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

purity of method in hilbert’s grundlagen 201At the end of the main part of the Ausarbeitung, there is the following statementabout unprovability results:An essential part of our investigation consisted in proofs of the unprovability ofcertain propositions; in conclusion, we recall that proofs of this kind play a largerole in modern mathematics, and have shown themselves to be fruitful. One onlyhas to think of the squaring of the circle, of the solution of equations of the fifthdegree by extracting roots, Poincaré’s theorem that there are no unique integralsexcept for the known ones, etc. (Hilbert ( ∗ 1899, p.169), p. 392 of Hallett andMajer (2004))⁴In his own notes for these lectures, Hilbert writes:The subjects we deal with here are old, originating with Euclid: the principleof the proof of unprovability is modern and arises first with two problems, thesquaring of the circle and the Parallel Axiom. However, we wish toset this as a modern principle: One should not stand aside when something inmathematics does not succeed; one should only be satisfied when we have gainedinsight into its unprovability. Most fruitful and deepest principle in mathematics.(Hilbert ( ∗ 1898/1899, p.106), p. 284 in Hallett and Majer (2004))Moreover, earlier in his notes Hilbert had written the following (theparticular example concerned will be discussed in Section 8.4.1; for themoment the details do not matter):Thus here for the first time we subject the means of carrying out a proof to a criticalanalysis. It is modern everywhere to guarantee the purity of method. Indeed, thisis quite in order. In many cases our understanding is not satisfied when, in a proofof a proposition of arithmetic, we appeal to geometry, or in proving a geometricaltruth we draw on function theory.⁵But Hilbert immediately goes on to say this:Nevertheless, drawing on differently constituted means has frequently a deeperand justified ground, and this has uncovered beautiful and fruitful relations; e.g.the prime number problem and the ζ(x) function, potential theory and analyticfunctions, etc. In any case, one should never leave such an occurrence of themutual interaction of different domains unattended. (Hilbert ( ∗ 1898/1899,p.30),p. 237 of Hallett and Majer (2004))Thus, even while he acknowledges the epistemological disquiet behindmany purity questions, Hilbert admits that ‘impure’ mixtures might point⁴ Bibliographical items with the date preceded by an asterisk were unpublished by Hilbert.⁵ This remark has a special interest in view of one of the examples I will present later, namely thatconcerning the analysis of the proof of the elementary Isoceles Triangle Theorem.