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

432 alasdair urquhartemerge from physics.’ Atiyah’s second approach is to ‘try to understand thephysics involved and enter into a dialogue with physicists concerned. This hasgreat potential benefits since we mathematicians can get behind the scenes andsee something of the stage machinery.’ It is the dialogue that emerges from thissecond approach that gives rise to the uneasiness of Jaffe and Quinn.The third approach mentioned by Atiyah is ‘to try to develop the physicson a rigorous basis so as to give a formal justification to the conclusions.’This corresponds quite closely to the second approach that I described inChapter 15, and of which I gave several examples above. The drawback ofthis third approach, according to Atiyah, is that ‘it is sometimes too slow tokeep up with the action. Depending on the maturity of the physical theoryand the technical difficulties involved, the gap between what is mathematicallyprovable and what is of current interest to physicists can be immense.’ Thefourth and most visionary idea is to ‘try to understand the deeper meanings ofthe physics-mathematics connection. Rather than view mathematics as a toolto establish physical theories, or physics as a way of pointing to mathematicaltruths, we can try to dig more deeply into the relation between them.’Whatever the attitude that one adopts as a mathematician or logician, it isclear that there is an immense amount of work to be done in this borderlinearea. The difficulties, both conceptual and mathematical, are severe, but theprizes to be gained through new understanding, are potentially immense.The last section of this chapter is devoted to the description of an areawhere the process of assimilation is incomplete, and which consequently posesdifficult problems of interpretation.16.7 The replica methodIn this section, I describe an anomalous object in the raw, so to speak. Inthe physics literature it is known as the Sherrington–Kirkpatrick (SK) modelof spin glasses; it belongs to the class of models known generically as meanfield models. It is the exact analogue in the field of disordered systems of theCurie–Weiss model of ferromagnetism that I described in my introductorychapter.The SK model, like the Curie–Weiss model, is a caricature of certain realphysical systems. The key feature of such systems is that they involve largecollections of magnetic spin variables, where the interaction between thesevariables is not consistently ferromagnetic (favouring consistent orientation)but rather randomly ferromagnetic and antiferromagnetic (favouring opposite