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

purity of method in hilbert’s grundlagen 2198.3.2 Non-elementary metamathematicsThe second very important thing to recognize about Hilbert’s foundationalenterprise as developed in his study of geometry follows on from this. Whenlooking for base theories, the natural thing is to choose those over which onehas a fine degree of control, perhaps because they have only countably manyelements which can be individually described.I mentioned above that Hilbert tried to separate synthetic geometry fromanalytic geometry, but the separation is only at the mathematical level; atthe metamathematical level, the link is retained. For one thing, as we haveremarked, analytic geometry is taken as the measure of synthetic geometry (forexample, via the demand for completeness). But more importantly here, realand complex analysis in its broadest sense is taken as the fundamental tool inthe logical analysis of synthetic geometry. Since synthetic geometry has as oneof its goals the aim of matching analytic geometry, then it should be clear thatordinary analytic geometry based on the complete ordered field of the realnumbers provides a model for (disinterpreted) synthetic geometry. It is not alarge leap from this to expect that one can find (or fabricate) substructures ofthe reals (or wider analysis) which will correspond to the variations of axiomsand central propositions of synthetic geometry, not least because analyticgeometry gives point-by-point and line-by-line control over the geometricalstructure. For instance, in the standard arrangement, lines are given by simplelinear functions of the number pairs giving the coordinates of points. But inprinciple, a vast range of other functions could be chosen, showing one kind ofbehaviour within a certain region, and a quite different behaviours outside thatregion. Indeed, this is the lesson taught by the various models of non-Euclideangeometry. A fundamental presupposition of Hilbert’s investigation, therefore,is the presence of the full panoply of analytic techniques. Thus, while Hilbert’saxiomatization of geometry distances itself from the analytic developments ofthe 19th century, the full range of analytic geometry is made available, notto prove results in the theory itself, but to prove results about the theory, andin particular to throw light upon the underlying source of knowledge. I willattempt to draw out the points made here in the examples given in Section 8.4.8.3.3 FoundationsThere is another sense in which Hilbert’s project is radically different from otherfoundational projects at the time, projects with a Euclidean flavour, and this isthat Hilbert does not automatically seek a more primitive conceptual level. Ofcourse, this might be done for specific reasons in certain circumstances, andof course important mathematical information might be gained from doing