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

purity of method in hilbert’s grundlagen 249its consequences, or examining its foundations in the way that Frege, forinstance, does.Thus, for Hilbert’s investigations in geometry, ‘purity of method’ analysis inthe standard sense is elaborated into the ‘analysis of intuition’. This resolves intotwo separate investigations, one at the intuitive level, and one at the abstractlevel, levels which frequently interact and instruct each other. Furthermore,extracting this information often itself involves a detour into the abstract. Oneexample is given by the investigation of Desargues’s Planar Theorem, wherethe use of the structure of segment fields requires first an abstract axiomatizationof ordered fields. In particular, as the examples treated here make abundantlyclear, higher mathematics is used to instruct or adumbrate intuition, or at thevery least to instruct us about it and what it entails.The second conclusion concerns the notion of the ‘elementary’ or ‘primitive’with respect to a domain of knowledge. The examples we have consideredshow that often we have to adopt a non-elementary point of view in order toachieve results about apparently elementary theorems. Hilbert often stressed,just as Klein did, that elementary mathematics must be studied from an advancedstandpoint. Certainly as far as Hilbert’s work is concerned, this is much morethan a pedagogical point, although Hilbert often stressed this, too.⁵⁰ For onething, the examples considered show that genuine knowledge concerning theelementary domain can flow from such investigations, and they also show thatapparently elementary propositions contain within themselves non-elementaryconsequences, often in a coded form.⁵¹ Furthermore, consideration of theintuitive and elementary is used to generate results at the abstract level; oneexample of this was the result about the abstract theory of fields sketched inSection 8.4.3.But there is a further question about appropriateness. Investigation ofindependence inevitably involves mathematics as broadly construed as possible,since it involves the construction of models, indeed, requires the precisedescription which is only afforded by mathematical models. Given this, onemight ask whether there is an appropriate limit on the mathematics which canbe used for the analysis of the intuitive. There is an obvious practical limitation:in constructing models, one naturally uses those branches of mathematics whichare most familiar, and which will afford the finest control over the modelswe construct. In Hilbert’s case, resort to higher analysis is especially natural,given the extensive theoretical development of analytic geometry in the 19th⁵⁰ See e.g. the introductory remarks in Hilbert’s Ferienkurs for 1896 (Hilbert, ∗ 1896), in Hallett andMajer (2004, Chapter3).⁵¹ There is surely here more than an analogy with the ‘hidden higher-order content’ stressed byIsaacson in connection with the Gödel incompleteness phenomena for arithmetic. See Isaacson (1987).