Danielle Macbeth Naturalism in the Philosophy of Mathematics

Haverford College [email protected]

Mathematicians distinguish in practice between those consistent systems, structures, and concepts that are of mathematical interest and those that are not; and this is problem for the would-be naturalist.1 If, as the practice of math- ematicians suggests, mathematics has its own subject matter, it is one that is abstract, that is, platonistic, but Platonism is anathema to the naturalist because and insofar as it cannot be made scientifically respectable. Unfortunately, the naturalist also cannot reject the idea that mathematics has its own subject mat- ter because that would be inappropriately to legislate to mathematics; the natu- ralist philosopher has no authority to determine what is, or is not, mathematics.2 And if the naturalist focuses not on what mathematicians do, on the distinc- tions they draw in practice, but instead on what mathematicians say, namely, that mathematically speaking any consistent system, structure, or concept is as good as any other, that the differences that are discernable in their practice are due only to personal preference or aesthetic considerations,3 then a quite dif- ferent but equally serious difficulty arises. If the distinctions that mathemati- cians draw in practice between “real” mathematics and everything else are to

1 For purposes here a naturalist is someone who rejects superstitious appeals to anything super- or non-natural, and rejects the conclusions of philosophical arguments when those conclusions conflict with what, on other grounds, it clearly appears rational to acknowledge. What our nat- uralist is not committed to is the thesis that there are no distinctively philosophical problems or distinctively philosophical means of addressing them. I take the topic of this essay to be a dis- tinctively philosophical problem, one that is addressed here in a distinctively philosophical way. 2 Both Penelope Maddy and John Burgess, although they are largely Quinean in their natural- ism, take Quine to task for legislating to mathematics. See Penelope Maddy, Naturalism in Mathematics (Oxford: Clarendon Press, 1997), as well as her “Three Forms of Naturalism” in The Oxford Handbook of Philosophy of Mathematics and Logic, ed. Stewart Shapiro (New York: Oxford University Press, 2005), which discusses both her views and Burgess’s in relation to Quine’s naturalism. 3 As Davis and Hersh report, “most writers on the subject seem to agree that the typical work- ing mathematician is a Platonist on weekdays and a formalist on Sundays. That is, when he is doing mathematics he is convinced that he is dealing with an objective reality whose proper- ties he is attempting to determine. But then, when challenged to give a philosophical account of this reality, he finds it easiest to pretend that he does not believe in it after all.” P. J. Davis and R. Hersh, The Mathematical Experience (New York: Penguin Books, 1983), p. 321. 88 Danielle Macbeth be explained not by appeal to objective mathematical facts but instead by appeal to subjective considerations such as the aesthetic or other preferences of mathematicians, then as Steiner has argued, the extraordinary successful- ness of applications of mathematics in the natural sciences is explicable only on anthropocentric grounds. But anthropocentrism, like Platonism, is inconsis- tent with naturalism; there is no good reason to hold that, as Steiner puts it, “the human race is in some way privileged, central to the scheme of things”.4 Is it, then, simply impossible to be a naturalist about the practice of mathematics? Mathematics appears to have its own proper content, its own distinctively mathematical subject matter. Mathematicians in their work tend to take this at face value, and it is corroborated by their mathematical experience.5 Philosophers tend to treat this as a mere appearance; mathematics, they tend to think, has no subject matter, no objects, of its own. (Interestingly, at the turn of the last century the roles were reversed: the mathematicians were the formalists, the philosophers the realists. We will return to this.) Either way, naturalism in the philosophy of mathematics seems to be fatally compromised. We can, however, put aside, at least for the moment, the issues raised by the application of mathematics in the natural sciences (on the assumption that mathematics has no content of its own) and focus on the dilemma: either mathematics has no content of its own and really is nothing more than a kind of a formal game with symbols, or naturalism, at least in the philosophy of mathematics, is false. More specifically, we need to consider the assumption that underlies this dilemma, the idea that mathematical content is incompatible with naturalism. Obviously, if that assumption is false then the application problem is solved as well. Although it can seem to us manifest that the idea of a distinctively mathe- matical subject matter is incompatible with naturalism, it is worth making explicit just why these two notions seem to be so deeply in tension with one another. Benacerraf’s discussion in “Mathematical Truth”, although not directed at precisely this issue, suggests an explanation.6 His interest is in two conflict- ing accounts of mathematical truth: what he calls the standard view according to which mathematical truth is to be conceived Tarski-style, in terms of refer- ence and satisfaction, that is, by appeal to abstract objects, and “combinatoric”

4 Mark Steiner, The Applicability of Mathematics as a Philosophical Problem (Cambridge, Mass.: Harvard University Press, 1998), p. 55. 5 Mathematicians regularly report that thinking about a problem in mathematics often involves imagining mathematical objects, and indeed imagining them doing things, that is, changing in various ways. See, for example, Reuben Hersh’s description in “Wings, not Foundations!”, in Essays on the Foundations of Mathematics and Logic, ed. G. Sica (Monza, Italy: Polimetrica International Scientific Publisher, 2005). 6 Paul Benacerraf, “Mathematical Truth”, originally in the Journal of Philosophy 70 (1973): 661–680, reprinted in Philosophy of Mathematics: Selected Readings (2nd edn.), ed. Paul Benacerraf and Hilary Putnam (Cambridge: Cambridge University Press, 1983).