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

226 michael hallettYQOXPFig. 8.3. Model for the failure of Desargues’s Theorem used in the Grundlagen derGeometrie; diagram adapted from p.53.fully projective, and takes place before the congruence axioms are even stated.In the Grundlagen, all the axioms are set out before there is any significantdevelopment. The statement of Desargues’s Theorem given there involvesparallels, and the proof from the full spatial axioms (and the Parallel Axiom) iscalled upon, though not given (p. 49). Hilbert then remarks that one can giveasimpleplanar proof provided a central result from the theory of proportionsis used. Since congruence is involved, this means in effect that congruence canreplace the spatial assumptions involved in the usual proof. Hilbert then givesanother planar model (see Fig. 8.3), thus creating a non-Desarguean affinegeometry, in which Desargues’s Theorem and the appropriate congruenceassumption (the Triangle Congruence Axiom, IV 6 of the Grundlagen, III 6of the lectures) both fail. The investigation is thus different, and certainlyadds important information to that of the lectures.²⁹ But the same points aswere made above hold. Hilbert again takes an ordinary analytic plane, but thistime one with a certain ‘distorting’ ellipse around the origin. Here, straightlines which would normally pass through the origin are ‘distorted’ by the²⁹ As against this, though, note that the whole fascinating discussion of the import of Desargues’sTheorem, of ‘Reinheit der Methode’, etc. is quite lacking in the first and subsequent editions of theGrundlagen.