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

16Mathematics and Physics:Strategies of AssimilationALASDAIR URQUHART16.1 IntroductionPhilosophers of mathematics and logicians have traditionally shown interestin the foundational controversies of the early part of the 20th century,when non-constructive methods were frowned upon by mathematicians likeBrouwer and Weyl. As far as the community of working mathematicians isconcerned, this seems to be a dead issue. The vast majority of mathematiciansuse non-constructive methods quite freely, and seldom seem to worry abouttheir validity. Nevertheless, debates over the methods of proof appropriate inmathematics are by no means dead.The irruption of the methods of physicists into the realm of pure mathematics,such as Edward Witten’s application of topological quantum field theoryto the theory of knot invariants, recently gave rise to a heated controversy inthe pages of the American Mathematical Bulletin. The mathematical physicistsArthur Jaffe and Frank Quinn pointed to some dangers in the use of speculativemethods in mathematics, which they dub ‘theoretical mathematics’ by analogywith theoretical physics (Jaffe and Quinn, 1993). Among the dangers that theysee are: a failure to distinguish between conjecture and mathematical speculation,with a resulting drop in the reliability of the mathematical literature, thecreation of ‘dead areas’ in research when full credit is claimed by vigoroustheorizers, and unfair distribution of rewards when theoreticians are accordedthe lion’s share of credit for results that are only proved in a rigorous fashionmuch later.Jaffe and Quinn’s article gave rise to a vigorous set of responses (Thurston,1994; Atiyahet al., 1994) from a number of distinguished mathematicians,including Michael Atiyah, Armand Borel, James Glimm, Morris Hirsch,