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

418 alasdair urquhartSaunders Mac Lane, Benoit Mandelbrot, Karen Uhlenbeck, René Thom andWilliam Thurston. The reactions of both mathematicians and physicists to theintentionally provocative article are remarkably varied, and repay close study.One of the most interesting rejoinders came from Sir Michael Atiyah, whoremarked:My fundamental objection is that Jaffe and Quinn present a sanitized view ofmathematics which condemns the subject to an arthritic old age ... if mathematicsis to rejuvenate itself and break exciting new ground it will have to allow for theexploration of new ideas and techniques which, in their creative phase, are likelyto be as dubious as in some of the great eras of the past. Perhaps we now havehigh standards of proof to aim at but, in the early stages of new developments,we must be prepared to act in more buccaneering style (Atiyah et al., 1994, p.1)Saunders Mac Lane made a vigorous reply to these startling comments ofAtiyah, remarking that ‘a buccaneer is a pirate, and a pirate is often engagedin stealing. There may be such mathematicians now. ... We do not need suchstyles in mathematics’ (Mac Lane, 1997, p.150).There is a striking article (Cartier, 2000) by the distinguished Frenchmathematician Pierre Cartier that touches on many of the themes of the debatefollowing the publication of the Jaffe–Quinn article. In the introduction to hispaper, Cartier writes:The implicit philosophical belief of the working mathematician is today theHilbert–Bourbaki formalism. Ideally, one works within a closed system: thebasicprinciples are clearly enunciated once for all, including (that is an additionof twentieth century science) the formal rules of logical reasoning clothedin mathematical form. The basic principles include precise definitions of allmathematical objects ... My thesis is: there is another way of doing mathematics,equally successful, and the two methods should supplement each other and not fight. Thisother way bears various names: symbolic method, operational calculus, operatortheory ... Euler was the first to use such methods in his extensive study of infiniteseries, convergent as well as divergent. ... But the modern master was R. Feynmanwho used his diagrams, his disentangling of operators, his path integrals ... Themethod consists in stretching the formulas to their extreme consequences,resorting to some internal feeling of coherence and harmony. (Cartier, 2000,p.6)These remarks of Cartier are all the more startling if we remember that in the1950s, he was a core member of the Bourbaki group, contributing perhaps 200pages a year to the collective work.A superficial interpretation of the words of Atiyah and Cartier would bethat they are advocating a loosening of the standards of mathematical rigour.However, a careful examination of their articles reveals a more nuanced