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

250 michael hallettcentury, which among other things produced the intricate analytic descriptionsof non-Euclidean geometry (models), and also involved the treatment andextension of intuitively based geometrical ideas by highly unintuitive means.It is also worth noting that the reason behind the very development of analyticgeometry was to be able to solve problems posed by synthetic geometry in ananalytic way, and to construe the solutions synthetically. There is a clear sensethat this is also what much of Hilbert’s work with analytic models does.But there is a philosophical reason which goes along with this. Part of thepoint of Hilbert’s axiomatization of geometry is to remove it to an abstractsphere ‘at the conceptual level’ where it is indeed divorced from its ‘natural’interpretation in the imperfectly understood structure of space, becoming inthe process a self-standing theory. The ideal for Hilbert in this respect was thetheory of numbers, and, by extension, analysis. The important philosophicalpoint about the theory of numbers is that for Hilbert it was entirely ‘aproduct of the mind’; it is not in origin an empirical, or empirically inspired,theory like geometry, and can be considered in some sense as already on the‘conceptual level’. Thus, Hilbert shared Gauss’ view of the difference in statusof geometry and number theory. In short, these geometrical investigations andthe consequent extension of geometrical knowledge presuppose arithmetic atsome level.That elementary arithmetic is explicitly presupposed is made clear at thebeginning of Hilbert’s 1898/1899 lectures:It is of importance to fix precisely the starting point of our investigation: Weconsider as given the laws of pure logic and full arithmetic. (On the relationship betweenlogic and arithmetic, cf. Dedekind, ‘Was sind und was sollen die Zahlen?’ [i.e.Dedekind (1888)].) Our question will then be: What propositions must we ‘adjoin’to the domain defined in order to obtain Euclidean geometry? (Ausarbeitung, p.2, p.303in Hallett and Majer (2004).)⁵²This use of number theory and analysis reflects two important philosophicalpositions which Hilbert held at that time concerning arithmetic and analysis.First, just prior to this (e.g. circa 1896; seetheFerienkurs mentioned in n.50),he seems to have held a version of the ‘Dirichlet thesis’ that all of higheranalysis will in some sense ‘reduce’ to the theory of natural numbers, athesis which is stated without challenge in the Vorwort to Dedekind’s 1888monograph. Secondly, there is clear indication that he thought of arithmeticas conceptually prior to geometry. This is illustrated in his 1905 lectures.⁵² There is no corresponding passage in Hilbert’s own lecture notes, suggesting perhaps that Hilbertbecame aware a little later that something ought to be said about the foundation for the mathematicspresupposed in the investigation of geometry.