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

414 alasdair urquhartAt least two strategies can be observed in the process of absorbing physicalideas into the fabric of mathematics. One is to treat the ideas of the physicistsas purely heuristic in nature, and to establish their mathematical conjectures byconventional means. A second and more radical idea is to attempt to reworkthe ideas of the physicists and thereby construct new conceptual schemes thatvalidate at least some of their calculations. It is this second idea towards whichDyson points in the quotation above. In the research contribution followingthis introductory chapter, I shall discuss a number of relevant cases from thehistory of mathematics, as well as mentioning some open areas where thisprocess of mathematical assimilation is still fragmentary or incomplete.15.4 Further readingThe topic of the interaction between mathematics and physics is vast, and thischapter discusses only a single aspect of this interaction. In §15.1, mean fieldmodels were discussed as examples of models in physics that seem to contradictolder received views on scientific explanation. The book by Robert W.Batterman (2002) contains detailed discussion of many other models of asimilar type. He points out (Batterman, 2002,p.13) that science often requiresmethods that ‘eliminate both detail, and, in some sense, precision’. He describessuch methods as ‘asymptotic methods’, and the type of reasoning involved inthem as ‘asymptotic reasoning’. He draws very interesting and rather heterodoxconclusions from this fact concerning models of scientific explanation andtheory reduction, as well as the notion of emergent properties. (See alsothe very interesting review Belot (2005) and the reply by Batterman (2005).)Similar themes in the context of condensed matter physics are discussed in theunusually stimulating monograph by Martin H. Krieger (1996).In the conclusion to a widely cited essay John von Neumann (1947) wrote:At a great distance from its empirical source, or after much ‘abstract’ inbreeding,a mathematical subject is in danger of degeneration. At the inception the style isusually classical; when it shows signs of becoming baroque, then the danger signalis up. ... Whenever this stage is reached, the only remedy seems to me to be therejuvenating return to the source: the reinjection of more or less empirical ideas.I am convinced that this was a necessary condition to conserve the freshness andthe vitality of the subject and that this will remain equally true in the future.Mathematics is now in the midst of a period of rapid growth throughthe absorption of a multitude of new ideas from physics. This renewedinteraction between the two fields offers, in addition to the problems already