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

mathematics and physics: strategies of assimilation 421disturb this basic picture, since, although category theory does in fact providea more convenient framework than set theory in many cases, there are somegeneral techniques available for reconciling category-theoretic methods withthose of set theory (Blass, 1984).One does not have to go far in the literature of physics to discover that nosuch consensus on the notion of proof exists in that area. Work in physics runsthe gamut from fully rigorous proofs of theorems (most of the literature ingeneral relativity falls into this category) to purely formal manipulations backedup by computer simulations, as in the case of a large part of the literaturein condensed matter physics. The important area of quantum field theoryoccupies a kind of intermediate position, where some of the computationscan be made fully rigorous, as a result of the hard work of constructive fieldtheorists (Glimm and Jaffe, 1987; Johnson and Lapidus, 2000), whereas thecomputations that really matter to the physicists, such as those in quantumelectrodynamics and quantum chromodynamics, still seem to languish in amathematical limbo, being defined only in terms of ill-defined perturbationexpansions about the theory of the free field.The number of physicists who write papers that satisfy the usual standardsof mathematical rigour is small, perhaps only a few per cent of the total numberof working physicists. However, their work is among the great achievements ofthe physics of the last century. I might mention here the work of Lars Onsageron the two-dimensional Ising model, and that of Dyson, Lieb, and Thirringon the stability of matter. These are examples of beautiful mathematics as wellas beautiful physics, and it is purely a historical accident that this work is notusually considered among the major mathematical accomplishments of the lastcentury.It is easy to understand that classically trained mathematicians might regardwith horror this unruly world of purely formal computations, in which itis hard to lay hold of mathematical bedrock in the maelstrom of ill-definedinfinite series and objects whose nature is far from clear. The mathematicianYuri Manin conveys this bewilderment very tellingly:The author, by training a mathematician, once delivered four lectures to studentsunder the title ‘How a mathematician should study physics’. In the lectures he saidthat modern theoretical physics is a luxuriant, totally Rabelaisian, vigorous worldof ideas, and a mathematician can find in it everything to satiate himself exceptthe order to which he is accustomed. Therefore a good method for attuningoneself to the active study of physics is to pretend that you are attempting toinduce this very order in it. (Manin, 1981, p.x)The remainder of this article will be devoted to the topic of just howmathematicians attempt to induce this order.