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

436 alasdair urquhartHopfield model was inspired by ideas from spin glass theory. A large selectionof such applications is described in the latter part of the book (Mézard et al.,1987). In each case, the physicists were able to draw surprising and apparentlycorrect conclusions that were not attainable by conventional approaches. Thus,the evidence so far indicates that the physicists have uncovered a broad domainof phenomena, all involving disorder of some kind, that can be described usingthe nonrigorous tools that they have developed and refined since the 1970s.So, the evidence is that there is some kind of mathematical sense lurkingbehind the replica method, just as there was in the case of the Dirac deltafunction. Although the physicists had some initial misgivings about the method,a quick glance at the literature of the theory of disordered systems shows thatin many cases, it is the first tool for which they reach when investigating a newsystem involving disorder and frustration.Nevertheless, a rigorous foundation for the method has so far eludedthe mathematicians. Remarkable and striking work has been accomplishedby probabilists working within the usual mathematical framework of largedeviations, concentration inequalities, and the like, and Michel Talagrand, inparticular, has made very substantial progress towards verifying the predictionsof the physicists. A survey of his work and an introduction to this important andintriguing area is to be found in his monograph (Talagrand, 2003b). Recently,Talagrand, building on the work of Francisco Guerra, achieved a breakthroughby proving Parisi’s formula for the free energy of the SK model (Talagrand,2003a, 2006).In spite of these major successes, and the verification of the physicists’ work,the replica method itself languishes in mathematical limbo. Talagrand remarks:At the present time, it is difficult to see in the physicist’s replica method more thana way to guess the correct formula. Moreover, the search for arguments makingthis method legitimate ‘a priori’ does not look to this author as a promising areaof investigation. (Talagrand, 2003b, p.195)In spite of these sceptical remarks, the example of the Dirac delta function, theumbral calculus, and the theory of infinitesimals should encourage us to hopethat the remarkable success of this method is more than a mere mathematicalcoincidence. Much remains to be done.16.8 A plea for nonrigorous mathematicsThere are numerous arguments for admitting ‘nonrigorous’ mathematics intothe field of philosophy of mathematics. Even if we identify mathematics