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

190 michael detlefsentheorem on the basis of an analysis of the concept of prime without appealing toextraneous techniques.Rota (1997b, 115)Parallels to the Aristotelian and neo-Aristotelian conceptions of purity couldhardly be more striking, and they suggest that purity of a broadly Aristoteliantype continues to function as an ideal among contemporary mathematicians.This is also suggested by the notice that was taken of Selberg’s and Erdös’ proofs,notice which resulted in Selberg’s being awarded the Fields Medal in 1950.Thetheorem itself was old.²⁰ What was new was the elementary proof, where ‘elementary’,in this case, meant something like what we mean by ‘topically pure’.The proof took what had been a mystery (the relationship of the densityof the primes to the Riemann zeta function, as presented in the proofs ofHadamard and de la Vallée Poussin) into something rooted in the concept ofa prime number itself.²¹ The result was thought to be remarkable.Such views of the significance of the elementary proof of the prime numbertheorem have been voiced by mathematicians other than Rota. Thus, in areview of Selberg’s and Erdös’ work, A. E. Ingham described their proofs as‘not depending on ... ideas remote from the problem itself’ (Ingham, 1949,595). Indeed, well before their work he challenged the use of analytic methodsin proving such theorems on the grounds that they ‘introduce ideas veryremote from the original problem’ (Ingham, 1932, 5). In a similar spirit,H. G. Diamond praised the use of elementary methods in solving problemsconcerning the distribution of the primes more generally because ‘they donot require the introduction of ideas so remote from the arithmetic questionsunder consideration’ (Diamond, 1982, 556).The elementary proof of the prime number theorem is thus a potentillustration of the continuing concern for broadly Aristotelian purity amongmathematicians. It is, however, by no means unique. In fact, purity seemsto be a common concern among combinatorists generally. A good exampleis a recent paper by Stanton and Zeilberger in which they motivate theirproject by observing that ‘[t]o a true combinatorialist, a combinatorial resultis not properly proved until it receives a direct combinatorial proof’ (StantonandZeilberger, 1989, 39). They also offer an interesting suggestion concerning thebenefit of such proof—namely, the added ‘insight’ it gives to the combinatorist.The idea seems to be that (i) a specialist is most likely to do somethingsignificant with any solution to a problem in her specialty, that (ii) this²⁰ As Rota put it, it ‘had been cooked in several sauces’ (Rota, 1997b, 114).²¹ Or this concept plus, perhaps, certain related concepts necessary in order to see it as a genuinelycombinatory problem.