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

mathematics and physics: strategies of assimilation 419picture. Most of Cartier’s paper is devoted to showing how some of the mostoutrageous claims of Leonard Euler and other formally inclined mathematicianscan be given a sensible interpretation, even such crazy-looking assertions as‘∞! = √ 2π’ (Cartier, 2000, p.64). Similarly, Atiyah and Thurston, thoughdeclaring themselves in opposition to the attitudes of Jaffe and Quinn, whichthey see as overly restrictive and authoritarian, do not really disagree with themon the question of separating rigorously established results from mathematicalspeculation. Thus Atiyah remarks: ‘I find myself agreeing with much of thedetail of the Jaffe–Quinn argument, especially the importance of distinguishingbetween results based on rigorous proofs and those which have a heuristicbasis’ (Atiyah et al., 1994, p.1), and Thurston says: ‘I am not advocating anyweakening of our community standard of proof; I am trying to describe howthe process really works. Careful proofs that will stand up to scrutiny are veryimportant’ (Thurston, 1994, p.9).Nevertheless, it is clear that the debate reflects some major changes takingplace in the field of mathematics, changes that are reflected in the concludingremarks of Cartier in a 1997 interview with Marjorie Senechal:When I began in mathematics the main task of a mathematician was to bringorder and make a synthesis of existing material, to create what Thomas Kuhncalled normal science. Mathematics, in the forties and fifties, was undergoing whatKuhn calls a solidification period. In a given science there are times when youhave to take all the existing material and create a unified terminology, unifiedstandards, and train people in a unified style. The purpose of mathematics, in thefifties and sixties, was that, to create a new era of normal science. Now we areagain at the beginning of a new revolution. Mathematics is undergoing majorchanges. We don’t know exactly where it will go. It is not yet time to make asynthesis of all these things—maybe in twenty or thirty years it will be time fora new Bourbaki. I consider myself very fortunate to have had two lives, a life ofnormal science and a life of scientific revolution. (Senechal, 1998)It is clear that mathematics, after a period of inner consolidation andabstraction, is now open to influences on many sides, and the resultingopening-up induces anxieties on numerous issues. One of the anxieties relatesto the classification of researchers as mathematicians. Benoit Mandelbrot clearlyconsiders himself to be a mathematician, but rails against the evil influenceof Bourbaki, and the ‘typical members of the AMS,’ whom he stigmatizes as‘Charles mathematicians’ (since the AMS headquarters is on Charles Street,Providence, RI). According to Mandelbrot, Jaffe and Quinn propose to set upa police state within ‘Charles mathematics’, and a world cop beyond its borders(Atiyah et al., 1994, p.16). Charles mathematicians practice a sterile form ofresearch, and were the bane of great creative figures such as Henri Poincaré