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

purity as an ideal of proof 189conflict has not, however, driven purity into exile. Indeed, it remains acommon ideal of proof today. Somewhat more accurately, there are differentideals of (and, not coincidentally, motives for) purity today, and these arereflected in actual mathematical practice. I’ll now briefly survey the moreimportant of these.7.3.1 Topical purityA natural point of departure is the prime number theorem¹⁹ and the wellknownsearch for its ‘elementary’ proof, a search which culminated in themore or less independently developed proofs of Selberg and Erdös in 1948(see Selberg (1949) andErdös (1949)). The theorem was, of course, proveda half century earlier (1896) by non-elementary means, again independently,by Hadamard and de la Vallée Poussin. Both their proofs relied heavily onmethods from complex analysis.The proofs of Selberg and Erdös avoided such methods, and it was thisavoidance which, at least in Selberg’s view (Selberg, 1949, 305), qualified themas elementary.Gian-Carlo Rota gave a useful digest of the developments leading up toSelberg’s and Erdös’ proofs in Rota (1997b). Among other things, he pointedout that it should be seen as a natural continuation of earlier work of NorbertWiener’s, work which suggested that there might be a ‘conceptual underpinningto the distribution of the primes’ (Rota, 1997b, 115, emphasis added). This,he said, was the primary motivation for looking for an elementary proofsince it encouraged the idea that the prime number theorem might havea proof that proceeds from the analysis of the concept of prime numberitself.Rota thus proposed the following understanding of the notion of elementaryproof in the setting of the prime number theorem.What does it mean to say that a proof is ‘elementary?’ In the case of the primenumber theorem, it means that an argument is given that shows the ‘analyticinevitability’ (in the Kantian sense of the expression) of the prime number¹⁹ The prime number theorem is the theorem that for any real number x, the number of primesnot exceeding x (commonly expressed by the so-called ‘prime counting’ function and written ‘π(x)’)is asymptotic toxln x .( )The reader should recall that for all x, y, lnx = y iff e y = x, wheree= lim n→∞ 1 +1 n.Tosaynthat π(x) is asymptotic toxxis to say that the ratio of π(x) to approaches 1 as x approachesln x ln xinfinity. As Selberg pointed out in his paper, it is possible to eliminate all references to limits in theproof he gave. That done, he also noted, the appeal to limits in the statement of the theorem wouldalso have to change.