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

248 michael hallettit in the passages quoted in the Introduction. And at some level, this was aconcern of Hilbert’s. There are many examples in Hilbert’s 1898/1899 lectureson Euclidean geometry where Hilbert is concerned to show that some implicitassumption made by Euclid or his successors is in fact dispensable; some ofthese were cited in Section 8.2. But the examples we have concentrated on inSection 8.4 show that the focus on eliminating ‘inappropriate’ assumptions wasonly one facet of Hilbert’s work on geometry; the cases examined are all oneswhere the appropriateness of an assumption is initially questioned, but whereit is shown that it is indeed required. Among other things, this forces a revisionof what is taken to be ‘appropriate’. It is part of the lesson taught by theexamples that in general this cannot be left to intuitive or informal assessment,for instance, the intuitive assessment of the complexity of the concepts used inthe assumption in question.Furthermore, full investigation of geometry requires its axiomatization,and proper examination of this requires that it be cut loose from its naturalepistemological roots, or, at the very least, no longer immovably tied tothem. According to Hilbert’s new conception of mathematics, an importantpart of geometrical knowledge is knowledge which is quite independent ofinterpretation, knowledge of the logical relationships between the various partsof the theory, the way the axioms combine to prove theorems, the reverserelationships between the theorems and the axioms, and so on, all componentswe have seen in the examples. And in garnering this sort of geometricalknowledge, there is not the restriction to the ‘appropriate’ which we see inthe ‘Euclidean’ part of Hilbert’s concerns. What is invoked in pursuing thisknowledge might be some highly elaborate theory, as it is in the analysis ofthe Isoceles Triangle Theorem, a theory far removed from the ‘appropriate’intuitive roots of geometry. Even in the cases of the fairly simple models of theanalytic plane used to demonstrate the failure of Desargues’s Planar Theorem,the models though visualizable are far from straightforwardly ‘intuitive’.One might be tempted to say that the knowledge so achieved is notgeometrical knowledge, but rather purely formal logical knowledge or (asit would be usually put now) meta-geometrical knowledge. But althoughthis designation is convenient in some respects, it is undoubtedly misleading.As we have seen, the ‘meta’-geometrical results have a direct bearing onwhat is taken to be geometrical knowledge of the most basic intuitivekind; in particular it can reveal a great deal about the content of intuitivegeometrical knowledge. In short, it effects an alteration in geometricalknowledge, and must therefore be considered to be a source of geometricalknowledge. To repeat: for Hilbert, meta-mathematical investigation ofa theory is as much a part of the study of a theory as is working out