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

purity of method in hilbert’s grundlagen 209divides geometry into three domains: (1) intuitive geometry, which includes‘school geometry’, projective geometry, and what he calls ‘analysis situs’; (2)axiomatic geometry; and (3) analytic geometry. Axiomatic geometry, accordingto Hilbert, ‘investigates which axioms are used in the garnering of the facts inintuitive geometry, and sets up systematically for comparison those geometriesin which various of these axioms are omitted’, and its main importance is ‘epistemological’.¹²The description of axiomatic geometry is a reasonably good,rough description of much of what Hilbert actually carries out systematically inthe period from 1898 to 1903. He does indeed investigate the ‘facts’ obtainedfrom intuition; in the Grundlagen (p. 3) he actually describes his project asbeing fundamentally a ‘logical analysis of our spatial intuition’ (Hallett andMajer, 2004,p.436), a description which also appears in the Ausarbeitung of the1898/1899 lecture notes (p. 2), where he says ‘we can outline our task as constitutinga logical analysis of our faculty of intuition [Anschauungsvermögens]’, and where‘the question of whether spatial intuition has an apriori or empirical characteris not hereby elucidated’ (see Hallett and Majer 2004, p.303). Hilbert doesconsider the geometries one obtains when various axioms are ‘set aside’, non-Euclidean geometry, non-Archimedean geometry, non-Pythagorean geometry,etc., and we will see some examples later on. Such investigationsnecessarily embrace questions of what can be, and cannot be, proved on thebasis of certain central propositions, either axioms or central theorems. Thatthe main benefit of doing this is said by Hilbert to be ‘epistemological’ is alsounderstandable. If empirical investigation and geometrical intuition are thefirst sources of geometrical knowledge, then Hilbert’s dissection of Euclideangeometry is indeed an ‘analysis’ of this source, revealing the propositionsresponsible for various central parts of our intuitive geometrical knowledge.In short, surely some of Hilbert’s work can be seen as stemming froma rather straighforward epistemological concern with the purity of method,namely showing that P (or some theoretical development T ) can be deducedsolely using some specified axioms (or more generally − Ɣ), and this isdesigned at least to explore the epistemological underpinnings of the axioms.¹³This kind of investigation fits with the traditional conception of axiomatic¹² See Hilbert ( ∗ 1891, 3–5), pp. 21–22 in Hallett and Majer (2004).¹³ Another geometrical example outside the framework of Euclidean geometry might be Hilbert’sanalysis of Lie’s work, where Lie is criticized for the way he uses the full theory of differentiablefunctions to analyze the concept of motion. Hilbert showed that much weaker assumptions than Lie’s(assumptions approximating more to the ‘ancient Euclid’, as he puts it) will suffice, Lie’s assumptionsbeing ‘foreign to the subject matter, and because of this superfluous’. For further details, see Hallettand Majer (2004, 9).