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

412 alasdair urquhartof Bourbaki-style mathematics, and have turned back towards more concreteproblems, often using the powerful machinery developed during the period ofabstraction to solve long standing classical problems, as the recent solution ofthe Fermat problem by Andrew Wiles illustrates.The fresh interactions between mathematicians and physicists have given riseto numerous volumes of proceedings and tutorials in which the two communitieshave attempted to convey the new insights that they developed during theperiod of separation. A good example is provided by the series of meetings atLes Houches bringing together number theorists and mathematically mindedphysicists (Luck et al., 1970; Waldschmidt et al., 1992; Cartieret al., 2006).Another sign of the times is the bulky set of two volumes containing tutorialson quantum fields and strings especially aimed at mathematicians (Deligne etal., 1999), whose stated goal is ‘to create and convey an understanding, in termscongenial to mathematicians, of some fundamental notions of physics, such asquantum field theory, supersymmetry and string theory’. Interactions betweenmathematics and physics are taking place across a broad front, in areas suchas low-dimensional topology, conformal field theory, random matrix theory,number theory, higher dimensional geometry, and even theoretical computerscience. This is an exciting time!15.3 Exploring the boundaryOf course, when two partners get together again after a long separation, thereare often difficulties and friction. The reunion of mathematics and physics is noexception. The extraordinarily fertile language of the physicists, centred aroundthe Feynman integral, presents enormous difficulties to a purely mathematicalunderstanding. More generally, physicists pay little attention to mathematicalrigour in their calculations, and often regard the mathematicians’ insistence onwell-defined quantities and objects as useless pedantry, holding up progress onbasic questions.Kevin Davey (2003) raises the question of whether mathematical rigour isnecessary in physics, and (with appropriate reservations) concludes that it is not.Basing his analysis on the examples of the Dirac delta function and Feynman’spath integrals, he points out that physicists are not troubled by the lack ofrigour in their own reasoning, because they restrict the inferences involvingquestionable methods. However, this solution cannot satisfy mathematicianswho are interested in adapting ideas from physics in solving their own problems.The inferential restrictions employed by the physicists usually take the form of