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

222 michael hallettSo, to sum up the points in these last two sub-sections: for Hilbert, wedo not generally seek ‘more primitive’ conceptual levels from which a theorycan be finally deduced; moreover, often conceptually ‘higher’ mathematicsis intrinsically necessary for the investigation of conceptual schemes, evenelementary ones.8.4 The ‘purity of method’ reconsideredLet us now return to the way that ‘purity of method’ is dealt with in Hilbert’sgeometrical work. I will consider three examples, concerning respectivelyDesargues’s Theorem, the Isoceles Triangle Theorem and the Three ChordTheorem, all of them extremely elementary geometrical results, and all threeof which touch centrally on the intuitive ‘facts’ behind geometry. This is noaccident. We have seen that for Hilbert the main source of knowledge behindtraditional geometry is a mixture of intuition and empirical investigation(experiment), a mixture ultimately behind the successful axiomatization. But,starting with informal ‘purity’ questions, Hilbert’s metamathematical analysisof the ‘facts’ uses higher mathematics, which in turn informs elementarygeometrical knowledge. None of the examples treated is fully represented inthe original 1899 version of the Grundlagen. The central result on Desargues’sTheorem (Section 8.4.1) is in the Grundlagen, but what leads up to this result,namely, the philosophical reflection and analysis undertaken in the 1898/1899notes, is suppressed; the analysis of the Three Chord Theorem (Section 8.4.3)isan important part of the 1898/1899 lectures, but only the abstract mathematicalresult, and not the analysis itself, appears in the Grundlagen; and the analysisof the Isoceles Triangle Theorem (Section 8.4.2) makes no appearance, beingfirst dealt with in the 1902 lectures.8.4.1 Desargues’s TheoremThe first example I want to consider where purity of method and the analysis ofintuition play a significant role is in Hilbert’s treatment of Desargues’s Theoremin elementary projective geometry. Suppose given two triangles ABC andA ′ B ′ C ′ , not in the same plane, which are so arranged that the lines AA ′ , BB ′ ,CC ′ meet at a point. Desargues’s Theorem then says that the three points ofintersection generated by the three pairs of straight lines AB and A ′ B ′ , BC andB ′ C ′ , AC and A ′ C ′ themselves lie on a straight line. Intuition might be said toplay a role from the beginning, since it is very easy to ‘see’ the correctness ofthe theorem; the intersection points must all lie in the planes of both triangles,