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

204 michael hallettIrequireintuition and experimentation, just as with the founding of physical laws,where also the subject matter [Materie] is given through the senses.In fact, therefore, the oldest geometry arises from contemplation of things in space,as they are given in daily life, and like all science at the beginning had posedproblems of practical importance. It also rests on the simplest kind of experimentation thatone can perform, namely on drawing. (Hilbert( ∗ 1891, Introduction, p. 7), p. 23in Hallett and Majer (2004))But he goes on to say that Euclidean geometry had ‘ ... an essential defect: ithad no general method, without which a fruitful further development of the scienceis impossible’ (ibid., p.8). This defect was rectified through the inventionof analytic (Cartesian) geometry, which indeed provided a powerful, unifiedmethod. Nevertheless, this brought its own disadvantages:As important as this step forward was, and as wonderful as the successes were,nevertheless geometry as such in the end suffered under the one-sided developmentof this method. One calculated exclusively, without having any intuition of whatwas calculated. Onelostthesense for the geometrical figure, andforthe geometricalconstruction. (Hibert( ∗ 1891, p.10), p. 24 in Hallett and Majer (2004))In what follows, Hilbert makes it clear that he sees the movement in the 19thcentury to promote synthetic geometry as at least in part a reaction to this.This movement concentrated on projective geometry (what was often called‘Geometrie der Lage’), but Hilbert’s aim was to reformulate and restructure fullEuclidean geometry itself as far as possible in an essentially synthetic way. Indoing so, he develops geometry in a modern axiom system building up from thesimplest possible projective framework (an incidence and order geometry), andarriving at full Euclidean geometry with, for example, the standard results froma Euclidean theory of congruence, of proportions, area (or surface measure),and parallels, all of which is sometimes called by Hilbert ‘school geometry’,and developed (where possible) before continuity is broached. This syntheticrestructuring is much clearer in the 1898/1899 lectures which preceded theGrundlagen than it is in the Grundlagen itself.The tone of the remarks from 1891 leaves the impression that, accordingto Hilbert, Euclidean geometry represents knowledge of a certain kind, animpression which is strengthened by Hilbert’s repeated declaration that anaxiomatisation in any area of science always begins with a certain domain of‘facts [Tatsachen]’. By this, Hilbert does not mean just facts in the sense ofempirical facts or even established truths, though such things might be included,but simply what over time has come to be accepted, for example, from anaccumulation of proofs or observations. Geometry, of course, is the centralexample: there are empirical investigations, over 2,000 years of mathematical