Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

6 paolo mancosuphilosophers of mathematics and traditional epistemologists and ontologists ofmathematics felt that the ‘mavericks’ were throwing away the baby with thebathwater.Within the traditional background of analytic philosophy of mathematics,and abstracting from Kitcher’s case, the most important direction in connectionto mathematical practice is that represented by Maddy’s naturalism. Roughly,one could see in Quine’s critique of the analytic/synthetic distinction adecisive step for considering mathematics methodologically on a par withnatural science. This is especially clear in a letter to Woodger, written in1942, where Quine comments on the consequences brought about by his (andTarski’s) refusal to accept the Carnapian distinction between the analytic andthe synthetic. Quine wrote:Last year logic throve. Carnap, Tarski and I had many vigorous sessions together,joined also, in the first semester, by Russell. Mostly it was a matter of Tarski andme against Carnap, to this effect. (a) C[arnap]’s professedly fundamental cleavagebetween the analytic and the synthetic is an empty phrase (cf. my ‘‘Truth byconvention’’), and (b) consequently the concepts of logic and mathematics are asdeserving of an empiricist or positivistic critique as are those of physics. (quotedin Mancosu (2005); my emphasis)The spin Quine gave to the empiricist critique of logic and mathematicsin the early 1940s was that of probing how far one could push a nominalisticconception of mathematics. But Quine was also conscious of the limits ofnominalism and was led, reluctantly, to accept a form of Platonism based onthe indispensability, in the natural sciences, of quantifying over some of theabstract entities of mathematics (see Mancosu (Forthcoming) for an account ofQuine’s nominalistic engagement).However, Quine’s attention to mathematics was always directed at itslogical structure and he showed no particular interest in other aspects ofmathematical practice. Still, there were other ways to pursue the possibilitiesthat Quine’s teachings had opened. In Section 3 of this introduction I willdiscuss the consequences Maddy has drawn from the Quinean position. Letme mention as an aside that the analogy between mathematics and physics wasalso something that emerged from thinkers who were completely opposed tological empiricism or Quinean empiricism, most notably Gödel. We will seehow Maddy combines both the influence of Quine and Gödel. Her case is ofinterest, for her work (unlike that of the ‘mavericks’) originates from an activeengagement with the foundationalist tradition in set theory.The general spirit of the tradition originating from Lakatos as well asMaddy’s naturalism requires extensive attention to mathematical practice. Thisis not to say that classical foundational programs were removed from such