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

purity of method in hilbert’s grundlagen 203mathematics’, and partly explain therefore his interest in purity questions.But it would be misleading to suggest that Hilbert regards ‘purity’ questionsas merely heuristic in this way, at least judging by his foundational work ongeometry, which Hilbert took to be the paradigm of foundational analysis.Purity of method investigations, in so far as they concern not just openproblems but also the analysis of the sources of mathematical knowledge,was fundamentally important to Hilbert. But to understand exactly how, itis necessary to recognize the novelty of Hilbert’s approach to foundationalissues, and to see the considerable effect this had on the type of mathematicalknowledge obtained through a purity investigation. There is a sense inwhich mixture of mathematical domains is intrinsic to Hilbert’s investigations;on the one hand, higher-level mathematics is essential to foundationalinvestigation of theories, even relatively low-level ones, and this higher levelalso instructs the fundamental source of geometrical knowledge. On theother hand, the examples given in Sections 8.4.1–8.4.3 make it clear that,while ‘purity’ results do throw light on on the question of the appropriatesources of knowledge for geometry, parallel to this are abstract mathematical/logicalresults at what Hilbert calls the ‘conceptual level’. Not only doesthis level largely emancipate mathematics from the epistemological constraintsof the ‘appropriate’, but it is an essential part of what is attained by a‘purity’ result. But there is a twist, which we will consider at the end inSection 8.5.8.2 The foundational projectBefore we go further, it is worth pointing out a number of things aboutHilbert’s approach to geometry.In the first place, Hilbert’s axiomatic presentation as it appeared in theGrundlagen of 1899 (and, to some extent, the preceding lectures) builds on thetradition of synthetic geometry, a tradition which saw a strong revival throughthe work of Monge and von Staudt in the middle of the 19th century. Inparticular, Hilbert attempted to avoid where possible the direct intrusion ofnumerical elements. Underlying this, at least in part, was a view that geometryis of empirical and intuitive origin, and concerns ‘the properties of things inspace’.⁶ This is concisely summarized in Hilbert’s introduction to his 1891lectures on projective geometry:⁶ This view is what underlies the Vorlesungen of Pasch from 1882. It is, of course, of much olderprovenance.