MATHEMATICAL NATURALISM: AN ANTHROPOLOGICAL PERSPECTIVE

The Southern Journal of Philosophy (1989) Volume XXVII, No. 3

MATHEMATICAL NATURALISM: AN ANTHROPOLOGICAL PERSPECTIVE Stephen Pollard Robert Bates Graber Northeast Missouri State University

Dazzled by the rich particularity of individual cultures, anthropology in the early 20th century had bogged down in “a programmatic avoidance of theoretical syntheses.”’ It was the cultural evolutionist Leslie A. White (1900-1975) who, more than anyone else, called anthropology back to the synoptic concerns that had quickened the discipline in the late 19th century. In this paper, we shall argue that one of White’s lesser known essays makes a timely contribution toward a viable ontology for Philip Kitcher’s “naturalistic” philosophy of mathematics. We shall also briefly sketch one contribution which anthropology (especially anthropology of a Whitean bent) might make to Kitcher’s analysis of rationality.

1. Naturalistic Metaphysics

In this section, we consider the views of Kitcher and White on a fundamental question of metaphysics: what conditions are necessary for the existence of mathematical objects? The following table will help to organize our discussion. We let the variable ‘t’ range over temporal regions.

I. Objects of Experience A. Non-modal accounts (being requires being perceived)

1. b, t (x exists at t - 3 y (y perceives x at t)) B. Modal accounts (being requires being perceivable)

1. b, t (x exists at t - 3 y 0 (y perceives x at t)) 2. Vx, t (x exists at t - 0 3 y (y perceives x at t))

11. Objects of Thought A. Non-modal accounts (being requires being thought)

Stephen Pollard is Assistant Professor of Philosophy at Northeast Missouri State University. His publications deal mainly with formal theories of properties and the philosophy of mathematics.

Robert Bates Graber is Associate Professor of Anthropology & Sociology at Northeast Missouri State University. His publications deal mainly with psychoanalytic and demographic anthropology.

427

1. Vx, t (x exists at t - 3 y (y thinks of x at t)) B. Modal accounts (being requires being thinkable)

1. Vx, t (x exists at t - 3 y 0 (y thinks of x at t)) 2. Vx, t (x exists at t - 0 3 y (y thinks of x at t))

We distinguish between objects of experience and objects of thought. We recognize three basic types of account of the conditions necessary for the existence of objects of each sort. The options listed above are really account-schemes, for they involve notions of perception, thought, and possibility which demand specification. Doctrines which satisfy the scheme IA1 are forms of Berkeleian idealism: existing at t requires being perceived at t. Doctrines which satisfy IB1 are Kantian in character: existing at t requires being potentially perceived at t by an actual subject; so, in the absence of actual beings to serve as potential perceivers, no objects of experience would exist. Doctrines which satisfy IB2 approximate common sense realism: existing at t requires being potentially perceived at t by a subject which may itself be merely potential; so, in the absence of actual beings to serve as potential perceivers, objects of experience could still exist-as long as the existence of perceivers remained compatible with, say, the laws of nature.

Views concerning objects of thought can also be fitted into three basic categories. According to non-modal accounts, an object of thought exists only when someone is actually thinking of it (Cf. IIAl.). According to modal accounts, an object of thought exists only when it is available to be thought. We might say that something is available to be thought only if there actually are beings capable of activities which count as thinking of that thing (Cf. IIB1.). On the other hand, we might say that something is available to be thought if it is possible for there to be beings who engage in activities which count as thinking of that thing (Cf. IIB2.). According to views of this last sort, the existence of objects of thought will not depend on the vicissitudes of actual thinkers-all that matters is that suitable thinkers remain possible.

We shall soon see that Leslie White proposes a doctrine of the form IIB1. First, however, we turn to the metaphysical doctrines of Philip Kitcher-for we believe that the virtues of White’s approach will appear more clearly after a critical appraisal of Kitcher’s views. Our consideration of Kitcher will also help us to locate a contemporary niche for White’s rather dated work on mathematical reality.

In his book The Nature of Mathematical Knowledge and in his recent paper “Mathematical Naturalism,” Kitcher

428

suggests that we, “ . . . treat mathematics as a n idealized science of human operations.” He suggests that, “One way to articulate the content of the science is to conceive of mathematics as a collection of stories about the performances of a n ideal subject to whom we attribute powers in the hope of illuminating the abilities we have to structure our environment.”2 He dubs this approach “natural is t ic constructivism.”

As Kitcher himself recognizes, h i s natural is t ic constructivism is conveniently regarded as a modal account of mat he ma tic^.^ In fact, Kitcher’s approach is a close relative of our IB2 or IIB2. By way of illustration, let us see how Kitcher would analyze a n existence claim about, say, X , (the first infinite cardinal). To every alleged mathematical object there corresponds, according to Kitcher’s view, a type of action. To an infinite cardinal there might correspond acts of “collection” of a certain sort. Accordingly, let f( X , ) be the type of action associated with X , . Then, according to Kitcher, we have:

X , exists - 0 3 y (y performs a n action of type f( ‘x, )).

As Charles Parsons has noted, Kitcher owes us a n account of the modal operator ‘ 0 ’. Since naturalistic constructivism is meant to ground mathematics in certain features of the physical world, ‘ 0 ’ must not express mere logical possibility: that would almost entirely sever the link between the ideal agent and the world of actual agents. Yet, on the other hand, ‘ 0 ’ cannot express any familiar sort of nomic possibility since Kitcher, “ . . . requires the possibility of a n agent . . . whose capacities go well beyond what the laws of nature permit.”4

In “Mathematical Naturalism,” Kitcher offers a pragmatist reading of ‘ 0 ’ according to which the powers that are properly attributed to the ideal agent are those which would be so attributed if mathematics were to reach the ideal limit of its rational development. That is:

0 3 y (y performs a n action of type f ( X , )) - The ideal version of mathematics would, if it came into existence, contain the statement that 3 y (y performs an action of type

An obvious criticism of this formulation is that it features an unwarranted definite article: we can justifiably speak of the ideal version of mathematics only when we have established that rational mathematical inquiry does indeed tend toward a unique limit. We would be better justified in saying:

f ( x, 1).

429

0 3 y (y performs a n action of type f ( x0 )) - Some ideal version of mathematics would, if it came into existence, contain the statement that 3 y (y performs a n action of type f ( KO )).

Here ‘ 0’ is viewed as a kind of existential quantifier. The possible worlds over which ‘ 0 ’ ranges are ideal versions of mathematics. Truth in a possible world is taken to mean occurrence in a perfected formulation of mathematics.

How successful is this approach? Kitcher’s account of ‘ 0 ’ would be a genuine clarification only if the notion of a n ideal version of mathematics were clearer than that of an ideal mathematical agent. We doubt that it is. We confess not to have any idea what millenarian mathematics will look like and, so, feel ourselves unable to make out the powers of the ideal agent(@ on this basis. (A mathematician of the 18th century could hardly have guessed what mathematics would be like just 200 years later. Do we have any better chance of guessing what mathematics will be like at the end of time?) We do believe, however, that a somewhat more helpful explication of ‘ 0 ’ can be gleaned from Kitcher’s own pragmatist remarks.

Kitcher’s basic point is that the stories we tell about an ideal mathematical agent are to be evaluated on the basis of their potential contribution to the general enterprise of ra t ional inquiry. To enter ta in a proposition of t he form 0 cp (where cp at t r ibutes a n action to an ideal mathematical agent) is to speculate about the contribution which our endorsement of cp could make to the attainment of our scientific or practical goals given enough time, resources, good will, and intelligence. We speak of the contributions whichcpcould ( ra ther t h a n will) make because the warrantedness of cp is presumably not to be undermined by future accidents. We might be warranted in embracing cp even if all future mathematicians are overcome by sloth, accomplish nothing, and so allow cp no opportunity to contribute to the progress of rational inquiry. What matters is not the contribution which cp does make, but rather the contribution it could make given favorable circumstances. Thus we have:

0 3 y (y performs a n action of type f ( KO )) - It could eventually contribute to the attainment of our scientific or practical goals to suppose that 3 y (y performs an action of

That is, a necessary condition for the possibility of a n ideal mathematical construction is that a belief in the actuality

type f ( xo )).

430

of that construction be a potential contributor to the progress of rational inquiry.

Now if the link between X , and f( KO ) is of the sort Kitcher desires (if his talk of f( X , ) does indeed provide an analysis of our talk of X , ), then the pragmatic contribution of supposing that 3 y (y performs an action of type X , ) should be identical to the pragmatic contribution of supposing that X , exists. So we have:

It could eventually contribute to the attainment of our scientific or practical goals to suppose that 3 y (y performs an action of type f( X , ) ) - It could eventually contribute to the attainment of our scientific or practical goals to suppose that X , exists.

But this yields:

X , exists - It could eventually contribute to the attainment of our scientific or practical goals to suppose that X , exists.

And we take this to mean:

X , exists -0 3 t, t’ (It contributes at t’ to the attainment of our scientific or practical goals to suppose at t that X , exists).

(We include two temporal indices because the contribution made by a proposition might lag behind its endorsement by the mathematical community.) On this view, a necessary condition for the truth of a mathematical statement is that an endorsement of the statement be a potential contributor to the progress of rational inquiry.

This formulation seems preferable to Kitcher’s talk of ideal versions of mathematics: speculation about a proposition’s potential contribution to rational inquiry has a better chance of being well grounded than speculation about the details of millenarian mathematics. Yet two concerns still occur to us. First, we wonder whether the modal language involved here is any clearer than that directly connected with Kitcher’s discussions of ideal agents. And, second, if the notion of a potential contributor to rational inquiry is genuinely clearer than that of a potential mathematical agent, we wonder why the ideal agent should be dragged in at all-the contribution of ideal-agent-talk to Kitcher’s own inquiry would be hard to discern. (Why not be content with saying: X exists only if an endorsement of X , ’s existence is a potentiaf contributor to rational inquiry?) Of these two concerns, the first is the

431

most serious. For it invites doubts about whether Kitcher’s modalism can be reconciled with his naturalism.

We believe, in fact, that Parsons has identified the weakest point in Kitcher’s allegedly naturalistic outlook: Kitcher offers a modal account of mathematical reality, but does so in such a way that the modalities involved resist naturalistic analysis. (At any rate, Kitcher has not himself offered such an analysis.) A philosophical account which is both modal and genuinely naturalistic will probably have to borrow its notion of potentiality from some branch of rational empirical inquiry. At least, this would seem the most promising way of rooting the modal vocabulary in the empirical world. Kitcher has not taken this path. But an empirically rooted modal account of mathematical reality is suggested by White. We turn now to a consideration of White’s approach.

White attempts to discover the “locus” of mathematical reality. Minimally, this involves identifying conditions which are necessary for the existence of mathematical entities and for the truth of mathematical theorems. When White claims that the reality of mathematics, “ . . . is cultural: the sort of reality possessed by a code of etiquette, traffic regulations, the rules of baseball, the English language or rules of grammar,”5 we take this to mean, in part, that the existence of mathematical entities requires the existence of certain cultural conditions. What sort of conditions might these be?

White insists that mathematics, as a cultural phenomenon, be regarded as a force which shapes the mathematical activities of individuals: “Each individual is born into a pre- existing organization of beliefs, tools, customs a n d institutions. These culture traits shape and mould each person’s life, give it content and direction. Mathematics is, of course, one of the streams in the total culture. It acts upon individuals in varying degree, and they respond according to their constitutions. Mathematics [i.e., mathematical activity] is the organic behavior response to the mathematical culture.”6 Of particular interest to White is a feature which we might call “cultural enablement.” Mathematical breakthroughs are not achievements of individuals sealed off from their cultural milieu. Breakthroughs are rendered possible by the particular state of the mathematical culture: “ . . . it is easy to exaggerate the role of superior brains in cultural advance. It is not merely superiority of brains that counts. There must be a juxtaposition of brains with the interactive, synthesizing cultural process. If the cultural elements are lacking, superior brains will be of no avail.”’

432

Objects of thought exist only if they are available to be thought. A mathematical entity’s availability to be thought at a particular time is grounded in the particular state of the mathematical culture at that time. The availability of X , to be discovered by Georg Cantor consisted in the presence of cultural factors which enabled Cantor to perform activities which mathematicians count as such a discovery. So the existence of X , , its availability to be discovered or, more generally, to be thought, depended upon the enabling capacities of late 19th century mathematical culture. The continued existence of X , depends upon the continued presence of those enabling capacities. We express this as follows:

Vt ( KO exists at t - 3 y, c (c enables y to think of X , at t)). Here c ranges over cultures, while y ranges over the members of those cultures. So our Whitean thesis states that X , exists at a certain time only if there is a culture which enables some of its members to think of X , at that time. The crucial factor here is that the enabling mechanisms be in place-not that they actually issue in appropriate acts of thinking.

Modal accounts of mathematical objects accord with well entrenched intuitions insofar as they allow such objects to endure even when no one is actively thinking of them. The position we have extracted from White has precisely this feature-because it is, in fact, a modal account of the type IIB1. The modal operator ‘ 0 ’ is to be read as “some culture enables it to be the case that.” More specifically:

Wr, t ( O ( y thinks of x at t) - 3 c (c enables y to think of x at t)).

An advantage of this particular form of modalism is that it employs a notion of potentiality which is firmly rooted in a branch of rational empirical inquiry: namely, anthropology.

Let us stress that the notions of culture and cultural enablement are central to contemporary anthropology. Cultural anthropologists have found much about which to disagree; the very definition of culture is a vexed subject, to say nothing of the fundamental disagreements over how best to go about studying it. Furthermore, the moment one undertakes to identify quite precisely a culture or set of cultures, one is beset by a host of problems. Yet nearly all anthropologists, across the whole range of theoretical diversity, accept, as meaningful, propositions about some culture in particular or cultures in general. Indeed, implicit

433

endorsement of quantification over cultures remains intact even as the gravest doubt is expressed as to the possibility of scientific generalizations about culture. In one of his last publications, White’s archrival, Franz Boas, wrote that, “ . . . on account of the uniqueness of cultural phenomena and their complexity nothing will ever be found that deserves the name of a law excepting those psychological, biologically determined characteristics which are common to all cultures and appear in a multitude of forms according to the particular culture in which they manifest themselves.”8 Like Descartes discovering that he could not doubt without thinking, anthropologists deep in doubt quantify over cultures in spite of themselves and thus, from a Quinean point of view, reveal their commitment to the existence of these occasionally elusive, but currently indispensable, entities.

Nor is the notion of cultural enablement peripheral to anthropology as we know it. Interpreting culture as the enabler of thought helps to account for several features of culture change. Innovations can be regarded as new combinations of old culture elements. Thus, the larger the store of pre- existing culture elements, the greater the number of possible new combinations; indeed, possible combinations increase geometrically with the number of elements. This helps account for the acceleration of culture change during the course of cultural evolution.9 Furthermore, the otherwise mysterious tendency for geniuses to be clustered in certain epochs, rather than being distributed randomly in time, might be explained by the enabling capacity of culture. An individual’s chance to be a “genius” is very much a function of patterns of cultural development. “The culture process is not an even and uniform flow. There are initial stages of development, periods of steady growth, peaks of culmination, plateaus of continuity and repetition, revolutionary upheavals a n d innovations, disruption, disintegration, and decline.” lo For those who “chance to be born at the time and place where streams of culture are converging and fusing into a final complete synthesis,”” culture enables the thinking of great new thoughts, thereby producing, so to speak, a bumper crop of geniuses.

We conclude, then, that a modal account of a Whitean sort has a virtue which Kitcher (qua pragmatist) should appreciate: such an account fashions its explication of ‘ 0 ’ out of elements which promise to contribute significantly to the progress of rational empirical inquiry (in this case, inquiry into human collectivities). We turn now to another area in which a Whitean outlook shapes our assessment of Kitcher’s program. We shall

434

discuss a fruitful way of weighing the rationality of mathematical subcultures.

2. Communal Rationality

According to Philip Kitcher, the epistemology of mathematics, insofar as it has been pursued at all, has too often been retarded by a fetishistic concentration on formal foundational research. Kitcher argues that it would be more fruitful for epistemologists to adopt a diachronic perspective. The epistemic warrantedness of mathematical theorems is to be viewed as an inheritance passed down through the generations: “Our present body of mathematical beliefs is justified in virtue of its relation to a prior body of beliefs; that prior body of beliefs is justified in virtue of its relation to a yet earlier corpus; and so it goes.”12 Kitcher speculates that this chain is grounded in an Edenic state of mathematical knowledge warranted by perceptual experience. The fundamental task of a “naturalistic” epistemologist is to show that the subsequent career of Edenic mathematics can be reconstructed as a series of rational episodes of self- transcendence on the road from Sense Certainty to Absolute Knowledge. The rationality of a metamorphic episode is here understood to consist in its preservation of prior epistemic warrantedneea.

Kitcher’s “naturalism” leads him to focus not on static foundational systems, but rather on the evolutionary transformations undergone by communities of concrete mathematical inquirers. For Kitcher, the units of mathematical change are not formal (or formalizable) theories, but rather communal constraints on mathematical activity. A constellation of such constraints is known, in Kitcher’s terminology, as a practice. So, “The problem of accounting for the growth of mathematical knowledge becomes that of understanding what makes a transition from a practice . . . to a n immediately succeeding practice . . . a rational transition.” l3 Kitcher suggests that a mathematical practice has five key components: “ . . . a language employed by the mathematicians whose practice it is, a set of statements accepted by those mathematicians, a set of questions that they regard as important and as currently unsolved, a set of reasonings that they use to justify the statements they accept, and a set of mathematical views embodying their ideas about how mathematics should be done, the ordering of mathematical disciplines, and so forth.”14

435

This conception of a practice shares crucial elements with the anthropologist’s conception of culture. A shared language, shared beliefs, a shared understanding of which ends are worth pursuing, shared reasonings legitimating the status quo, and shared norms governing appropriate behavior are all ingredients of culture as classically conceived.15 Furthermore, both cultures and practices are regarded as a shared “heredity” transmitted from generation to generation in social (rather than genetic) ways. Finally, as Leslie White (and many others) have noted, culture can be broken down not only geographically into specific “cultures,” but also topically into certain subdivisions. Conveniently enough, White sometimes chose mathematics to exemplify the latter: “ . . . culture may be regarded as a one or as a many, as a n all-inclusive system-the culture of mankind as a whole-or as a n indefinite number of subsystems of two different kinds: (1) the cultures of peoples or regions, and (2) subdivisions such as writing, mathematics, currency, metallurgy, social organizations, etc. Mathematics, language, writ ing, architecture, social organization, etc., may each be considered as a one or a many also; one may work out the evolution of mathematics as a whole, or a number of lines of development may be distinguished.”16 Working out the evolution of mathematical subcultures (in very much this Whitean sense) is central to Kitcher’s epistemological program.

It would not be surprising, then, if the anthropological literature on cultural dynamics (“culture change”) were to illuminate practices in their dynamic aspect (what Kitcher terms “interpractice transition”). We shall examine one possible source of such illumination. Anthropologists, most notably White, stress a facet of culture change which Kitcher has not emphasized: namely, the positive contribution which practices, or mathematical subcultures, make to their own transformation. Just as “paradigms” in the natural sciences are widely thought to contribute to scientific change primarily through their own degeneration, so Kitcher focuses on how,

. . . major modifications of mathematical practice were achieved on the basis of tensions within prior practices.”17 On this view, practices contribute to the evolutionary process in a largely negative way: they make change a matter of urgency by threatening to collapse or, less catastrophically, by indicating problems for whose solution they do not themselves provide a framework. White identifies a more positive evolutionary role for cultural systems: they can render possible (even inevitable) transitions into new systems and, furthermore, t he superseded systems can be directly

6 6

436

responsible for key features of their successors. On White’s view, culture change is not a matter of heroic individuals struggling to save precious cargo on a sinking ship. Rather it is a matter of energetic communal processes directing and undergirding the contribution which individuals make to cultural evolution. This outlook on culture change prompts us to introduce a notion of communal rationality which Kitcher seems to have overlooked.

Kitcher’s Hegelian Naturalism causes him to devote considerable attention to the rationality of interpractice transitions (i.e., to those evolutionary characteristics which tend to preserve epistemic warrantedness).18 Furthermore, having repented of an earlier fixation on the rationality of individual inquirers, Kitcher has recently recognized a form of communal rationality: a community of inquirers is rational insofar as, “ . . . the distribution of practices within it maximizes the probability that the community will ultimately attain its . . . ends.”lg Kitcher has in mind, for example, that communities within which a wide range of puzzle solving techniques are simultaneously deployed are (at least under certain circumstances) more likely to produce solutions than are homogeneous communities. In this way, community ends will be attained at the expense of many individual aspirations, since many community members will find themselves saddled with techniques which ultimately prove infertile.

We have no particular objection to Kitcher’s discussion of rationality-as far as it goes. Yet, influenced by White, we feel that Kitcher has ignored a crucially important species of communal rationality. We suggest that a mathematical practice or subculture is rational insofar as it promotes and structures its own rational metamorphosis. That is, taking it for granted that there is some respectable notion of the rationality of interpractice transitions, we propose that practices be considered rational insofar as they structure and motivate rational transitions to points beyond themselves. In precisely this sense, Vihtean logistice speciosa is more rational than classical dPL8pvTlKfj and h O y l U T l K f j . This example deserves some elaboration.

Under the influence of Greek logistic and Arabic algebra, Francois Vihte (1540-1605) developed a “calculus of forms” or logistice speciosa. His key insight was that a n art of calculation could be developed whose objects are neither determinate pluralities of concrete objects nor denatured, but nonetheless cardinally determinate, pluralities of units. Vihte realized that logistic could adopt an even more abstract perspective, concerning itself with the universal properties

437

which numbers have regardless of their particular position in the number sequence.

As Jacob Klein has shown, this new perspective precipitated a radical transformation of the Greek notions of dpmp~+ (numerable) and dpmpbs (number).21 ViBte’s work helped to undermine the Greek view that numbers are either pluralities of given objects or abstract representatives of such pluralities. And it helped to foster the modern presumption that our access to numbers somehow depends on our mastery of techniques of symbolic manipulation. Thanks in part to Vihte, we no longer regard the plural ostension of concrete individuals as the fundamental mode of acquaintance with the objects of logistic (or, as we would say, algebra). Instead, logistic itself has become the primary vehicle for acquaintance with its own objects.

That we would today hesitate to ascribe knowledge of the natural numbers to anyone inept at arithmetical calculation and unable to solve number theoretic problems is a sign of how little hold the Greek concept of number has on us. From the Greek point of view, to be conscious of several objects at once is to be acquainted with a numerable. But to be acquainted with numerables is to be only one act of abstraction away from a minimal knowledge of numbers. So, just as someone innocent of ornithology can nonetheless be well acquainted with birds, someone unschooled in arithmetic and logistic need not be blind to pluralities of “ones” or to their abstract counterparts, pluralities of units. By contrast, while we today might not be terribly clear about what numbers are, we do seem fairly confident about closely linking knowledge of them with mastery of arithmetical techniques.

The move away from d pmpqrbu and d pmpbs opened the way for the algebraic treatment of numbers other than the naturals. It was not until the 4th century A.D. that a Greek ventured to regard fractions as numbers in their own right rather than as ratios. And even this brave soul (Diophantus) would have no truck with equations whose roots are negative or irrational. Yet once the notion that numbers are pluralities of units ceases to hold sway (as it did, we should note, among the Arabs well before Viete), the “weirdness” or “impossibility” of such equations fades. For negatives and irrationals are “number- like” in precisely the way which has come to be of the greatest importance-that is , they a re subject to algebraic manipulation and, more positively, play a useful role in the systematization of our mathematical calculi.

From the time of ViBte on, the acceptance by mathematicians of new species of numbers came to be

438

influenced less and less by the concrete contents of direct experience and more and more by the internal standards and requirements of mathematics and mathematical physics. By the 1880’s, Cantor could argue with some success that the introduction of numbers within mathematics is to be governed only by certain standards of consistency, intelligibility, and coherence. This feeling of creative freedom aided the development of set theory not only because it left Cantor uninhibited about introducing transfinite numbers, but also because it helped to supply him with a positive mathematical reason for doing so. Prior to the birth of transfinite arithmetic, Karl Weierstrass, Richard Dedekind, and Cantor himself formulated rigorous theories of arithmetical continua- continua constituted not by some sort of seamless “flow,” but by utterly discrete numbers. It was the complex structure of these continua of distinct points which led Cantor to introduce both his infinite ordinals and his infinite cardinals (the former in connection with his uniqueness theorems for Fourier series representation and the latter in connection with his novel proof of Liouville’s theorem about the distribution of transcendentals).

Classical Greek number theory had the virtue of rooting the concept of number quite directly in perceptual experience. But this had the unfortunate effect of inhibiting the application of algebraic operations to the inhabitants of domains with order type other than W. The Vihtean algebraic revolution motivated and structured precisely such transformations of mathematical culture. So, in our dynamical Whitean sense, mathematical practices since the 16th century have enjoyed a higher level of rationality than Greek number theory. From this point of view, the explosion of mathematical activity in the Modern Period is hardly surprising. Similarly, we predict that the complete triumph of an abstract, structuralist conception of the set theoretic hierarchies (a conception which would sever the notion of mathematical set from the notion of common sense plurality, just as the notion of number has been severed from that of numerable) would stimulate fundamental set theoretic research. We conclude that a Whitean approach to cultural dynamics points toward a notion of rationality which promises to illuminate key aspects of mathematical progress. We suggest t ha t a mathematical subculture is to be counted rational to the extent that it carries within itself the seeds of its own rational metamorphosis.

439

NOTES

1 Hams, 1968, p. 250. * Kitcher, 1988, p. 313. 3 Cf. Kitcher, 1984, pp. 120-122. 4 Parsons, 1986, p. 134.

White, 1947, p. 303. Ibid., p. 298.

7 Ibid., p. 299. “Contrariwise,” White (1947, p. 299) continues, “when the cultural elements are present, the discovery or invention becomes so inevitable that it takes place independently in two or three nervous systems at once.” Are certain culture elements not merely necessary, but also sufficient, for the occurrence of a certain discovery or invention? The apparent frequency with which people working independently, but within a shared cultural context, discover or invent the same thing is suggestive, but hardly decisive. (For a long list of instances see Ogburn et al., 1929.) For one thing, establishing the “sameness” of the independent instances is sufficiently problematic to provide an opening for those disposed to underscore individual creativity rather than cultural process. (See Patinkin, 1983; and cf. Graber, 1985.) Furthermore, even assuming a large absolute number of genuine instances, we must concede that this evidence covers but a small proportion of all innovations. It is conceivable that a certain innovation could require a personality so rare that the necessary culture elements, though present, would lie forever unassembled. For these reasons, we content ourselves with the claim that certain culture elements render certain innovations possible-or, more strongly, that they promote and structure certain innovations without, however, necessitating them.

8 Boas, 1966, p. 311 (italics added). The passage dates from 1939. 9 Lenski et al., 1987, pp. 64-65. lo White, 1949, p. 217. 11 Zbid. l2 Kitcher, 1988, p. 299. l 3 Kitcher, 1984, p. 164. l 4 Kitcher, 1988, p. 299 cf. also Kitcher, 1984, p, 163. 15 See, for example, Kroeber et al., 1952. l6 White, 1959, pp. 30-31. l7 Kitcher, 1988, p. 318. 18 Cf. Kitcher, 1984, ch. 9. 19 Kitcher, 1988, p. 306. 20 The following discussion is drawn from Pollard’s forthcoming

Philosophical Introduction to Set Theory. We mentioned in our discussion of cultural enablement that a large store of culture elements entails a large number of combinatory possibilities. Obviously, not all possible combinations will be of practical significance; and of those that are, most will have no great role in triggering culture change. A few, however, will prove to be fundamental innovations, radically affecting subsequent culture (Lenski et al., 1987, pp. 66-67). Mechanical inventions such as the steam engine, the automobile, and the silicon chip come immediately to mind. But attitudes and insights can have equally dramatic impacts. A fundamental attitudinal innovation is the very valuing of innovation. When a culture has come to value innovation itself, it not only renders new thoughts possible, but may explicitly encourage them (Lenski et al., 1987, p. 67). Admittedly, this encouragement is highly selective: creative thoughts about many social and cultural phenomena are not only directly suppressed on the grounds that the subject matter is “controversial,” but also indirectly forestalled by formal and informal means of political and religious enculturation (Hams, 1988,

440

pp. 385-388). Still, to whatever extent and in whatever realms valuation of innovation becomes a culture element, we see a specific basis for White’s general claim that “culture makes itself” (White, 1949, p. 340). For an example of a mathematical insight which served as a fundamental innovation, supporting and channeling the subsequent evolution of mathematical subcultures, see the following discussion of the Vietean revolution.

21 Cf. Klein, 1968.

REFERENCES

Boas, F., 1966, Race, Language and Culture (Free Press, New York). Graber, R. B., 1985, “A Foolproof Method for Disposing of Multiple Discoveries: Comment on Patinkin,” American Journal of Sociology 90,

Harris, M., 1968, The Rise of Anthropological Theory (Crowell, New York). Harris, M., 1988, Culture, People, Nature (Harper & Row, New York). Kitcher, P., 1984, The Nature of Mathematical Knowledge (Oxford University Press, New York). Kitcher, P., 1988, “Mathematical Naturalism,” in History and Philosophy of Modern Mathematics, W. Aspray and P. Kitcher, eds., Minneapolis: University of Minnesota Press, pp. 293-325. Klein, J., 1968, Greek Mathematical Thought and the Origin of Algebra (M.I.T. Press, Cambridge, MA). Kroeber, A. L. and Kluckhohn, C., 1952, Culture: A Critical Review of Concepts and Definitions (Vintage Books, New York). Lenski, G. and Lenski, J., 1987, Human Societies (McGraw-Hill, New York). Ogburn, W. F. and Thomas, D., 1929, “Are Inventions Inevitable?,” Political Science Quarterly 37, pp. 83-98. Parsons, C., 1986, “Review of P. Kitcher’s The Nature of Mathematical Knowledge,” Philosophical Review 95, pp. 129-137. Patinkin, D., 1983, “Multiple Discoveries and the Central Message,” American Journal of Sociology 89, pp. 306-323. White, L. A., 1947, “The locus of Mathematical Reality: an anthropological footnote,” Philosophy of Science 14, pp. 289-303. Reprinted in White, 1949,

White, L. A., 1949, The Science of Culture (Farrar-Straus, New York). White, L. A., 1959, The Evolution of Culture (McGraw-Hill, New York).

pp. 902-903.

pp. 282-302.

44 1