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

purity of method in hilbert’s grundlagen 211evidence in favour of the Parallel Axiom was called into question. ... In this way,there developed the view that the essential thing in the axiomatic method does notconsist in the securing of absolute certainty, which is transmitted to the theoremsby means of logic, but in this, that the investigation of the logical interconnectionsis separated off from the question of the actual truth of the axioms.¹⁵This Hilbert emphasized as being the ‘main service’ of the axiomatic method.The interest is therefore much more logical than epistemological.The remark of Hilbert’s just quoted leads to a second sense in whichHilbert’s work is new, and which has not yet become clear: part of the point ofHilbert’s investigation is to effect an emancipation from the sources of knowledgeprovided by the ‘facts’ or by intuition. As I hope will be shown, this plays asignificant part in understanding the role of purity of method investigations inHilbert’s investigation of geometry.The radicalness of Hilbert’s approach has three important features, to bedealt with briefly in the next section.8.3 Independence and metamathematicalinvestigation8.3.1 InterpretationIt is clear that in his study of geometry, Hilbert’s focus is less on questions ofprovability from a circumscribed stock of principles than on unprovability, inother words, on independence. The basic technique which Hilbert adoptedfor this investigation is that of modelling, more strictly, of translating thetheory to be investigated into another mathematical theory. For this, it isessential (and Hilbert is very clear about this) that the primitive conceptsemployed are not tied to their usual fixed meanings; they must be free forreinterpretation. Indeed,inhis1898/1899 lectures Hilbert stresses that the mostdifficult part in carrying out the investigation will be separating the basic termsfrom their usual, intuitive associations.¹⁶ It follows that the axioms cease tohave fixed meaning, and thus cease to be, for someone like Frege, genuineaxioms at all.This was not just a matter of expediency for Hilbert, done for the sake of theindependence proofs; along with it goes a new picture of mature mathematics,¹⁵ See also Bernays (1922, 95). Bernays’s remarks are very reminiscent of the long passage fromHilbert ( ∗ 1921/1922) just quoted. Bernays was the Ausarbeiter for the lecture notes.¹⁶ See p. 7 of the 1898/1899 lectures (Hallett and Majer, 2004, 223).