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

mathematics and physics: strategies of assimilation 435the system, and then let n (the number of replicas) go to zero. This is alreadyhard to understand, but things get worse.The physicists predict that there is a critical temperature T c at which thesystem undergoes a phase transition. Above T c , the system is in a single purestate, while below T c the system undergoes a transition into an infinity ofpure states, and the system undergoes a continuous splitting of states downto zero temperature. Above T c , the physicists consider that they are in the‘replica-symmetric’ regime, while below there is ‘replica symmetry breaking’.The extent of the gap between the mathematics and physics community canbe gauged by the fact that the latter considers the replica-symmetric regimetrivial, while the former considers it difficult, and the mathematicians haveonly succeeded in verifying some of the physicists’ predictions with a greatdeal of difficult effort.It is below the critical temperature that things get really strange. Here thegenerally accepted solution is due to the outstanding Italian physicist GiorgioParisi. Parisi introduced what he called a ‘replica symmetry breaking matrix’.This is an n × n matrix that apparently is supposed to correspond to thedistances between pure states. The problem is, since n goes to zero, it is a zeroby zero matrix. Here I can only quote from a famous paper of Parisi from 1980:We face the rather difficult problem of parametrising an n × n matrix in thelimit n = 0. To work directly in zero-dimensional space is rather difficult ... Itis evident that the number of parametrisations is unbounded and the space ofO ⊗ O matrices with these definitions is an infinite dimensional space. (Parisi,1980; Mézard et al., 1987, p.166)This appears to be mathematical nonsense. But is it complete nonsense? Itappears not, because in the first place, the physicists’ numbers check out againstnumerical simulations, and in the second place, many of the predictions madeby the physicists using the replica method have been borne out by rigorousproofs. The predictions here are not just numerical predictions, such as theone I mentioned above as a solution to the administrator’s problem, but alsoquite specific and detailed formulas for key quantities such as the free energy.Even more importantly, the methods developed by the physicists havebeen applied in a surprising number of different areas, some of them quitefar removed from the original physical systems that inspired them. Thus thereplica method has been applied in the theory of combinatorial optimizationproblems, such as the matching problem, the travelling salesman problem,and the graph partitioning problem (this is the problem we described asthe administrator’s problem above). Other applications have been found in thetheory of evolution, neural networks, and the theory of memories, where the