that there are worlds beyond our own – transcendental worlds, supernatural worlds, worlds of souls, spirits and ghosts, gods, devils, and angels, heavens and hells. There are no such worlds. They are symbolic of social categories and heavens and hells. There are no such worlds. They are symbolic of social categories and classifications in our earthly societies and cultures. There is nothing beyond our material, organic, and social world. Death is final; there is no soul, there is no life after death. It is also possible that the so-called “many worlds interpretation” in quantum mechanics is contaminated by this fallacy as the result

of mathegrammatical illusions. The world, the universe, may be more complex than we can know or imagine, but that complexity does not include transcendental or supernatural features. Stated positively, this is Durkheim’s Law

 

Bibliography

 

Aleksandrov, A.D., A.N. Kolmogorov and M.A. Lavrent’ev (eds.) (1969), Mathematics: It’s Content, Methods, and Meanings, Cambridge, MA: MIT Press (orig. publ. in Russian in 1956). [A product of some the greatest mathematicians of the twentieth century, this is a readable review of the basic fields of mathematics. The authors are aware of the fact that mathematics is a human and a social creation and they do not mystify it or present it as a revelation of Platonic discourses.]

 

Almond, G., M. Chodorow, and R.H. Pearce (1985), Progress and its Discontents, Berkeley: University of California Press. [The twentieth century has not been kind to the idea of progress. Intellectuals across the spectrum of the disciplines are no longer that there are any viable grounds for the claim that science and technology have improved our lives morally or materially. The term “progress” is no longer meaningful empirically but is now about our aspirations and a “compelling obligation.”]

 

Ascher, M. (2004), Mathematics Elsewhere: An Exploration of Ideas Across Cultures, Princeton: Princeton University Press. [An introduction to mathematical ideas of peoples from a variety of small- scale and traditional cultures that challenges our conception of what mathematics is. Traditional cultures have mathematical ideas that are far more substantial and sophisticated than has been traditionally acknowledged.]

 

Baber, Z. (1996), The Science of Empire: Scientific Knowledges, Civilization, and Colonial Rule in India. Albany NY: SUNY Press. [Baber analyzes the reciprocal interactions between science technology and society in India from antiquity to modern times. The author analyzes institutional factors, including pre- colonial trading circuits, in the transfer of science and technology from India to other civilizational centers. He also explains the role of modern science and technology as factors in consolidating British rule in India.]

 

Ball, W. W. R. (1960), A Short Account of the History of Mathematics. New York: Dover. Unabridged and altered version of the fourth edition published in 1908. [A classic contribution to the history of mathematics.]

 

Bell, E. T. (1937/1986) Men of Mathematics, New York: [Simon and Schuster Touchstone Books. Bell is a mathematician and lacks the rigorous standards of the professional historians of science. But this book, for all of its idealism and focus on the dramatic, has been very inspiring for young people interested in mathematics and offers some insights into what it’s like to be a mathematician.]

 

Bellos, A. (2010), Here’s Looking at Euclid. New York: The Free Press. [This is a book designed to inspire and surprise. Bellos, who studied mathematics and philosophy at Oxford, takes the reader on a tour of mathematics based on his skills as a writer, traveler, and interviewer.]

 

Benesch, W. (1992), “Comparative Logics and the Comparative Study of Civilizations,” Comparative Civilizations Review, No. 27, Fall 1992, 88-105. [Benesch views “civilization as a state of mind.” Drawing heavily on Indian civilizational features, and especially the culture of the Jain for comparative materials, Benesch identifies four primary logics: object logics, subject logics, situational logics, and aspect logics. Benesch’s typology can be expanded to encompass propositional and predicate logic, first and higher order logics, modal logics, temporal logics, probability logics, multi-valued logics, negation logics, and quantum logic. Using another classification scheme we get: relevant logic (defined by some mathematicians as classical logic practiced with “good taste;” constructive logic (known in some of the literature by the less descriptive term "intuitionistic logic;" fuzzy logic, and comparative logic. The way to understand the proliferation of logics is to recognize that what we understand in everyday life and in a good deal of everyday science as the one universal logic is that set of rules and rules of inference that

reflect the workings of the macro-level physical world. In the worlds of the very small and very large, and the very fast and very slow, and across different levels of reality, different logics play out.]

 

Benveniste, É. (1966), Civilisation. “Contribution à l'histoire du mot” (Civilisation. Contribution to the history of the word), 1954, published in Problèmes de linguistique générale, Paris: Editions Gallimard ,

pp. 336-345 (translated by Mary Elizabeth Meek as Problems in general linguistics, 2 vols. 1971). [The author, a semiotician, traces the development of the concept “civilization” from its first occurrence in Adam Ferguson’s 1767 treatise on the history of civil society.]

 

Boos, B. and M. Niss, eds. (1979), Mathematics and the Real World, Boston: Birkhauser. [The editors recognize that to the extent that “pure” mathematics has gone too far and increasingly fails to interact with “real world” sciences and engineering disciplines, to that extent has it diminished its practical effectiveness. Mathematics that increasingly turns in on itself increasingly negates Wigner’s (1960) “unreasonable effectiveness” claim.]

 

Boutroux, P. (1919), L’ideal scientifique mathematiciens dans l’antiquité et dans les temps modernes. , Paris: Presses Universitaire. [Boutroux (1880-1922) was the son of the famous French philosopher Émile Boutroux and Aline Catherine Eugénie Boutroux, He was related to the statesman Raymond Poincaré¸the physicist Lucien Poincaré, and noted mathematician Henri Poincaré. Boutroux identifies three stages in the history of mathematics: the aesthetic, contemplate mathematics of the Greeks, the synthetic conception of Cartesian algebra, and what he saw as the incoherence of the mathematics of his time. There is a progressive theme in this history, an improving understanding of a higher reality. His didactic goals were to assert that progress in the history of science was a function of interactions between all the sciences, and to argue that problems should dictate approaches to solutions.]

 

Boyer, C. (1968), A History of Mathematics, Wiley, New York. [The novelist David F. Wallace called Boyer “the Gibbon of math history.” This book is to the history of mathematics what Paul Samuelson’s book on economics is to the study of economics. The reference of choice in the history of mathematics is now in its 3^rd edition (2011, updated by Uta Merzbach).]

 

Cajori, F. (1894), A History of Mathematics. New York: Macmillan & Co.. [Cajori’s (1859-1930) book was first published in 1893 and has gone through several editions. It is a reliable, readable treatment of the history of mathematics from antiquity to the early 1900s.]

 

Cajori, F. (1929/1993), A History of Mathematical Notations. Dover, New York. [This book is still today considered with good reason to be unsurpassed. Demonstrates how notation changes with changes in the shape of civilization. Here we discover, for example, how the Greeks, Romans, and Hebrews counted.]

 

Cantor, M. (1907), Vorlesungen uber die Geschichte del' Mathematik,. [Anastatischer Neudruck, Leipzig. Moritz Cantor (1829-1920) was an historian of mathematics who studied under some the giants of mathematics, including notably Gauss, and this book in four volumes is not just comprehensive but is considered a founding document in the history of mathematics as a critical, methodologically sound field of study.]

 

Clark, A. (2010), Supersizing the Mind: Embodiment, Action, and Cognitive Extension, New York: Oxford University Press. [Clark is yet another of a small but increasing number of authors who are focusing in on the problems and paradoxes that arise when we make the brain the source of our thinking and consciousness, and the source of our morals and our beliefs. In fact it is becoming increasingly clear that cognition is a complex result of tangled networks that criss-cross the boundaries of brain, body, and world. Mind is not bound by the brain; consciousness, as Nietzsche already intuited, is a network of relationships.]

 

Collins, R. (1998), The Sociology of Philosophies. Cambridge, MA: Harvard University Press. [A comprehensive social history of world philosophy in the context of global intellectual life. Collins traces the development of philosophical thought in China, Japan, India, ancient Greece, the medieval Islamic and Jewish world, medieval Christendom, and modern Europe. The result is an empirically grounded theory of ideas as the product of social networks. One of the most important contributions to modern sociology by a leading theorist.]

Collins, R. and S. Restivo. (1983), "Robber Barons and Politicians in Mathematics," The Canadian Journal of Sociology 8, 2 (Spring 1983), pp. 199-227. [Two of the leading contributors to the sociology of science offer an alternative to Kuhn’s theory of scientific change. Major scandals in mathematics are shown to be associated with shifts in the organizational structure of mathematical work. The article demonstrates the power of conflict theory as an explanatory approach in the sociology of mathematics.] Courant, R. and H. Robbins. (1996), What is Mathematics? New York: Oxford University Press (rev. by Ian Stewart, orig. publ. 1906). [A classic contribution to our understanding of mathematics as an enterprise of practice rather than philosophy.]

 

D’Ambrosio, U. (2006), Ethnomathematics, Rotterdam: Sense Publishers. An introduction to the concept of ethnomathematics by the founding father of the field.

 

Davis, P.J. and R. Hersh. (1981), The Mathematical Experience. Boston: Birkhauser. [Two mathematicians bring sociological, psychological, and philosophical perspectives to this popular exposition of the nature and relevance of mathematics.]

 

Dostoevsky, F. (1864), "Notes from the Underground," pp. 107-2-10 in The Best Short Stories of Dostoevsky. New York, n.d. (orig. publ, in Russian). [This classic piece of literature can be read as a contribution to the sociology of mathematics; the author’s discussion of 2+2=4 and 2+2=5 shows how arithmetic can be used to symbolize ideologies. Should be compared with how these equalities are treated by George Orwell in 1984.]

 

Dorrie, H. (1965), One Hundred Great Problems of Elementary Mathematics,Their History and Solution. New York: Dover. [This book was originally published in 1932 under the title Triumph der Mathematik. Fascinating well selected problems but not really “elementary.” An intriguing 2000 year survey but not for those without a strong background in mathematics.]

 

Durkheim, E. (912/1995), The Elementary Forms of Religious Life. New York: The Free Press 1 (trans. Karen Fields). [One of the most important studies in the history of scholarship. Durkheim crystallizes the idea of who and what God is and explains the function of religion in society. Criticisms abound, but they do not change the world transforming potential of this book.]

 

Eglash, R. (1999), African Fractals, Piscataway, NJ: Rutgers University Press. [Patterns across cultures are characterized by specific design themes. In Europe and America, cities are often laid out in a grid pattern of straight streets and right-angle corners. In contrast, traditional African settlements tend to use fractal structures--circles of circles of circular dwellings, rectangular walls enclosing ever-smaller rectangles, and streets in which broad avenues branch down to tiny footpaths with striking geometric repetition. These indigenous fractals are not limited to architecture; their recursive patterns echo throughout many disparate African designs and knowledge systems.]

 

Gasking, D. (1956), "Mathematics and the World," pp. 1708-1722 in J.R. Newman (ed.), The World of Mathematics. New York: Simon & Schuster. [Originally published in 1940, this is an exemplar of conventionalist philosophy of mathematics. Gasking views mathematics as true by virtue of linguistic conventions. Mathematics on this view is not empirical in the way, for example, that Wittgenstein argues it is or in the way it is understood today in social studies of mathematics.]

 

Geertz, C. (1983), “Art as a Cultural System,” pp. 94-120 in C. Geertz, Local Knowledge, New York: Basic Books. [Art, like speech, myth and other cultural systems are situated, ideationally connected to society; in a word cultural systems are socially constructed.]

 

Hersh, R. (1999), What is Mathematics, Really? New York: Oxford University Press. [Mathematics for the modern reader by a mathematician who appreciates the new sociology of mathematics developed by Bloor on the one hand and Restivo on the other.]

 

Horton, R. (1997), Patterns of Thought in Africa and the West: Essays on Magic, Science and Religion, Cambridge; Cambridge University Press. [The distinguished anthropologist and philosopher was at the center of debates that crystallized in the 1950s and 1960s concerning the nature of and relationship between “primitive” (read, primarily, African) and “advanced” (read “the West”) systems of knowledge

and belief. In this collection of his essays, Horton addresses the debates by exploring African beliefs, rituals, and cosmologies in the context of open and closed systems thinking. His objective is to demonstrate the kinship between primitive and modern thought and between science and religion more generally. Among the most important provocations for this and related debates are studies of Zande (n. Azande) logic. Were they “pre-logical,” as some claimed (e.g., in my view arguably Levy-Bruhl) or perfectly coherent in context (as Evans-Pritchard argued)? In the early years of the science studies movement David Bloor took up this question and argued for a cultural relativism that provoked new interest in this topic. This literature is relevant to my topic but requires more direct attention than I can give it here.]

 

Kavolis, V. (1985), “Civilizational Analysis as a Sociology of Culture,” Sociological Theory 85, 31, Spring 1985, 29-38. [The comparative civilizational analysis of any phenomenon, Kavolis argues, requires that we spell out the structure of each level of sociocultural organization and the connections and resources linking all the levels that bear on the particular phenomenon in a particular civilization. We don’t require an elaborate analytical apparatus but it helps to carry an awareness of such an apparatus in the study of a particular problem in terms of the relations between general civilizational structures, processes, and issues. Civilizational analysis is about the distinctive varieties of and changes in human experience, and the universe of symbolic designs in social behavior.]

 

Kline, M. (1962), Mathematics: A Cultural Approach, Reading, Massachusetts: Addison-Wesley. [A good introduction, even where dated factually and ideologically, to mathematics in relation to the humanities and the search for truth.]

 

Kramer, E. E. (1970), The Nature and Growth of Modern Mathematics. New York: Hawthorne Books. [Kramer discusses the lives and contributions of prominent mathematicians from Pythagoras and Newton to the modern period. Mathematical concepts such as binary operations, point-set topology, post-relativity geometries, optimization and decision processes ergodic theorems epsilon-delta arithmetization, and integral equations are discussed with admirable clarity.]

 

Lenski, G. (1974), Human Societies. New York: McGraw-Hill. [A cultural ecology of the evolution of societies. This is a major textbook contribution to macrosociology, now in its 11^th edition from Paradigm Publishers in Boulder, Colorado, 2008 co-authored by P. Nolan.]

 

Mannheim, K. (1936), Ideology and Utopia. Eugene, Oregon: Harvest Publishers. [This is a complex treatise in the context of Mannheim’s legacy in relationship to European and America sociology. It is a founding document in the emergence of sociology but especially of the sociology of knowledge. In terms of its relationship to the substance of this essay, the significance of this book is that while situating knowledge in its social, cultural, and historical contexts, Mannheim exempts the formal science from his analysis. There cannot be, he claims in these pages, a sociology of 2+2=4. This idea carried into the emergence of the sociology of science in the 1930s and was not seriously challenged until the birth of the science studies movement in the late 1960s.]

 

McClain, E.G. (1976), The Myth of Invariance: The Origins of the Gods, Mathematics and Music from the Rg Veda to Plato. York Beach, ME: Nicolas-Hays, Inc. [McClain’s thesis is that in the ancient civilizations music was a science that bridged the gap between the everyday world and the divine. The invariance of music contrasted sharply with the variability of the everyday world. Music expresses and motivates mathematics. Music as a science is revealed by studying the mathematical relationships between musical notes. In this context music is properly understood as an expression of and the motive for mathematical study. The "key" to unlocking this science comes from a study of the mathematical relationships between various musical notes. The Pythagorean notion of number was in fact more general and more widespread and an essential feature of the very idea of culture and civilization. The presentation is controversial in some details but we are learning more and more about the centrality of musicality in humans and this book speaks to that idea.]

 

Merton, R.K. (1968), Social Theory and Social Structure, enlarged ed., New York: The Free Press. [Merton founded and dominated the sociology of science with his students from the late 1930s to the late 1960s. The Mertonian paradigm, consonant with Mannheim’s sociology of knowledge, focused on the

social system of science – for example, norms, values, the reward system, stratification in science, age- grading – but exempted scientific knowledge per se from sociological scrutiny.]

 

Merton, R. K. (1961), , "Singletons and Multiples in Scientific Discovery: a Chapter in the Sociology of Science," Proceedings of the American Philosophical Society, 105: 470–86. Reprinted in R. K. Merton, The Sociology of Science: Theoretical and Empirical Investigations, Chicago, University of Chicago Press, 1973, pp. 343–70.

 

Merton, R. K. (1973), "Resistance to the Systematic Study of Multiple Discoveries in Science," European Journal of Sociology, 4:237–82, 1963. Reprinted in R.K. Merton, The Sociology of Science: Theoretical and Empirical Investigations, Chicago, University of Chicago Press , pp. 371–82. [This and the following paper introduce Merton’s idea that all discoveries are in principle multiples. These papers are milestones in the Merton corpus.]

 

Merton, R.K. (1958), “The Matthew Effect in Science,” Science, 159(3810): 56-63, January 5, 1968. [Widely cited in the sociology of science, the Matthew effect, or the principle of accumulated advantage, refers to situations in which the rich get richer and the poor get poor. Merton named the effect after Matthew 25: 29 (NRSV Bible). In science as in other activities, power and economic or social capital can be leveraged to gain additional power and capital. This is one the causes of mis-eponymy along with historical amnesia and random and deliberate acts of misattribution.]

 

Mesquita,M., S. Restivo and U. D'Ambrosio (2011), Asphalt Children and City Streets: A Life, a City, and a Case Study of History, Culture, and Ethnomathematics in São Paulo, Rotterdam: Sense Publishers. [An innovative plural voiced ethnography and history of street children that attends to how mathematics plays into their survival strategies.]

 

Morris-Suzuki, T. (1994), The Technological Transformation of Japan: From the Seventeenth to the Twenty-first Century, Cambridge: Cambridge University Press. [Japan did not miraculously leap into the technological forefront of twentieth century societies. Its rise to superpower status, as this book makes clear, is rooted in its history. This is the first general English language history of technology in modern Japan. One of the significant features of this book is its consideration of the social costs of rapid changes in technology.]

 

Nasr, S. H. (2007), Science and Civilization in Islam. Chicago: Kazi Publications. [The first one volume English language book on Islamic science from the Muslim perspective. Hossein explains the place of science in Muslim culture as he unfolds its content and spirit.]

 

Needham, J. (1959), Science and Civilization in China. Volume III: Mathematics and the Sciences of the Heavens and the Earth. Cambridge: Cambridge University Press. [One of several volumes in the monumental study that uncovered the hidden history of science and technology in China. Needham documented that not only did China have a history in science and technology, it was the leading civilizational center of science and technology in the world between the early Christian period in Palestine and 1500 CE. The explanatory framework is Marxist cultural ecology.]

 

Neugebauer, O. (1952), The Exact Sciences in Antiquity. Princeton: Princeton University Press. [It took a long time for historians of science to overcome the ideology of the once and always Greek miracle. Neugebauer contributes to the demise of this myth in this non-technical discussion of the influence of Egyptian and Babylonian mathematics and astronomy on the Hellenistic world. An early look into the sophistication of ancient Babylonian mathematics.]

 

Noë, A. (2010), Out of Our Heads: Why You Are Not Your Brain, and Other Lessons from the Biology of Consciousness, New York: Hill and Wang. [Makes an important contribution to getting away from classical ideas about the primacy of the brain in consciousness. His approach is radically social but in a strange way that makes biology, rather than sociology, the science of the social. But just because of this twisted logic, he furthers the interdisciplinary agenda of figuring out a non-reductionist way to link biology and society.]

Polster, B. and M. Ross, (2011), “Pythagoras’s Theorem ain’t Pythagoras’s,” h t t p : / / education . theage . co m . au / cmspage . php ? I ntid = 147 & intversion = 79, March 7, 2011.

 

Resnikoff, H.L. and R.O. Wells, Jr. (2011), Mathematics and Civilization. New York: Dover (paperback). [This book was originally published by Holt, Rinehart, and Winston in 1973 and later published by Dover with supplemental materials. Still a good general introduction to the reciprocal relations between mathematics and human culture with an emphasis on the technical mathematics. No great demands are made on the mathematical aptitudes of readers, and the more sophisticated reader will find some of the treatment, especially on the calculus, technically deficient.]

 

Restivo, S. (1979), "Joseph Needham and the Comparative Sociology of Chinese and Modern Science: A Critical Perspective," pp. 25-51 in R.A. Jones and H. Kuklick (eds.), Research in the Sociology of Knowledge, Sciences, and Art, Vol. II. JAI Press, Greenwich. [The most extensive journal length critical survey of Needham’s monumental history. Supports and extends his sociological materialism based on a human ecological perspective.]

 

Restivo, S. (1981), "Mathematics and the Limits of Sociology of Knowledge," Social Science Information, V. 20, 4/5: 679-701. [The new sociology of science associated with the science studies movement that emerged in the late 1960s challenged the status of mathematics as the arbiter of the limits of the sociology of science and knowledge. This is one of the early examples of the challenge by one of the founders of the modern sociology of mathematics.]

 

Restivo, S. (2001/1992), Mathematics in Society and History. New York. [This is the first book devoted completely to the sociology of mathematics as a subfield of sociology. The author reviews precursors, Spengler’s thesis on numbers and culture, mathematical traditions in different civilizations, and develops an original conception of the sociology of pure mathematics.]

 

Restivo, S. (1993), “The Social Life of Mathematics,” pp. 247-278 in S. Restivo, J.P. van Bendegem, and

R. Fischer (eds.), Math Worlds: Philosophical and Social Studies of Mathematics and Mathematics Education, Albany, NY: SUNY Press. [A collection of essays, including a set of papers by German scholars previously unavailable in English, focused on recent developments in the study and teaching of mathematics. The authors are all guided by the idea that mathematical knowledge must be grounded in and reflect the realities of mathematical practice.]

 

Restivo, S. (1994), Science, Society, and Values: Toward a Sociology of Objectivity. Bethlehem PA: Lehigh University Press. [This book introduces Restivo’s main contributions to the sociology of science between 1966 and the early 1990s. Based on his work in the ethnography of science, the history of science in China and the West, his social problems approach to understanding modern science, and other contributions, he develops a sociological perspective on objectivity.]

 

Restivo, S. (2007), “Mathematics,” Monza: Polimetrica. June, 2007 The Language of Science (ISSN 1971-1352). [An overview of the sociology of mathematics written for an online encyclopedia.]

 

Restivo, S. (2011), Red, Black, and Objective: Science, Sociology, and Anarchism, Surrey: Ashgate Publishers. [This book explores the implications of the science studies movement for science and society in the context of an anarchist tradition. The particular tradition the author has in mind here makes anarchism one of the sociological sciences. Here he follows Peter Kropotkin. The book is grounded in the empirical studies carried out over the last forty years by researchers in science studies (and more broadly science and technology studies). The author’s perspective is at once empirical, normative, and policy- oriented.]

 

Restivo, S. and H. Karp (1974), "Ecological Factors in the Emergence of Modern Science," pp. 123-142 in S. Restivo and C.K. Vanderpool (eds.), Comparative Studies in Science and Society. Columbus, OH: C. Merrill. [The authors theoretically link organizational and institutional structures to their ecological contexts and apply this theory to the problem of why modern science emerged in the West and not in China.]

Restivo, S. and R. Collins (1982), “Mathematics and Civilization,” The Centennial Review Vol. XXVI, No. 3 (Summer 1982), pp. 277-301. [An introduction to the comparative historical sociology of mathematics that modifies Spengler’s ideas on numbers and culture in the context of sociological theory.]

 

Restivo, S. and J. Croissant (2008), “Social Constructionism in Science and Technology Studies,” pp. 213-229 in J.A. Holstein & J.F. Gubrium, eds., Handbook of Constructionist Research, New York: Guilford. [The authors aim is to clarify the widespread misconceptions, misapplications, and misconstruals of this term which they identify as the fundamental theorem of sociology. They are at pains to argue that the term does not imply or entail any form of relativism; it is compatible with a realistic sociology that recognizes objectivity and truth as real. They are real however in an institutional sense. This view, as Durkheim pointed out more than one hundred years ago, is consistent with the idea that there is a reality “outside of us,” but we do not have access to a ding an sich.]

 

Reyna, S.P. (2007), Connections: Brain, Mind and Culture in Social Anthropology, New York: Taylor and Francis. [Reyna returns to the Boasian (Franz Boas) roots of anthropology to creatively construct a new paradigm for connecting the biological and cultural domains without submitting to the Sirens of reductionism.]

 

Rosental, C. (2008), Weaving Self-Evidence: A Sociology of Logic, Princeton: Princeton University Press. [Rosental traces the history of a theorem in the foundations of fuzzy logic to demonstrate the inherently social nature of logic. He describes the process by which logical propositions are produced, disseminated, and established as truths.]

 

Roszak, T. (1995), The Making of a Counter-Culture: Reflections on the Technocratic Society and its Youthful Opposition, with a new introduction, Berkeley; University of California Press (orig. publ. 1969). [Published in the middle of the 1960s sociocultural revolution, this book spoke directly to those who would become “the children of the 60s” while simultaneously bewildering their parents. Roszak coined the term “counter-culture” and damned the technocracy that was at the heart of the problems the protesters were angry about. A literate effort to explain the disaffection of young people and the young at heart during this tumultuous period.]

 

Sahlins, M. and E. Service, eds. (1960), Evolution and Culture, Ann Arbor: University of Michigan Press (co-authored by T.G. Harding, D. Kaplan, M.D. Sahlins, and E.R. Service). [In the hands of these authors, evolutionary anthropology becomes a predictive tool that can be applied to theorizing the future of human societies. Current events on the world scene, including the political and economic rise of China and the troubled status of the United States would not have surprised these anthropologists. This is a classic and still relevant contribution to our understanding of culture and cultural change writ large.]

 

Schechter, Eric (2005), Classical and Nonclassical Logics: An Introduction to the Mathematics of Propositions, Princeton: Princeton University Press. [Classical logic--the logic crystallized in the early twentieth century by Frege, Russell, and others--is computationally the simplest of the major logics, and adequate for the needs of most mathematicians. But it is just one of the many kinds of reasoning in everyday thought. Schechter introduces classical logic alongside constructive, relevant, comparative, and other nonclassical logics. This is the first textbook to make this subject accessible to beginners.]

 

Sedlacek, T. (2011), Economics of Good and Evil, New York, Oxford University Press. [Sedlacek understands economics as a social, cultural, and historical phenomenon. It is a product of our civilization not a pure, value free science. In this sense, Sedlacek lines up with contemporary students of the sociology of science and mathematics. Economics, at the end of the day, is about “good and evil.” In viewing economics as a moral enterprise he reminds us that the author of The Wealth of Nations, Adam Smith, is also the author of The Theory of Moral Sentiments.]

 

Selin, H. (ed.) (2001), Mathematics Across Cultures, New York: Springer. [A survey of Islamic, Chinese, Native American, Aboriginal Australian, Inca, Egyptian, and African mathematics, with essays on rationality, logic and mathematics. Shows how science and math practice is situated in its cultural context.]

Smith, D. E. (1958), History of Mathematics. New York: Dover. [A classical two volume introduction to the history of mathematics from ancient Egypt to modern times, very lucid survey including biographical notes and chronology.]

 

Sohn-Rethel, A. (1975), "Science as Alienated Consciousness." Radical Science Journal, Nos. 2/3: 65-

101. [Working from inside a sophisticated Marxist paradigm, Sohn-Rethel discusses science as an alienated and alienating form of knowledge and knowledge building. He is known for linking Kant and Marx in his work on epistemology.]

 

Spengler, O. (1926), The Decline of the West. New York: A. Knopf. [Spengler’s readers append adjectives like “audacious,” “profound,” “magnificent,” “exciting,” and “dazzlingly” to this book which flaws and all is one of the most amazing efforts in human history to capture humanity’s march through space and time. It’s relevance for the topic at hand is that Spengler’s theses are grounded in a radically cultural understanding of the relationship between culture and mathematics.]

 

Stigler, Stephen S. (1980), ‘Stigler’s Law of Eponymy’, Trans. N. Y. Acad. Sci. (2) 39, 147–157.

 

Leon Stover (1974), The Cultural Ecology of Chinese Civilization. New York: Signet. [An innovative interpretation of peasants and elites in what Stover refers to as a “once and always Bronze Age culture.” A paradigm for understanding the nature of Chinese culture and the continuities between the age of the dynasties and the modern era from Sun Yat Sen and Chiang Kai Shek to Maoism and modernism.]

 

Struik, D. (1967), A Concise History of Mathematics,. New York: Dover Publications. [The fourth revised edition of this classic was published by Dover in 1987. Struik, a Dutch mathematician and Marxist theory, was a professor at MIT for most of his professional career and to my knowledge the first person to identify the sociology of mathematics as a field of study. This book is a very readable introduction to the history of mathematics, concise as advertised but with a lot of substance poured into the books roughly 230 pages. Struik covers the period from the ancient world to the early twentieth century.]

 

Sugimoto, Masayoshi and D.L. Swain, (1978), Science & Culture in Traditional Japan. Cambridge MA: MIT Press. [Between 600 and 1854 CE, Japan was impacted by a first and second Chinese cultural wave and the first Western Cultural Wave in the nineteenth century. The authors focus on how these cultural waves set the stage for the development of an indigenous science and technology.]

 

Verran, Helen. (1992), Science and an African Logic. Chicago: The University of Chicago Press. [An empirical study that supports the idea of mathematics and logics as culturally situated. Quantity is not always absolute (as in 2=2=4) but sometimes relational, as in Yoruba. Verran’s experience and research as a teacher in Nigeria is the basis for this important contribution to the sociology of mathematics and ethnomathematics.]

 

Wigner, E. (1960), “On the Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Communications on Pure and Applied Mathematics , 13: 1–14. [A classic paper in defense of the idea that the mathematical structure of a physics theory often points the way to further advances in that theory and even to empirical predictions. It is basically an argument in support of “pure” mathematics.]

 

Wright, Ronald (2004), A Short History of Progress, Philadelphia: Da Capo Press. [Looking over the long history of humanity, Wright sees not the unfolding a linear evolution of progress but rather a series of “progress traps.” He reveals a history of “progress and disasters” that should serve as a warning to humanity and especially to those people who assume that progress is an inevitable and positive manifestation of human exceptionalism.]

 

Zaslavsky, C. (1999), Africa Counts: Number and Pattern in African Cultures, 3^rd ed. Chicago: Lawrence Hill Books (orig. publ. 1973 by Prindle, Weber, and Schmidt). [This is one of the earliest efforts to document the experience of mathematics in a non-Western culture and to view it in a positive civilizational perspective.]

 

Zeleza, Paul Tiyambe and Ibulaimu Kakoma (2005), Science and Technology in Africa. Trenton NJ: Africa World Press. [The authors deal with scientific and technology literacy, production, and

consumption in modern Africa. The focus is on developments in information technology and biotechnology in the context of The Knowledge Society in a globalizing context.]

 

Biographical Sketch

 

Dr. Sal Restivo is widely recognized as one of the founders of the field of Science and Technology Studies (STS), a pioneer in ethnographic studies of science, a founder of the modern sociology of mathematics, a contributor to public sociology and a prominent figure in the radical science movement of the 1960s. Dr. Restivo was Professor of Sociology, Science Studies, and Information Technology in the Department of Science and Technology Studies, at Rensselaer Polytechnic Institute in Troy, New York until his retirement in June 2012. He is Special Lecture Professor in STS at Northeastern University in Shenyang, China; a former Special Professor of Mathematics, Education, and Society at Nottingham University in Great Britain; and a former Hixon/Riggs Professor of Science, Technology, and Society at Harvey Mudd College. In 2012, he was a Senior Postdoctoral Fellow at the University of Ghent in Belgium. He is a founding member (1975) of and a former president (1994/95) of the Society for Social Studies of Science.

 

 

 

 

 

 

 

 

 

 

 

Sal Restivo, (2012), MATHEMATICS, CIVILIZATION, AND PROGRESS, in

History of Mathematics, [Eds.UNESCO-EOLSS Joint Commitee], in Encyclopedia of Life Support Systems (EOLSS), Developed under the Auspices of the UNESCO, Eolss Publishers, Paris, France, [http://www.eolss.net]

 

MATHEMATICS, CIVILIZATION, AND PROGRESS

 

Sal Restivo

Department of Intercultural Communication and Interaction, University of Ghent, Ghent Belgium

 

Keywords: abstraction, arithmetic, Chinese mathematics, civilization, ethnomathematics, European mathematics, generalization, geometry, Greek mathematics, Indian mathematics, mathematical workers, mathematicians, mathematics, non-Euclidean geometry, Platonism, proofs, progress, puzzle solving, social constructionism, sociology, transcendental

 

Contents

 

1.  Prologue: Interrogating the terms of our discourse

2.  Introduction: mathematics and civilization

3.  Mathematics and Civilization: Case Studies

4.  The Social Roots of Mathematics

5.  Puzzles and Proofs

6.  Generalization (“Abstraction”) and Self-consciousness

7.  Conclusion Acknowledgement Glossary Bibliography Biographical Sketch

 

Summary

 

This chapter provides an overview of the comparative history of mathematics in the context of the ideas of civilization and progress. The narrative is guided by the idea that mathematics is a social practice and not a set of ideas revealed through some sort of heavenly or Platonic discourse. This implies that mathematics has a normative dimension in terms of its cultural and professional settings. The chapter begins with a critical review of the terms of the discourse, namely mathematics, civilization, and progress. Following some general remarks on mathematics and civilization, the reader is guided through case studies of mathematics in its civilizational and cultural contexts including mathematics in China, India, and Greece, as well as modern Europe. A more analytical section follows, summarizing the idea of the social roots of mathematics, the development and functions of puzzle solving and proofs, and generalization as a way to think about “abstraction” in concrete terms. The narrative follows an historical social science perspective that draws on the ideas of classical and contemporary social theorists from Durkheim and Spengler to Randall Collins and Sal Restivo.

 

1.  Prologue: Interrogating the Terms of Our Discourse

 

Let us begin by briefly interrogating the three terms that make up the title of this essay. First, we should ask, borrowing the title of the book by Reuben Hersh (1999): What is mathematics, really? Mathematics has been shrouded in mystery and halos for most of

its history. The reason for this is that it has seemed impossible to account for the nature and successes of mathematics without granting it some sort of transcendental status. Classically, this is most dramatically expressed in the Platonic notion of mathematics.

 

Briefly, what we call Platonism in mathematics refers to Plato’s theory of Forms. Skirting the complexities of scholarly discourse, Plato is associated with the idea that there are “Forms” or “ideals” that are transcendent and pure. These immaterial Forms exist in a realm outside of our everyday space and time. They are the pure types of the ideas and concepts we manage in our everyday world.

 

This over-simplifies Plato but is consistent with a long tradition in the history and philosophy of mathematics. Consider, for example, the way some scholars have viewed the development of non-Euclidean geometries (NEGs). The mathematician Dirk Struik (1967: 167), for example, described that development as “remarkable” in two respects. First, he claimed, the ideas emerged independently in Gȍttingen, Budapest, and Kazan; second, they emerged on the periphery of the world mathematical community (most notably in the case of Kazan and to a lesser extent Budapest). And the distinguished historian of mathematics, Carl Boyer (1968: 585) characterized the case as one of “startling…simultaneity.”

 

These reflect classical Platonic, transcendental views of mathematics. One even finds such views in the forms of the sociology of knowledge and science developed from the 1920s on in the works of Karl Mannheim and Robert K. Merton and their followers. Mannheim, for example, wrote in 1936 that 2+2 = 4 exists outside of history; and Merton championed a sociology of science that focused on the social system of science and not on scientific knowledge which he claimed lay outside of the influences of society and culture.

 

His ambivalence about this is reflected in his critical reply to G.N. Clark’s criticism of Boris Hessen’s historical materialism of Newton’s Principia (Merton, 1967: 661-663). Clark opposed Hessen’s (Marxist) political economy of the Principia with a defense of Newton’s “purely” scientific motives. Merton argued that individual motivations do not change the structural facts of the matter and in this case they support Hessen’s argument. This doesn’t reach to the core of the social construction of scientific knowledge but it does demonstrate at least an appreciation for the contextual foundations of that knowledge.

 

There are a couple of curiosities in the case of non-Euclidean geometry (NEGs). Even a cursory review of the facts reveals that NEGs have a history that begins already with Euclid’s earliest commentators, runs over the centuries through names like Saccheri, Lambert, Klügel, and Legendre, and culminates in the works of Lobachevsky (1793- 1856), Reimann (1826-1866), and J. Bolyai (1802-1860). The concerns over Euclid’s parallels postulate moved geometers eventually to the systematic development of NEGs. The issue was that the parallels postulate, the fifth postulate in Euclid’s system, did not possess the axiomatic self-evidence of the first four postulates, and it could not be derived from the first four. The three creators of NEGs were by no means isolated and working independently. All were connected to Gauss (1777-1855) who had been working on NEGs since the late 1700s.

J. Bolyai was the son of one of Gauss’ friends, W. Bolyai. Gauss and his friend Bolyai were at the University of Göttingen where the parallels postulate was the subject of lectures by Kastner and a number of dissertations. Reimann was Gauss’ dissertation student. And as for Lobachevsky, he did indeed work at a university on the periphery of the European mathematical community, the University of Kazan. However, the university was staffed by distinguished German professors, including Gauss’ teacher,

J.M. Bartels. J. Bolyai developed ideas on non-Euclidean geometries (NEGs) as early as 1823. His “The Science of Absolute Space” was published ten years later in a book written by his father. Lobachevsky published on the foundations of geometry from 1825 on. Reimann’s Habilitationschrift was on the foundations of geometry. Gauss, we know, wrote about NEGs in letters to W. Bolyai (December 17, 1799), Taurinus (November 8, 1824), and to Besel (January 27, 1829). He also wrote about NEGs in published notes from 1831 on. There are two short reviews on NEGs in Göttingische Anziegen in 1816 and 1822. One has to wonder why in the face of the facts of the case Struik and Boyer chose to view things as “remarkable” and “startling.”

 

Classically, the story of the development of NEGs was told in the context of “pure” mathematics. Thus, to take the case of Riemann as an example, the story was that he constructed the generalization of elliptic geometry as a purely mathematical exercise. The idea that there was a concrete possibility of practical applications for this exercise was not a consideration. In the light of a more realistic sociological and network analysis, Riemann’s work along with that of Gauss, Lobachevsky, Bolyai, Helmoltz, and Clifford, the story of NEGs takes on a different shape.

 

To some extent, they all agreed that Euclidean geometry was an unimpeachable system of ideal space and logic. It could be read as a game played in accordance with a set of formal rules. In fact, however, they interrogated Euclidean geometry in terms of whether it was a valid representation of “actual space.” This should be tested not by mathematics, not what is within the confines of the social world of mathematics per se, but should be tested scientifically – by observation and some mode of experimentation.

 

The sociological generalization this leads to is that if you are given a “genius” or a startling event, search for a social network –cherchez le réseau. No one has made the case for social networks as the roots of ideas more powerfully than Randall Collins (1998). The rationale here should become clearer over the course of this chapter.

 

Even more curious in the case of the sociology of knowledge is the fact that already in his The Elementary Forms of Religious Life published in French in 1912, Emile Durkheim had linked the social construction of religion and the gods to the social construction of logical concepts. Durkheim’s program in the rejection of transcendence languished until the emergence of the science studies movement in the late 1960s and the works of David Bloor, Donald MacKenzie, and Sal Restivo in the sociology of mathematics.

 

It is interesting that a focus on practice as opposed to cognition was already adumbrated in Courant’s and Robbins’ classic “What is Mathematics?” (1906/1995). We must turn to active experience, not philosophy, they wrote, to answer the question “What is mathematics”? They challenged the idea of mathematics as nothing more than a set of

consistent conclusions and postulates produced by the “free will” of mathematicians. Forty years later, Davis and Hersh (1981) wrote an introduction to “the mathematical experience” for a general readership that already reflected the influence of the emergent sociology of mathematics. They eschewed Platonism in favor of grounding the meaning of mathematics in “the shared understanding of human beings…” Their ideas reflect a kind of weak sociology of mathematics that still privileges the mind and the individual as the creative founts of a real objective mathematics.

 

Almost twenty years later, Hersh, now clearly well-read in the sociology of mathematics, wrote “What is Mathematics, Really?” (1997). The allusion to Courant and Robbins is not an accident. Hersh does not find their definition of mathematics satisfactory. In spite of his emphasis on the social nature of mathematics, Hersh views this anti-Platonic anti-foundationalist perspective as a philosophical humanism. While he makes some significant progress by comparison to his work with Davis, by conflating and confusing philosophical and sociological discourses, he ends up once again defending a weak sociology of mathematics. The modern sociology of mathematics associated with the science and technology studies movement that emerged in the late 1960s has established mathematics as a human construction, a social construction. Mathematics is manufactured by humans on the earth out of the material and symbolic resources at their disposal in their local environments. While traditional philosophical and sociological discourses have become estranged, especially in the arena of science studies, there are efforts abroad to reconcile the two disciplines consistent with the interdisciplinary turn in contemporary research and theory. In this sense, one can consider Hersh’s philosophical humanism a step in that direction.

 

The second term in my title, “civilization” is not without its controversial features. In the ancient world, “civilized” peoples contrasted themselves with “barbarians;” in modern times, “civilized” has been opposed to “primitive” or “savage.” The relevance of the concept of civilization to the topic of mathematics lies in its association with the idea of progress. More to the point is the fact that different civilizations (variously “nations,” “societies,” and “cultures”) are associated with different mathematical traditions (v. Restivo, 1992, 23-88). To the extent that humans have developed in ways that can be captured in the ideas of “evolution” and “progress,” mathematics, in conjunction with science and technology more generally, is assumed to have contributed positively to and benefited from those developments. The Scottish philosopher Adam Ferguson (1723-1816) is credited by Benveniste (1954) with introducing the term “civilization” in its modern sense into the English language in his Essay on the History of Civil Society (1767), and perhaps as early as 1759. The term also appears in the works of Boswell (1772), Adam Smith (1776), and John Millar (1771). Mirabeau (1757) introduces the term in French in his L'Ami des hommes ou traité de la population. Just as the individual grows from infancy to adulthood, Ferguson wrote, the species advances from “rudeness to civilization.” Set at the pinnacle of forms of society, civilizations are characterized by complexity, hierarchies, dynamism, advanced levels of rationality, social progress, and a statist form of government. Rousseau, by contrast, viewed civilization as opposed to human nature.

 

If we adopt Ferguson’s view of civilization, then clearly mathematics has been both a result of the emergence and development of civilization and a contributor to that

development. If on the other hand we adopt Rousseau’s viewpoint, the virtues of science and mathematics and the very idea of “civilization” are made severely problematic.

 

What about progress, the very idea? Arguably, the idea comes into Western and world culture in the Old Testament with its conception of linear time and a God that moves through time with humans (e.g., Sedlacek, 2011: 47). The idea of scientific and technological progress was fueled by the seventeenth century advances in science and literature by such cultural giants as Galileo, Newton, Descartes, Molière, and Racine. The idea of social progress was added later. Early in the eighteenth century, the Abbé de Saint Pierre advocated establishing political and ethical academies to promote social progress. Saint Pierre and Turgot influenced the Encyclopedists. The great Encyclopédie was produced by a group of eighteenth century philosophers under the direction of Denis Diderot.

 

It defines the Enlightenment program of promoting reason and unified knowledge. It was at this point that social progress became mated to the values of industrialization and incorporated into the ideology of the bourgeoisie. Scientific, technological, and social progress were all aspects of the ideology of industrial civilization. Veblen, for example, argued that the various sciences could be distinguished in terms of their proximity to the domain of technology. Thus, the physical sciences were closest to that domain, even integral with it, whereas such areas as political theory and economics were farther afield. We have entered an era of machine discipline unlike any in human history. And now we stand on the threshold of machines that will discipline us with conscious awareness and values, including social and sociable robots (the so-called robosapiens), and cyborgs.

 

There have been attempts to identify a type of progress that is independent of material or technological criteria (see, for example, the discussion in Almond, Chodorow, and Pearce, 1985, and the classic criticisms in Roszak, 1969/1995). For many ancient as well as modern thinkers, the idea of progress has always been problematic. We are right to be concerned about the actual and potential impacts of our new bio- and nano- technologies. But one finds similar concerns in Plato’s Phaedrus. There, in the dialogue between Theuth and king Thamus concerning the new technology of writing, Theuth makes promising predictions about the impact of writing.

 

The king claims to be in a better position to do what in effect is a “technology assessment,” and concludes that writing will have the opposite of the effects predicted by Theuth. The cultural meaning of science has fared no better. Where the Rousseaus and the Roszaks saw danger and alienation in science, the Francis Bacons and Bronowskis saw civilization and progress. When the biochemist J.B.S. Haldane wrote about a future of human happiness built on the application of science, Bertrand Russell replied with a vision of science used to promote power and privilege rather than to improve the human condition. St. Augustine worried about the invention of machines of destruction; Spengler predicted that humans would be annihilated by Faustian man. Fontenelle, in the first modern secular treatise on progress published in 1688 argued that science was the clearest and most reliable path to progress. Rousseau, by contrast,

argued that science and the arts have corrupted our minds. The author will draw attention to some additional examples in his conclusion in this chapter.

 

By its intimate association with the very foundations of science, mathematics does not escape this ambivalence. But it stands apart from science in terms of its stronger association with human progress. In the seventeenth and eighteenth centuries, a wave of positivism fueled by Newton’s achievements evoked nothing but the promise of progress among mathematicians of that period. The historian Florian Cajori (1894: 4) had no question about the connection between mathematics and human progress. For Alex Bellos (2010: ix), mathematics is (“arguably”) the foundation of all human progress.

 

Progress, then, can be viewed in terms of “amelioration” or “improvement” in a social or ethical sense. Are we more advanced than cultures that are less dominated by machines and machine ideology? How do we measure the primacy of humans and ecologies and how do we sustain them in any given culture? Can we bring them to fruition and nourish them in any culture, or are some more friendly to the primacy of humans and ecologies than others? These issues are really matters of degree associated with the degree to which individuation of the self (and then the myth of individualism, selfishness, and greed) has progressed in any given society. Furthermore, the degree of awareness of and attention to ethics, values, and social justice has to come into consideration here. It is impossible to even discuss the idea of progress without engaging ideas about and the value of the person, freedom, and democracy.

 

It may be possible to define progress in a way that takes it out of the realm of hopes, wishes, and dreams and plants it more firmly on a meaningful (and even perhaps measurable) foundation. Following Gerhard Lenski (1974: 59), progress can be defined as the process by which human beings raise the upper limit of their capacity for perceiving, conceptualizing, accumulating, processing, mobilizing, distributing, and utilizing information, resources, and energy in the adaptive-evolutionary process. The relationship between adaptation and evolution is a paradoxical one. On the one hand, survival depends on the capacity to adapt to surroundings; on the other hand, adaptation involves increasing specialization and decreasing evolutionary potential. Adaptation is a dead end. As a given entity adapts to a given set of conditions, it specializes to the point that it begins to lose any capacity for adapting to significant changes in those conditions. The anthropologists Sahlins and Service (1960: 95-97) summarize these ideas as follows:

 

Principle of Stabilization: specific evolution (the increase in adaptive specialization by a given system) is ultimately self-limiting.

 

General evolution (progressive advance measured in absolute terms rather than in terms of degrees of adaptation in particular environments) occurs because of the emergence of new, relatively unspecialized forms.

 

Law of Evolutionary Potential: increasing specialization narrows adaptive potential. The more specialized and adaptive a mechanism or form is at any given point in

evolutionary history, the smaller is its potential for adapting to new situations and passing on to a new stage of development.

 

We can add here the Law of Adaptive Levels: adaptation occurs at different levels across various life orders and systems and occurs at different speeds in different spatial arenas. This law draws attention to the complexity of adaptation and the general processes of variation and selection. Adaptation suggests an active agent in a stable environment. But active agents can and do change their environments in ways that make different demands on the adapting agents. Looked at another way, environments have agential like dynamics. Law of Agent-Environment Entanglement.

 

Perhaps the most important aspect of the ideology of science is that it is (in its mythical pure form) completely independent of technology. This serves among other things to deflect social criticism from science onto technology and to justify the separation of science from concerns about ethics and values. Interestingly, this idea seems to be more readily appreciated in general by third world intellectuals than by the Brahmin scholars of the West and their emulators. Careful study of the history of contemporary Western science has demonstrated the intimate connection between what we often distinguish as science and technology. It has also revealed the intimate connection between technoscience research and development and the production, maintenance, and use of the means (and the most advanced means) of violence in society. Not only that, but this is true in general for the most advanced systems of knowledge in at least every society that has reached a level of complexity that gives rise to a system of social stratification.

 

Contradictions and ambivalence about science, technology, and progress may be built into the very core of our cultural machinery. Agricultural activities in the ancient Near East reduced vast forests to open plains, and wind erosion and over-grazing turned those areas into deserts. Deforestation in ancient China led to the development of the loess plateau. Loess sediment gives the Yellow River (nicknamed “China’s Sorrow”) its signature color and flooding pattern. Was deforestation necessary for building China into the greatest civilizational area on earth between the first and sixteenth centuries of the common era? Or were there conservation principles that the ancient Chinese could have relied on without detracting from their cultural development? There is some evidence that at least some of the deforestation they caused could have been avoided. The deforestation experiences of China, Rome, and other civilizational areas of the ancient world are being repeated today and offer cautionary tales for an era characterized by many hard to monitor emerging and converging technologies, that is, technocultural systems.

 

At the end of the day, it should be clear that progress is not easy to define, and that it is even harder to point to examples of progress that resist critical interrogation. How can we sustain the idea of progress in the face of the widespread ecological, environmental, and human destruction that has characterized the industrial age? The fact is that the destruction and danger we see all around us is integrally connected to the very things we use to mark the progress of humanity. For these reasons, we must be cautious when considering whether any of the sciences, engineering disciplines, or mathematics have contributed to or served as signposts of progress. Mathematics, like all systems of knowledge, does not exist in a vacuum. It is always connected to social institutions and

under the control of the most powerful institutions in any given society. All of this may put too much of the onus on the sciences and technology when what we are dealing with is culture in general. Is it possible that cultures by their very natures inevitably destroy planets?

 

It should be clear from this brief introduction that the terms of our title, “mathematics,” “civilization,” and “progress” are all imbued with some level of ambivalence and uncertainty. It remains to be seen whether in the rest of this chapter we can find our way to greater certainty about the meaning and implications of these terms.

 

2.  Introduction: Mathematics and Civilization

 

Karl Mannheim (1893-1947) and Oswald Spengler (1880-1936) defend diametrically opposed positions on the possibility of a sociology of mathematics. Mannheim argues that mathematics is exempt from sociocultural and historical explanations; mathematics is not an ideology, and mathematical truths are not culturally relative. This view has been reinforced by Pythagoreans and Platonists who believe that mathematical truths are eternal objects that exist independently of the flux of historical experience and outside of time and space. Most historians, philosophers, and sociologists of science have traditionally adopted a basically Mannheimian view of mathematics.

 

Spengler, on the other hand, holds that each culture has its own conception of number. Spengler's notion of the "soul” of a civilization cannot provide the basis for an adequate sociological analysis. However, Spengler's goal of explaining mathematics in terms of the particular social and historical forms in which it is produced is sociologically viable. Spengler’s argument is summarized in two statements: (1) “There is not, and cannot be, number as such." There are several number-worlds as there are several cultures; and (2) "There is no Mathematik but only mathematics." Spengler's objective in his analysis of "number" is to show how a crystallized culture demonstrates its idea of the human condition, of what it means to be human. The "peculiar position" of mathematics rests on the fact that it is at once science, art, and metaphysics. It is safe to assume, taking some liberties, that Spengler’s use of “Culture” is more or less commensurate with what we commonly mean by “civilization.”

 

The author takes some additional liberties here with Spengler’s spiritualized materialism in order to avoid some of the idiosyncrasies of his vocabulary. Number, like God, represents the ultimate meaning of the natural world. And like myth, number originated in naming, an act that gives humans power over features of their experience and environments. Nature, the numerable, is contrasted with history, the aggregate of all things that have no relationship to number. Note that in this moment it appears that Mannheim and Spengler might be at one on the nature of mathematics. While Spengler clearly sees a closer connection between mathematics and culture than Mannheim does, he does as the following paragraph illustrates harbor some ambivalence.

 

Spengler argues against treating earlier mathematical events as stages in the development of "mathematics”. This is consistent with his thesis on the incommensurability of Cultures and with his cyclical view of historical change. His general schema of Classical and Western styles and stages in "Culture," "number," and

"mind" is essentially an analysis of world views. This is reflected in Spengler's attempt to correlate mathematical and other sociocultural "styles.” For example, he argues that Gothic cathedrals and Doric temples are “mathematics in stone." Spengler is aware of the problem of the limits of a "naturalistic" approach to number and pessimistic about a solution. It is impossible, he writes, to distinguish between cultural features that are independent of time and space, and those that follow from the forms of culture manufactured by humans.

 

Finally, Spengler claims that a deep religious intuition is behind the greatest creative acts of mathematicians. Number thought is not merely a matter of knowledge and experience, it is a "view of the universe," that is, a world view. The second claim Spengler makes is that a "high mathematical endowment" may exist without any "mathematical science.” For example, the discovery of the boomerang can only be attributed to people having a sense of mathematics that we must recognize as a reflection of the higher geometry.

 

Sociologists of mathematics have been bold enough about challenging the Platonic conception of number, but they have hesitated to follow Spengler. His ideas must seem mad to scholars and laypersons, and specialists and non-specialists alike, to whom the truth of number relations appears to be self-evident. And yet, the "necessary truth" of numbers has been challenged by mathematical insiders and outsiders. One of the outsiders is Dostoevsky. In his Notes from Underground (1864/1918), Dostoevsky argues that 2+2 = 4 is not life but death, impudent, a farce. It’s nonetheless “excellent” and we must give it its due, but then we must recognize that 2+2=5 is also sometimes “a most charming little thing”. Contrast Dostoevsky’s perspective with Orwell’s (1949) use of these two equalities in 1984. For Orwell, 2+2=4 stands for freedom and liberty; 2+2=5 stands for Big Brother totalitarianism. Dostoevsky uses 2+2=4 to stand for everyday routines and tradition; 2+2=5 represents creativity.

 

Dostoevsky's remarks are not merely a matter of literary privilege. Mathematicians and historians and philosophers of mathematics have also challenged the conventional wisdom on number. We should not expect ordinary arithmetic to apply in every physical situation; whether it does or does not has to be based on our experience in different physical situations. In other words, whether or not 2+2=4 is always an empirical question. Where we have long term experiences with situations in which 2+2=4 we are justified in considering those situations closed to further interrogation, that is, we are justified in taking the equality for granted. This is not a warrant for universalizing mathematics uncritically.

 

Studies in ethnomathematics have helped to reinforce and ground the notion that mathematics and logics are culturally situated (cf. Benesch, 1992). African mathematics has posed problems for European intellectuals since at least the publication of Robin Horton’s (1997) studies on patterns of thought in Africa and the West beginning in the 1960s. The earliest findings and interpretations suggested a different way of reasoning and a different logic in African cultures by comparison with the Europeans. Ethnomathematics has helped to sort out the early discourses and ground differences in cultural patterns rather than in mental proclivities. Malagasy divination rituals, for example, rely on complex algebraic algorithms. Some peoples use calendars far more

abstract and elegant than those used in European cultures, notably the Chinese and the Maya. Certain concepts about time and equality that Westerners assumed to be universal in fact vary across cultures. The Basque notion of equivalence, for example, is a dynamic and temporal one not adequately captured by the familiar equal sign. Other ideas taken to be the exclusive province of professionally trained Western mathematicians are, in fact, shared by people in many societies. (see D’Ambrosio, 2006; Mesquita, Restivo, and D’Ambrosio, 2011; Eglash, 1999; Ascher, 2004; Verran, 1992).

 

There is, in brief, a rationale for pursuing the Spenglerian program for a sociology of mathematics based on the views of at least some mathematicians, historians of mathematics, and observers of numbers. The author refers here to authors such as Dostoevsky and Orwell, mathematicians such as Dirk Struik and Chandler Davis, philosophers of mathematics such as Paul Ernest and Leone Burton, and historians of mathematics such as Morris Kline.

 

Sociologists are obliged to interrogate mathematics in a comparative perspective: how does mathematics develop at different times and places; who are the noted mathematicians, what are the social positions they held and how were they related to one another; and what are the social conditions within and outside of mathematical communities as they go through phases of growth, stagnation, and decline. The degree of "community" among mathematicians, the level of specialization, the extent of institutionalization and the relative autonomy of the social activity of mathematics, it should be stressed, are variable across time and space.

 

It is possible to narrate the history of mathematics as a more or less linear unfolding that gives the appearance of an inevitable “logical evolution.” However, the evidence is that there are a number of variations among the types of mathematics produced in different cultures. The latter "horizontal" variations are prima .facie evidence for the Spengler thesis. But what about the long-term trends? These too are socially determined, and in two different senses. First, and in a weaker sense, the "longitudinal" development of mathematics does not occur without interruptions, nor does it unfold in a single cultural context. Mathematicians move along a certain path at some times and not others. This implies among other things that the concepts of truth and what counts as a proof in mathematics vary over time. A sequence in mathematical development or in the network of mathematicians will stop, start, stop, and start again over a period of a hundred or a thousand years. What drives these processes? And why do particular mathematicians at particular times and places make the major advances and not others? More to the point, how is it that some mathematicians and not others who might be better candidates get credit for discoveries and inventions? This question opens up an opportunity to consider eponymy and mis-eponymy in mathematics.

 

In the sciences, eponymy refers to the process of naming an invention or discovery after the person who made the invention or discovery. Some well-known examples are Ohm’s law, the Pythagorean theorem, and Pascal’s triangle. In 1980, the statistics professor Stephen Stigler (1980) formulated "Stigler’s law of eponymy:” stated in general terms it states that no discovery or invention is named after its original discoverer or inventor. Stigler then attributed “Stigler’s Law” to the sociologist of science Robert K. Merton (1961, 1963; and on the related Matthew effect, see Merton,

1968), thus making Stigler’s Law self-exemplifying (further exemplifying Merton’s fondness for the self-exemplifying hypothesis). Merton had already formulated the idea that all discoveries are in principle multiples. Consider the following examples: Pythagoras’ theorem was already known in Mesopotamia at least a thousand years earlier (crediting Pythagoras with a proof is entirely speculative). The Chinese had Pascal’s triangle at least 320 years before he was born. L’Hôpital’s Rule is probably due to Johann Bernoulli; Cardano’s formula for solving cubic equations comes from Tartaglia; Pell’s equation is due to Fermat; Benford’s Law is due to Simon Newcomb; and the Möbius strip could just as easily be named for Listing (Polster and Moss, 2011).

 

Following Spengler and more recent developments in the sociology of mathematics, it is possible to identify social factors that affect the variations, interruptions, progressions, and retrogressions characteristic of the longitudinal development of mathematics. There is a stronger sense in which the longitudinal development of mathematics towards more "advanced" forms is socially determined. The longitudinal development of mathematics reveals the social aspects of mathematical work. Much of professionalized mathematics is created in response to stimuli from within the mathematical community, especially as mathematicians go about playing competitive games with one another. The development of higher and higher levels of “abstraction” (see immediately below), for example, reflects the increasing self-consciousness of mathematicians about their own operations. This in turn reflects higher levels of specialization and institutional autonomy among mathematicians. The Spengler thesis is true in a very strong form: “number”, and all that it stands for metaphorically, is a socially created activity, or more technically, a social construction (see the detailed explanation of this much abused term in Restivo and Croissant (2008).

 

We should consider replacing the term “abstraction” with the term “generalization.” The reason is that what we commonly understand as abstractions are simply concrete forms constructed under conditions of professionalization and disciplinary closure. As mathematics becomes more organized and disciplined, mathematicians build new levels of mathematics on the grounds of earlier mathematical forms. This removes mathematics further and further ceteris paribus from the everyday world and gives rise to the idea of abstraction. But it is important to understand that we are dealing with new levels of concreteness. The term “abstraction” makes us vulnerable to the myths and ideologies of purity and even to ideas about heavens and gods. It is in my view better to eliminate it from our vocabulary in the sense that it is used in the sciences and mathematics.

 

3.  Mathematics and Civilization: Case Studies

 

The world history of mathematics has not unfolded in a unilinear, unidirectional manner. The Greeks, for example, took a step backward from the Babylonian achievements in notation. Different types of mathematical systems have developed in different parts of the world; and rival forms of mathematics have sometimes developed within societies and professional networks. Hindu mathematics, especially in the period before the influx of Greek astronomy (ca. 400CE) placed unique emphasis upon large numbers. Geometry, arithmetic, number theory, and algebra were ignored in favor of the use of numbers in "sociological" schemes. The Upanishads (ca. 700 to 500BCE) are full

of numerical (or more accurately, numerological) descriptions: 72,000 arteries; 36,360, or 36,000 syllables; the 33, 303, or 3306 gods; the 5, 6, or 12 basic elements out of which the world is composed. The wisdom of the Buddha is illustrated by the gigantic numbers he can count out (on the order of 8 times 23 series of 107), and his magnificence is shown by the huge number of Bodhisattvas and other celestial beings who gather to set the scenes for his various sutras. The Hindu cosmology includes a cyclical view of time that enumerates great blocks of years called yugas. There are four yugas ranging from 432,000 to 1,728,000 years, all of which together make up one thousandth of a kalpa or 4,320,000,000 years.

 

This emphasis upon immense, cosmological numbers, gives a distinctively Hindu view of the near-infinite stretches of being that surround the empirical world. It seems almost inevitable that the Hindus should have invented zero (sunya, emptiness, in Sanskrit). The concept sunya, developed about 100CE, was the central concept in Madhyamika Buddhist mysticism, and preceded the invention of the mathematical zero about 600CE. Brahmagupta published a number of rules governing the use of zero and negative numbers in his Brahmasputha Siddhanta (ca. 630CE). Classical Indian world views are permeated with a “mathematics of transcendence.” Numbers were used as a technology for transcending experience not as a mathematics that was directed toward rationalistic generalization. Numbers were used rhetorically to mystify, impress, and awe. In general, numbers were used numerologically rather than mathematically. The social roots of this distinctive mathematical system lie in the particularly exalted status of Indian religious specialists. A sociologist of mathematics would search for the roots of the Hindu emphasis on large numbers in the great variety of ethnic groups making up Indian society, institutionalized in the ramifications of the caste system.

 

The cosmological significance of Chinese mathematics has an ideographic bias. Numbers, and higher mathematical expressions, are written as concrete pictures. The system of hexagrams that make up the I Ching, the ancient book of divination, was continuously reinterpreted in successive Chinese cosmologies as the basic form of the changing universe. Chinese arithmetic and algebra were always worked out in positional notation. Different algebraic unknowns, for example, could be represented by counting sticks laid out in different directions from a central point. Chinese algebra, at its height around 1300CE, could be used to represent fairly complex equations, and included some notion of determinants (i.e., the pattern of coefficients). But it could not be developed in the direction of increasingly general rules. The ideographs (and the social conditions of their use) helped preserve the everyday roots of mathematics.

 

Why did Chinese mathematics take this form? Probably for some of the same reasons that account for the maintenance of ideographic writing among Chinese intellectuals. Both gave a concrete aesthetic emphasis to Chinese culture. The ideographic form had technical limitations that a more generalized form - an alphabet, a more mechanical mathematical symbolism - would have overcome. Ideographs are hard to learn; they require a great deal of memorization. But these limitations may in fact have been the reason why Chinese intellectuals preferred them. A difficult notation is a social advantage to a group attempting to monopolize intellectual positions. This may be contrasted with the algorithmic imperative characteristic of periods of rapid commercial expansion.

Writing and mathematics were highly esoteric skills in the ancient civilizations when they were first developed. Those who possessed these skills were almost exclusively state or religious dignitaries. Writing and mathematical notation tended therefore to be retained in forms that were very difficult to read and interpret, except by those who could spend a long time in acquiring familiarity with them. Sanskrit, for example, was written without vowels and without spaces between the words. Egyptian writing was similarly conservative. Chinese writing and mathematics are notable because archaic styles lasted much longer than anywhere else. The over-riding cultural issue might be that China was, as Leon Stover argued (1974: 24-25), a “once and always Bronze Age culture,” the only primary civilization to develop its Bronze Age to the fullest.

 

The development of ideographs and mathematical notation in China was in the direction of greater aggregative complexity and aesthetic elaboration, not of simplification and generalization. The Chinese literati thus managed to make their tools progressively more difficult to acquire. This is in keeping with the unusually high social position of Chinese intellectuals. They maintained their status through an examination system that was used to select officials in many dynasties. Many students of mathematics have contended that a “good” notation is a condition for progress in mathematics. The question for China or for any given society or mathematical community is: why wasn’t a more appropriate symbolism invented at some particular point in the history of mathematics? To answer that question we should envision a struggle between monopolizing and democratizing forces over access to writing and mathematics.

 

Monopolistic groups were strong in highly centralized administrations such as ancient Egypt, the Mesopotamian states, and China. Democratizing forces won the upper hand in decentralized situations, and/or under social conditions where there was a great deal of private business activity – as in ancient (especially Ionian) Greece, and certain periods in ancient and medieval India. The predominance of these forces was to varying degrees opposed by counter-forces. Greek mathematics also had some conservative elements, especially in the Alexandrian period when difficult rhetorical forms of exposition limited the development of algebra. The specific character of mathematics in given world cultures is due to the differential incidence of such conditions.

 

Greek mathematics is distinguished by its emphasis on geometry, generalized puzzles, and formal logical proofs. This is the intellectual lineage of modern Western mathematics. But the history of Greek and European mathematics also shows a divergent type that rose to prominence following the establishment of the classical form. During the Alexandrian period, another form of arithmetic was developed that was used neither for practical calculations nor for puzzle-contests. This was a type of numerology that used the real relations among numbers to reveal a mystical cosmology. The system was connected with verbal symbolism through a set of correspondences between numbers and letters of the Hebrew or Greek alphabets. Any word could be transformed into a related number that in turn would reveal mathematical relations to other words.

 

The social conditions involved in the creation and development of this alternative mathematics are connected with religious movements. Numerology is related to Hebrew Cabbalism, Christian Gnosticism, and the Neo-Pythagorean revival associated with Philo of Alexandria, ca. 20BCE-50CE). The most prominent expositor of this new

mathematics was Nichomachus (ca.100CE). Like Philo, he was a Hellenistic Jew (living in Syria). It was in this Jewish-Greek intellectual milieu of the Levant that the major religious movements of the time were organized.

 

There are variants even in modern European mathematics. There are conflicts between alternative notational systems in the 1500s and 1600s; and a century-long battle between the followers of Newton and those of Leibniz over the calculus. In the nineteenth century, a major dispute arose between Riemann, Dedekind, Cantor, Klein, and Hilbert and critics such as Kronecker and Brouwer. This split continued and widened in the twentieth century. The result was the emergence of schools in conflict over the foundations of mathematics. The main competitors in this arena were the logicists, the formalists, and the intuitionists

 

4.  The Social Roots of Mathematics

 

The social activities of everyday life in all the ancient civilizations gave rise to arithmetic and geometry, the two major modes of mathematical work. Each of these modes is associated with specific types of social activity. The development of arithmetic is stimulated by problems in accounting, taxation, stock-piling, and commerce; and by religious, magical, and artistic concerns in astronomy, in the construction of altars and temples, in the design of musical instruments, and in divination. Geometry is the product of problems that arise in measurement, land surveying, construction and engineering in general. Arithmetic and geometrical systems appear in conjunction with the emergence of literacy in all the earliest civilizations – China, India, Mesopotamia, Egypt, and Greece. These mathematical systems are, to varying degrees in the different civilizations, products of independent invention and diffusion. Note that while it is analytically “simple” to distinguish geometrical and arithmetical systems and methods it is not always so simple to do this in practice.

 

While we have found arithmetic, geometry, number work, and general mathematics in cultures throughout recorded history, special conditions were required for the emergence and crystallization of the discipline of mathematics. The general human ecological conditions for the emergence of modern science are discussed in Restivo and Karp (1974; and see Restivo, 1979, and 1994: 29-48). A combination of organizational and institutional factors rooted in a human ecology was required to foster and sustain the development of mathematical communities with generational continuity. As those conditions crystallized in Western Europe beginning in the 1500s and earlier, the discipline of mathematics emerged when sets of arithmetic and geometrical problems were assembled for purposes of codification and teaching, and to facilitate mathematical studies. Assembling problems was an important step toward unifying mathematics and stimulating generalization.

 

One of the most important steps in unifying and disciplining mathematics arose from efforts to state general rules for solving all problems of a given type. A further step could be taken once problems were arranged so that they could be treated in more general terms. Problems that had arisen in practical settings could now be transformed into hypothetical puzzles, and problems could be invented without explicit reference to practical issues. The three famous puzzles proposed by Greek geometers of the 5th and

4th centuries BCE are among the earliest examples of such puzzles: to double the volume of a cube (duplication of the cube), to construct a square with the same area as a given circle (quadrature of the circle), and to divide a given angle into three equal parts (trisection of the angle). Such problems were related to the non-mathematical riddles religious oracles commonly posed for one another. One account of the origin of the problem of duplicating the cube, for example, is that the oracle at Delos, in reply to an appeal from the Athenians concerning the plague of 430 BCE, recommended doubling the size of the altar of Apollo. The altar was a cube. The early Hindu literature already refers to problems about the size and shape of altars, and these may have been transmitted to Greece by the Pythagoreans, a secret religio-political society. The problem is also a translation into spatial geometric algebra of the Babylonian cubic

equation X ^{3 =}V .

The duplication, quadrature, and trisection problems were popular with the Sophists, who made a specialty out of debates of all kinds. A generation or two later, Plato introduced the constraint that the only valid solutions to these problems were those in which only an unmarked straightedge and a compass were used. This meant that special mechanical devices for geometrical forms could not be used in mathematical competitions. The result was stiffer competitive conditions and an emphasis on intellectual means and “gentlemanly” norms.        Plato's Academy was organized to help an elite group of intellectuals gain political power; and it represented the opposition of an aristocracy to democratization and commercialization. It is not surprising that this elite group of intellectuals developed an ideology of extreme intellectual purity, glorifying the extreme separation of hand and brain in the slave economy of classical Greece.

 

The three famous Greek puzzles and other problems became the basis for mathematical games (i.e., competitions) of challenge-and-response. Various forms of these games are important throughout most of the subsequent history of Western mathematics. Prior to the nineteenth and twentieth centuries, however, the challenge-and-response competitions were often initiated, endorsed, or rewarded by patrons, scientific academies, and governments. Prizes were sometimes offered for solutions to practical problems. Economic concerns as well as governmental prestige were often mixed in with the struggles for intellectual preeminence. At about the same time that they initiated mathematical contests, the Greek mathematicians took two further steps that led to new mathematical forms. They stipulated that a formal, logical mode of argument must be used in solving problems; this represented a further development of earlier methods of proof. And by extending this idea they created systems of interrelated proofs. This culminated in the Elements of Euclid shortly after 300 BCE. In addition to a collection of problems, Euclid presented an explicit body of generalizations in the form of definitions, postulates, and axioms. Euclid, like Aristotle, did not use the term "axiom" but something closer to "common notion." They both self-consciously worked at codifying past human experiences. The process of "systematization-and- generalization” is one of the two major paths to new mathematical forms. The other major path is an "empirical" one.

 

The empirical path to new mathematical forms involves applying existing mathematical concepts and methods to new areas of experience. Most of the early Greek geometrical

puzzles, for example, concerned flat figures. But the methods of plane geometry could be easily extended to solid geometry, and then to the properties of spheres or of conic sections; the work on conic sections eventually led to work on curves of various shapes. The intermittent periods of creativity in Alexandrian mathematics (especially from 300 to 200 BCE and 150-200 CE) were largely devoted to these extensions. No new level of generalization (with the exception of trigonometry, considered below) was achieved, but a number of new specialties appeared.

 

The history of arithmetic shows some of the same processes that occur in the history of geometry. The effort to find general rules for solving numerical problems led gradually to what we now call algebra. Here again we see mathematicians developing the practice of posing problems primarily to challenge other mathematicians. For example, there is this famous problem, attributed to Archimedes (287 to 212 BCE): find the number of bulls and cows of various colors in a herd, if the number of white cows is one third plus one quarter of the total number of black cattle; the number of black bulls is one quarter plus one fifth the number of the spotted bulls in excess of the number of brown bulls, etc. Such problems, involving unknown quantities, led over a very long period to the introduction of various kinds of notations and symbolisms.

 

These took quite different directions in ancient and medieval China and India, the Arab world and later in medieval and Renaissance Europe. The creation of a highly generalized symbolism which could be mechanically manipulated to find solutions did not appear until the late1500s and 1600s in Europe.

 

Over this period, and to different degrees in different parts of the world, algebra underwent an empirical extension. Problems were deliberately created to increase the number of unknowns, and to raise them to successively higher powers. Equations of the form ax + b = c gave way to those on the order of ax⁴ + by³ + cz² = g . The complexity of these, of course, could be extended indefinitely (Vieta in the1580s, for example, was

challenged to solve an equation involving x⁴⁵ ); but the extensions also gave rise to

efforts to find general rules for solving higher order equations. In other words, empirical extensions tended to promote generalized extensions. At the same time, arithmetic was developing in other directions.

 

What is generally called elementary arithmetic (solving numerical problems in, for example, addition, subtraction, multiplication, and division) continued to stimulate efforts to find general rules for solving particular problems. There was tremendous variation from one system of numerical symbols and calculating rules to another in terms of the ease or difficulty with which they could be applied to solving practical problems. Most of the ancient forms of notation made working with large numbers, fractions, or complex operations like division or the extraction of roots difficult; the exposition of problems was usually carried out in words. A great deal of mathematical creativity went into the development of notational systems that could be readily manipulated. Among the most important of these innovations were the invention of decimal place notation and the zero sign in India; the standardization of positional methods for writing multiplication and division (in Europe ca. 1600); and the invention of logarithms by the Scotsman Napier in 1614, for use in astronomy, navigation, and commerce.

A different development in arithmetic led to what we now call “number theory.” This focused on the properties of numbers themselves. As early as Eratosthenes (ca. 230 BCE), efforts were made to find a general method for identifying prime numbers. There were also various propositions about how numbers are composed of other numbers. The Pythagorean work on “triangular’ and “square” numbers anticipated Fermat’s theorem that every prime number of the form 4n +1 is a sum of two squares. Number theory was particularly popular in the Alexandrian period in an occultist, cabalistic form. In its more standard puzzle-solving form, it has remained popular among mathematicians from the Renaissance through the modern period.

 

One more branch of mathematics, based on a combination of arithmetic and geometry, developed in the Alexandrian period. Measurements of angles and lines, and the calculation of their ratios, led to the creation of trigonometry, notably by Hipparchus (ca. 140 BCE) and Menelaus (ca, 100 BCE). Trigonometry spread to medieval India and the Arab world, and in Renaissance Europe provided the basis for Napier's development of logarithms.

 

The overall picture so far, then, shows mathematics arising from practical geometry and arithmetic. The development of general mathematical puzzles and the extension of mathematics to new areas led to the emergence of new fields. Geometry became increasingly systematic, and progressively applied to plane and solid figures, to conics, and eventually to trigonometry. Arithmetic gave rise to algebra in successfully more complex forms (based on practical calculating systems), and to number theory. The creation of new fields continued in modern Europe. They grew out of the processes of generalizing (commonly understood in terms of increasing levels of abstraction). New fields were furthermore the result of extending results to new empirical areas, and the combination of existing mathematical fields into hybrid fields. The combination of algebra with a new coordinate representation in geometry by Descartes and Fermat produced analytic geometry.

 

Consideration of the problems of motion and the study of curves gave rise to the calculus in the 1600s. Calculus was then applied to successively more complex functions (empirical extension); and eventually (in the 1800s) it was generalized into a theory concerning such things as the rules for solving equations, and the properties of all functions (generalized extension). It should be noted that the drive towards creating new fields by generalization and extension seems to be characteristic of highly competitive periods. Geometry itself experienced a rapid series of branching around 1800 and thereafter, the best known being the non-Euclidean geometries. But there was also the creation of descriptive geometry by Monge, projective geometry by Poncelet, higher analytical geometry by Plucker, modern synthetic geometry by Steiner and Von Staudt, and topology by Mobius, Klein, and Poincare. In the late nineteenth and early twentieth centuries, Klein, Hilbert, and Cartan unified these different geometries. This unification occurs prominently in Klein’s Erlangen Program and its generalization in Cartan’s program which was designed to place the unification into the framework of Riemannian geometry.

 

In algebra, there was a parallel set of developments after 1800.The effort to find a general solution for the quintic and other higher-order equations led to the creation of

the theory of groups by Abel, Galois, Cauchy, and others. This theory focused on an abstract pattern among the coefficients of equations, and opened up a new area of inquiry in higher mathematics. “Abstract” algebras were created by Boole, Cayley, Sylvester, Hamilton, and Grassman. All of these new tools were applied to other branches of mathematics. Dedekind applied set theory to the calculus, Cantor applied it to the concept of infinity, and others applied it to topology, number theory, and geometry. These developments led to the creation of yet another even more general field toward the end of the nineteenth century, “foundations.” "Foundations" focused on the nature of mathematical objects themselves and with the rules by which mathematics should be carried out. Foundations research has been the focus of a number of opposing schools, and has led to what are probably the most intense controversies in the history of mathematics.

 

The basic forms of mathematics, arithmetic and geometry arise from practical problems in construction, taxation, administration, astronomy, and commerce. Moreover, the stimulus of practical concerns does not simply disappear once mathematics is launched. For example, the basic forms of arithmetic, including the number system, developed over a very long period, during which virtually the sole interest in improvement was to facilitate practical calculations. The same can be said for the invention of logarithms, and much of the development of trigonometry. Other advanced forms of mathematics were also stimulated by efforts to solve practical problems. The development of the calculus was linked to problems in ballistics and navigational astronomy. Newton’s mechanics makes an interesting case study in this regard. In a hallmark paper in the history and sociology of science presented at the Second International Congress of the History of Science in London in 1931, Boris Hessen (1893-1936) situated Newton’s work in the technological problems of his time, his social class position, and Newton’s lack of familiarity with the steam engine and the conservation of energy principle. Hessen’s views contradicted the view that Newton was divinely placed at a particular historical juncture and that his genius was the fountain out of which his physics sprung. The present author noted earlier R.K. Merton’s defense of Hessen’s sociology of science against the idealistic and motivational explanations offered by G.N. Clark. Merton’s argument implicitly opposes Carlyles’s Great Man theory, Alfred North Whitehead’s mystical explanation of Newton’s successes (he was born in the same year that Galileo died), and Alexander Pope’s divine providence conjecture: “Nature and nature’s laws lay hid in night; God said ‘Let Newton be!’ and all was light.”

 

Descriptive geometry and Fourier's analysis answered problems in the production of new machinery in the industrial revolution. Practical concerns do not tell the whole story of mathematics, but they are one component that continuously shapes its history. This suggests a general principle: an increase in the amount, type, intensity, or scope of practical concerns in a society will stimulate mathematical activity. The relationship between economic concerns and mathematics is especially strong; commercial growth tends to be very stimulating for mathematics. Mathematical innovations will also tend to occur when there is a shift to new productive technologies (and when there are shifts to new technologies of warfare and transportation, and shifts to more intensive administrative modes of organization). This implies a link between the development of modern European mathematics and the development of modern industrial technological societies (loosely, “capitalism” because the term “capitalism” does not refer to an actual

economic system but rather expresses an economic ideology). Since this is one factor among several, it does not imply that mathematics must come to an end in non-capitalist societies. It does, however, suggest that the form and content of mathematics (within the constraints noted by Spengler) as we know it today is a product of specific lines of cultural development. One could say that modern mathematics, like modern science, is part of the knowledge system generated by and supportive of capitalism.

The roots of mathematics in practical concerns are more apparent in some cases than in others. For example, the history of Chinese mathematics is primarily a history of an inductive “mathematics of survival." Its origins can be traced from the myth of Yü the Great Engineer's discovery of a magic square on the back of a Lo River tortoise (ca. 500 BCE). China’s mathematical evolution reaches its high point in the late Sung and early Yuan dynasties with the publication of Chu Shih-Chieh's "Precious Mirror of the Four Elements," written in 1313. Chinese mathematics never ventured far from problems of everyday life such as taxation, barter, canal and dike construction, surveying, warfare, and property matters. Chinese mathematical workers could not organize an autonomous mathematical community, and consequently failed to establish the level of generational continuity that is a necessary condition for long-term mathematical development. This helps to explain why the Chinese did not develop the more general forms of higher mathematics.

 

Conditions in ancient Greece were more favorable for generalizing mathematics. The commercial expansion in Greece in the 600s BCE stimulated mathematical growth. Learned merchants practiced and taught mathematical arts, and master-student relationships across generations fostered mathematical progress. Political and economic changes in Greek civilization led to the development of an increasingly elitist and self- perpetuating intellectual community, culminating in the oligarchic conditions and intellectual elitism of Plato's time. The achievements of the "thinking Greeks" depended on a division of labor that divorced hand and brain. The "thinkers" had the "leisure" to reflect on and elaborate mathematics. The class structure of the slave-based society that developed in the post-Ionian period conditioned the development of classical mathematics. Arithmetic was left to the slaves who carried out most commercial transactions, and householders for whom simple calculations were a part of everyday life. The elite intellectual class, by contrast, courted geometry which was considered democratic and more readily adapted to the interests of the ruling classes than arithmetic. What we know as "Greek mathematics" is a product of the classical period.

 

The development of specialties within the division of labor, left unchecked, tends to foster virtuosity. Such specialization tends to increase the specialists' distance from the order and distribution of everyday phenomena and to increase the importance of human- created phenomena, especially symbols. The result is an increase in the level of generalization and the development of ideologies of purity. This is essentially what occurred in classical Greece. Hand and brain slowly reunited following Plato's death; there is already evidence of an increased interest in linking mathematical and practical concerns in Aristotle. In the Alexandrian period, hand and brain were more or less united, but the ideology of purity retained some vitality. This is notably illustrated by Archimedes, whose work clearly exhibited a unity of hand and brain but whose philosophy echoed Platonist purity.

The decline of Greek commercial culture was accompanied by the decline of Greek mathematical culture. The achievements of Archimedes, which brought Greek mathematics to the threshold of the calculus, mark the high point of Greek mathematics. When mathematics was revived in the European commercial revolution (beginning haltingly as early as the twelfth century, the Gothic cathedrals signaling the coming economic revolution by serving as accumulating devices for labor, resources, and machinery), many aspects of the Greek case were recapitulated. European mathematics moved on in the direction of the calculus, rooted in problems of motion. It picked up, in other words, essentially where Archimedes had left off, and under the influence of the Archimedean (and more generally, Greek) writings as they were recovered and translated (and here, of course, the Arabic-Islamic transmissions were critical). By 1676, Newton was writing about mathematical quantities "described by continual motion."

 

The concept of function, central to practically all seventeenth and eighteenth century mathematics, was derived from studies of motion. Newton and Leibniz helped to reduce the basic problems addressed in the development of the calculus – rates of change, tangents, maxima and minima, and summations – to differentiation and anti- differentiation. Infinitesimals, nurtured earlier in the debates of theologians and the scholastics, entered into the process of production. Highly general intellectual ideas in the Euclidean realm of the straight, the flat, and the uniform gave way to the ideas of a dynamic world of guns and machinery, and global navigation and commerce characterized by skews, curves, and accelerations. The search for algorithms, time- saving rules for solving problems, is evident in the writings of the inventors of the calculus (e.g. in Leibniz's "De geometria recondita et analysi indivisibilium atque infinitorium" of 1686).

 

As the industrial “machine" of capitalist society was fashioned, so was the “machine of the calculus." Descartes' analytic geometry, the other great contribution to the development of pre-modern European mathematics, was also characterized by an algorithmic imperative.

 

It was, in spite of the conflicts between Cartesians and Newtonians, from the very beginning in constant association with the development of the Newtonian-Leibnizian calculus. Let us note parenthetically that Newton and Leibniz invented two different calculuses. Newton’s “method of fluxions” was much more indebted to classical geometry (and especially to Archimedes) than is commonly supposed, and Leibniz’s “differential calculus” used a much better notation (Restivo, 1992: 134).

 

The historian of mathematics Boutroux (1919) characterized Descartes’ analytic geometry as an industrial process; it transformed mathematical research into “manufacturing.” The idea that the calculus is linked to the emergence of capitalism (that is, early industrialization) is further suggested by the Japanese case. When the Japanese established a monetary economy and experienced a commercial revolution in the seventeenth century, they also worked out a "native calculus." This was not entirely indigenous since they had probably come into contact with Europe’s calculus from European contacts at their ports of entry.

5.  Puzzles and Proofs

 

Mathematical workers and mathematicians, from the earliest times onward, and especially in the West, have posed puzzles for one another. This practice tends to make mathematics a competitive game. Some periods have been dominated by public challenges such as those that the Emperor Frederick's court mathematician posed to Leonardo Fibonacci (ca. 1200), those that Tartaglia and Cardano posed for one another in sixteenth century Italy, or those that led to such high acclaim for Vieta at the French court in the 1570s. Such puzzle-contests have been important for several reasons. They often involved pushing mathematics into more general realms. Mathematicians would try to invent problems which were unknown in practical life in order to stump their opponents. The search for general solutions to equations, such as those that Tartaglia found for cubic equations and Vieta found for the reduction of equations from one form to another, was directly motivated by these contests.

 

The emphasis on proofs which has characterized various periods in the development of mathematics was partly due to a heightening of the competitiveness in these contests. Greek mathematicians rationalized the concept and method of proof at a time when mathematics was popular among the elite class of philosophers and there was a lot of competition for power and attention in the intellectual arena. This was the same period during which the wandering Sophists challenged one another to debating contests and in doing so began to develop canons of logic. This is completely analogous to the development in mathematics, in terms of both cause and effect. The analogy turns into a virtual identity when we realize that many of the mathematicians of the time were Sophists, and that many of the formal schools that were organized in the classical period (e.g., the Academy) used prowess in mathematics as a grounds for claiming superiority over competing institutions. Stressing proofs was a way of clarifying the rules of the game and escalating the intensity of competition. In general, competitive puzzle- contests are probably responsible for much of the inventiveness characteristic of Western mathematics. This analysis should not obscure the economic stimulus to the initial development of proofs.

 

Thales, the philosopher-merchant (who might have been a composite or imaginary construction), is credited with carrying the idea of a proof to a more general level than the Babylonians and Egyptians. Thales symbolically and iconically personifies the need among the Ionians of his era to develop a comprehensive and organized understanding of physical reality and successful computational methods in the context of the increasingly well-organized economy that they were products of and helped to fashion. Thales' proofs were probably crude extensions of Babylonian or Egyptian "rules" for checking results. In any case, the process of constructing proofs was rationalized over the next three hundred years and eventually led to Euclidean-type proofs.

 

Concern for proof has varied a great deal in the history of mathematics. The Chinese and Hindu mathematicians ignored proofs almost entirely; indeed, they would often present problems without solutions, or with incorrect solutions. These practices were the result of a relatively uncompetitive situation in mathematics in these societies. The social density of mathematicians in these societies was rather low; we rarely hear of more than a few mathematicians working at the same time, whereas in Greece and

Europe the numbers in creative periods are quite high. Most of the Oriental mathematicians were government officials, and thus were insulated from outside competition, while most of the ancient Greek and modern mathematicians were private individuals or teachers in competitive itinerant or formal educational systems. In the Islamic-Arabic world, there was a flurry of mathematical activity in the period 800- 1000 (and later to some extent).There was some concern for proofs (in the works of Tahbit Ibn Qurra, for example), but this was much more limited than in classical Greece. The Greek works they translated stimulated an awareness of and interest in proofs among the Islamic-Arabic mathematicians.

 

The limited emphasis on proofs reflects the fact that their community (or network) was not as densely populated as the Greek mathematical community, competition was not as intense, master-student chains and schools were not as well organized, and generational continuity was limited. Similarly, the episodic history of mathematics in India constantly interrupted generational continuity, the Bronze Age dynastic history of mathematics in China undermined specialization free of the centripetal force of the Emperor’s bureaucracy, and the delimited mathematical renaissance of Japan in the seventeenth century ended abruptly with the consolidation of Tokogawa power (Restivo, 1992: 22-60). Only Europe from the 1500s on was able to sustain generational continuity on a level no other civilization had achieved.

 

In modern Europe, the emphasis on proofs has grown steadily. In the 1600s, Fermat presented his theorems without proofs, and in the 1700s, Euler offered proofs that were not very rigorous. The early 1800s saw a shift towards more rigorous standards of proof; earlier solutions were rejected, not because they were incorrect, but because the reasoning behind them was not sufficiently universal and comprehensive. This went along with a massive increase in the number of people engaged in mathematics (which in turn was the result of the expansion of educational systems, especially in Germany and France and other social changes related to the rationalization of social and economic life). Both this shift towards rigor, and the earlier invention of proofs, had important effects on the nature of mathematics. Both pressed mathematics toward new levels of generalization: proofs had to invoke more general elements than particular numerical examples, and rigorous proofs stimulated the systematic consideration of the nature of mathematics in the nineteenth century.

 

6.  Generalization (“Abstraction”) and Self-Consciousness

 

The "main line" development of Western puzzle-solving mathematics is characterized by an increasing awareness that levels of generalization have been created by the mathematicians themselves. Mathematicians moved beyond a naïve realism about mathematical objects (sometimes real in the material sense, sometimes real in the Platonic sense) when they gradually began to use negative numbers instead of dropping negative roots of equations (as Hindu, Arab, and medieval European mathematicians had done). Later they came to recognize that imaginary numbers (an unfortunate and distracting nomenclature) could be used despite their apparent absurdity. Gauss established a new basis for modern algebra by creating a representational system for complex numbers. Nineteenth century higher mathematics took off from this point.

Mathematicians finally realized that they were not tied to common-sense representations of the world, but that mathematical concepts and systems could be deliberately created. These creations were not ab novo but situated in the cultural objects of the increasingly well-organized mathematical community. The new, more general geometries (including projective and non-Euclidean geometries) popularized the point, and stimulated the creation of new algebras and more generalized forms of analysis. The objects with which modern mathematics deals, however, are real in a particular sense. They are not simply things, as was once naively believed; they are, rather, operations, activities that mathematicians carry out. The imaginary number i is a shorthand for an activity, the operation of extracting a square root from a negative number. This operation, of course, cannot be carried out. But mathematicians had long been used to working backwards from solutions-not-yet-found, to the premises, by symbolizing the solution using an arbitrary designation (e.g., x ). This symbol represented the result of an imaginary operation. The imaginary number i, then, could be used as the basis for other mathematical operations, even though the operation of producing it could never actually be performed. The ordinary arithmetic operations, the concept of a function, the concept of a group - all of these are operations of different degrees of complexity. A natural whole number itself is not a thing but an operation - the operation of counting (and perhaps also other operations whose nature modern mathematicians are untangling). Modern mathematics has proceeded by taking its operations as its units. These are crystallized into new symbols which can then be manipulated as if they were things. In fact, within the social world of mathematics these symbols are things, the matters of fact, and the material resources of the human ecology of the mathematics community (see Alexksandrov, Kolmogorov, and Lavrent’ev, 1963 for a thoroughly materialistic and realistic history of mathematics).

 

A process of reification has gone on in conjunction with the emergence of the notion that generalized operations (what are commonly referred to as “abstractions”) are socially created by mathematicians. Thus mathematics has built upon itself hierarchically by treating operations as entities upon which other operations can be performed. The Western trend in symbolism, then, is not an "accidental" feature of Western mathematical uniqueness; the symbolism was created precisely because the mathematical community was pushing towards this degree of self-consciousness and reflexivity. What I am pointing to here is an emerging awareness among the more reflective mathematicians and philosophers of mathematics that crystallized in the late nineteenth and early twentieth century. That awareness that the creative construction of mathematics came from the hands and brains of situated human beings was enough to fuel the more recent efforts to ground mathematics in the material world of activities and experience. It has not, however, been enough to eliminate all vestiges of Platonic thinking among mathematicians.

 

Mathematics, like other modern activities, has been affected by specialization on a level unknown in earlier historical periods. As a result, the "causal power" of mathematics itself in the relationships between mathematical and other social activities has steadily increased. Mathematical ideas have increasingly become the generative basis for new mathematical ideas. The work setting and institutional context of mathematical activity has become a social foundation of a higher order than the social foundation of subsistence productive activity. Mathematics continues to be socially rooted within the

mathematical community; it is especially important to recognize the social nature of the symbols mathematicians create for communication within their own ranks.

 

This perspective throws new light on an old problem reflected in the title of a famous paper by Eugene Wigner (1960) on “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” The utility of “pure” mathematics for the physical and natural science is not just a coincidence; it must reflect some larger and deeper truth about both mathematics and physics. In fact, the “coincidence” reflects a constant interplay between mathematics and physics. As soon as this is interrupted by the professional and bureaucratic closure of the mathematics community, we can expect the cycles of effective applications to become increasingly rare. This is not just a sociological conclusion based on the theory of closure and autonomy in social systems (Restivo, 1992: 171-175; Restivo, 1993: 263-267) but something that has been recognized by professional mathematicians (e.g., Boos and Niss, eds., 1979).

 

The development of Western higher mathematics, then, is a social and a cultural development. The objects with which mathematicians deal are activities of mathematicians. In building upon the operations already in existence, and making them symbolic entities on which further operations can be performed, mathematicians are self-consciously building upon previous activities in their intellectual community. Mathematics thus embodies its own social and cultural history, and uses it as the base upon which its current community activities are constructed. Western mathematics thus depends upon a particular kind of long-term organization of the intellectual community. This is an organization in which strong links are maintained across generations, and in a highly self-conscious and competitive form. The new attempts to competitively consume the old. The important linkages of teachers and pupils typically found among European mathematicians, together with strong external competition among different mathematical "lineages," have been the social basis for this pattern (on the general theory of lineages and networks as the progenitors of ideas, see Collins, 1998). Once the pattern of competitive self-consciousness was established, subsequent rounds of competition could only escalate the degree of self-reflection and inventiveness among mathematicians. Out of this situation arose the hyper-reflexive concerns of twentieth- century foundations research. It is important to keep in the mind the self-conscious creativity of mathematicians is inseparable from the crystallization of an autonomous mathematical community.

 

7.  Conclusion

 

All thought, in its early stages, begins as action. The actions which you [King Arthur] have been wading through have been ideas, clumsy ones of

course, but they had to be established as a foundation before we could begin to think in earnest.

T.H. White’s Merlyn the Magician What, “In the beginning was the Word?” Absurd.

Then maybe it should say “In the beginning was the Mind?” Or better “…there was Force?”

Yet something warns me as I grasp the pen,

 

 

Act.”

That my translation must be changed again.

The spirit helps me. Now it is exact. I write: “In the beginning was the

 

Goethe’s Faust

 

The history of mathematics can be situated socially and culturally. This task requires a sociology both of the external institutional and organizational conditions of the societies within which mathematical activities are situated, and a sociology of the internal organization of the mathematical activities within communities and social networks of mathematicians. The notion of "internal and external factors" is an analytic device. The Spenglerian idea of mathematics as a world view is not, in the end, compatible with a strict adherence to internal-external analysis. The mathematics of any particular time embodies its own social history. This process becomes increasingly intense as and to the extent that mathematical activity becomes and remains more clearly differentiated from other social activities and more autonomous. But "autonomy" simply means that mathematicians communicate more intensively with each other than with outsiders. It does not mean that mathematicians are more removed from social determinants or that they have unimpeded access to "objective reality." Their activities remain at all times coupled to the social activities of insiders and outsiders, and thus unfold in an environment of multiple social, cultural and historical determinants. This is the rationale for a Spenglerian approach to the sociology of mathematics. That this is becoming more accepted is indicated by a clear turn to practice, experience, and shared meaning in the philosophy of mathematics, the philosophy of mathematics education, and among reflective mathematicians.

 

We are no longer entranced by the idea that the power of mathematics lies in formal relations among meaningless symbols, nor are we as ready as in the past to take seriously Platonic and foundationalist perspectives on mathematics. We do, however, need to be more radical in our sociological imagination if we are going to release ourselves from the strong hold that philosophy has on our intellectual lives. Philosophy, indeed, can be viewed as a general Platonism and equally detrimental in its classical forms and agendas to our efforts to ground mathematics (as well as science and logic) in social life. It is to philosophy in its more recent turn to practice that we must look if we are going to salvage philosophy as a credible intellectual activity. The strengths of philosophy as a guide to reasoned speculation, thought experiments, and ethical stands will be heightened by the turn to practice and the empirical arena. This is how we can save philosophy as an analytical tool for understanding mathematics. And here is where philosophy is drawn into the interdisciplinary boundary breaking modalities of late twentieth and twenty first century intellectual life. We are witnessing the demise of the traditional disciplines as we have known them for more than one hundred years. As the new inter-disciplines come to the fore, it begins to appear that we are witnessing the emergence of a second generation natural philosophy. For the moment, let us bracket this development and focus on mathematics and sociology as distinct, viable disciplinary projects. Both of these disciplines are undergoing interdisciplinary developments that are contributing to the emergence of a neo-natural philosophy.

 

Once again, then, what is mathematics? Technical talk about mathematics – trying to understand mathematics in terms of mathematics or mathematical philosophy – has the

effect of isolating mathematics from practice, experience, and shared meaning; it tends to “spiritualize” the technical. It is important to understand technical talk as social talk, to recognize that mathematics and mathematical objects are not (to borrow terms from the anthropologist Clifford Geertz' (1983: 94-120) “concatenations of pure form,” “parades of syntactic variations,” or sets of “structural transformations.” To address the question “What is mathematics?” is to reveal a sensibility, a collective formation, a worldview, a form of life. This implies that we can understand mathematics and mathematical objects in terms of a natural history, or an ethnography of a cultural system. We can only answer this question by immersing ourselves in the social worlds in which mathematicians work, in their networks of cooperating and conflicting human beings. It is these “math worlds” that produce mathematics, not individual mathematicians or mathematicians’ minds or brains. It is easy to interpret this perspective as somehow “mystical,” or as implying a “super-organic” entity. A realistic interpretation requires understanding human individuals as social things and their ideas as the “voice” of their social networks.

 

Mathematics, mathematical objects, and mathematicians themselves are manufactured out of the social ecology of everyday interactions, the locally available social, material, and symbolic interpersonally meaningful resources. All of what has been written in the last two paragraphs is captured by the short hand phrase, “the social construction of mathematics.” This phrase and the concept it conveys are widely misunderstood. It is not a philosophical statement or claim but rather a statement of the fundamental theorem of a sociology broadly conceived, a sociology being transformed by the interdisciplinary movements of our era. Everything we do and think is a product of our social ecologies. Our thoughts and actions are not simple products of revelation, genetics, biology, or mind or brain. To put it the simplest terms, all of our cultural productions come out of our social interactions in the context of sets of locally available material and symbolic resources. The idea of the social seems to be transparent, but in fact it is one of the most profound discoveries about the natural world, a discovery that still eludes the majority of our intellectuals and scholars. The interdisciplinary imperative can sustain this idea of the social even while it brings it into closer and closer association with our biology (cf. Clark, 2010, Noë, 2010, and Reyna, 2007).

 

Mathematics is a human, and thus a social, creation rooted in the materials and symbols of our everyday lives. It is earthbound and rooted in human labor. We can account for the Platonic angels and devils that accompany mathematics everywhere in two ways. First, there are certain human universals and environmental overlaps across the variety of our material environments, the physics, biology, and chemistry of life, culture, space, and time that can account for certain “universalistic” features of mathematics. Everywhere in everyday life, putting two apples together with two apples gives us phenomenologically four apples. But the generalization that 2+2 = 4 is culturally glossed and means something very different in Plato, Leibniz, Peano, and Russell and Whitehead. The earthbound everyday world of apples is commensurable for Plato and Russell and Whitehead; that world has not changed in the millennia that separate their lives. However, the discipline and then the profession of mathematics has changed dramatically across the centuries and created new incommensurable experiences within the social world of mathematics. The professionalization of mathematics gives rise to the phenomenon of mathematics giving rise to mathematics, an outcome that reinforces

the idea of a mathematics independent of work, space-time, and culture. Mathematics is always and everywhere culturally, historically, and locally embedded. There is, to recall Spengler, only mathematics and not Mathematik. There is, however, number work that carries across all cultures. Culture always intrudes in our mathematics; but while there is no Mathematik, there are ways for us to translate and communicate commensurable number and math work across cultures.

 

The concept-phrase “mathematics is a social construction” must be unpacked in order to give us what we see when we look at working mathematicians and the products of their work. We need to describe how mathematicians come to be mathematicians, the conditions under which mathematicians work, their work sites, the materials they work with, and the things they produce. This comes down to describing their culture – their material culture (tools, techniques, and products), their social culture (patterns of organization – social networks and structures, patterns of social interaction, rituals, norms, values, ideas, concepts, theories, and beliefs), and their symbolic culture (the reservoir of past and present symbolic resources that they manipulate in order to manufacture equations, theorems, proofs, and so on). This implies that in order to understand mathematics at all, we must carry out ethnographies – studies of mathematicians in action. To say, furthermore, that “mathematics is a social construction” is to say that the products of mathematics – mathematical objects – embody the social relations of mathematics. They are not free standing, culturally or historically independent, Platonic objects. To view a mathematical object is to view a social history of mathematicians at work. It is in this sense that mathematical objects are real. Before there is mathematics there is number work; before there are professional mathematicians there are number workers and then mathematics workers.

 

Arithmetic, geometry, and the higher mathematics are produced originally by number or mathematical workers and later on by variously disciplined and ultimately professional mathematicians. Ethnographies and historical sociologies of mathematics must, to be complete, situate mathematics cultures in their wider social, cultural, historical, and global settings. They must also attend to issues of power, class, gender, ethnicity, and status inside and outside more or less well-defined mathematical communities.

 

There is a hidden interrogation in the interrogation of mathematics that undermines its claims to transcendence and purity. That hidden interrogation is the interrogation of the very idea of the transcendent, of a Platonic realm of ideas. It is not too much of a stretch to see that this sort of interrogation will sooner or later have us interrogating religion and the gods. This is not the place to follow this line of inquiry. However, it is important to note that any deep understanding of the nature of mathematics as a this- worldly phenomenon is necessarily linked to the possibility of bringing religion and the gods down to earth. One only has to consider that the sociologist Emile Durkheim (1912/1995) concludes his remarkable study of the social construction of religion and the gods by arguing that logical concepts are, like religion and the gods, collective representations, this-worldly social constructs. Durkheim manifests the unfolding of the sociological enterprise as an exercise in the rejection of the transcendental.

 

Finally, let us revisit the ideas of civilization and progress. As we saw earlier, we could try to place ourselves on a continuum of civilization and progress between the polar

positions of Rousseau and Ferguson; everything about human society and culture as we have known it (especially since the coming of modern science and technology and the industrial revolution) denies our humanity, or everything trumpets its triumphs. This inevitably enmeshes us in a conflictful conversation without stop signs.

 

We could try to adopt Lenski’s notion of progress introduced earlier and bring the idea of civilization into his framework as a form of social and cultural organization. This would give us a more “scientific” foundation to rest on, but not one without its own ethical and value biases and implications. In the end, we are probably safest at this point if we recognize that mathematics is one of the many tools humans have fashioned to help them wind their way through the complex tapestries of life, and then one of humanity’s most useful as well as awe inspiring tools, even if more often than not actualized in the service of state power and ruling elites as “weapons of math destruction.” Moreover, it is one of the achievements that Nietzsche would have recognized as part of the fragile reason and sense of freedom humans have purchased at great cost.

 

Every entity recognized in history as a “civilization” from ancient Sumer and Babylonia to contemporary China has fed mathematical traditions into the ocean of a mathematical heritage that is in principle a part of humanity’s world culture. We can value the mathematics that has been developed out of and applied to the best of our activities as humans, activities that have been liberating, supported social justice, and in general made our planet a better and safer place on which to pursue the betterment of our species and its social and natural ecologies. We are by many measures and accounts at a cross-roads in human and planetary history, perhaps the cross-roads that opens a road just a couple of hundred years into the industrial-technological global society to annihilation on various levels and scales. It may be that our fate as a species and a part of a global and planetary ecology doomed to annihilation on a scale of millennia is now sealed on a scale of centuries or even years. Our survival, and especially our survival with some decent quality of life distributed across the planet and its life forms, will depend on long-term thinking that is at once broad, deep, and wise. All of our civilizational tools will be needed in perhaps our last chance in the short run to “save” ourselves and our planet.

 

Ours is an era when the very idea of progress, under attack throughout the twentieth century as wars, holocausts, ecological disasters, and radioactive fallout crushed romantic and idealistic dreams of a better world, has given way to concerns about unintended consequences, precautionary principles, and technology assessments. How dangerous it sounds today to hear the echoes of the British Labor Party’s celebration in the mid-1960s of “the white heat of technology revolution.” And yet we must ask if we are still subject to the hypnotic seductions of the technological lottery. Imagine a group of distinguished scientists brought together to speculate on “the next hundred years,” and to speculate optimistically because the idea (read “ideology’) of science was by definition inclined to optimism. And consider that when just such an event was organized to celebrate the centennial of Joseph E. Seagram & Sons (purveyors of whiskey and related spirited drinks) in 1957, they all choked on their optimism as they offered their speculations. Distinguished speakers such as Nobelist geneticist Herman J. Muller, geochemist Harrison Brown, economist and college president John Weir, and

infamous rocket scientist Wernher Von Braun found it necessary to qualify their optimism. Muller said the future would be rosy if we could avoid war, dictatorship, overpopulation, or fanaticism; Brown began by saying “If we survive the next century; Weir began with “If man survives…” Braun, in the most bizarre opening sentence, said “I believe the intercontinental ballistic missile is actually merely a humble beginning of much greater things to come.”

 

No matter. It is beyond doubt that mathematics will play a key role in our efforts, successful or not, realistic or delusional, to realize civilization and progress in terms that even Rousseau might embrace. This is not meant to lock into mathematics as simply a means to quantifying problem formulation and problem solving across all arenas of human experience. It is meant mathematics as a humanistic mode of knowing. It is not only or at all to Plato’s view of the role of mathematics for his guardian-rulers we must turn to (keeping in mind that for all his resistance to poetry and metaphor, Plato was a master of both). The author wants rather to endorse something like Nietzsche’s notion of mathematics. That is, the rigor and refinement of mathematics must be brought into all of our endeavors but not because this is necessarily the best way to apprehend things (recall Nietzsche’s views on the limits of science). We should want to do this in order to become more aware of our relationships as human beings to the things in our world. Mathematics does not give us the most general or the ultimate form of knowledge but opens a path (only asymptotically, at best) to such a goal. This view of mathematics has been promoted for almost half a century by the humanistic mathematics movement. If civilization has a future, if progress can become at least a Lenskian possibility, a humanistic mathematics will have to take hold in our efforts to identify, specify, and solve the problems of planetary survival we face today.

 

Acknowledgments

 

The author wants to acknowledge my collaboration with and indebtedness to Randall Collins of the University of Pennsylvania with whom he first tackled the problem of mathematics and civilization in the early 1980s. Randy is an important inspiration for the perspective developed here. He also wants to thank his collaborators and colleagues in the sociology of mathematics over the many years in which he worked in this field: Wenda Bauchspies, Deborah Sloan, Monica Mesquita, Ubiratan D’Ambrosio, the late Dirk Struik, the late Joseph Needham, the late Leone Burton, Paul Ernest, David Bloor, Donald MacKenzie, Jean Paul Bendegem, and Jens Hoyrup,

 

Glossary

 

Abstraction                : Classically, a process by which higher level concepts are derived from the usage and classification of literal ("real" or "concrete") concepts, first principles, or other methods; used as a noun, a concept that acts as a super-categorical noun for all subordinate concepts, and connects any related concepts as a group, field, or category. Abstractions may be formed by reducing the information content of a concept or an observable phenomenon, typically to retain only information which is relevant for a particular purpose. For example, abstracting a

leather soccer ball to the more general idea of a ball retains only the information on general ball attributes and behavior, eliminating the other characteristics of that particular ball. In this chapter, abstractions are replaced by the term “generalization” and given a concrete sense. In this sense, what we normally refer to as abstractions are the concrete terms of an autonomous professional community of practice.

Arithmetic                  : From the Greek word ἀριθμός, arithmos “number”; historically, the oldest and most elementary branch of mathematics, used for everything from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of operations that combine numbers. In common usage, it refers to the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers. The term “higher arithmetic” as distinguished from elementary arithmetic, is used in professional mathematics to refer to more advanced practices related to number theory.

Civilization                 : A relatively high level of cultural and technological development; specifically associated with the development of writing and written records in the ancient world. Classically, “civilized” peoples contrasted themselves with “barbarians;” in modern times, “civilized” has been opposed to “primitive” or “savage.” Viewed in positive and optimistic terms, civilizations are characterized by complexity, hierarchies, dynamism, advanced levels of rationality, social progress, and a statist form of government. Viewed in negative pessimistic terms, notably by the philosopher Rousseau, civilization is conceived to be opposed to human nature.

Ethnomathematics : The study of the relationship between mathematics and culture. This research brings to light the mathematical and logical traditions of non-literate, but it in general the study of the mathematical and logical practices of specific cultures. It refers to a broad cluster of ideas ranging from distinct numerical and mathematical systems to multicultural mathematics education.

Formalists                  : Formalists, following the mathematical philosophy of one of the most influential leaders of this school, David Hilbert (1862- 1943), treat mathematics as a “game.” In its most extreme version, formalism claims that mathematics is not about anything but rather sets of rules of inference that can be applied to given “strings” (axioms) to generate new strings. You can, for example, use the “game” Euclidean geometry (which is viewed as some strings) to generate a new string such as the Pythagorean theorem. This is roughly the equivalent of proving the theorem in classical mathematics. See also intuitionists and logicists.

Foundationalism  : In epistemology (theories of knowledge) the idea that there are basic (foundational) beliefs, assumptions, etc. that are the grounds for beliefs in general. Basic beliefs justify other beliefs. Basic beliefs are said to be self-evident or self-justifying. Basic beliefs

can also derive their warrant from sensory experience. Anti- foundationalists have a problem with giving an uncontroversial or principled account of which beliefs are self-evident or indubitable and see foundationalism as a form of an unexamined a priori or even as a God surrogate. from the Ancient Greek

Geometry                   : From the Ancient Greek γεωμετρία; geo- "earth", -metria "measurement;” the branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, Euclid (3rd century BCE) put it into an axiomatic form,

Intuitionists                : The intuitionists approached mathematics as the result of constructive cognition. Humans make mathematics and logic as mental constructs by applying internally consistent methods.

L.E.J. Brouwer (1881-1996) is considered the father of modern intuitionism. Intuitionists view the truth of mathematics as a subjective claim. Brouwer rejected realist/Platonist ideas about the truth or reality of mathematical objects. Intuitionism substitutes constructability for abstract truth and is the provocation for the transition from the proof theory of truth to the model theory of truth in modern mathematics. See also logicists and formalists.

Logicists                     : The logicists believed that mathematics was an extension of logic and therefore that all or at least some of mathematics was reducible to logic. Richard Dedekind (1831-1916) and Gottlob Frege (1848-1925) are considered the founders of this school. The logicist culminated in the monumental Principia Mathematica (published in three volumes in 1910, 1912, and 1913, and in a second edition in 1927 by Cambridge University Press )by Bertrand Russell (1872-1970) and Alfred North Whitehead (1861-1947) . Logicism survives today in Zermelo- Frankael set theory or one of its variations (such as Zermelo- Frankael set theory with the axiom of choice, or ZFC). Most of mathematics is believed to be reducible to the logical foundations provided by the axioms of ZF, ZFC, and derivatives. See also formalism and intuitionism.

Mathematical worker

: (Also number worker). To be distinguished from “mathematician,” someone who works with numbers and “does” mathematics as a member of a specialized occupational or professional class. The mathematical worker works with numbers as a member of a relatively informal and unorganized specialty outside of a formal credentializing system. The two terms are sometimes used interchangeably in the text for convenience but readers alerted to the distinction should be able to identify which term is most appropriate from the context.

Mathematics              : In its standard dictionary sense, the science of numbers and their operations,  interrelations,  combinations,  generalizations,  and

 

 

 

 

 

 

 

 

 

Non-Euclidean Geometry

abstractions and of space configurations and their structure, measurement, transformations, and generalizations . Classically, “mathematics” refers to the two fundamental ways in which humans work with numbers, arithmetic and geometry. In the Platonic view, this number work reveals a transcendental realm of ideal numbers revealed to humans as they develop over time. More recently the Platonic view has been opposed by the idea that mathematics is a social practice; it is manufactured by humans on the earth out of the material and symbolic resources at their disposal in their local environments.

: Refers to the geometries generated by denying Euclid’s fifth postulate, the parallels postulate. Consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line. In Euclidean geometry the lines remain at a constant distance from each other even if extended to infinity, and are known as parallels. In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels. In elliptic geometry the lines "curve toward" each other and eventually intersect. Spherical geometry is the geometry of the two- dimensional surface of a sphere.

Objectivity                 : This term is traditionally associated with ideas about value- neutrality, observations and interpretations uncolored by personal biases, and methods that are systematically developed and applied and not based on the idiosyncrasies of individual scientists. In this context, an objective statement is based on replicable research processes, and beyond fundamental dispute. In the sociology of science, objectivity has been shown to be an achievement of social groups and thus situated in “cultures of objectivity” or “objectivity communities.

Progress                     : In general terms, the forward development and betterment of humankind over time. The idea of scientific and technological progress was fueled by the seventeenth century advances in science and literature by Galileo, Newton, Descartes, Molière, and Racine. The idea of social progress was added later. The wars, holocausts, and human and environmental degradations characteristic of the twentieth century contributed to the development of critiques of the very idea of progress.

Proof                           : As a noun, evidence or an argument establishing or helping to establish a fact or the truth of a statement; the cogency of evidence that compels rational acceptance of a truth or a fact; the process or an instance of establishing the validity of a statement especially by derivation from other statements in accordance with principles of reasoning. As in the case other traditionally taken for granted ideas in the sciences, proof has become the subject of empirically grounded interrogations by sociologists of science who now refer proof to proof communities, and demonstrate that

 

Social Constructionism

 

 

 

 

Sociology of knowledge

what counts as a proof varies across time, space, and cultures.

: The fundamental theorem or central dogma of sociology, it refers to the fact that the only means humans have available for invention and discovery are their interactions with each other using the material resources of their environments and the symbolic resources of their cultures. It is fully compatible with a critical realism that assumes a “reality out there” that can only be known through the lenses of society and culture.

: The study of the relationship between human thought and the social, cultural, and historical contexts within which it arises, and of the effects prevailing ideas have on societies; deals with broad fundamental questions about the extent and limits of social influences on individual's lives and the social-cultural basics of our knowledge about the world When applied to the study of science in particular, an important specialty in sociology known as the sociology of science.

Spengler thesis   : Oswald Spengler argued that that there is no “Mathematik” but only “mathematics.” Mathematical forms are related to particular cultures. The weak form of the thesis is that there are as many mathematics as there are cultures. The strong form of the thesis is that mathematics is socially constructed.

Technoscience            : Variously, the idea the science and technology are intricately interrelated; the separation of “science” and “technology” may be analytically useful under some conditions but the idea that the two ideas are conceptually distinct is more ideological than substantive. This idea gives rise to such hybrid concepts as technocultural and technosocial.

Transcendental (transcendent; transcendence)

: Transcendence, transcendent, and transcendental are words that refer to an object (or a property of an object) as being comparatively beyond that of other objects. Such objects (or properties) transcend other objects (or properties) in some way. In philosophy, transcendence refers to climbing or going beyond some philosophical concept or limit In nineteenth century American, transcendentalism was developed within a religio- philosophical movement that claimed there is an ideal spiritual state that 'transcends' the physical and empirical. Following Restivo (2011 170) The Transcendental Fallacy (also known as the theologian’s fallacy) is that there is a world or that there are worlds beyond our own – transcendental worlds, supernatural worlds, worlds of souls, spirits and ghosts, gods, devils, and angels, heavens and hells. There are no such worlds. They are symbolic of social categories and heavens and hells. There are no such worlds. They are symbolic of social categories and classifications in our earthly societies and cultures. There is nothing beyond our material, organic, and social world. Death is final; there is no soul, there is no life after death. It is also possible that the so-called “many worlds interpretation” in quantum mechanics is contaminated by this fallacy as the result

of mathegrammatical illusions. The world, the universe, may be more complex than we can know or imagine, but that complexity does not include transcendental or supernatural features. Stated positively, this is Durkheim’s Law

 

Bibliography

 

Aleksandrov, A.D., A.N. Kolmogorov and M.A. Lavrent’ev (eds.) (1969), Mathematics: It’s Content, Methods, and Meanings, Cambridge, MA: MIT Press (orig. publ. in Russian in 1956). [A product of some the greatest mathematicians of the twentieth century, this is a readable review of the basic fields of mathematics. The authors are aware of the fact that mathematics is a human and a social creation and they do not mystify it or present it as a revelation of Platonic discourses.]

 

Almond, G., M. Chodorow, and R.H. Pearce (1985), Progress and its Discontents, Berkeley: University of California Press. [The twentieth century has not been kind to the idea of progress. Intellectuals across the spectrum of the disciplines are no longer that there are any viable grounds for the claim that science and technology have improved our lives morally or materially. The term “progress” is no longer meaningful empirically but is now about our aspirations and a “compelling obligation.”]

 

Ascher, M. (2004), Mathematics Elsewhere: An Exploration of Ideas Across Cultures, Princeton: Princeton University Press. [An introduction to mathematical ideas of peoples from a variety of small- scale and traditional cultures that challenges our conception of what mathematics is. Traditional cultures have mathematical ideas that are far more substantial and sophisticated than has been traditionally acknowledged.]

 

Baber, Z. (1996), The Science of Empire: Scientific Knowledges, Civilization, and Colonial Rule in India. Albany NY: SUNY Press. [Baber analyzes the reciprocal interactions between science technology and society in India from antiquity to modern times. The author analyzes institutional factors, including pre- colonial trading circuits, in the transfer of science and technology from India to other civilizational centers. He also explains the role of modern science and technology as factors in consolidating British rule in India.]

 

Ball, W. W. R. (1960), A Short Account of the History of Mathematics. New York: Dover. Unabridged and altered version of the fourth edition published in 1908. [A classic contribution to the history of mathematics.]

 

Bell, E. T. (1937/1986) Men of Mathematics, New York: [Simon and Schuster Touchstone Books. Bell is a mathematician and lacks the rigorous standards of the professional historians of science. But this book, for all of its idealism and focus on the dramatic, has been very inspiring for young people interested in mathematics and offers some insights into what it’s like to be a mathematician.]

 

Bellos, A. (2010), Here’s Looking at Euclid. New York: The Free Press. [This is a book designed to inspire and surprise. Bellos, who studied mathematics and philosophy at Oxford, takes the reader on a tour of mathematics based on his skills as a writer, traveler, and interviewer.]

 

Benesch, W. (1992), “Comparative Logics and the Comparative Study of Civilizations,” Comparative Civilizations Review, No. 27, Fall 1992, 88-105. [Benesch views “civilization as a state of mind.” Drawing heavily on Indian civilizational features, and especially the culture of the Jain for comparative materials, Benesch identifies four primary logics: object logics, subject logics, situational logics, and aspect logics. Benesch’s typology can be expanded to encompass propositional and predicate logic, first and higher order logics, modal logics, temporal logics, probability logics, multi-valued logics, negation logics, and quantum logic. Using another classification scheme we get: relevant logic (defined by some mathematicians as classical logic practiced with “good taste;” constructive logic (known in some of the literature by the less descriptive term "intuitionistic logic;" fuzzy logic, and comparative logic. The way to understand the proliferation of logics is to recognize that what we understand in everyday life and in a good deal of everyday science as the one universal logic is that set of rules and rules of inference that

reflect the workings of the macro-level physical world. In the worlds of the very small and very large, and the very fast and very slow, and across different levels of reality, different logics play out.]

 

Benveniste, É. (1966), Civilisation. “Contribution à l'histoire du mot” (Civilisation. Contribution to the history of the word), 1954, published in Problèmes de linguistique générale, Paris: Editions Gallimard ,

pp. 336-345 (translated by Mary Elizabeth Meek as Problems in general linguistics, 2 vols. 1971). [The author, a semiotician, traces the development of the concept “civilization” from its first occurrence in Adam Ferguson’s 1767 treatise on the history of civil society.]

 

Boos, B. and M. Niss, eds. (1979), Mathematics and the Real World, Boston: Birkhauser. [The editors recognize that to the extent that “pure” mathematics has gone too far and increasingly fails to interact with “real world” sciences and engineering disciplines, to that extent has it diminished its practical effectiveness. Mathematics that increasingly turns in on itself increasingly negates Wigner’s (1960) “unreasonable effectiveness” claim.]

 

Boutroux, P. (1919), L’ideal scientifique mathematiciens dans l’antiquité et dans les temps modernes. , Paris: Presses Universitaire. [Boutroux (1880-1922) was the son of the famous French philosopher Émile Boutroux and Aline Catherine Eugénie Boutroux, He was related to the statesman Raymond Poincaré¸the physicist Lucien Poincaré, and noted mathematician Henri Poincaré. Boutroux identifies three stages in the history of mathematics: the aesthetic, contemplate mathematics of the Greeks, the synthetic conception of Cartesian algebra, and what he saw as the incoherence of the mathematics of his time. There is a progressive theme in this history, an improving understanding of a higher reality. His didactic goals were to assert that progress in the history of science was a function of interactions between all the sciences, and to argue that problems should dictate approaches to solutions.]

 

Boyer, C. (1968), A History of Mathematics, Wiley, New York. [The novelist David F. Wallace called Boyer “the Gibbon of math history.” This book is to the history of mathematics what Paul Samuelson’s book on economics is to the study of economics. The reference of choice in the history of mathematics is now in its 3^rd edition (2011, updated by Uta Merzbach).]

 

Cajori, F. (1894), A History of Mathematics. New York: Macmillan & Co.. [Cajori’s (1859-1930) book was first published in 1893 and has gone through several editions. It is a reliable, readable treatment of the history of mathematics from antiquity to the early 1900s.]

 

Cajori, F. (1929/1993), A History of Mathematical Notations. Dover, New York. [This book is still today considered with good reason to be unsurpassed. Demonstrates how notation changes with changes in the shape of civilization. Here we discover, for example, how the Greeks, Romans, and Hebrews counted.]

 

Cantor, M. (1907), Vorlesungen uber die Geschichte del' Mathematik,. [Anastatischer Neudruck, Leipzig. Moritz Cantor (1829-1920) was an historian of mathematics who studied under some the giants of mathematics, including notably Gauss, and this book in four volumes is not just comprehensive but is considered a founding document in the history of mathematics as a critical, methodologically sound field of study.]

 

Clark, A. (2010), Supersizing the Mind: Embodiment, Action, and Cognitive Extension, New York: Oxford University Press. [Clark is yet another of a small but increasing number of authors who are focusing in on the problems and paradoxes that arise when we make the brain the source of our thinking and consciousness, and the source of our morals and our beliefs. In fact it is becoming increasingly clear that cognition is a complex result of tangled networks that criss-cross the boundaries of brain, body, and world. Mind is not bound by the brain; consciousness, as Nietzsche already intuited, is a network of relationships.]

 

Collins, R. (1998), The Sociology of Philosophies. Cambridge, MA: Harvard University Press. [A comprehensive social history of world philosophy in the context of global intellectual life. Collins traces the development of philosophical thought in China, Japan, India, ancient Greece, the medieval Islamic and Jewish world, medieval Christendom, and modern Europe. The result is an empirically grounded theory of ideas as the product of social networks. One of the most important contributions to modern sociology by a leading theorist.]

Collins, R. and S. Restivo. (1983), "Robber Barons and Politicians in Mathematics," The Canadian Journal of Sociology 8, 2 (Spring 1983), pp. 199-227. [Two of the leading contributors to the sociology of science offer an alternative to Kuhn’s theory of scientific change. Major scandals in mathematics are shown to be associated with shifts in the organizational structure of mathematical work. The article demonstrates the power of conflict theory as an explanatory approach in the sociology of mathematics.] Courant, R. and H. Robbins. (1996), What is Mathematics? New York: Oxford University Press (rev. by Ian Stewart, orig. publ. 1906). [A classic contribution to our understanding of mathematics as an enterprise of practice rather than philosophy.]

 

D’Ambrosio, U. (2006), Ethnomathematics, Rotterdam: Sense Publishers. An introduction to the concept of ethnomathematics by the founding father of the field.

 

Davis, P.J. and R. Hersh. (1981), The Mathematical Experience. Boston: Birkhauser. [Two mathematicians bring sociological, psychological, and philosophical perspectives to this popular exposition of the nature and relevance of mathematics.]

 

Dostoevsky, F. (1864), "Notes from the Underground," pp. 107-2-10 in The Best Short Stories of Dostoevsky. New York, n.d. (orig. publ, in Russian). [This classic piece of literature can be read as a contribution to the sociology of mathematics; the author’s discussion of 2+2=4 and 2+2=5 shows how arithmetic can be used to symbolize ideologies. Should be compared with how these equalities are treated by George Orwell in 1984.]

 

Dorrie, H. (1965), One Hundred Great Problems of Elementary Mathematics,Their History and Solution. New York: Dover. [This book was originally published in 1932 under the title Triumph der Mathematik. Fascinating well selected problems but not really “elementary.” An intriguing 2000 year survey but not for those without a strong background in mathematics.]

 

Durkheim, E. (912/1995), The Elementary Forms of Religious Life. New York: The Free Press 1 (trans. Karen Fields). [One of the most important studies in the history of scholarship. Durkheim crystallizes the idea of who and what God is and explains the function of religion in society. Criticisms abound, but they do not change the world transforming potential of this book.]

 

Eglash, R. (1999), African Fractals, Piscataway, NJ: Rutgers University Press. [Patterns across cultures are characterized by specific design themes. In Europe and America, cities are often laid out in a grid pattern of straight streets and right-angle corners. In contrast, traditional African settlements tend to use fractal structures--circles of circles of circular dwellings, rectangular walls enclosing ever-smaller rectangles, and streets in which broad avenues branch down to tiny footpaths with striking geometric repetition. These indigenous fractals are not limited to architecture; their recursive patterns echo throughout many disparate African designs and knowledge systems.]

 

Gasking, D. (1956), "Mathematics and the World," pp. 1708-1722 in J.R. Newman (ed.), The World of Mathematics. New York: Simon & Schuster. [Originally published in 1940, this is an exemplar of conventionalist philosophy of mathematics. Gasking views mathematics as true by virtue of linguistic conventions. Mathematics on this view is not empirical in the way, for example, that Wittgenstein argues it is or in the way it is understood today in social studies of mathematics.]

 

Geertz, C. (1983), “Art as a Cultural System,” pp. 94-120 in C. Geertz, Local Knowledge, New York: Basic Books. [Art, like speech, myth and other cultural systems are situated, ideationally connected to society; in a word cultural systems are socially constructed.]

 

Hersh, R. (1999), What is Mathematics, Really? New York: Oxford University Press. [Mathematics for the modern reader by a mathematician who appreciates the new sociology of mathematics developed by Bloor on the one hand and Restivo on the other.]

 

Horton, R. (1997), Patterns of Thought in Africa and the West: Essays on Magic, Science and Religion, Cambridge; Cambridge University Press. [The distinguished anthropologist and philosopher was at the center of debates that crystallized in the 1950s and 1960s concerning the nature of and relationship between “primitive” (read, primarily, African) and “advanced” (read “the West”) systems of knowledge

and belief. In this collection of his essays, Horton addresses the debates by exploring African beliefs, rituals, and cosmologies in the context of open and closed systems thinking. His objective is to demonstrate the kinship between primitive and modern thought and between science and religion more generally. Among the most important provocations for this and related debates are studies of Zande (n. Azande) logic. Were they “pre-logical,” as some claimed (e.g., in my view arguably Levy-Bruhl) or perfectly coherent in context (as Evans-Pritchard argued)? In the early years of the science studies movement David Bloor took up this question and argued for a cultural relativism that provoked new interest in this topic. This literature is relevant to my topic but requires more direct attention than I can give it here.]

 

Kavolis, V. (1985), “Civilizational Analysis as a Sociology of Culture,” Sociological Theory 85, 31, Spring 1985, 29-38. [The comparative civilizational analysis of any phenomenon, Kavolis argues, requires that we spell out the structure of each level of sociocultural organization and the connections and resources linking all the levels that bear on the particular phenomenon in a particular civilization. We don’t require an elaborate analytical apparatus but it helps to carry an awareness of such an apparatus in the study of a particular problem in terms of the relations between general civilizational structures, processes, and issues. Civilizational analysis is about the distinctive varieties of and changes in human experience, and the universe of symbolic designs in social behavior.]

 

Kline, M. (1962), Mathematics: A Cultural Approach, Reading, Massachusetts: Addison-Wesley. [A good introduction, even where dated factually and ideologically, to mathematics in relation to the humanities and the search for truth.]

 

Kramer, E. E. (1970), The Nature and Growth of Modern Mathematics. New York: Hawthorne Books. [Kramer discusses the lives and contributions of prominent mathematicians from Pythagoras and Newton to the modern period. Mathematical concepts such as binary operations, point-set topology, post-relativity geometries, optimization and decision processes ergodic theorems epsilon-delta arithmetization, and integral equations are discussed with admirable clarity.]

 

Lenski, G. (1974), Human Societies. New York: McGraw-Hill. [A cultural ecology of the evolution of societies. This is a major textbook contribution to macrosociology, now in its 11^th edition from Paradigm Publishers in Boulder, Colorado, 2008 co-authored by P. Nolan.]

 

Mannheim, K. (1936), Ideology and Utopia. Eugene, Oregon: Harvest Publishers. [This is a complex treatise in the context of Mannheim’s legacy in relationship to European and America sociology. It is a founding document in the emergence of sociology but especially of the sociology of knowledge. In terms of its relationship to the substance of this essay, the significance of this book is that while situating knowledge in its social, cultural, and historical contexts, Mannheim exempts the formal science from his analysis. There cannot be, he claims in these pages, a sociology of 2+2=4. This idea carried into the emergence of the sociology of science in the 1930s and was not seriously challenged until the birth of the science studies movement in the late 1960s.]

 

McClain, E.G. (1976), The Myth of Invariance: The Origins of the Gods, Mathematics and Music from the Rg Veda to Plato. York Beach, ME: Nicolas-Hays, Inc. [McClain’s thesis is that in the ancient civilizations music was a science that bridged the gap between the everyday world and the divine. The invariance of music contrasted sharply with the variability of the everyday world. Music expresses and motivates mathematics. Music as a science is revealed by studying the mathematical relationships between musical notes. In this context music is properly understood as an expression of and the motive for mathematical study. The "key" to unlocking this science comes from a study of the mathematical relationships between various musical notes. The Pythagorean notion of number was in fact more general and more widespread and an essential feature of the very idea of culture and civilization. The presentation is controversial in some details but we are learning more and more about the centrality of musicality in humans and this book speaks to that idea.]

 

Merton, R.K. (1968), Social Theory and Social Structure, enlarged ed., New York: The Free Press. [Merton founded and dominated the sociology of science with his students from the late 1930s to the late 1960s. The Mertonian paradigm, consonant with Mannheim’s sociology of knowledge, focused on the

social system of science – for example, norms, values, the reward system, stratification in science, age- grading – but exempted scientific knowledge per se from sociological scrutiny.]

 

Merton, R. K. (1961), , "Singletons and Multiples in Scientific Discovery: a Chapter in the Sociology of Science," Proceedings of the American Philosophical Society, 105: 470–86. Reprinted in R. K. Merton, The Sociology of Science: Theoretical and Empirical Investigations, Chicago, University of Chicago Press, 1973, pp. 343–70.

 

Merton, R. K. (1973), "Resistance to the Systematic Study of Multiple Discoveries in Science," European Journal of Sociology, 4:237–82, 1963. Reprinted in R.K. Merton, The Sociology of Science: Theoretical and Empirical Investigations, Chicago, University of Chicago Press , pp. 371–82. [This and the following paper introduce Merton’s idea that all discoveries are in principle multiples. These papers are milestones in the Merton corpus.]

 

Merton, R.K. (1958), “The Matthew Effect in Science,” Science, 159(3810): 56-63, January 5, 1968. [Widely cited in the sociology of science, the Matthew effect, or the principle of accumulated advantage, refers to situations in which the rich get richer and the poor get poor. Merton named the effect after Matthew 25: 29 (NRSV Bible). In science as in other activities, power and economic or social capital can be leveraged to gain additional power and capital. This is one the causes of mis-eponymy along with historical amnesia and random and deliberate acts of misattribution.]

 

Mesquita,M., S. Restivo and U. D'Ambrosio (2011), Asphalt Children and City Streets: A Life, a City, and a Case Study of History, Culture, and Ethnomathematics in São Paulo, Rotterdam: Sense Publishers. [An innovative plural voiced ethnography and history of street children that attends to how mathematics plays into their survival strategies.]

 

Morris-Suzuki, T. (1994), The Technological Transformation of Japan: From the Seventeenth to the Twenty-first Century, Cambridge: Cambridge University Press. [Japan did not miraculously leap into the technological forefront of twentieth century societies. Its rise to superpower status, as this book makes clear, is rooted in its history. This is the first general English language history of technology in modern Japan. One of the significant features of this book is its consideration of the social costs of rapid changes in technology.]

 

Nasr, S. H. (2007), Science and Civilization in Islam. Chicago: Kazi Publications. [The first one volume English language book on Islamic science from the Muslim perspective. Hossein explains the place of science in Muslim culture as he unfolds its content and spirit.]

 

Needham, J. (1959), Science and Civilization in China. Volume III: Mathematics and the Sciences of the Heavens and the Earth. Cambridge: Cambridge University Press. [One of several volumes in the monumental study that uncovered the hidden history of science and technology in China. Needham documented that not only did China have a history in science and technology, it was the leading civilizational center of science and technology in the world between the early Christian period in Palestine and 1500 CE. The explanatory framework is Marxist cultural ecology.]

 

Neugebauer, O. (1952), The Exact Sciences in Antiquity. Princeton: Princeton University Press. [It took a long time for historians of science to overcome the ideology of the once and always Greek miracle. Neugebauer contributes to the demise of this myth in this non-technical discussion of the influence of Egyptian and Babylonian mathematics and astronomy on the Hellenistic world. An early look into the sophistication of ancient Babylonian mathematics.]

 

Noë, A. (2010), Out of Our Heads: Why You Are Not Your Brain, and Other Lessons from the Biology of Consciousness, New York: Hill and Wang. [Makes an important contribution to getting away from classical ideas about the primacy of the brain in consciousness. His approach is radically social but in a strange way that makes biology, rather than sociology, the science of the social. But just because of this twisted logic, he furthers the interdisciplinary agenda of figuring out a non-reductionist way to link biology and society.]

Polster, B. and M. Ross, (2011), “Pythagoras’s Theorem ain’t Pythagoras’s,” h t t p : / / education . theage . co m . au / cmspage . php ? I ntid = 147 & intversion = 79, March 7, 2011.

 

Resnikoff, H.L. and R.O. Wells, Jr. (2011), Mathematics and Civilization. New York: Dover (paperback). [This book was originally published by Holt, Rinehart, and Winston in 1973 and later published by Dover with supplemental materials. Still a good general introduction to the reciprocal relations between mathematics and human culture with an emphasis on the technical mathematics. No great demands are made on the mathematical aptitudes of readers, and the more sophisticated reader will find some of the treatment, especially on the calculus, technically deficient.]

 

Restivo, S. (1979), "Joseph Needham and the Comparative Sociology of Chinese and Modern Science: A Critical Perspective," pp. 25-51 in R.A. Jones and H. Kuklick (eds.), Research in the Sociology of Knowledge, Sciences, and Art, Vol. II. JAI Press, Greenwich. [The most extensive journal length critical survey of Needham’s monumental history. Supports and extends his sociological materialism based on a human ecological perspective.]

 

Restivo, S. (1981), "Mathematics and the Limits of Sociology of Knowledge," Social Science Information, V. 20, 4/5: 679-701. [The new sociology of science associated with the science studies movement that emerged in the late 1960s challenged the status of mathematics as the arbiter of the limits of the sociology of science and knowledge. This is one of the early examples of the challenge by one of the founders of the modern sociology of mathematics.]

 

Restivo, S. (2001/1992), Mathematics in Society and History. New York. [This is the first book devoted completely to the sociology of mathematics as a subfield of sociology. The author reviews precursors, Spengler’s thesis on numbers and culture, mathematical traditions in different civilizations, and develops an original conception of the sociology of pure mathematics.]

 

Restivo, S. (1993), “The Social Life of Mathematics,” pp. 247-278 in S. Restivo, J.P. van Bendegem, and

R. Fischer (eds.), Math Worlds: Philosophical and Social Studies of Mathematics and Mathematics Education, Albany, NY: SUNY Press. [A collection of essays, including a set of papers by German scholars previously unavailable in English, focused on recent developments in the study and teaching of mathematics. The authors are all guided by the idea that mathematical knowledge must be grounded in and reflect the realities of mathematical practice.]

 

Restivo, S. (1994), Science, Society, and Values: Toward a Sociology of Objectivity. Bethlehem PA: Lehigh University Press. [This book introduces Restivo’s main contributions to the sociology of science between 1966 and the early 1990s. Based on his work in the ethnography of science, the history of science in China and the West, his social problems approach to understanding modern science, and other contributions, he develops a sociological perspective on objectivity.]

 

Restivo, S. (2007), “Mathematics,” Monza: Polimetrica. June, 2007 The Language of Science (ISSN 1971-1352). [An overview of the sociology of mathematics written for an online encyclopedia.]

 

Restivo, S. (2011), Red, Black, and Objective: Science, Sociology, and Anarchism, Surrey: Ashgate Publishers. [This book explores the implications of the science studies movement for science and society in the context of an anarchist tradition. The particular tradition the author has in mind here makes anarchism one of the sociological sciences. Here he follows Peter Kropotkin. The book is grounded in the empirical studies carried out over the last forty years by researchers in science studies (and more broadly science and technology studies). The author’s perspective is at once empirical, normative, and policy- oriented.]

 

Restivo, S. and H. Karp (1974), "Ecological Factors in the Emergence of Modern Science," pp. 123-142 in S. Restivo and C.K. Vanderpool (eds.), Comparative Studies in Science and Society. Columbus, OH: C. Merrill. [The authors theoretically link organizational and institutional structures to their ecological contexts and apply this theory to the problem of why modern science emerged in the West and not in China.]

Restivo, S. and R. Collins (1982), “Mathematics and Civilization,” The Centennial Review Vol. XXVI, No. 3 (Summer 1982), pp. 277-301. [An introduction to the comparative historical sociology of mathematics that modifies Spengler’s ideas on numbers and culture in the context of sociological theory.]

 

Restivo, S. and J. Croissant (2008), “Social Constructionism in Science and Technology Studies,” pp. 213-229 in J.A. Holstein & J.F. Gubrium, eds., Handbook of Constructionist Research, New York: Guilford. [The authors aim is to clarify the widespread misconceptions, misapplications, and misconstruals of this term which they identify as the fundamental theorem of sociology. They are at pains to argue that the term does not imply or entail any form of relativism; it is compatible with a realistic sociology that recognizes objectivity and truth as real. They are real however in an institutional sense. This view, as Durkheim pointed out more than one hundred years ago, is consistent with the idea that there is a reality “outside of us,” but we do not have access to a ding an sich.]

 

Reyna, S.P. (2007), Connections: Brain, Mind and Culture in Social Anthropology, New York: Taylor and Francis. [Reyna returns to the Boasian (Franz Boas) roots of anthropology to creatively construct a new paradigm for connecting the biological and cultural domains without submitting to the Sirens of reductionism.]

 

Rosental, C. (2008), Weaving Self-Evidence: A Sociology of Logic, Princeton: Princeton University Press. [Rosental traces the history of a theorem in the foundations of fuzzy logic to demonstrate the inherently social nature of logic. He describes the process by which logical propositions are produced, disseminated, and established as truths.]

 

Roszak, T. (1995), The Making of a Counter-Culture: Reflections on the Technocratic Society and its Youthful Opposition, with a new introduction, Berkeley; University of California Press (orig. publ. 1969). [Published in the middle of the 1960s sociocultural revolution, this book spoke directly to those who would become “the children of the 60s” while simultaneously bewildering their parents. Roszak coined the term “counter-culture” and damned the technocracy that was at the heart of the problems the protesters were angry about. A literate effort to explain the disaffection of young people and the young at heart during this tumultuous period.]

 

Sahlins, M. and E. Service, eds. (1960), Evolution and Culture, Ann Arbor: University of Michigan Press (co-authored by T.G. Harding, D. Kaplan, M.D. Sahlins, and E.R. Service). [In the hands of these authors, evolutionary anthropology becomes a predictive tool that can be applied to theorizing the future of human societies. Current events on the world scene, including the political and economic rise of China and the troubled status of the United States would not have surprised these anthropologists. This is a classic and still relevant contribution to our understanding of culture and cultural change writ large.]

 

Schechter, Eric (2005), Classical and Nonclassical Logics: An Introduction to the Mathematics of Propositions, Princeton: Princeton University Press. [Classical logic--the logic crystallized in the early twentieth century by Frege, Russell, and others--is computationally the simplest of the major logics, and adequate for the needs of most mathematicians. But it is just one of the many kinds of reasoning in everyday thought. Schechter introduces classical logic alongside constructive, relevant, comparative, and other nonclassical logics. This is the first textbook to make this subject accessible to beginners.]

 

Sedlacek, T. (2011), Economics of Good and Evil, New York, Oxford University Press. [Sedlacek understands economics as a social, cultural, and historical phenomenon. It is a product of our civilization not a pure, value free science. In this sense, Sedlacek lines up with contemporary students of the sociology of science and mathematics. Economics, at the end of the day, is about “good and evil.” In viewing economics as a moral enterprise he reminds us that the author of The Wealth of Nations, Adam Smith, is also the author of The Theory of Moral Sentiments.]

 

Selin, H. (ed.) (2001), Mathematics Across Cultures, New York: Springer. [A survey of Islamic, Chinese, Native American, Aboriginal Australian, Inca, Egyptian, and African mathematics, with essays on rationality, logic and mathematics. Shows how science and math practice is situated in its cultural context.]

Smith, D. E. (1958), History of Mathematics. New York: Dover. [A classical two volume introduction to the history of mathematics from ancient Egypt to modern times, very lucid survey including biographical notes and chronology.]

 

Sohn-Rethel, A. (1975), "Science as Alienated Consciousness." Radical Science Journal, Nos. 2/3: 65-

101. [Working from inside a sophisticated Marxist paradigm, Sohn-Rethel discusses science as an alienated and alienating form of knowledge and knowledge building. He is known for linking Kant and Marx in his work on epistemology.]

 

Spengler, O. (1926), The Decline of the West. New York: A. Knopf. [Spengler’s readers append adjectives like “audacious,” “profound,” “magnificent,” “exciting,” and “dazzlingly” to this book which flaws and all is one of the most amazing efforts in human history to capture humanity’s march through space and time. It’s relevance for the topic at hand is that Spengler’s theses are grounded in a radically cultural understanding of the relationship between culture and mathematics.]

 

Stigler, Stephen S. (1980), ‘Stigler’s Law of Eponymy’, Trans. N. Y. Acad. Sci. (2) 39, 147–157.

 

Leon Stover (1974), The Cultural Ecology of Chinese Civilization. New York: Signet. [An innovative interpretation of peasants and elites in what Stover refers to as a “once and always Bronze Age culture.” A paradigm for understanding the nature of Chinese culture and the continuities between the age of the dynasties and the modern era from Sun Yat Sen and Chiang Kai Shek to Maoism and modernism.]

 

Struik, D. (1967), A Concise History of Mathematics,. New York: Dover Publications. [The fourth revised edition of this classic was published by Dover in 1987. Struik, a Dutch mathematician and Marxist theory, was a professor at MIT for most of his professional career and to my knowledge the first person to identify the sociology of mathematics as a field of study. This book is a very readable introduction to the history of mathematics, concise as advertised but with a lot of substance poured into the books roughly 230 pages. Struik covers the period from the ancient world to the early twentieth century.]

 

Sugimoto, Masayoshi and D.L. Swain, (1978), Science & Culture in Traditional Japan. Cambridge MA: MIT Press. [Between 600 and 1854 CE, Japan was impacted by a first and second Chinese cultural wave and the first Western Cultural Wave in the nineteenth century. The authors focus on how these cultural waves set the stage for the development of an indigenous science and technology.]

 

Verran, Helen. (1992), Science and an African Logic. Chicago: The University of Chicago Press. [An empirical study that supports the idea of mathematics and logics as culturally situated. Quantity is not always absolute (as in 2=2=4) but sometimes relational, as in Yoruba. Verran’s experience and research as a teacher in Nigeria is the basis for this important contribution to the sociology of mathematics and ethnomathematics.]

 

Wigner, E. (1960), “On the Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Communications on Pure and Applied Mathematics , 13: 1–14. [A classic paper in defense of the idea that the mathematical structure of a physics theory often points the way to further advances in that theory and even to empirical predictions. It is basically an argument in support of “pure” mathematics.]

 

Wright, Ronald (2004), A Short History of Progress, Philadelphia: Da Capo Press. [Looking over the long history of humanity, Wright sees not the unfolding a linear evolution of progress but rather a series of “progress traps.” He reveals a history of “progress and disasters” that should serve as a warning to humanity and especially to those people who assume that progress is an inevitable and positive manifestation of human exceptionalism.]

 

Zaslavsky, C. (1999), Africa Counts: Number and Pattern in African Cultures, 3^rd ed. Chicago: Lawrence Hill Books (orig. publ. 1973 by Prindle, Weber, and Schmidt). [This is one of the earliest efforts to document the experience of mathematics in a non-Western culture and to view it in a positive civilizational perspective.]

 

Zeleza, Paul Tiyambe and Ibulaimu Kakoma (2005), Science and Technology in Africa. Trenton NJ: Africa World Press. [The authors deal with scientific and technology literacy, production, and

consumption in modern Africa. The focus is on developments in information technology and biotechnology in the context of The Knowledge Society in a globalizing context.]

 

Biographical Sketch

 

Dr. Sal Restivo is widely recognized as one of the founders of the field of Science and Technology Studies (STS), a pioneer in ethnographic studies of science, a founder of the modern sociology of mathematics, a contributor to public sociology and a prominent figure in the radical science movement of the 1960s. Dr. Restivo was Professor of Sociology, Science Studies, and Information Technology in the Department of Science and Technology Studies, at Rensselaer Polytechnic Institute in Troy, New York until his retirement in June 2012. He is Special Lecture Professor in STS at Northeastern University in Shenyang, China; a former Special Professor of Mathematics, Education, and Society at Nottingham University in Great Britain; and a former Hixon/Riggs Professor of Science, Technology, and Society at Harvey Mudd College. In 2012, he was a Senior Postdoctoral Fellow at the University of Ghent in Belgium. He is a founding member (1975) of and a former president (1994/95) of the Society for Social Studies of Science.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sal Restivo, (2012), MATHEMATICS, CIVILIZATION, AND PROGRESS, in

History of Mathematics, [Eds.UNESCO-EOLSS Joint Commitee], in Encyclopedia of Life Support Systems (EOLSS), Developed under the Auspices of the UNESCO, Eolss Publishers, Paris, France, [http://www.eolss.net]

 

MATHEMATICS, CIVILIZATION, AND PROGRESS

 

Sal Restivo

Department of Intercultural Communication and Interaction, University of Ghent, Ghent Belgium

 

Keywords: abstraction, arithmetic, Chinese mathematics, civilization, ethnomathematics, European mathematics, generalization, geometry, Greek mathematics, Indian mathematics, mathematical workers, mathematicians, mathematics, non-Euclidean geometry, Platonism, proofs, progress, puzzle solving, social constructionism, sociology, transcendental

 

Contents

 

1.  Prologue: Interrogating the terms of our discourse

2.  Introduction: mathematics and civilization

3.  Mathematics and Civilization: Case Studies

4.  The Social Roots of Mathematics

5.  Puzzles and Proofs

6.  Generalization (“Abstraction”) and Self-consciousness

7.  Conclusion Acknowledgement Glossary Bibliography Biographical Sketch

 

Summary

 

This chapter provides an overview of the comparative history of mathematics in the context of the ideas of civilization and progress. The narrative is guided by the idea that mathematics is a social practice and not a set of ideas revealed through some sort of heavenly or Platonic discourse. This implies that mathematics has a normative dimension in terms of its cultural and professional settings. The chapter begins with a critical review of the terms of the discourse, namely mathematics, civilization, and progress. Following some general remarks on mathematics and civilization, the reader is guided through case studies of mathematics in its civilizational and cultural contexts including mathematics in China, India, and Greece, as well as modern Europe. A more analytical section follows, summarizing the idea of the social roots of mathematics, the development and functions of puzzle solving and proofs, and generalization as a way to think about “abstraction” in concrete terms. The narrative follows an historical social science perspective that draws on the ideas of classical and contemporary social theorists from Durkheim and Spengler to Randall Collins and Sal Restivo.

 

1.  Prologue: Interrogating the Terms of Our Discourse

 

Let us begin by briefly interrogating the three terms that make up the title of this essay. First, we should ask, borrowing the title of the book by Reuben Hersh (1999): What is mathematics, really? Mathematics has been shrouded in mystery and halos for most of

its history. The reason for this is that it has seemed impossible to account for the nature and successes of mathematics without granting it some sort of transcendental status. Classically, this is most dramatically expressed in the Platonic notion of mathematics.

 

Briefly, what we call Platonism in mathematics refers to Plato’s theory of Forms. Skirting the complexities of scholarly discourse, Plato is associated with the idea that there are “Forms” or “ideals” that are transcendent and pure. These immaterial Forms exist in a realm outside of our everyday space and time. They are the pure types of the ideas and concepts we manage in our everyday world.

 

This over-simplifies Plato but is consistent with a long tradition in the history and philosophy of mathematics. Consider, for example, the way some scholars have viewed the development of non-Euclidean geometries (NEGs). The mathematician Dirk Struik (1967: 167), for example, described that development as “remarkable” in two respects. First, he claimed, the ideas emerged independently in Gȍttingen, Budapest, and Kazan; second, they emerged on the periphery of the world mathematical community (most notably in the case of Kazan and to a lesser extent Budapest). And the distinguished historian of mathematics, Carl Boyer (1968: 585) characterized the case as one of “startling…simultaneity.”

 

These reflect classical Platonic, transcendental views of mathematics. One even finds such views in the forms of the sociology of knowledge and science developed from the 1920s on in the works of Karl Mannheim and Robert K. Merton and their followers. Mannheim, for example, wrote in 1936 that 2+2 = 4 exists outside of history; and Merton championed a sociology of science that focused on the social system of science and not on scientific knowledge which he claimed lay outside of the influences of society and culture.

 

His ambivalence about this is reflected in his critical reply to G.N. Clark’s criticism of Boris Hessen’s historical materialism of Newton’s Principia (Merton, 1967: 661-663). Clark opposed Hessen’s (Marxist) political economy of the Principia with a defense of Newton’s “purely” scientific motives. Merton argued that individual motivations do not change the structural facts of the matter and in this case they support Hessen’s argument. This doesn’t reach to the core of the social construction of scientific knowledge but it does demonstrate at least an appreciation for the contextual foundations of that knowledge.

 

There are a couple of curiosities in the case of non-Euclidean geometry (NEGs). Even a cursory review of the facts reveals that NEGs have a history that begins already with Euclid’s earliest commentators, runs over the centuries through names like Saccheri, Lambert, Klügel, and Legendre, and culminates in the works of Lobachevsky (1793- 1856), Reimann (1826-1866), and J. Bolyai (1802-1860). The concerns over Euclid’s parallels postulate moved geometers eventually to the systematic development of NEGs. The issue was that the parallels postulate, the fifth postulate in Euclid’s system, did not possess the axiomatic self-evidence of the first four postulates, and it could not be derived from the first four. The three creators of NEGs were by no means isolated and working independently. All were connected to Gauss (1777-1855) who had been working on NEGs since the late 1700s.

J. Bolyai was the son of one of Gauss’ friends, W. Bolyai. Gauss and his friend Bolyai were at the University of Göttingen where the parallels postulate was the subject of lectures by Kastner and a number of dissertations. Reimann was Gauss’ dissertation student. And as for Lobachevsky, he did indeed work at a university on the periphery of the European mathematical community, the University of Kazan. However, the university was staffed by distinguished German professors, including Gauss’ teacher,

J.M. Bartels. J. Bolyai developed ideas on non-Euclidean geometries (NEGs) as early as 1823. His “The Science of Absolute Space” was published ten years later in a book written by his father. Lobachevsky published on the foundations of geometry from 1825 on. Reimann’s Habilitationschrift was on the foundations of geometry. Gauss, we know, wrote about NEGs in letters to W. Bolyai (December 17, 1799), Taurinus (November 8, 1824), and to Besel (January 27, 1829). He also wrote about NEGs in published notes from 1831 on. There are two short reviews on NEGs in Göttingische Anziegen in 1816 and 1822. One has to wonder why in the face of the facts of the case Struik and Boyer chose to view things as “remarkable” and “startling.”

 

Classically, the story of the development of NEGs was told in the context of “pure” mathematics. Thus, to take the case of Riemann as an example, the story was that he constructed the generalization of elliptic geometry as a purely mathematical exercise. The idea that there was a concrete possibility of practical applications for this exercise was not a consideration. In the light of a more realistic sociological and network analysis, Riemann’s work along with that of Gauss, Lobachevsky, Bolyai, Helmoltz, and Clifford, the story of NEGs takes on a different shape.

 

To some extent, they all agreed that Euclidean geometry was an unimpeachable system of ideal space and logic. It could be read as a game played in accordance with a set of formal rules. In fact, however, they interrogated Euclidean geometry in terms of whether it was a valid representation of “actual space.” This should be tested not by mathematics, not what is within the confines of the social world of mathematics per se, but should be tested scientifically – by observation and some mode of experimentation.

 

The sociological generalization this leads to is that if you are given a “genius” or a startling event, search for a social network –cherchez le réseau. No one has made the case for social networks as the roots of ideas more powerfully than Randall Collins (1998). The rationale here should become clearer over the course of this chapter.

 

Even more curious in the case of the sociology of knowledge is the fact that already in his The Elementary Forms of Religious Life published in French in 1912, Emile Durkheim had linked the social construction of religion and the gods to the social construction of logical concepts. Durkheim’s program in the rejection of transcendence languished until the emergence of the science studies movement in the late 1960s and the works of David Bloor, Donald MacKenzie, and Sal Restivo in the sociology of mathematics.

 

It is interesting that a focus on practice as opposed to cognition was already adumbrated in Courant’s and Robbins’ classic “What is Mathematics?” (1906/1995). We must turn to active experience, not philosophy, they wrote, to answer the question “What is mathematics”? They challenged the idea of mathematics as nothing more than a set of

consistent conclusions and postulates produced by the “free will” of mathematicians. Forty years later, Davis and Hersh (1981) wrote an introduction to “the mathematical experience” for a general readership that already reflected the influence of the emergent sociology of mathematics. They eschewed Platonism in favor of grounding the meaning of mathematics in “the shared understanding of human beings…” Their ideas reflect a kind of weak sociology of mathematics that still privileges the mind and the individual as the creative founts of a real objective mathematics.

 

Almost twenty years later, Hersh, now clearly well-read in the sociology of mathematics, wrote “What is Mathematics, Really?” (1997). The allusion to Courant and Robbins is not an accident. Hersh does not find their definition of mathematics satisfactory. In spite of his emphasis on the social nature of mathematics, Hersh views this anti-Platonic anti-foundationalist perspective as a philosophical humanism. While he makes some significant progress by comparison to his work with Davis, by conflating and confusing philosophical and sociological discourses, he ends up once again defending a weak sociology of mathematics. The modern sociology of mathematics associated with the science and technology studies movement that emerged in the late 1960s has established mathematics as a human construction, a social construction. Mathematics is manufactured by humans on the earth out of the material and symbolic resources at their disposal in their local environments. While traditional philosophical and sociological discourses have become estranged, especially in the arena of science studies, there are efforts abroad to reconcile the two disciplines consistent with the interdisciplinary turn in contemporary research and theory. In this sense, one can consider Hersh’s philosophical humanism a step in that direction.

 

The second term in my title, “civilization” is not without its controversial features. In the ancient world, “civilized” peoples contrasted themselves with “barbarians;” in modern times, “civilized” has been opposed to “primitive” or “savage.” The relevance of the concept of civilization to the topic of mathematics lies in its association with the idea of progress. More to the point is the fact that different civilizations (variously “nations,” “societies,” and “cultures”) are associated with different mathematical traditions (v. Restivo, 1992, 23-88). To the extent that humans have developed in ways that can be captured in the ideas of “evolution” and “progress,” mathematics, in conjunction with science and technology more generally, is assumed to have contributed positively to and benefited from those developments. The Scottish philosopher Adam Ferguson (1723-1816) is credited by Benveniste (1954) with introducing the term “civilization” in its modern sense into the English language in his Essay on the History of Civil Society (1767), and perhaps as early as 1759. The term also appears in the works of Boswell (1772), Adam Smith (1776), and John Millar (1771). Mirabeau (1757) introduces the term in French in his L'Ami des hommes ou traité de la population. Just as the individual grows from infancy to adulthood, Ferguson wrote, the species advances from “rudeness to civilization.” Set at the pinnacle of forms of society, civilizations are characterized by complexity, hierarchies, dynamism, advanced levels of rationality, social progress, and a statist form of government. Rousseau, by contrast, viewed civilization as opposed to human nature.

 

If we adopt Ferguson’s view of civilization, then clearly mathematics has been both a result of the emergence and development of civilization and a contributor to that

development. If on the other hand we adopt Rousseau’s viewpoint, the virtues of science and mathematics and the very idea of “civilization” are made severely problematic.

 

What about progress, the very idea? Arguably, the idea comes into Western and world culture in the Old Testament with its conception of linear time and a God that moves through time with humans (e.g., Sedlacek, 2011: 47). The idea of scientific and technological progress was fueled by the seventeenth century advances in science and literature by such cultural giants as Galileo, Newton, Descartes, Molière, and Racine. The idea of social progress was added later. Early in the eighteenth century, the Abbé de Saint Pierre advocated establishing political and ethical academies to promote social progress. Saint Pierre and Turgot influenced the Encyclopedists. The great Encyclopédie was produced by a group of eighteenth century philosophers under the direction of Denis Diderot.

 

It defines the Enlightenment program of promoting reason and unified knowledge. It was at this point that social progress became mated to the values of industrialization and incorporated into the ideology of the bourgeoisie. Scientific, technological, and social progress were all aspects of the ideology of industrial civilization. Veblen, for example, argued that the various sciences could be distinguished in terms of their proximity to the domain of technology. Thus, the physical sciences were closest to that domain, even integral with it, whereas such areas as political theory and economics were farther afield. We have entered an era of machine discipline unlike any in human history. And now we stand on the threshold of machines that will discipline us with conscious awareness and values, including social and sociable robots (the so-called robosapiens), and cyborgs.

 

There have been attempts to identify a type of progress that is independent of material or technological criteria (see, for example, the discussion in Almond, Chodorow, and Pearce, 1985, and the classic criticisms in Roszak, 1969/1995). For many ancient as well as modern thinkers, the idea of progress has always been problematic. We are right to be concerned about the actual and potential impacts of our new bio- and nano- technologies. But one finds similar concerns in Plato’s Phaedrus. There, in the dialogue between Theuth and king Thamus concerning the new technology of writing, Theuth makes promising predictions about the impact of writing.

 

The king claims to be in a better position to do what in effect is a “technology assessment,” and concludes that writing will have the opposite of the effects predicted by Theuth. The cultural meaning of science has fared no better. Where the Rousseaus and the Roszaks saw danger and alienation in science, the Francis Bacons and Bronowskis saw civilization and progress. When the biochemist J.B.S. Haldane wrote about a future of human happiness built on the application of science, Bertrand Russell replied with a vision of science used to promote power and privilege rather than to improve the human condition. St. Augustine worried about the invention of machines of destruction; Spengler predicted that humans would be annihilated by Faustian man. Fontenelle, in the first modern secular treatise on progress published in 1688 argued that science was the clearest and most reliable path to progress. Rousseau, by contrast,

argued that science and the arts have corrupted our minds. The author will draw attention to some additional examples in his conclusion in this chapter.

 

By its intimate association with the very foundations of science, mathematics does not escape this ambivalence. But it stands apart from science in terms of its stronger association with human progress. In the seventeenth and eighteenth centuries, a wave of positivism fueled by Newton’s achievements evoked nothing but the promise of progress among mathematicians of that period. The historian Florian Cajori (1894: 4) had no question about the connection between mathematics and human progress. For Alex Bellos (2010: ix), mathematics is (“arguably”) the foundation of all human progress.

 

Progress, then, can be viewed in terms of “amelioration” or “improvement” in a social or ethical sense. Are we more advanced than cultures that are less dominated by machines and machine ideology? How do we measure the primacy of humans and ecologies and how do we sustain them in any given culture? Can we bring them to fruition and nourish them in any culture, or are some more friendly to the primacy of humans and ecologies than others? These issues are really matters of degree associated with the degree to which individuation of the self (and then the myth of individualism, selfishness, and greed) has progressed in any given society. Furthermore, the degree of awareness of and attention to ethics, values, and social justice has to come into consideration here. It is impossible to even discuss the idea of progress without engaging ideas about and the value of the person, freedom, and democracy.

 

It may be possible to define progress in a way that takes it out of the realm of hopes, wishes, and dreams and plants it more firmly on a meaningful (and even perhaps measurable) foundation. Following Gerhard Lenski (1974: 59), progress can be defined as the process by which human beings raise the upper limit of their capacity for perceiving, conceptualizing, accumulating, processing, mobilizing, distributing, and utilizing information, resources, and energy in the adaptive-evolutionary process. The relationship between adaptation and evolution is a paradoxical one. On the one hand, survival depends on the capacity to adapt to surroundings; on the other hand, adaptation involves increasing specialization and decreasing evolutionary potential. Adaptation is a dead end. As a given entity adapts to a given set of conditions, it specializes to the point that it begins to lose any capacity for adapting to significant changes in those conditions. The anthropologists Sahlins and Service (1960: 95-97) summarize these ideas as follows:

 

Principle of Stabilization: specific evolution (the increase in adaptive specialization by a given system) is ultimately self-limiting.

 

General evolution (progressive advance measured in absolute terms rather than in terms of degrees of adaptation in particular environments) occurs because of the emergence of new, relatively unspecialized forms.

 

Law of Evolutionary Potential: increasing specialization narrows adaptive potential. The more specialized and adaptive a mechanism or form is at any given point in

evolutionary history, the smaller is its potential for adapting to new situations and passing on to a new stage of development.

 

We can add here the Law of Adaptive Levels: adaptation occurs at different levels across various life orders and systems and occurs at different speeds in different spatial arenas. This law draws attention to the complexity of adaptation and the general processes of variation and selection. Adaptation suggests an active agent in a stable environment. But active agents can and do change their environments in ways that make different demands on the adapting agents. Looked at another way, environments have agential like dynamics. Law of Agent-Environment Entanglement.

 

Perhaps the most important aspect of the ideology of science is that it is (in its mythical pure form) completely independent of technology. This serves among other things to deflect social criticism from science onto technology and to justify the separation of science from concerns about ethics and values. Interestingly, this idea seems to be more readily appreciated in general by third world intellectuals than by the Brahmin scholars of the West and their emulators. Careful study of the history of contemporary Western science has demonstrated the intimate connection between what we often distinguish as science and technology. It has also revealed the intimate connection between technoscience research and development and the production, maintenance, and use of the means (and the most advanced means) of violence in society. Not only that, but this is true in general for the most advanced systems of knowledge in at least every society that has reached a level of complexity that gives rise to a system of social stratification.

 

Contradictions and ambivalence about science, technology, and progress may be built into the very core of our cultural machinery. Agricultural activities in the ancient Near East reduced vast forests to open plains, and wind erosion and over-grazing turned those areas into deserts. Deforestation in ancient China led to the development of the loess plateau. Loess sediment gives the Yellow River (nicknamed “China’s Sorrow”) its signature color and flooding pattern. Was deforestation necessary for building China into the greatest civilizational area on earth between the first and sixteenth centuries of the common era? Or were there conservation principles that the ancient Chinese could have relied on without detracting from their cultural development? There is some evidence that at least some of the deforestation they caused could have been avoided. The deforestation experiences of China, Rome, and other civilizational areas of the ancient world are being repeated today and offer cautionary tales for an era characterized by many hard to monitor emerging and converging technologies, that is, technocultural systems.

 

At the end of the day, it should be clear that progress is not easy to define, and that it is even harder to point to examples of progress that resist critical interrogation. How can we sustain the idea of progress in the face of the widespread ecological, environmental, and human destruction that has characterized the industrial age? The fact is that the destruction and danger we see all around us is integrally connected to the very things we use to mark the progress of humanity. For these reasons, we must be cautious when considering whether any of the sciences, engineering disciplines, or mathematics have contributed to or served as signposts of progress. Mathematics, like all systems of knowledge, does not exist in a vacuum. It is always connected to social institutions and

under the control of the most powerful institutions in any given society. All of this may put too much of the onus on the sciences and technology when what we are dealing with is culture in general. Is it possible that cultures by their very natures inevitably destroy planets?

 

It should be clear from this brief introduction that the terms of our title, “mathematics,” “civilization,” and “progress” are all imbued with some level of ambivalence and uncertainty. It remains to be seen whether in the rest of this chapter we can find our way to greater certainty about the meaning and implications of these terms.

 

2.  Introduction: Mathematics and Civilization

 

Karl Mannheim (1893-1947) and Oswald Spengler (1880-1936) defend diametrically opposed positions on the possibility of a sociology of mathematics. Mannheim argues that mathematics is exempt from sociocultural and historical explanations; mathematics is not an ideology, and mathematical truths are not culturally relative. This view has been reinforced by Pythagoreans and Platonists who believe that mathematical truths are eternal objects that exist independently of the flux of historical experience and outside of time and space. Most historians, philosophers, and sociologists of science have traditionally adopted a basically Mannheimian view of mathematics.

 

Spengler, on the other hand, holds that each culture has its own conception of number. Spengler's notion of the "soul” of a civilization cannot provide the basis for an adequate sociological analysis. However, Spengler's goal of explaining mathematics in terms of the particular social and historical forms in which it is produced is sociologically viable. Spengler’s argument is summarized in two statements: (1) “There is not, and cannot be, number as such." There are several number-worlds as there are several cultures; and (2) "There is no Mathematik but only mathematics." Spengler's objective in his analysis of "number" is to show how a crystallized culture demonstrates its idea of the human condition, of what it means to be human. The "peculiar position" of mathematics rests on the fact that it is at once science, art, and metaphysics. It is safe to assume, taking some liberties, that Spengler’s use of “Culture” is more or less commensurate with what we commonly mean by “civilization.”

 

The author takes some additional liberties here with Spengler’s spiritualized materialism in order to avoid some of the idiosyncrasies of his vocabulary. Number, like God, represents the ultimate meaning of the natural world. And like myth, number originated in naming, an act that gives humans power over features of their experience and environments. Nature, the numerable, is contrasted with history, the aggregate of all things that have no relationship to number. Note that in this moment it appears that Mannheim and Spengler might be at one on the nature of mathematics. While Spengler clearly sees a closer connection between mathematics and culture than Mannheim does, he does as the following paragraph illustrates harbor some ambivalence.

 

Spengler argues against treating earlier mathematical events as stages in the development of "mathematics”. This is consistent with his thesis on the incommensurability of Cultures and with his cyclical view of historical change. His general schema of Classical and Western styles and stages in "Culture," "number," and

"mind" is essentially an analysis of world views. This is reflected in Spengler's attempt to correlate mathematical and other sociocultural "styles.” For example, he argues that Gothic cathedrals and Doric temples are “mathematics in stone." Spengler is aware of the problem of the limits of a "naturalistic" approach to number and pessimistic about a solution. It is impossible, he writes, to distinguish between cultural features that are independent of time and space, and those that follow from the forms of culture manufactured by humans.

 

Finally, Spengler claims that a deep religious intuition is behind the greatest creative acts of mathematicians. Number thought is not merely a matter of knowledge and experience, it is a "view of the universe," that is, a world view. The second claim Spengler makes is that a "high mathematical endowment" may exist without any "mathematical science.” For example, the discovery of the boomerang can only be attributed to people having a sense of mathematics that we must recognize as a reflection of the higher geometry.

 

Sociologists of mathematics have been bold enough about challenging the Platonic conception of number, but they have hesitated to follow Spengler. His ideas must seem mad to scholars and laypersons, and specialists and non-specialists alike, to whom the truth of number relations appears to be self-evident. And yet, the "necessary truth" of numbers has been challenged by mathematical insiders and outsiders. One of the outsiders is Dostoevsky. In his Notes from Underground (1864/1918), Dostoevsky argues that 2+2 = 4 is not life but death, impudent, a farce. It’s nonetheless “excellent” and we must give it its due, but then we must recognize that 2+2=5 is also sometimes “a most charming little thing”. Contrast Dostoevsky’s perspective with Orwell’s (1949) use of these two equalities in 1984. For Orwell, 2+2=4 stands for freedom and liberty; 2+2=5 stands for Big Brother totalitarianism. Dostoevsky uses 2+2=4 to stand for everyday routines and tradition; 2+2=5 represents creativity.

 

Dostoevsky's remarks are not merely a matter of literary privilege. Mathematicians and historians and philosophers of mathematics have also challenged the conventional wisdom on number. We should not expect ordinary arithmetic to apply in every physical situation; whether it does or does not has to be based on our experience in different physical situations. In other words, whether or not 2+2=4 is always an empirical question. Where we have long term experiences with situations in which 2+2=4 we are justified in considering those situations closed to further interrogation, that is, we are justified in taking the equality for granted. This is not a warrant for universalizing mathematics uncritically.

 

Studies in ethnomathematics have helped to reinforce and ground the notion that mathematics and logics are culturally situated (cf. Benesch, 1992). African mathematics has posed problems for European intellectuals since at least the publication of Robin Horton’s (1997) studies on patterns of thought in Africa and the West beginning in the 1960s. The earliest findings and interpretations suggested a different way of reasoning and a different logic in African cultures by comparison with the Europeans. Ethnomathematics has helped to sort out the early discourses and ground differences in cultural patterns rather than in mental proclivities. Malagasy divination rituals, for example, rely on complex algebraic algorithms. Some peoples use calendars far more

abstract and elegant than those used in European cultures, notably the Chinese and the Maya. Certain concepts about time and equality that Westerners assumed to be universal in fact vary across cultures. The Basque notion of equivalence, for example, is a dynamic and temporal one not adequately captured by the familiar equal sign. Other ideas taken to be the exclusive province of professionally trained Western mathematicians are, in fact, shared by people in many societies. (see D’Ambrosio, 2006; Mesquita, Restivo, and D’Ambrosio, 2011; Eglash, 1999; Ascher, 2004; Verran, 1992).

 

There is, in brief, a rationale for pursuing the Spenglerian program for a sociology of mathematics based on the views of at least some mathematicians, historians of mathematics, and observers of numbers. The author refers here to authors such as Dostoevsky and Orwell, mathematicians such as Dirk Struik and Chandler Davis, philosophers of mathematics such as Paul Ernest and Leone Burton, and historians of mathematics such as Morris Kline.

 

Sociologists are obliged to interrogate mathematics in a comparative perspective: how does mathematics develop at different times and places; who are the noted mathematicians, what are the social positions they held and how were they related to one another; and what are the social conditions within and outside of mathematical communities as they go through phases of growth, stagnation, and decline. The degree of "community" among mathematicians, the level of specialization, the extent of institutionalization and the relative autonomy of the social activity of mathematics, it should be stressed, are variable across time and space.

 

It is possible to narrate the history of mathematics as a more or less linear unfolding that gives the appearance of an inevitable “logical evolution.” However, the evidence is that there are a number of variations among the types of mathematics produced in different cultures. The latter "horizontal" variations are prima .facie evidence for the Spengler thesis. But what about the long-term trends? These too are socially determined, and in two different senses. First, and in a weaker sense, the "longitudinal" development of mathematics does not occur without interruptions, nor does it unfold in a single cultural context. Mathematicians move along a certain path at some times and not others. This implies among other things that the concepts of truth and what counts as a proof in mathematics vary over time. A sequence in mathematical development or in the network of mathematicians will stop, start, stop, and start again over a period of a hundred or a thousand years. What drives these processes? And why do particular mathematicians at particular times and places make the major advances and not others? More to the point, how is it that some mathematicians and not others who might be better candidates get credit for discoveries and inventions? This question opens up an opportunity to consider eponymy and mis-eponymy in mathematics.

 

In the sciences, eponymy refers to the process of naming an invention or discovery after the person who made the invention or discovery. Some well-known examples are Ohm’s law, the Pythagorean theorem, and Pascal’s triangle. In 1980, the statistics professor Stephen Stigler (1980) formulated "Stigler’s law of eponymy:” stated in general terms it states that no discovery or invention is named after its original discoverer or inventor. Stigler then attributed “Stigler’s Law” to the sociologist of science Robert K. Merton (1961, 1963; and on the related Matthew effect, see Merton,

1968), thus making Stigler’s Law self-exemplifying (further exemplifying Merton’s fondness for the self-exemplifying hypothesis). Merton had already formulated the idea that all discoveries are in principle multiples. Consider the following examples: Pythagoras’ theorem was already known in Mesopotamia at least a thousand years earlier (crediting Pythagoras with a proof is entirely speculative). The Chinese had Pascal’s triangle at least 320 years before he was born. L’Hôpital’s Rule is probably due to Johann Bernoulli; Cardano’s formula for solving cubic equations comes from Tartaglia; Pell’s equation is due to Fermat; Benford’s Law is due to Simon Newcomb; and the Möbius strip could just as easily be named for Listing (Polster and Moss, 2011).

 

Following Spengler and more recent developments in the sociology of mathematics, it is possible to identify social factors that affect the variations, interruptions, progressions, and retrogressions characteristic of the longitudinal development of mathematics. There is a stronger sense in which the longitudinal development of mathematics towards more "advanced" forms is socially determined. The longitudinal development of mathematics reveals the social aspects of mathematical work. Much of professionalized mathematics is created in response to stimuli from within the mathematical community, especially as mathematicians go about playing competitive games with one another. The development of higher and higher levels of “abstraction” (see immediately below), for example, reflects the increasing self-consciousness of mathematicians about their own operations. This in turn reflects higher levels of specialization and institutional autonomy among mathematicians. The Spengler thesis is true in a very strong form: “number”, and all that it stands for metaphorically, is a socially created activity, or more technically, a social construction (see the detailed explanation of this much abused term in Restivo and Croissant (2008).

 

We should consider replacing the term “abstraction” with the term “generalization.” The reason is that what we commonly understand as abstractions are simply concrete forms constructed under conditions of professionalization and disciplinary closure. As mathematics becomes more organized and disciplined, mathematicians build new levels of mathematics on the grounds of earlier mathematical forms. This removes mathematics further and further ceteris paribus from the everyday world and gives rise to the idea of abstraction. But it is important to understand that we are dealing with new levels of concreteness. The term “abstraction” makes us vulnerable to the myths and ideologies of purity and even to ideas about heavens and gods. It is in my view better to eliminate it from our vocabulary in the sense that it is used in the sciences and mathematics.

 

3.  Mathematics and Civilization: Case Studies

 

The world history of mathematics has not unfolded in a unilinear, unidirectional manner. The Greeks, for example, took a step backward from the Babylonian achievements in notation. Different types of mathematical systems have developed in different parts of the world; and rival forms of mathematics have sometimes developed within societies and professional networks. Hindu mathematics, especially in the period before the influx of Greek astronomy (ca. 400CE) placed unique emphasis upon large numbers. Geometry, arithmetic, number theory, and algebra were ignored in favor of the use of numbers in "sociological" schemes. The Upanishads (ca. 700 to 500BCE) are full

of numerical (or more accurately, numerological) descriptions: 72,000 arteries; 36,360, or 36,000 syllables; the 33, 303, or 3306 gods; the 5, 6, or 12 basic elements out of which the world is composed. The wisdom of the Buddha is illustrated by the gigantic numbers he can count out (on the order of 8 times 23 series of 107), and his magnificence is shown by the huge number of Bodhisattvas and other celestial beings who gather to set the scenes for his various sutras. The Hindu cosmology includes a cyclical view of time that enumerates great blocks of years called yugas. There are four yugas ranging from 432,000 to 1,728,000 years, all of which together make up one thousandth of a kalpa or 4,320,000,000 years.

 

This emphasis upon immense, cosmological numbers, gives a distinctively Hindu view of the near-infinite stretches of being that surround the empirical world. It seems almost inevitable that the Hindus should have invented zero (sunya, emptiness, in Sanskrit). The concept sunya, developed about 100CE, was the central concept in Madhyamika Buddhist mysticism, and preceded the invention of the mathematical zero about 600CE. Brahmagupta published a number of rules governing the use of zero and negative numbers in his Brahmasputha Siddhanta (ca. 630CE). Classical Indian world views are permeated with a “mathematics of transcendence.” Numbers were used as a technology for transcending experience not as a mathematics that was directed toward rationalistic generalization. Numbers were used rhetorically to mystify, impress, and awe. In general, numbers were used numerologically rather than mathematically. The social roots of this distinctive mathematical system lie in the particularly exalted status of Indian religious specialists. A sociologist of mathematics would search for the roots of the Hindu emphasis on large numbers in the great variety of ethnic groups making up Indian society, institutionalized in the ramifications of the caste system.

 

The cosmological significance of Chinese mathematics has an ideographic bias. Numbers, and higher mathematical expressions, are written as concrete pictures. The system of hexagrams that make up the I Ching, the ancient book of divination, was continuously reinterpreted in successive Chinese cosmologies as the basic form of the changing universe. Chinese arithmetic and algebra were always worked out in positional notation. Different algebraic unknowns, for example, could be represented by counting sticks laid out in different directions from a central point. Chinese algebra, at its height around 1300CE, could be used to represent fairly complex equations, and included some notion of determinants (i.e., the pattern of coefficients). But it could not be developed in the direction of increasingly general rules. The ideographs (and the social conditions of their use) helped preserve the everyday roots of mathematics.

 

Why did Chinese mathematics take this form? Probably for some of the same reasons that account for the maintenance of ideographic writing among Chinese intellectuals. Both gave a concrete aesthetic emphasis to Chinese culture. The ideographic form had technical limitations that a more generalized form - an alphabet, a more mechanical mathematical symbolism - would have overcome. Ideographs are hard to learn; they require a great deal of memorization. But these limitations may in fact have been the reason why Chinese intellectuals preferred them. A difficult notation is a social advantage to a group attempting to monopolize intellectual positions. This may be contrasted with the algorithmic imperative characteristic of periods of rapid commercial expansion.

Writing and mathematics were highly esoteric skills in the ancient civilizations when they were first developed. Those who possessed these skills were almost exclusively state or religious dignitaries. Writing and mathematical notation tended therefore to be retained in forms that were very difficult to read and interpret, except by those who could spend a long time in acquiring familiarity with them. Sanskrit, for example, was written without vowels and without spaces between the words. Egyptian writing was similarly conservative. Chinese writing and mathematics are notable because archaic styles lasted much longer than anywhere else. The over-riding cultural issue might be that China was, as Leon Stover argued (1974: 24-25), a “once and always Bronze Age culture,” the only primary civilization to develop its Bronze Age to the fullest.

 

The development of ideographs and mathematical notation in China was in the direction of greater aggregative complexity and aesthetic elaboration, not of simplification and generalization. The Chinese literati thus managed to make their tools progressively more difficult to acquire. This is in keeping with the unusually high social position of Chinese intellectuals. They maintained their status through an examination system that was used to select officials in many dynasties. Many students of mathematics have contended that a “good” notation is a condition for progress in mathematics. The question for China or for any given society or mathematical community is: why wasn’t a more appropriate symbolism invented at some particular point in the history of mathematics? To answer that question we should envision a struggle between monopolizing and democratizing forces over access to writing and mathematics.

 

Monopolistic groups were strong in highly centralized administrations such as ancient Egypt, the Mesopotamian states, and China. Democratizing forces won the upper hand in decentralized situations, and/or under social conditions where there was a great deal of private business activity – as in ancient (especially Ionian) Greece, and certain periods in ancient and medieval India. The predominance of these forces was to varying degrees opposed by counter-forces. Greek mathematics also had some conservative elements, especially in the Alexandrian period when difficult rhetorical forms of exposition limited the development of algebra. The specific character of mathematics in given world cultures is due to the differential incidence of such conditions.

 

Greek mathematics is distinguished by its emphasis on geometry, generalized puzzles, and formal logical proofs. This is the intellectual lineage of modern Western mathematics. But the history of Greek and European mathematics also shows a divergent type that rose to prominence following the establishment of the classical form. During the Alexandrian period, another form of arithmetic was developed that was used neither for practical calculations nor for puzzle-contests. This was a type of numerology that used the real relations among numbers to reveal a mystical cosmology. The system was connected with verbal symbolism through a set of correspondences between numbers and letters of the Hebrew or Greek alphabets. Any word could be transformed into a related number that in turn would reveal mathematical relations to other words.

 

The social conditions involved in the creation and development of this alternative mathematics are connected with religious movements. Numerology is related to Hebrew Cabbalism, Christian Gnosticism, and the Neo-Pythagorean revival associated with Philo of Alexandria, ca. 20BCE-50CE). The most prominent expositor of this new

mathematics was Nichomachus (ca.100CE). Like Philo, he was a Hellenistic Jew (living in Syria). It was in this Jewish-Greek intellectual milieu of the Levant that the major religious movements of the time were organized.

 

There are variants even in modern European mathematics. There are conflicts between alternative notational systems in the 1500s and 1600s; and a century-long battle between the followers of Newton and those of Leibniz over the calculus. In the nineteenth century, a major dispute arose between Riemann, Dedekind, Cantor, Klein, and Hilbert and critics such as Kronecker and Brouwer. This split continued and widened in the twentieth century. The result was the emergence of schools in conflict over the foundations of mathematics. The main competitors in this arena were the logicists, the formalists, and the intuitionists

 

4.  The Social Roots of Mathematics

 

The social activities of everyday life in all the ancient civilizations gave rise to arithmetic and geometry, the two major modes of mathematical work. Each of these modes is associated with specific types of social activity. The development of arithmetic is stimulated by problems in accounting, taxation, stock-piling, and commerce; and by religious, magical, and artistic concerns in astronomy, in the construction of altars and temples, in the design of musical instruments, and in divination. Geometry is the product of problems that arise in measurement, land surveying, construction and engineering in general. Arithmetic and geometrical systems appear in conjunction with the emergence of literacy in all the earliest civilizations – China, India, Mesopotamia, Egypt, and Greece. These mathematical systems are, to varying degrees in the different civilizations, products of independent invention and diffusion. Note that while it is analytically “simple” to distinguish geometrical and arithmetical systems and methods it is not always so simple to do this in practice.

 

While we have found arithmetic, geometry, number work, and general mathematics in cultures throughout recorded history, special conditions were required for the emergence and crystallization of the discipline of mathematics. The general human ecological conditions for the emergence of modern science are discussed in Restivo and Karp (1974; and see Restivo, 1979, and 1994: 29-48). A combination of organizational and institutional factors rooted in a human ecology was required to foster and sustain the development of mathematical communities with generational continuity. As those conditions crystallized in Western Europe beginning in the 1500s and earlier, the discipline of mathematics emerged when sets of arithmetic and geometrical problems were assembled for purposes of codification and teaching, and to facilitate mathematical studies. Assembling problems was an important step toward unifying mathematics and stimulating generalization.

 

One of the most important steps in unifying and disciplining mathematics arose from efforts to state general rules for solving all problems of a given type. A further step could be taken once problems were arranged so that they could be treated in more general terms. Problems that had arisen in practical settings could now be transformed into hypothetical puzzles, and problems could be invented without explicit reference to practical issues. The three famous puzzles proposed by Greek geometers of the 5th and

4th centuries BCE are among the earliest examples of such puzzles: to double the volume of a cube (duplication of the cube), to construct a square with the same area as a given circle (quadrature of the circle), and to divide a given angle into three equal parts (trisection of the angle). Such problems were related to the non-mathematical riddles religious oracles commonly posed for one another. One account of the origin of the problem of duplicating the cube, for example, is that the oracle at Delos, in reply to an appeal from the Athenians concerning the plague of 430 BCE, recommended doubling the size of the altar of Apollo. The altar was a cube. The early Hindu literature already refers to problems about the size and shape of altars, and these may have been transmitted to Greece by the Pythagoreans, a secret religio-political society. The problem is also a translation into spatial geometric algebra of the Babylonian cubic

equation X ^{3 =}V .

The duplication, quadrature, and trisection problems were popular with the Sophists, who made a specialty out of debates of all kinds. A generation or two later, Plato introduced the constraint that the only valid solutions to these problems were those in which only an unmarked straightedge and a compass were used. This meant that special mechanical devices for geometrical forms could not be used in mathematical competitions. The result was stiffer competitive conditions and an emphasis on intellectual means and “gentlemanly” norms.        Plato's Academy was organized to help an elite group of intellectuals gain political power; and it represented the opposition of an aristocracy to democratization and commercialization. It is not surprising that this elite group of intellectuals developed an ideology of extreme intellectual purity, glorifying the extreme separation of hand and brain in the slave economy of classical Greece.

 

The three famous Greek puzzles and other problems became the basis for mathematical games (i.e., competitions) of challenge-and-response. Various forms of these games are important throughout most of the subsequent history of Western mathematics. Prior to the nineteenth and twentieth centuries, however, the challenge-and-response competitions were often initiated, endorsed, or rewarded by patrons, scientific academies, and governments. Prizes were sometimes offered for solutions to practical problems. Economic concerns as well as governmental prestige were often mixed in with the struggles for intellectual preeminence. At about the same time that they initiated mathematical contests, the Greek mathematicians took two further steps that led to new mathematical forms. They stipulated that a formal, logical mode of argument must be used in solving problems; this represented a further development of earlier methods of proof. And by extending this idea they created systems of interrelated proofs. This culminated in the Elements of Euclid shortly after 300 BCE. In addition to a collection of problems, Euclid presented an explicit body of generalizations in the form of definitions, postulates, and axioms. Euclid, like Aristotle, did not use the term "axiom" but something closer to "common notion." They both self-consciously worked at codifying past human experiences. The process of "systematization-and- generalization” is one of the two major paths to new mathematical forms. The other major path is an "empirical" one.

 

The empirical path to new mathematical forms involves applying existing mathematical concepts and methods to new areas of experience. Most of the early Greek geometrical

puzzles, for example, concerned flat figures. But the methods of plane geometry could be easily extended to solid geometry, and then to the properties of spheres or of conic sections; the work on conic sections eventually led to work on curves of various shapes. The intermittent periods of creativity in Alexandrian mathematics (especially from 300 to 200 BCE and 150-200 CE) were largely devoted to these extensions. No new level of generalization (with the exception of trigonometry, considered below) was achieved, but a number of new specialties appeared.

 

The history of arithmetic shows some of the same processes that occur in the history of geometry. The effort to find general rules for solving numerical problems led gradually to what we now call algebra. Here again we see mathematicians developing the practice of posing problems primarily to challenge other mathematicians. For example, there is this famous problem, attributed to Archimedes (287 to 212 BCE): find the number of bulls and cows of various colors in a herd, if the number of white cows is one third plus one quarter of the total number of black cattle; the number of black bulls is one quarter plus one fifth the number of the spotted bulls in excess of the number of brown bulls, etc. Such problems, involving unknown quantities, led over a very long period to the introduction of various kinds of notations and symbolisms.

 

These took quite different directions in ancient and medieval China and India, the Arab world and later in medieval and Renaissance Europe. The creation of a highly generalized symbolism which could be mechanically manipulated to find solutions did not appear until the late1500s and 1600s in Europe.

 

Over this period, and to different degrees in different parts of the world, algebra underwent an empirical extension. Problems were deliberately created to increase the number of unknowns, and to raise them to successively higher powers. Equations of the form ax + b = c gave way to those on the order of ax⁴ + by³ + cz² = g . The complexity of these, of course, could be extended indefinitely (Vieta in the1580s, for example, was

challenged to solve an equation involving x⁴⁵ ); but the extensions also gave rise to

efforts to find general rules for solving higher order equations. In other words, empirical extensions tended to promote generalized extensions. At the same time, arithmetic was developing in other directions.

 

What is generally called elementary arithmetic (solving numerical problems in, for example, addition, subtraction, multiplication, and division) continued to stimulate efforts to find general rules for solving particular problems. There was tremendous variation from one system of numerical symbols and calculating rules to another in terms of the ease or difficulty with which they could be applied to solving practical problems. Most of the ancient forms of notation made working with large numbers, fractions, or complex operations like division or the extraction of roots difficult; the exposition of problems was usually carried out in words. A great deal of mathematical creativity went into the development of notational systems that could be readily manipulated. Among the most important of these innovations were the invention of decimal place notation and the zero sign in India; the standardization of positional methods for writing multiplication and division (in Europe ca. 1600); and the invention of logarithms by the Scotsman Napier in 1614, for use in astronomy, navigation, and commerce.

A different development in arithmetic led to what we now call “number theory.” This focused on the properties of numbers themselves. As early as Eratosthenes (ca. 230 BCE), efforts were made to find a general method for identifying prime numbers. There were also various propositions about how numbers are composed of other numbers. The Pythagorean work on “triangular’ and “square” numbers anticipated Fermat’s theorem that every prime number of the form 4n +1 is a sum of two squares. Number theory was particularly popular in the Alexandrian period in an occultist, cabalistic form. In its more standard puzzle-solving form, it has remained popular among mathematicians from the Renaissance through the modern period.

 

One more branch of mathematics, based on a combination of arithmetic and geometry, developed in the Alexandrian period. Measurements of angles and lines, and the calculation of their ratios, led to the creation of trigonometry, notably by Hipparchus (ca. 140 BCE) and Menelaus (ca, 100 BCE). Trigonometry spread to medieval India and the Arab world, and in Renaissance Europe provided the basis for Napier's development of logarithms.

 

The overall picture so far, then, shows mathematics arising from practical geometry and arithmetic. The development of general mathematical puzzles and the extension of mathematics to new areas led to the emergence of new fields. Geometry became increasingly systematic, and progressively applied to plane and solid figures, to conics, and eventually to trigonometry. Arithmetic gave rise to algebra in successfully more complex forms (based on practical calculating systems), and to number theory. The creation of new fields continued in modern Europe. They grew out of the processes of generalizing (commonly understood in terms of increasing levels of abstraction). New fields were furthermore the result of extending results to new empirical areas, and the combination of existing mathematical fields into hybrid fields. The combination of algebra with a new coordinate representation in geometry by Descartes and Fermat produced analytic geometry.

 

Consideration of the problems of motion and the study of curves gave rise to the calculus in the 1600s. Calculus was then applied to successively more complex functions (empirical extension); and eventually (in the 1800s) it was generalized into a theory concerning such things as the rules for solving equations, and the properties of all functions (generalized extension). It should be noted that the drive towards creating new fields by generalization and extension seems to be characteristic of highly competitive periods. Geometry itself experienced a rapid series of branching around 1800 and thereafter, the best known being the non-Euclidean geometries. But there was also the creation of descriptive geometry by Monge, projective geometry by Poncelet, higher analytical geometry by Plucker, modern synthetic geometry by Steiner and Von Staudt, and topology by Mobius, Klein, and Poincare. In the late nineteenth and early twentieth centuries, Klein, Hilbert, and Cartan unified these different geometries. This unification occurs prominently in Klein’s Erlangen Program and its generalization in Cartan’s program which was designed to place the unification into the framework of Riemannian geometry.

 

In algebra, there was a parallel set of developments after 1800.The effort to find a general solution for the quintic and other higher-order equations led to the creation of

the theory of groups by Abel, Galois, Cauchy, and others. This theory focused on an abstract pattern among the coefficients of equations, and opened up a new area of inquiry in higher mathematics. “Abstract” algebras were created by Boole, Cayley, Sylvester, Hamilton, and Grassman. All of these new tools were applied to other branches of mathematics. Dedekind applied set theory to the calculus, Cantor applied it to the concept of infinity, and others applied it to topology, number theory, and geometry. These developments led to the creation of yet another even more general field toward the end of the nineteenth century, “foundations.” "Foundations" focused on the nature of mathematical objects themselves and with the rules by which mathematics should be carried out. Foundations research has been the focus of a number of opposing schools, and has led to what are probably the most intense controversies in the history of mathematics.

 

The basic forms of mathematics, arithmetic and geometry arise from practical problems in construction, taxation, administration, astronomy, and commerce. Moreover, the stimulus of practical concerns does not simply disappear once mathematics is launched. For example, the basic forms of arithmetic, including the number system, developed over a very long period, during which virtually the sole interest in improvement was to facilitate practical calculations. The same can be said for the invention of logarithms, and much of the development of trigonometry. Other advanced forms of mathematics were also stimulated by efforts to solve practical problems. The development of the calculus was linked to problems in ballistics and navigational astronomy. Newton’s mechanics makes an interesting case study in this regard. In a hallmark paper in the history and sociology of science presented at the Second International Congress of the History of Science in London in 1931, Boris Hessen (1893-1936) situated Newton’s work in the technological problems of his time, his social class position, and Newton’s lack of familiarity with the steam engine and the conservation of energy principle. Hessen’s views contradicted the view that Newton was divinely placed at a particular historical juncture and that his genius was the fountain out of which his physics sprung. The present author noted earlier R.K. Merton’s defense of Hessen’s sociology of science against the idealistic and motivational explanations offered by G.N. Clark. Merton’s argument implicitly opposes Carlyles’s Great Man theory, Alfred North Whitehead’s mystical explanation of Newton’s successes (he was born in the same year that Galileo died), and Alexander Pope’s divine providence conjecture: “Nature and nature’s laws lay hid in night; God said ‘Let Newton be!’ and all was light.”

 

Descriptive geometry and Fourier's analysis answered problems in the production of new machinery in the industrial revolution. Practical concerns do not tell the whole story of mathematics, but they are one component that continuously shapes its history. This suggests a general principle: an increase in the amount, type, intensity, or scope of practical concerns in a society will stimulate mathematical activity. The relationship between economic concerns and mathematics is especially strong; commercial growth tends to be very stimulating for mathematics. Mathematical innovations will also tend to occur when there is a shift to new productive technologies (and when there are shifts to new technologies of warfare and transportation, and shifts to more intensive administrative modes of organization). This implies a link between the development of modern European mathematics and the development of modern industrial technological societies (loosely, “capitalism” because the term “capitalism” does not refer to an actual

economic system but rather expresses an economic ideology). Since this is one factor among several, it does not imply that mathematics must come to an end in non-capitalist societies. It does, however, suggest that the form and content of mathematics (within the constraints noted by Spengler) as we know it today is a product of specific lines of cultural development. One could say that modern mathematics, like modern science, is part of the knowledge system generated by and supportive of capitalism.

The roots of mathematics in practical concerns are more apparent in some cases than in others. For example, the history of Chinese mathematics is primarily a history of an inductive “mathematics of survival." Its origins can be traced from the myth of Yü the Great Engineer's discovery of a magic square on the back of a Lo River tortoise (ca. 500 BCE). China’s mathematical evolution reaches its high point in the late Sung and early Yuan dynasties with the publication of Chu Shih-Chieh's "Precious Mirror of the Four Elements," written in 1313. Chinese mathematics never ventured far from problems of everyday life such as taxation, barter, canal and dike construction, surveying, warfare, and property matters. Chinese mathematical workers could not organize an autonomous mathematical community, and consequently failed to establish the level of generational continuity that is a necessary condition for long-term mathematical development. This helps to explain why the Chinese did not develop the more general forms of higher mathematics.

 

Conditions in ancient Greece were more favorable for generalizing mathematics. The commercial expansion in Greece in the 600s BCE stimulated mathematical growth. Learned merchants practiced and taught mathematical arts, and master-student relationships across generations fostered mathematical progress. Political and economic changes in Greek civilization led to the development of an increasingly elitist and self- perpetuating intellectual community, culminating in the oligarchic conditions and intellectual elitism of Plato's time. The achievements of the "thinking Greeks" depended on a division of labor that divorced hand and brain. The "thinkers" had the "leisure" to reflect on and elaborate mathematics. The class structure of the slave-based society that developed in the post-Ionian period conditioned the development of classical mathematics. Arithmetic was left to the slaves who carried out most commercial transactions, and householders for whom simple calculations were a part of everyday life. The elite intellectual class, by contrast, courted geometry which was considered democratic and more readily adapted to the interests of the ruling classes than arithmetic. What we know as "Greek mathematics" is a product of the classical period.

 

The development of specialties within the division of labor, left unchecked, tends to foster virtuosity. Such specialization tends to increase the specialists' distance from the order and distribution of everyday phenomena and to increase the importance of human- created phenomena, especially symbols. The result is an increase in the level of generalization and the development of ideologies of purity. This is essentially what occurred in classical Greece. Hand and brain slowly reunited following Plato's death; there is already evidence of an increased interest in linking mathematical and practical concerns in Aristotle. In the Alexandrian period, hand and brain were more or less united, but the ideology of purity retained some vitality. This is notably illustrated by Archimedes, whose work clearly exhibited a unity of hand and brain but whose philosophy echoed Platonist purity.

The decline of Greek commercial culture was accompanied by the decline of Greek mathematical culture. The achievements of Archimedes, which brought Greek mathematics to the threshold of the calculus, mark the high point of Greek mathematics. When mathematics was revived in the European commercial revolution (beginning haltingly as early as the twelfth century, the Gothic cathedrals signaling the coming economic revolution by serving as accumulating devices for labor, resources, and machinery), many aspects of the Greek case were recapitulated. European mathematics moved on in the direction of the calculus, rooted in problems of motion. It picked up, in other words, essentially where Archimedes had left off, and under the influence of the Archimedean (and more generally, Greek) writings as they were recovered and translated (and here, of course, the Arabic-Islamic transmissions were critical). By 1676, Newton was writing about mathematical quantities "described by continual motion."

 

The concept of function, central to practically all seventeenth and eighteenth century mathematics, was derived from studies of motion. Newton and Leibniz helped to reduce the basic problems addressed in the development of the calculus – rates of change, tangents, maxima and minima, and summations – to differentiation and anti- differentiation. Infinitesimals, nurtured earlier in the debates of theologians and the scholastics, entered into the process of production. Highly general intellectual ideas in the Euclidean realm of the straight, the flat, and the uniform gave way to the ideas of a dynamic world of guns and machinery, and global navigation and commerce characterized by skews, curves, and accelerations. The search for algorithms, time- saving rules for solving problems, is evident in the writings of the inventors of the calculus (e.g. in Leibniz's "De geometria recondita et analysi indivisibilium atque infinitorium" of 1686).

 

As the industrial “machine" of capitalist society was fashioned, so was the “machine of the calculus." Descartes' analytic geometry, the other great contribution to the development of pre-modern European mathematics, was also characterized by an algorithmic imperative.

 

It was, in spite of the conflicts between Cartesians and Newtonians, from the very beginning in constant association with the development of the Newtonian-Leibnizian calculus. Let us note parenthetically that Newton and Leibniz invented two different calculuses. Newton’s “method of fluxions” was much more indebted to classical geometry (and especially to Archimedes) than is commonly supposed, and Leibniz’s “differential calculus” used a much better notation (Restivo, 1992: 134).

 

The historian of mathematics Boutroux (1919) characterized Descartes’ analytic geometry as an industrial process; it transformed mathematical research into “manufacturing.” The idea that the calculus is linked to the emergence of capitalism (that is, early industrialization) is further suggested by the Japanese case. When the Japanese established a monetary economy and experienced a commercial revolution in the seventeenth century, they also worked out a "native calculus." This was not entirely indigenous since they had probably come into contact with Europe’s calculus from European contacts at their ports of entry.

5.  Puzzles and Proofs

 

Mathematical workers and mathematicians, from the earliest times onward, and especially in the West, have posed puzzles for one another. This practice tends to make mathematics a competitive game. Some periods have been dominated by public challenges such as those that the Emperor Frederick's court mathematician posed to Leonardo Fibonacci (ca. 1200), those that Tartaglia and Cardano posed for one another in sixteenth century Italy, or those that led to such high acclaim for Vieta at the French court in the 1570s. Such puzzle-contests have been important for several reasons. They often involved pushing mathematics into more general realms. Mathematicians would try to invent problems which were unknown in practical life in order to stump their opponents. The search for general solutions to equations, such as those that Tartaglia found for cubic equations and Vieta found for the reduction of equations from one form to another, was directly motivated by these contests.

 

The emphasis on proofs which has characterized various periods in the development of mathematics was partly due to a heightening of the competitiveness in these contests. Greek mathematicians rationalized the concept and method of proof at a time when mathematics was popular among the elite class of philosophers and there was a lot of competition for power and attention in the intellectual arena. This was the same period during which the wandering Sophists challenged one another to debating contests and in doing so began to develop canons of logic. This is completely analogous to the development in mathematics, in terms of both cause and effect. The analogy turns into a virtual identity when we realize that many of the mathematicians of the time were Sophists, and that many of the formal schools that were organized in the classical period (e.g., the Academy) used prowess in mathematics as a grounds for claiming superiority over competing institutions. Stressing proofs was a way of clarifying the rules of the game and escalating the intensity of competition. In general, competitive puzzle- contests are probably responsible for much of the inventiveness characteristic of Western mathematics. This analysis should not obscure the economic stimulus to the initial development of proofs.

 

Thales, the philosopher-merchant (who might have been a composite or imaginary construction), is credited with carrying the idea of a proof to a more general level than the Babylonians and Egyptians. Thales symbolically and iconically personifies the need among the Ionians of his era to develop a comprehensive and organized understanding of physical reality and successful computational methods in the context of the increasingly well-organized economy that they were products of and helped to fashion. Thales' proofs were probably crude extensions of Babylonian or Egyptian "rules" for checking results. In any case, the process of constructing proofs was rationalized over the next three hundred years and eventually led to Euclidean-type proofs.

 

Concern for proof has varied a great deal in the history of mathematics. The Chinese and Hindu mathematicians ignored proofs almost entirely; indeed, they would often present problems without solutions, or with incorrect solutions. These practices were the result of a relatively uncompetitive situation in mathematics in these societies. The social density of mathematicians in these societies was rather low; we rarely hear of more than a few mathematicians working at the same time, whereas in Greece and

Europe the numbers in creative periods are quite high. Most of the Oriental mathematicians were government officials, and thus were insulated from outside competition, while most of the ancient Greek and modern mathematicians were private individuals or teachers in competitive itinerant or formal educational systems. In the Islamic-Arabic world, there was a flurry of mathematical activity in the period 800- 1000 (and later to some extent).There was some concern for proofs (in the works of Tahbit Ibn Qurra, for example), but this was much more limited than in classical Greece. The Greek works they translated stimulated an awareness of and interest in proofs among the Islamic-Arabic mathematicians.

 

The limited emphasis on proofs reflects the fact that their community (or network) was not as densely populated as the Greek mathematical community, competition was not as intense, master-student chains and schools were not as well organized, and generational continuity was limited. Similarly, the episodic history of mathematics in India constantly interrupted generational continuity, the Bronze Age dynastic history of mathematics in China undermined specialization free of the centripetal force of the Emperor’s bureaucracy, and the delimited mathematical renaissance of Japan in the seventeenth century ended abruptly with the consolidation of Tokogawa power (Restivo, 1992: 22-60). Only Europe from the 1500s on was able to sustain generational continuity on a level no other civilization had achieved.

 

In modern Europe, the emphasis on proofs has grown steadily. In the 1600s, Fermat presented his theorems without proofs, and in the 1700s, Euler offered proofs that were not very rigorous. The early 1800s saw a shift towards more rigorous standards of proof; earlier solutions were rejected, not because they were incorrect, but because the reasoning behind them was not sufficiently universal and comprehensive. This went along with a massive increase in the number of people engaged in mathematics (which in turn was the result of the expansion of educational systems, especially in Germany and France and other social changes related to the rationalization of social and economic life). Both this shift towards rigor, and the earlier invention of proofs, had important effects on the nature of mathematics. Both pressed mathematics toward new levels of generalization: proofs had to invoke more general elements than particular numerical examples, and rigorous proofs stimulated the systematic consideration of the nature of mathematics in the nineteenth century.

 

6.  Generalization (“Abstraction”) and Self-Consciousness

 

The "main line" development of Western puzzle-solving mathematics is characterized by an increasing awareness that levels of generalization have been created by the mathematicians themselves. Mathematicians moved beyond a naïve realism about mathematical objects (sometimes real in the material sense, sometimes real in the Platonic sense) when they gradually began to use negative numbers instead of dropping negative roots of equations (as Hindu, Arab, and medieval European mathematicians had done). Later they came to recognize that imaginary numbers (an unfortunate and distracting nomenclature) could be used despite their apparent absurdity. Gauss established a new basis for modern algebra by creating a representational system for complex numbers. Nineteenth century higher mathematics took off from this point.

Mathematicians finally realized that they were not tied to common-sense representations of the world, but that mathematical concepts and systems could be deliberately created. These creations were not ab novo but situated in the cultural objects of the increasingly well-organized mathematical community. The new, more general geometries (including projective and non-Euclidean geometries) popularized the point, and stimulated the creation of new algebras and more generalized forms of analysis. The objects with which modern mathematics deals, however, are real in a particular sense. They are not simply things, as was once naively believed; they are, rather, operations, activities that mathematicians carry out. The imaginary number i is a shorthand for an activity, the operation of extracting a square root from a negative number. This operation, of course, cannot be carried out. But mathematicians had long been used to working backwards from solutions-not-yet-found, to the premises, by symbolizing the solution using an arbitrary designation (e.g., x ). This symbol represented the result of an imaginary operation. The imaginary number i, then, could be used as the basis for other mathematical operations, even though the operation of producing it could never actually be performed. The ordinary arithmetic operations, the concept of a function, the concept of a group - all of these are operations of different degrees of complexity. A natural whole number itself is not a thing but an operation - the operation of counting (and perhaps also other operations whose nature modern mathematicians are untangling). Modern mathematics has proceeded by taking its operations as its units. These are crystallized into new symbols which can then be manipulated as if they were things. In fact, within the social world of mathematics these symbols are things, the matters of fact, and the material resources of the human ecology of the mathematics community (see Alexksandrov, Kolmogorov, and Lavrent’ev, 1963 for a thoroughly materialistic and realistic history of mathematics).

 

A process of reification has gone on in conjunction with the emergence of the notion that generalized operations (what are commonly referred to as “abstractions”) are socially created by mathematicians. Thus mathematics has built upon itself hierarchically by treating operations as entities upon which other operations can be performed. The Western trend in symbolism, then, is not an "accidental" feature of Western mathematical uniqueness; the symbolism was created precisely because the mathematical community was pushing towards this degree of self-consciousness and reflexivity. What I am pointing to here is an emerging awareness among the more reflective mathematicians and philosophers of mathematics that crystallized in the late nineteenth and early twentieth century. That awareness that the creative construction of mathematics came from the hands and brains of situated human beings was enough to fuel the more recent efforts to ground mathematics in the material world of activities and experience. It has not, however, been enough to eliminate all vestiges of Platonic thinking among mathematicians.

 

Mathematics, like other modern activities, has been affected by specialization on a level unknown in earlier historical periods. As a result, the "causal power" of mathematics itself in the relationships between mathematical and other social activities has steadily increased. Mathematical ideas have increasingly become the generative basis for new mathematical ideas. The work setting and institutional context of mathematical activity has become a social foundation of a higher order than the social foundation of subsistence productive activity. Mathematics continues to be socially rooted within the

mathematical community; it is especially important to recognize the social nature of the symbols mathematicians create for communication within their own ranks.

 

This perspective throws new light on an old problem reflected in the title of a famous paper by Eugene Wigner (1960) on “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” The utility of “pure” mathematics for the physical and natural science is not just a coincidence; it must reflect some larger and deeper truth about both mathematics and physics. In fact, the “coincidence” reflects a constant interplay between mathematics and physics. As soon as this is interrupted by the professional and bureaucratic closure of the mathematics community, we can expect the cycles of effective applications to become increasingly rare. This is not just a sociological conclusion based on the theory of closure and autonomy in social systems (Restivo, 1992: 171-175; Restivo, 1993: 263-267) but something that has been recognized by professional mathematicians (e.g., Boos and Niss, eds., 1979).

 

The development of Western higher mathematics, then, is a social and a cultural development. The objects with which mathematicians deal are activities of mathematicians. In building upon the operations already in existence, and making them symbolic entities on which further operations can be performed, mathematicians are self-consciously building upon previous activities in their intellectual community. Mathematics thus embodies its own social and cultural history, and uses it as the base upon which its current community activities are constructed. Western mathematics thus depends upon a particular kind of long-term organization of the intellectual community. This is an organization in which strong links are maintained across generations, and in a highly self-conscious and competitive form. The new attempts to competitively consume the old. The important linkages of teachers and pupils typically found among European mathematicians, together with strong external competition among different mathematical "lineages," have been the social basis for this pattern (on the general theory of lineages and networks as the progenitors of ideas, see Collins, 1998). Once the pattern of competitive self-consciousness was established, subsequent rounds of competition could only escalate the degree of self-reflection and inventiveness among mathematicians. Out of this situation arose the hyper-reflexive concerns of twentieth- century foundations research. It is important to keep in the mind the self-conscious creativity of mathematicians is inseparable from the crystallization of an autonomous mathematical community.

 

7.  Conclusion

 

All thought, in its early stages, begins as action. The actions which you [King Arthur] have been wading through have been ideas, clumsy ones of

course, but they had to be established as a foundation before we could begin to think in earnest.

T.H. White’s Merlyn the Magician What, “In the beginning was the Word?” Absurd.

Then maybe it should say “In the beginning was the Mind?” Or better “…there was Force?”

Yet something warns me as I grasp the pen,

 

 

Act.”

That my translation must be changed again.

The spirit helps me. Now it is exact. I write: “In the beginning was the

 

Goethe’s Faust

 

The history of mathematics can be situated socially and culturally. This task requires a sociology both of the external institutional and organizational conditions of the societies within which mathematical activities are situated, and a sociology of the internal organization of the mathematical activities within communities and social networks of mathematicians. The notion of "internal and external factors" is an analytic device. The Spenglerian idea of mathematics as a world view is not, in the end, compatible with a strict adherence to internal-external analysis. The mathematics of any particular time embodies its own social history. This process becomes increasingly intense as and to the extent that mathematical activity becomes and remains more clearly differentiated from other social activities and more autonomous. But "autonomy" simply means that mathematicians communicate more intensively with each other than with outsiders. It does not mean that mathematicians are more removed from social determinants or that they have unimpeded access to "objective reality." Their activities remain at all times coupled to the social activities of insiders and outsiders, and thus unfold in an environment of multiple social, cultural and historical determinants. This is the rationale for a Spenglerian approach to the sociology of mathematics. That this is becoming more accepted is indicated by a clear turn to practice, experience, and shared meaning in the philosophy of mathematics, the philosophy of mathematics education, and among reflective mathematicians.

 

We are no longer entranced by the idea that the power of mathematics lies in formal relations among meaningless symbols, nor are we as ready as in the past to take seriously Platonic and foundationalist perspectives on mathematics. We do, however, need to be more radical in our sociological imagination if we are going to release ourselves from the strong hold that philosophy has on our intellectual lives. Philosophy, indeed, can be viewed as a general Platonism and equally detrimental in its classical forms and agendas to our efforts to ground mathematics (as well as science and logic) in social life. It is to philosophy in its more recent turn to practice that we must look if we are going to salvage philosophy as a credible intellectual activity. The strengths of philosophy as a guide to reasoned speculation, thought experiments, and ethical stands will be heightened by the turn to practice and the empirical arena. This is how we can save philosophy as an analytical tool for understanding mathematics. And here is where philosophy is drawn into the interdisciplinary boundary breaking modalities of late twentieth and twenty first century intellectual life. We are witnessing the demise of the traditional disciplines as we have known them for more than one hundred years. As the new inter-disciplines come to the fore, it begins to appear that we are witnessing the emergence of a second generation natural philosophy. For the moment, let us bracket this development and focus on mathematics and sociology as distinct, viable disciplinary projects. Both of these disciplines are undergoing interdisciplinary developments that are contributing to the emergence of a neo-natural philosophy.

 

Once again, then, what is mathematics? Technical talk about mathematics – trying to understand mathematics in terms of mathematics or mathematical philosophy – has the

effect of isolating mathematics from practice, experience, and shared meaning; it tends to “spiritualize” the technical. It is important to understand technical talk as social talk, to recognize that mathematics and mathematical objects are not (to borrow terms from the anthropologist Clifford Geertz' (1983: 94-120) “concatenations of pure form,” “parades of syntactic variations,” or sets of “structural transformations.” To address the question “What is mathematics?” is to reveal a sensibility, a collective formation, a worldview, a form of life. This implies that we can understand mathematics and mathematical objects in terms of a natural history, or an ethnography of a cultural system. We can only answer this question by immersing ourselves in the social worlds in which mathematicians work, in their networks of cooperating and conflicting human beings. It is these “math worlds” that produce mathematics, not individual mathematicians or mathematicians’ minds or brains. It is easy to interpret this perspective as somehow “mystical,” or as implying a “super-organic” entity. A realistic interpretation requires understanding human individuals as social things and their ideas as the “voice” of their social networks.

 

Mathematics, mathematical objects, and mathematicians themselves are manufactured out of the social ecology of everyday interactions, the locally available social, material, and symbolic interpersonally meaningful resources. All of what has been written in the last two paragraphs is captured by the short hand phrase, “the social construction of mathematics.” This phrase and the concept it conveys are widely misunderstood. It is not a philosophical statement or claim but rather a statement of the fundamental theorem of a sociology broadly conceived, a sociology being transformed by the interdisciplinary movements of our era. Everything we do and think is a product of our social ecologies. Our thoughts and actions are not simple products of revelation, genetics, biology, or mind or brain. To put it the simplest terms, all of our cultural productions come out of our social interactions in the context of sets of locally available material and symbolic resources. The idea of the social seems to be transparent, but in fact it is one of the most profound discoveries about the natural world, a discovery that still eludes the majority of our intellectuals and scholars. The interdisciplinary imperative can sustain this idea of the social even while it brings it into closer and closer association with our biology (cf. Clark, 2010, Noë, 2010, and Reyna, 2007).

 

Mathematics is a human, and thus a social, creation rooted in the materials and symbols of our everyday lives. It is earthbound and rooted in human labor. We can account for the Platonic angels and devils that accompany mathematics everywhere in two ways. First, there are certain human universals and environmental overlaps across the variety of our material environments, the physics, biology, and chemistry of life, culture, space, and time that can account for certain “universalistic” features of mathematics. Everywhere in everyday life, putting two apples together with two apples gives us phenomenologically four apples. But the generalization that 2+2 = 4 is culturally glossed and means something very different in Plato, Leibniz, Peano, and Russell and Whitehead. The earthbound everyday world of apples is commensurable for Plato and Russell and Whitehead; that world has not changed in the millennia that separate their lives. However, the discipline and then the profession of mathematics has changed dramatically across the centuries and created new incommensurable experiences within the social world of mathematics. The professionalization of mathematics gives rise to the phenomenon of mathematics giving rise to mathematics, an outcome that reinforces

the idea of a mathematics independent of work, space-time, and culture. Mathematics is always and everywhere culturally, historically, and locally embedded. There is, to recall Spengler, only mathematics and not Mathematik. There is, however, number work that carries across all cultures. Culture always intrudes in our mathematics; but while there is no Mathematik, there are ways for us to translate and communicate commensurable number and math work across cultures.

 

The concept-phrase “mathematics is a social construction” must be unpacked in order to give us what we see when we look at working mathematicians and the products of their work. We need to describe how mathematicians come to be mathematicians, the conditions under which mathematicians work, their work sites, the materials they work with, and the things they produce. This comes down to describing their culture – their material culture (tools, techniques, and products), their social culture (patterns of organization – social networks and structures, patterns of social interaction, rituals, norms, values, ideas, concepts, theories, and beliefs), and their symbolic culture (the reservoir of past and present symbolic resources that they manipulate in order to manufacture equations, theorems, proofs, and so on). This implies that in order to understand mathematics at all, we must carry out ethnographies – studies of mathematicians in action. To say, furthermore, that “mathematics is a social construction” is to say that the products of mathematics – mathematical objects – embody the social relations of mathematics. They are not free standing, culturally or historically independent, Platonic objects. To view a mathematical object is to view a social history of mathematicians at work. It is in this sense that mathematical objects are real. Before there is mathematics there is number work; before there are professional mathematicians there are number workers and then mathematics workers.

 

Arithmetic, geometry, and the higher mathematics are produced originally by number or mathematical workers and later on by variously disciplined and ultimately professional mathematicians. Ethnographies and historical sociologies of mathematics must, to be complete, situate mathematics cultures in their wider social, cultural, historical, and global settings. They must also attend to issues of power, class, gender, ethnicity, and status inside and outside more or less well-defined mathematical communities.

 

There is a hidden interrogation in the interrogation of mathematics that undermines its claims to transcendence and purity. That hidden interrogation is the interrogation of the very idea of the transcendent, of a Platonic realm of ideas. It is not too much of a stretch to see that this sort of interrogation will sooner or later have us interrogating religion and the gods. This is not the place to follow this line of inquiry. However, it is important to note that any deep understanding of the nature of mathematics as a this- worldly phenomenon is necessarily linked to the possibility of bringing religion and the gods down to earth. One only has to consider that the sociologist Emile Durkheim (1912/1995) concludes his remarkable study of the social construction of religion and the gods by arguing that logical concepts are, like religion and the gods, collective representations, this-worldly social constructs. Durkheim manifests the unfolding of the sociological enterprise as an exercise in the rejection of the transcendental.

 

Finally, let us revisit the ideas of civilization and progress. As we saw earlier, we could try to place ourselves on a continuum of civilization and progress between the polar

positions of Rousseau and Ferguson; everything about human society and culture as we have known it (especially since the coming of modern science and technology and the industrial revolution) denies our humanity, or everything trumpets its triumphs. This inevitably enmeshes us in a conflictful conversation without stop signs.

 

We could try to adopt Lenski’s notion of progress introduced earlier and bring the idea of civilization into his framework as a form of social and cultural organization. This would give us a more “scientific” foundation to rest on, but not one without its own ethical and value biases and implications. In the end, we are probably safest at this point if we recognize that mathematics is one of the many tools humans have fashioned to help them wind their way through the complex tapestries of life, and then one of humanity’s most useful as well as awe inspiring tools, even if more often than not actualized in the service of state power and ruling elites as “weapons of math destruction.” Moreover, it is one of the achievements that Nietzsche would have recognized as part of the fragile reason and sense of freedom humans have purchased at great cost.

 

Every entity recognized in history as a “civilization” from ancient Sumer and Babylonia to contemporary China has fed mathematical traditions into the ocean of a mathematical heritage that is in principle a part of humanity’s world culture. We can value the mathematics that has been developed out of and applied to the best of our activities as humans, activities that have been liberating, supported social justice, and in general made our planet a better and safer place on which to pursue the betterment of our species and its social and natural ecologies. We are by many measures and accounts at a cross-roads in human and planetary history, perhaps the cross-roads that opens a road just a couple of hundred years into the industrial-technological global society to annihilation on various levels and scales. It may be that our fate as a species and a part of a global and planetary ecology doomed to annihilation on a scale of millennia is now sealed on a scale of centuries or even years. Our survival, and especially our survival with some decent quality of life distributed across the planet and its life forms, will depend on long-term thinking that is at once broad, deep, and wise. All of our civilizational tools will be needed in perhaps our last chance in the short run to “save” ourselves and our planet.

 

Ours is an era when the very idea of progress, under attack throughout the twentieth century as wars, holocausts, ecological disasters, and radioactive fallout crushed romantic and idealistic dreams of a better world, has given way to concerns about unintended consequences, precautionary principles, and technology assessments. How dangerous it sounds today to hear the echoes of the British Labor Party’s celebration in the mid-1960s of “the white heat of technology revolution.” And yet we must ask if we are still subject to the hypnotic seductions of the technological lottery. Imagine a group of distinguished scientists brought together to speculate on “the next hundred years,” and to speculate optimistically because the idea (read “ideology’) of science was by definition inclined to optimism. And consider that when just such an event was organized to celebrate the centennial of Joseph E. Seagram & Sons (purveyors of whiskey and related spirited drinks) in 1957, they all choked on their optimism as they offered their speculations. Distinguished speakers such as Nobelist geneticist Herman J. Muller, geochemist Harrison Brown, economist and college president John Weir, and

infamous rocket scientist Wernher Von Braun found it necessary to qualify their optimism. Muller said the future would be rosy if we could avoid war, dictatorship, overpopulation, or fanaticism; Brown began by saying “If we survive the next century; Weir began with “If man survives…” Braun, in the most bizarre opening sentence, said “I believe the intercontinental ballistic missile is actually merely a humble beginning of much greater things to come.”

 

No matter. It is beyond doubt that mathematics will play a key role in our efforts, successful or not, realistic or delusional, to realize civilization and progress in terms that even Rousseau might embrace. This is not meant to lock into mathematics as simply a means to quantifying problem formulation and problem solving across all arenas of human experience. It is meant mathematics as a humanistic mode of knowing. It is not only or at all to Plato’s view of the role of mathematics for his guardian-rulers we must turn to (keeping in mind that for all his resistance to poetry and metaphor, Plato was a master of both). The author wants rather to endorse something like Nietzsche’s notion of mathematics. That is, the rigor and refinement of mathematics must be brought into all of our endeavors but not because this is necessarily the best way to apprehend things (recall Nietzsche’s views on the limits of science). We should want to do this in order to become more aware of our relationships as human beings to the things in our world. Mathematics does not give us the most general or the ultimate form of knowledge but opens a path (only asymptotically, at best) to such a goal. This view of mathematics has been promoted for almost half a century by the humanistic mathematics movement. If civilization has a future, if progress can become at least a Lenskian possibility, a humanistic mathematics will have to take hold in our efforts to identify, specify, and solve the problems of planetary survival we face today.

 

Acknowledgments

 

The author wants to acknowledge my collaboration with and indebtedness to Randall Collins of the University of Pennsylvania with whom he first tackled the problem of mathematics and civilization in the early 1980s. Randy is an important inspiration for the perspective developed here. He also wants to thank his collaborators and colleagues in the sociology of mathematics over the many years in which he worked in this field: Wenda Bauchspies, Deborah Sloan, Monica Mesquita, Ubiratan D’Ambrosio, the late Dirk Struik, the late Joseph Needham, the late Leone Burton, Paul Ernest, David Bloor, Donald MacKenzie, Jean Paul Bendegem, and Jens Hoyrup,

 

Glossary

 

Abstraction                : Classically, a process by which higher level concepts are derived from the usage and classification of literal ("real" or "concrete") concepts, first principles, or other methods; used as a noun, a concept that acts as a super-categorical noun for all subordinate concepts, and connects any related concepts as a group, field, or category. Abstractions may be formed by reducing the information content of a concept or an observable phenomenon, typically to retain only information which is relevant for a particular purpose. For example, abstracting a

leather soccer ball to the more general idea of a ball retains only the information on general ball attributes and behavior, eliminating the other characteristics of that particular ball. In this chapter, abstractions are replaced by the term “generalization” and given a concrete sense. In this sense, what we normally refer to as abstractions are the concrete terms of an autonomous professional community of practice.

Arithmetic                  : From the Greek word ἀριθμός, arithmos “number”; historically, the oldest and most elementary branch of mathematics, used for everything from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of operations that combine numbers. In common usage, it refers to the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers. The term “higher arithmetic” as distinguished from elementary arithmetic, is used in professional mathematics to refer to more advanced practices related to number theory.

Civilization                 : A relatively high level of cultural and technological development; specifically associated with the development of writing and written records in the ancient world. Classically, “civilized” peoples contrasted themselves with “barbarians;” in modern times, “civilized” has been opposed to “primitive” or “savage.” Viewed in positive and optimistic terms, civilizations are characterized by complexity, hierarchies, dynamism, advanced levels of rationality, social progress, and a statist form of government. Viewed in negative pessimistic terms, notably by the philosopher Rousseau, civilization is conceived to be opposed to human nature.

Ethnomathematics : The study of the relationship between mathematics and culture. This research brings to light the mathematical and logical traditions of non-literate, but it in general the study of the mathematical and logical practices of specific cultures. It refers to a broad cluster of ideas ranging from distinct numerical and mathematical systems to multicultural mathematics education.

Formalists                  : Formalists, following the mathematical philosophy of one of the most influential leaders of this school, David Hilbert (1862- 1943), treat mathematics as a “game.” In its most extreme version, formalism claims that mathematics is not about anything but rather sets of rules of inference that can be applied to given “strings” (axioms) to generate new strings. You can, for example, use the “game” Euclidean geometry (which is viewed as some strings) to generate a new string such as the Pythagorean theorem. This is roughly the equivalent of proving the theorem in classical mathematics. See also intuitionists and logicists.

Foundationalism  : In epistemology (theories of knowledge) the idea that there are basic (foundational) beliefs, assumptions, etc. that are the grounds for beliefs in general. Basic beliefs justify other beliefs. Basic beliefs are said to be self-evident or self-justifying. Basic beliefs

can also derive their warrant from sensory experience. Anti- foundationalists have a problem with giving an uncontroversial or principled account of which beliefs are self-evident or indubitable and see foundationalism as a form of an unexamined a priori or even as a God surrogate. from the Ancient Greek

Geometry                   : From the Ancient Greek γεωμετρία; geo- "earth", -metria "measurement;” the branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, Euclid (3rd century BCE) put it into an axiomatic form,

Intuitionists                : The intuitionists approached mathematics as the result of constructive cognition. Humans make mathematics and logic as mental constructs by applying internally consistent methods.

L.E.J. Brouwer (1881-1996) is considered the father of modern intuitionism. Intuitionists view the truth of mathematics as a subjective claim. Brouwer rejected realist/Platonist ideas about the truth or reality of mathematical objects. Intuitionism substitutes constructability for abstract truth and is the provocation for the transition from the proof theory of truth to the model theory of truth in modern mathematics. See also logicists and formalists.

Logicists                     : The logicists believed that mathematics was an extension of logic and therefore that all or at least some of mathematics was reducible to logic. Richard Dedekind (1831-1916) and Gottlob Frege (1848-1925) are considered the founders of this school. The logicist culminated in the monumental Principia Mathematica (published in three volumes in 1910, 1912, and 1913, and in a second edition in 1927 by Cambridge University Press )by Bertrand Russell (1872-1970) and Alfred North Whitehead (1861-1947) . Logicism survives today in Zermelo- Frankael set theory or one of its variations (such as Zermelo- Frankael set theory with the axiom of choice, or ZFC). Most of mathematics is believed to be reducible to the logical foundations provided by the axioms of ZF, ZFC, and derivatives. See also formalism and intuitionism.

Mathematical worker

: (Also number worker). To be distinguished from “mathematician,” someone who works with numbers and “does” mathematics as a member of a specialized occupational or professional class. The mathematical worker works with numbers as a member of a relatively informal and unorganized specialty outside of a formal credentializing system. The two terms are sometimes used interchangeably in the text for convenience but readers alerted to the distinction should be able to identify which term is most appropriate from the context.

Mathematics              : In its standard dictionary sense, the science of numbers and their operations,  interrelations,  combinations,  generalizations,  and

 

 

 

 

 

 

 

 

 

Non-Euclidean Geometry

abstractions and of space configurations and their structure, measurement, transformations, and generalizations . Classically, “mathematics” refers to the two fundamental ways in which humans work with numbers, arithmetic and geometry. In the Platonic view, this number work reveals a transcendental realm of ideal numbers revealed to humans as they develop over time. More recently the Platonic view has been opposed by the idea that mathematics is a social practice; it is manufactured by humans on the earth out of the material and symbolic resources at their disposal in their local environments.

: Refers to the geometries generated by denying Euclid’s fifth postulate, the parallels postulate. Consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line. In Euclidean geometry the lines remain at a constant distance from each other even if extended to infinity, and are known as parallels. In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels. In elliptic geometry the lines "curve toward" each other and eventually intersect. Spherical geometry is the geometry of the two- dimensional surface of a sphere.

Objectivity                 : This term is traditionally associated with ideas about value- neutrality, observations and interpretations uncolored by personal biases, and methods that are systematically developed and applied and not based on the idiosyncrasies of individual scientists. In this context, an objective statement is based on replicable research processes, and beyond fundamental dispute. In the sociology of science, objectivity has been shown to be an achievement of social groups and thus situated in “cultures of objectivity” or “objectivity communities.

Progress                     : In general terms, the forward development and betterment of humankind over time. The idea of scientific and technological progress was fueled by the seventeenth century advances in science and literature by Galileo, Newton, Descartes, Molière, and Racine. The idea of social progress was added later. The wars, holocausts, and human and environmental degradations characteristic of the twentieth century contributed to the development of critiques of the very idea of progress.

Proof                           : As a noun, evidence or an argument establishing or helping to establish a fact or the truth of a statement; the cogency of evidence that compels rational acceptance of a truth or a fact; the process or an instance of establishing the validity of a statement especially by derivation from other statements in accordance with principles of reasoning. As in the case other traditionally taken for granted ideas in the sciences, proof has become the subject of empirically grounded interrogations by sociologists of science who now refer proof to proof communities, and demonstrate that

 

Social Constructionism

 

 

 

 

Sociology of knowledge

what counts as a proof varies across time, space, and cultures.

: The fundamental theorem or central dogma of sociology, it refers to the fact that the only means humans have available for invention and discovery are their interactions with each other using the material resources of their environments and the symbolic resources of their cultures. It is fully compatible with a critical realism that assumes a “reality out there” that can only be known through the lenses of society and culture.

: The study of the relationship between human thought and the social, cultural, and historical contexts within which it arises, and of the effects prevailing ideas have on societies; deals with broad fundamental questions about the extent and limits of social influences on individual's lives and the social-cultural basics of our knowledge about the world When applied to the study of science in particular, an important specialty in sociology known as the sociology of science.

Spengler thesis   : Oswald Spengler argued that that there is no “Mathematik” but only “mathematics.” Mathematical forms are related to particular cultures. The weak form of the thesis is that there are as many mathematics as there are cultures. The strong form of the thesis is that mathematics is socially constructed.

Technoscience            : Variously, the idea the science and technology are intricately interrelated; the separation of “science” and “technology” may be analytically useful under some conditions but the idea that the two ideas are conceptually distinct is more ideological than substantive. This idea gives rise to such hybrid concepts as technocultural and technosocial.

Transcendental (transcendent; transcendence)

: Transcendence, transcendent, and transcendental are words that refer to an object (or a property of an object) as being comparatively beyond that of other objects. Such objects (or properties) transcend other objects (or properties) in some way. In philosophy, transcendence refers to climbing or going beyond some philosophical concept or limit In nineteenth century American, transcendentalism was developed within a religio- philosophical movement that claimed there is an ideal spiritual state that 'transcends' the physical and empirical. Following Restivo (2011 170) The Transcendental Fallacy (also known as the theologian’s fallacy) is that there is a world or that there are worlds beyond our own – transcendental worlds, supernatural worlds, worlds of souls, spirits and ghosts, gods, devils, and angels, heavens and hells. There are no such worlds. They are symbolic of social categories and heavens and hells. There are no such worlds. They are symbolic of social categories and classifications in our earthly societies and cultures. There is nothing beyond our material, organic, and social world. Death is final; there is no soul, there is no life after death. It is also possible that the so-called “many worlds interpretation” in quantum mechanics is contaminated by this fallacy as the result

of mathegrammatical illusions. The world, the universe, may be more complex than we can know or imagine, but that complexity does not include transcendental or supernatural features. Stated positively, this is Durkheim’s Law

 

Bibliography

 

Aleksandrov, A.D., A.N. Kolmogorov and M.A. Lavrent’ev (eds.) (1969), Mathematics: It’s Content, Methods, and Meanings, Cambridge, MA: MIT Press (orig. publ. in Russian in 1956). [A product of some the greatest mathematicians of the twentieth century, this is a readable review of the basic fields of mathematics. The authors are aware of the fact that mathematics is a human and a social creation and they do not mystify it or present it as a revelation of Platonic discourses.]

 

Almond, G., M. Chodorow, and R.H. Pearce (1985), Progress and its Discontents, Berkeley: University of California Press. [The twentieth century has not been kind to the idea of progress. Intellectuals across the spectrum of the disciplines are no longer that there are any viable grounds for the claim that science and technology have improved our lives morally or materially. The term “progress” is no longer meaningful empirically but is now about our aspirations and a “compelling obligation.”]

 

Ascher, M. (2004), Mathematics Elsewhere: An Exploration of Ideas Across Cultures, Princeton: Princeton University Press. [An introduction to mathematical ideas of peoples from a variety of small- scale and traditional cultures that challenges our conception of what mathematics is. Traditional cultures have mathematical ideas that are far more substantial and sophisticated than has been traditionally acknowledged.]

 

Baber, Z. (1996), The Science of Empire: Scientific Knowledges, Civilization, and Colonial Rule in India. Albany NY: SUNY Press. [Baber analyzes the reciprocal interactions between science technology and society in India from antiquity to modern times. The author analyzes institutional factors, including pre- colonial trading circuits, in the transfer of science and technology from India to other civilizational centers. He also explains the role of modern science and technology as factors in consolidating British rule in India.]

 

Ball, W. W. R. (1960), A Short Account of the History of Mathematics. New York: Dover. Unabridged and altered version of the fourth edition published in 1908. [A classic contribution to the history of mathematics.]

 

Bell, E. T. (1937/1986) Men of Mathematics, New York: [Simon and Schuster Touchstone Books. Bell is a mathematician and lacks the rigorous standards of the professional historians of science. But this book, for all of its idealism and focus on the dramatic, has been very inspiring for young people interested in mathematics and offers some insights into what it’s like to be a mathematician.]

 

Bellos, A. (2010), Here’s Looking at Euclid. New York: The Free Press. [This is a book designed to inspire and surprise. Bellos, who studied mathematics and philosophy at Oxford, takes the reader on a tour of mathematics based on his skills as a writer, traveler, and interviewer.]

 

Benesch, W. (1992), “Comparative Logics and the Comparative Study of Civilizations,” Comparative Civilizations Review, No. 27, Fall 1992, 88-105. [Benesch views “civilization as a state of mind.” Drawing heavily on Indian civilizational features, and especially the culture of the Jain for comparative materials, Benesch identifies four primary logics: object logics, subject logics, situational logics, and aspect logics. Benesch’s typology can be expanded to encompass propositional and predicate logic, first and higher order logics, modal logics, temporal logics, probability logics, multi-valued logics, negation logics, and quantum logic. Using another classification scheme we get: relevant logic (defined by some mathematicians as classical logic practiced with “good taste;” constructive logic (known in some of the literature by the less descriptive term "intuitionistic logic;" fuzzy logic, and comparative logic. The way to understand the proliferation of logics is to recognize that what we understand in everyday life and in a good deal of everyday science as the one universal logic is that set of rules and rules of inference that

reflect the workings of the macro-level physical world. In the worlds of the very small and very large, and the very fast and very slow, and across different levels of reality, different logics play out.]

 

Benveniste, É. (1966), Civilisation. “Contribution à l'histoire du mot” (Civilisation. Contribution to the history of the word), 1954, published in Problèmes de linguistique générale, Paris: Editions Gallimard ,

pp. 336-345 (translated by Mary Elizabeth Meek as Problems in general linguistics, 2 vols. 1971). [The author, a semiotician, traces the development of the concept “civilization” from its first occurrence in Adam Ferguson’s 1767 treatise on the history of civil society.]

 

Boos, B. and M. Niss, eds. (1979), Mathematics and the Real World, Boston: Birkhauser. [The editors recognize that to the extent that “pure” mathematics has gone too far and increasingly fails to interact with “real world” sciences and engineering disciplines, to that extent has it diminished its practical effectiveness. Mathematics that increasingly turns in on itself increasingly negates Wigner’s (1960) “unreasonable effectiveness” claim.]

 

Boutroux, P. (1919), L’ideal scientifique mathematiciens dans l’antiquité et dans les temps modernes. , Paris: Presses Universitaire. [Boutroux (1880-1922) was the son of the famous French philosopher Émile Boutroux and Aline Catherine Eugénie Boutroux, He was related to the statesman Raymond Poincaré¸the physicist Lucien Poincaré, and noted mathematician Henri Poincaré. Boutroux identifies three stages in the history of mathematics: the aesthetic, contemplate mathematics of the Greeks, the synthetic conception of Cartesian algebra, and what he saw as the incoherence of the mathematics of his time. There is a progressive theme in this history, an improving understanding of a higher reality. His didactic goals were to assert that progress in the history of science was a function of interactions between all the sciences, and to argue that problems should dictate approaches to solutions.]

 

Boyer, C. (1968), A History of Mathematics, Wiley, New York. [The novelist David F. Wallace called Boyer “the Gibbon of math history.” This book is to the history of mathematics what Paul Samuelson’s book on economics is to the study of economics. The reference of choice in the history of mathematics is now in its 3^rd edition (2011, updated by Uta Merzbach).]

 

Cajori, F. (1894), A History of Mathematics. New York: Macmillan & Co.. [Cajori’s (1859-1930) book was first published in 1893 and has gone through several editions. It is a reliable, readable treatment of the history of mathematics from antiquity to the early 1900s.]

 

Cajori, F. (1929/1993), A History of Mathematical Notations. Dover, New York. [This book is still today considered with good reason to be unsurpassed. Demonstrates how notation changes with changes in the shape of civilization. Here we discover, for example, how the Greeks, Romans, and Hebrews counted.]

 

Cantor, M. (1907), Vorlesungen uber die Geschichte del' Mathematik,. [Anastatischer Neudruck, Leipzig. Moritz Cantor (1829-1920) was an historian of mathematics who studied under some the giants of mathematics, including notably Gauss, and this book in four volumes is not just comprehensive but is considered a founding document in the history of mathematics as a critical, methodologically sound field of study.]

 

Clark, A. (2010), Supersizing the Mind: Embodiment, Action, and Cognitive Extension, New York: Oxford University Press. [Clark is yet another of a small but increasing number of authors who are focusing in on the problems and paradoxes that arise when we make the brain the source of our thinking and consciousness, and the source of our morals and our beliefs. In fact it is becoming increasingly clear that cognition is a complex result of tangled networks that criss-cross the boundaries of brain, body, and world. Mind is not bound by the brain; consciousness, as Nietzsche already intuited, is a network of relationships.]

 

Collins, R. (1998), The Sociology of Philosophies. Cambridge, MA: Harvard University Press. [A comprehensive social history of world philosophy in the context of global intellectual life. Collins traces the development of philosophical thought in China, Japan, India, ancient Greece, the medieval Islamic and Jewish world, medieval Christendom, and modern Europe. The result is an empirically grounded theory of ideas as the product of social networks. One of the most important contributions to modern sociology by a leading theorist.]

Collins, R. and S. Restivo. (1983), "Robber Barons and Politicians in Mathematics," The Canadian Journal of Sociology 8, 2 (Spring 1983), pp. 199-227. [Two of the leading contributors to the sociology of science offer an alternative to Kuhn’s theory of scientific change. Major scandals in mathematics are shown to be associated with shifts in the organizational structure of mathematical work. The article demonstrates the power of conflict theory as an explanatory approach in the sociology of mathematics.] Courant, R. and H. Robbins. (1996), What is Mathematics? New York: Oxford University Press (rev. by Ian Stewart, orig. publ. 1906). [A classic contribution to our understanding of mathematics as an enterprise of practice rather than philosophy.]

 

D’Ambrosio, U. (2006), Ethnomathematics, Rotterdam: Sense Publishers. An introduction to the concept of ethnomathematics by the founding father of the field.

 

Davis, P.J. and R. Hersh. (1981), The Mathematical Experience. Boston: Birkhauser. [Two mathematicians bring sociological, psychological, and philosophical perspectives to this popular exposition of the nature and relevance of mathematics.]

 

Dostoevsky, F. (1864), "Notes from the Underground," pp. 107-2-10 in The Best Short Stories of Dostoevsky. New York, n.d. (orig. publ, in Russian). [This classic piece of literature can be read as a contribution to the sociology of mathematics; the author’s discussion of 2+2=4 and 2+2=5 shows how arithmetic can be used to symbolize ideologies. Should be compared with how these equalities are treated by George Orwell in 1984.]

 

Dorrie, H. (1965), One Hundred Great Problems of Elementary Mathematics,Their History and Solution. New York: Dover. [This book was originally published in 1932 under the title Triumph der Mathematik. Fascinating well selected problems but not really “elementary.” An intriguing 2000 year survey but not for those without a strong background in mathematics.]

 

Durkheim, E. (912/1995), The Elementary Forms of Religious Life. New York: The Free Press 1 (trans. Karen Fields). [One of the most important studies in the history of scholarship. Durkheim crystallizes the idea of who and what God is and explains the function of religion in society. Criticisms abound, but they do not change the world transforming potential of this book.]

 

Eglash, R. (1999), African Fractals, Piscataway, NJ: Rutgers University Press. [Patterns across cultures are characterized by specific design themes. In Europe and America, cities are often laid out in a grid pattern of straight streets and right-angle corners. In contrast, traditional African settlements tend to use fractal structures--circles of circles of circular dwellings, rectangular walls enclosing ever-smaller rectangles, and streets in which broad avenues branch down to tiny footpaths with striking geometric repetition. These indigenous fractals are not limited to architecture; their recursive patterns echo throughout many disparate African designs and knowledge systems.]

 

Gasking, D. (1956), "Mathematics and the World," pp. 1708-1722 in J.R. Newman (ed.), The World of Mathematics. New York: Simon & Schuster. [Originally published in 1940, this is an exemplar of conventionalist philosophy of mathematics. Gasking views mathematics as true by virtue of linguistic conventions. Mathematics on this view is not empirical in the way, for example, that Wittgenstein argues it is or in the way it is understood today in social studies of mathematics.]

 

Geertz, C. (1983), “Art as a Cultural System,” pp. 94-120 in C. Geertz, Local Knowledge, New York: Basic Books. [Art, like speech, myth and other cultural systems are situated, ideationally connected to society; in a word cultural systems are socially constructed.]

 

Hersh, R. (1999), What is Mathematics, Really? New York: Oxford University Press. [Mathematics for the modern reader by a mathematician who appreciates the new sociology of mathematics developed by Bloor on the one hand and Restivo on the other.]

 

Horton, R. (1997), Patterns of Thought in Africa and the West: Essays on Magic, Science and Religion, Cambridge; Cambridge University Press. [The distinguished anthropologist and philosopher was at the center of debates that crystallized in the 1950s and 1960s concerning the nature of and relationship between “primitive” (read, primarily, African) and “advanced” (read “the West”) systems of knowledge

and belief. In this collection of his essays, Horton addresses the debates by exploring African beliefs, rituals, and cosmologies in the context of open and closed systems thinking. His objective is to demonstrate the kinship between primitive and modern thought and between science and religion more generally. Among the most important provocations for this and related debates are studies of Zande (n. Azande) logic. Were they “pre-logical,” as some claimed (e.g., in my view arguably Levy-Bruhl) or perfectly coherent in context (as Evans-Pritchard argued)? In the early years of the science studies movement David Bloor took up this question and argued for a cultural relativism that provoked new interest in this topic. This literature is relevant to my topic but requires more direct attention than I can give it here.]

 

Kavolis, V. (1985), “Civilizational Analysis as a Sociology of Culture,” Sociological Theory 85, 31, Spring 1985, 29-38. [The comparative civilizational analysis of any phenomenon, Kavolis argues, requires that we spell out the structure of each level of sociocultural organization and the connections and resources linking all the levels that bear on the particular phenomenon in a particular civilization. We don’t require an elaborate analytical apparatus but it helps to carry an awareness of such an apparatus in the study of a particular problem in terms of the relations between general civilizational structures, processes, and issues. Civilizational analysis is about the distinctive varieties of and changes in human experience, and the universe of symbolic designs in social behavior.]

 

Kline, M. (1962), Mathematics: A Cultural Approach, Reading, Massachusetts: Addison-Wesley. [A good introduction, even where dated factually and ideologically, to mathematics in relation to the humanities and the search for truth.]

 

Kramer, E. E. (1970), The Nature and Growth of Modern Mathematics. New York: Hawthorne Books. [Kramer discusses the lives and contributions of prominent mathematicians from Pythagoras and Newton to the modern period. Mathematical concepts such as binary operations, point-set topology, post-relativity geometries, optimization and decision processes ergodic theorems epsilon-delta arithmetization, and integral equations are discussed with admirable clarity.]

 

Lenski, G. (1974), Human Societies. New York: McGraw-Hill. [A cultural ecology of the evolution of societies. This is a major textbook contribution to macrosociology, now in its 11^th edition from Paradigm Publishers in Boulder, Colorado, 2008 co-authored by P. Nolan.]

 

Mannheim, K. (1936), Ideology and Utopia. Eugene, Oregon: Harvest Publishers. [This is a complex treatise in the context of Mannheim’s legacy in relationship to European and America sociology. It is a founding document in the emergence of sociology but especially of the sociology of knowledge. In terms of its relationship to the substance of this essay, the significance of this book is that while situating knowledge in its social, cultural, and historical contexts, Mannheim exempts the formal science from his analysis. There cannot be, he claims in these pages, a sociology of 2+2=4. This idea carried into the emergence of the sociology of science in the 1930s and was not seriously challenged until the birth of the science studies movement in the late 1960s.]

 

McClain, E.G. (1976), The Myth of Invariance: The Origins of the Gods, Mathematics and Music from the Rg Veda to Plato. York Beach, ME: Nicolas-Hays, Inc. [McClain’s thesis is that in the ancient civilizations music was a science that bridged the gap between the everyday world and the divine. The invariance of music contrasted sharply with the variability of the everyday world. Music expresses and motivates mathematics. Music as a science is revealed by studying the mathematical relationships between musical notes. In this context music is properly understood as an expression of and the motive for mathematical study. The "key" to unlocking this science comes from a study of the mathematical relationships between various musical notes. The Pythagorean notion of number was in fact more general and more widespread and an essential feature of the very idea of culture and civilization. The presentation is controversial in some details but we are learning more and more about the centrality of musicality in humans and this book speaks to that idea.]

 

Merton, R.K. (1968), Social Theory and Social Structure, enlarged ed., New York: The Free Press. [Merton founded and dominated the sociology of science with his students from the late 1930s to the late 1960s. The Mertonian paradigm, consonant with Mannheim’s sociology of knowledge, focused on the

social system of science – for example, norms, values, the reward system, stratification in science, age- grading – but exempted scientific knowledge per se from sociological scrutiny.]

 

Merton, R. K. (1961), , "Singletons and Multiples in Scientific Discovery: a Chapter in the Sociology of Science," Proceedings of the American Philosophical Society, 105: 470–86. Reprinted in R. K. Merton, The Sociology of Science: Theoretical and Empirical Investigations, Chicago, University of Chicago Press, 1973, pp. 343–70.

 

Merton, R. K. (1973), "Resistance to the Systematic Study of Multiple Discoveries in Science," European Journal of Sociology, 4:237–82, 1963. Reprinted in R.K. Merton, The Sociology of Science: Theoretical and Empirical Investigations, Chicago, University of Chicago Press , pp. 371–82. [This and the following paper introduce Merton’s idea that all discoveries are in principle multiples. These papers are milestones in the Merton corpus.]

 

Merton, R.K. (1958), “The Matthew Effect in Science,” Science, 159(3810): 56-63, January 5, 1968. [Widely cited in the sociology of science, the Matthew effect, or the principle of accumulated advantage, refers to situations in which the rich get richer and the poor get poor. Merton named the effect after Matthew 25: 29 (NRSV Bible). In science as in other activities, power and economic or social capital can be leveraged to gain additional power and capital. This is one the causes of mis-eponymy along with historical amnesia and random and deliberate acts of misattribution.]

 

Mesquita,M., S. Restivo and U. D'Ambrosio (2011), Asphalt Children and City Streets: A Life, a City, and a Case Study of History, Culture, and Ethnomathematics in São Paulo, Rotterdam: Sense Publishers. [An innovative plural voiced ethnography and history of street children that attends to how mathematics plays into their survival strategies.]

 

Morris-Suzuki, T. (1994), The Technological Transformation of Japan: From the Seventeenth to the Twenty-first Century, Cambridge: Cambridge University Press. [Japan did not miraculously leap into the technological forefront of twentieth century societies. Its rise to superpower status, as this book makes clear, is rooted in its history. This is the first general English language history of technology in modern Japan. One of the significant features of this book is its consideration of the social costs of rapid changes in technology.]

 

Nasr, S. H. (2007), Science and Civilization in Islam. Chicago: Kazi Publications. [The first one volume English language book on Islamic science from the Muslim perspective. Hossein explains the place of science in Muslim culture as he unfolds its content and spirit.]

 

Needham, J. (1959), Science and Civilization in China. Volume III: Mathematics and the Sciences of the Heavens and the Earth. Cambridge: Cambridge University Press. [One of several volumes in the monumental study that uncovered the hidden history of science and technology in China. Needham documented that not only did China have a history in science and technology, it was the leading civilizational center of science and technology in the world between the early Christian period in Palestine and 1500 CE. The explanatory framework is Marxist cultural ecology.]

 

Neugebauer, O. (1952), The Exact Sciences in Antiquity. Princeton: Princeton University Press. [It took a long time for historians of science to overcome the ideology of the once and always Greek miracle. Neugebauer contributes to the demise of this myth in this non-technical discussion of the influence of Egyptian and Babylonian mathematics and astronomy on the Hellenistic world. An early look into the sophistication of ancient Babylonian mathematics.]

 

Noë, A. (2010), Out of Our Heads: Why You Are Not Your Brain, and Other Lessons from the Biology of Consciousness, New York: Hill and Wang. [Makes an important contribution to getting away from classical ideas about the primacy of the brain in consciousness. His approach is radically social but in a strange way that makes biology, rather than sociology, the science of the social. But just because of this twisted logic, he furthers the interdisciplinary agenda of figuring out a non-reductionist way to link biology and society.]

Polster, B. and M. Ross, (2011), “Pythagoras’s Theorem ain’t Pythagoras’s,” h t t p : / / education . theage . co m . au / cmspage . php ? I ntid = 147 & intversion = 79, March 7, 2011.

 

Resnikoff, H.L. and R.O. Wells, Jr. (2011), Mathematics and Civilization. New York: Dover (paperback). [This book was originally published by Holt, Rinehart, and Winston in 1973 and later published by Dover with supplemental materials. Still a good general introduction to the reciprocal relations between mathematics and human culture with an emphasis on the technical mathematics. No great demands are made on the mathematical aptitudes of readers, and the more sophisticated reader will find some of the treatment, especially on the calculus, technically deficient.]

 

Restivo, S. (1979), "Joseph Needham and the Comparative Sociology of Chinese and Modern Science: A Critical Perspective," pp. 25-51 in R.A. Jones and H. Kuklick (eds.), Research in the Sociology of Knowledge, Sciences, and Art, Vol. II. JAI Press, Greenwich. [The most extensive journal length critical survey of Needham’s monumental history. Supports and extends his sociological materialism based on a human ecological perspective.]

 

Restivo, S. (1981), "Mathematics and the Limits of Sociology of Knowledge," Social Science Information, V. 20, 4/5: 679-701. [The new sociology of science associated with the science studies movement that emerged in the late 1960s challenged the status of mathematics as the arbiter of the limits of the sociology of science and knowledge. This is one of the early examples of the challenge by one of the founders of the modern sociology of mathematics.]

 

Restivo, S. (2001/1992), Mathematics in Society and History. New York. [This is the first book devoted completely to the sociology of mathematics as a subfield of sociology. The author reviews precursors, Spengler’s thesis on numbers and culture, mathematical traditions in different civilizations, and develops an original conception of the sociology of pure mathematics.]

 

Restivo, S. (1993), “The Social Life of Mathematics,” pp. 247-278 in S. Restivo, J.P. van Bendegem, and

R. Fischer (eds.), Math Worlds: Philosophical and Social Studies of Mathematics and Mathematics Education, Albany, NY: SUNY Press. [A collection of essays, including a set of papers by German scholars previously unavailable in English, focused on recent developments in the study and teaching of mathematics. The authors are all guided by the idea that mathematical knowledge must be grounded in and reflect the realities of mathematical practice.]

 

Restivo, S. (1994), Science, Society, and Values: Toward a Sociology of Objectivity. Bethlehem PA: Lehigh University Press. [This book introduces Restivo’s main contributions to the sociology of science between 1966 and the early 1990s. Based on his work in the ethnography of science, the history of science in China and the West, his social problems approach to understanding modern science, and other contributions, he develops a sociological perspective on objectivity.]

 

Restivo, S. (2007), “Mathematics,” Monza: Polimetrica. June, 2007 The Language of Science (ISSN 1971-1352). [An overview of the sociology of mathematics written for an online encyclopedia.]

 

Restivo, S. (2011), Red, Black, and Objective: Science, Sociology, and Anarchism, Surrey: Ashgate Publishers. [This book explores the implications of the science studies movement for science and society in the context of an anarchist tradition. The particular tradition the author has in mind here makes anarchism one of the sociological sciences. Here he follows Peter Kropotkin. The book is grounded in the empirical studies carried out over the last forty years by researchers in science studies (and more broadly science and technology studies). The author’s perspective is at once empirical, normative, and policy- oriented.]

 

Restivo, S. and H. Karp (1974), "Ecological Factors in the Emergence of Modern Science," pp. 123-142 in S. Restivo and C.K. Vanderpool (eds.), Comparative Studies in Science and Society. Columbus, OH: C. Merrill. [The authors theoretically link organizational and institutional structures to their ecological contexts and apply this theory to the problem of why modern science emerged in the West and not in China.]

Restivo, S. and R. Collins (1982), “Mathematics and Civilization,” The Centennial Review Vol. XXVI, No. 3 (Summer 1982), pp. 277-301. [An introduction to the comparative historical sociology of mathematics that modifies Spengler’s ideas on numbers and culture in the context of sociological theory.]

 

Restivo, S. and J. Croissant (2008), “Social Constructionism in Science and Technology Studies,” pp. 213-229 in J.A. Holstein & J.F. Gubrium, eds., Handbook of Constructionist Research, New York: Guilford. [The authors aim is to clarify the widespread misconceptions, misapplications, and misconstruals of this term which they identify as the fundamental theorem of sociology. They are at pains to argue that the term does not imply or entail any form of relativism; it is compatible with a realistic sociology that recognizes objectivity and truth as real. They are real however in an institutional sense. This view, as Durkheim pointed out more than one hundred years ago, is consistent with the idea that there is a reality “outside of us,” but we do not have access to a ding an sich.]

 

Reyna, S.P. (2007), Connections: Brain, Mind and Culture in Social Anthropology, New York: Taylor and Francis. [Reyna returns to the Boasian (Franz Boas) roots of anthropology to creatively construct a new paradigm for connecting the biological and cultural domains without submitting to the Sirens of reductionism.]

 

Rosental, C. (2008), Weaving Self-Evidence: A Sociology of Logic, Princeton: Princeton University Press. [Rosental traces the history of a theorem in the foundations of fuzzy logic to demonstrate the inherently social nature of logic. He describes the process by which logical propositions are produced, disseminated, and established as truths.]

 

Roszak, T. (1995), The Making of a Counter-Culture: Reflections on the Technocratic Society and its Youthful Opposition, with a new introduction, Berkeley; University of California Press (orig. publ. 1969). [Published in the middle of the 1960s sociocultural revolution, this book spoke directly to those who would become “the children of the 60s” while simultaneously bewildering their parents. Roszak coined the term “counter-culture” and damned the technocracy that was at the heart of the problems the protesters were angry about. A literate effort to explain the disaffection of young people and the young at heart during this tumultuous period.]

 

Sahlins, M. and E. Service, eds. (1960), Evolution and Culture, Ann Arbor: University of Michigan Press (co-authored by T.G. Harding, D. Kaplan, M.D. Sahlins, and E.R. Service). [In the hands of these authors, evolutionary anthropology becomes a predictive tool that can be applied to theorizing the future of human societies. Current events on the world scene, including the political and economic rise of China and the troubled status of the United States would not have surprised these anthropologists. This is a classic and still relevant contribution to our understanding of culture and cultural change writ large.]

 

Schechter, Eric (2005), Classical and Nonclassical Logics: An Introduction to the Mathematics of Propositions, Princeton: Princeton University Press. [Classical logic--the logic crystallized in the early twentieth century by Frege, Russell, and others--is computationally the simplest of the major logics, and adequate for the needs of most mathematicians. But it is just one of the many kinds of reasoning in everyday thought. Schechter introduces classical logic alongside constructive, relevant, comparative, and other nonclassical logics. This is the first textbook to make this subject accessible to beginners.]

 

Sedlacek, T. (2011), Economics of Good and Evil, New York, Oxford University Press. [Sedlacek understands economics as a social, cultural, and historical phenomenon. It is a product of our civilization not a pure, value free science. In this sense, Sedlacek lines up with contemporary students of the sociology of science and mathematics. Economics, at the end of the day, is about “good and evil.” In viewing economics as a moral enterprise he reminds us that the author of The Wealth of Nations, Adam Smith, is also the author of The Theory of Moral Sentiments.]

 

Selin, H. (ed.) (2001), Mathematics Across Cultures, New York: Springer. [A survey of Islamic, Chinese, Native American, Aboriginal Australian, Inca, Egyptian, and African mathematics, with essays on rationality, logic and mathematics. Shows how science and math practice is situated in its cultural context.]

Smith, D. E. (1958), History of Mathematics. New York: Dover. [A classical two volume introduction to the history of mathematics from ancient Egypt to modern times, very lucid survey including biographical notes and chronology.]

 

Sohn-Rethel, A. (1975), "Science as Alienated Consciousness." Radical Science Journal, Nos. 2/3: 65-

101. [Working from inside a sophisticated Marxist paradigm, Sohn-Rethel discusses science as an alienated and alienating form of knowledge and knowledge building. He is known for linking Kant and Marx in his work on epistemology.]

 

Spengler, O. (1926), The Decline of the West. New York: A. Knopf. [Spengler’s readers append adjectives like “audacious,” “profound,” “magnificent,” “exciting,” and “dazzlingly” to this book which flaws and all is one of the most amazing efforts in human history to capture humanity’s march through space and time. It’s relevance for the topic at hand is that Spengler’s theses are grounded in a radically cultural understanding of the relationship between culture and mathematics.]

 

Stigler, Stephen S. (1980), ‘Stigler’s Law of Eponymy’, Trans. N. Y. Acad. Sci. (2) 39, 147–157.

 

Leon Stover (1974), The Cultural Ecology of Chinese Civilization. New York: Signet. [An innovative interpretation of peasants and elites in what Stover refers to as a “once and always Bronze Age culture.” A paradigm for understanding the nature of Chinese culture and the continuities between the age of the dynasties and the modern era from Sun Yat Sen and Chiang Kai Shek to Maoism and modernism.]

 

Struik, D. (1967), A Concise History of Mathematics,. New York: Dover Publications. [The fourth revised edition of this classic was published by Dover in 1987. Struik, a Dutch mathematician and Marxist theory, was a professor at MIT for most of his professional career and to my knowledge the first person to identify the sociology of mathematics as a field of study. This book is a very readable introduction to the history of mathematics, concise as advertised but with a lot of substance poured into the books roughly 230 pages. Struik covers the period from the ancient world to the early twentieth century.]

 

Sugimoto, Masayoshi and D.L. Swain, (1978), Science & Culture in Traditional Japan. Cambridge MA: MIT Press. [Between 600 and 1854 CE, Japan was impacted by a first and second Chinese cultural wave and the first Western Cultural Wave in the nineteenth century. The authors focus on how these cultural waves set the stage for the development of an indigenous science and technology.]

 

Verran, Helen. (1992), Science and an African Logic. Chicago: The University of Chicago Press. [An empirical study that supports the idea of mathematics and logics as culturally situated. Quantity is not always absolute (as in 2=2=4) but sometimes relational, as in Yoruba. Verran’s experience and research as a teacher in Nigeria is the basis for this important contribution to the sociology of mathematics and ethnomathematics.]

 

Wigner, E. (1960), “On the Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Communications on Pure and Applied Mathematics , 13: 1–14. [A classic paper in defense of the idea that the mathematical structure of a physics theory often points the way to further advances in that theory and even to empirical predictions. It is basically an argument in support of “pure” mathematics.]

 

Wright, Ronald (2004), A Short History of Progress, Philadelphia: Da Capo Press. [Looking over the long history of humanity, Wright sees not the unfolding a linear evolution of progress but rather a series of “progress traps.” He reveals a history of “progress and disasters” that should serve as a warning to humanity and especially to those people who assume that progress is an inevitable and positive manifestation of human exceptionalism.]

 

Zaslavsky, C. (1999), Africa Counts: Number and Pattern in African Cultures, 3^rd ed. Chicago: Lawrence Hill Books (orig. publ. 1973 by Prindle, Weber, and Schmidt). [This is one of the earliest efforts to document the experience of mathematics in a non-Western culture and to view it in a positive civilizational perspective.]

 

Zeleza, Paul Tiyambe and Ibulaimu Kakoma (2005), Science and Technology in Africa. Trenton NJ: Africa World Press. [The authors deal with scientific and technology literacy, production, and

consumption in modern Africa. The focus is on developments in information technology and biotechnology in the context of The Knowledge Society in a globalizing context.]

 

Biographical Sketch

 

Dr. Sal Restivo is widely recognized as one of the founders of the field of Science and Technology Studies (STS), a pioneer in ethnographic studies of science, a founder of the modern sociology of mathematics, a contributor to public sociology and a prominent figure in the radical science movement of the 1960s. Dr. Restivo was Professor of Sociology, Science Studies, and Information Technology in the Department of Science and Technology Studies, at Rensselaer Polytechnic Institute in Troy, New York until his retirement in June 2012. He is Special Lecture Professor in STS at Northeastern University in Shenyang, China; a former Special Professor of Mathematics, Education, and Society at Nottingham University in Great Britain; and a former Hixon/Riggs Professor of Science, Technology, and Society at Harvey Mudd College. In 2012, he was a Senior Postdoctoral Fellow at the University of Ghent in Belgium. He is a founding member (1975) of and a former president (1994/95) of the Society for Social Studies of Science.