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

purity of method in hilbert’s grundlagen 231with physical manipulation and superposition in view, triangles of differentorientation can only be shown congruent by superposition after lifting andturning them in space. Thus, under such a physical interpretation, Hilbert’saxiom licenses ‘flipping’ as a kind of displacement, and not just ‘sliding’ withinthe same plane. Seen in this way, the proofs of the ITT all exploit such flipping,either directly or indirectly, even, it might be said, Hilbert’s in his ‘neutral’system. The original ‘purity of method’ question, namely, ‘Is flipping essentialto the proof?’, now has a direct correlate with respect to Hilbert’s system:Is the Triangle Congruence Axiom (which can be seen to license ‘flipping’)essential to the proof of ITT in the axiomatic system, divorced as it is from theintuitive/empirical perspective? In order to pose the question more precisely,Hilbert considers a weakened version of the Triangle Congruence Axiom, onewhich insists in effect that, before the side-angle-side criterion can be usedto license triangle congruence, the triangles be of the same orientation in theplane. This weakened axiom no longer underwrites the usual proofs of theITT, as a quick look at the Pappus/Hilbert argument indicates; can the ITTnevertheless be proved? As Hilbert puts it in his presentation of the work inhis 1905 lectures, the original ‘purity of method’ question is now transformedinto a question primarily about logical relations:There now arises the question of whether or not the [original] broader version ofthe [Triangle Congruence] Axiom contains a superfluous part, whether or not itcan be replaced by the restricted version, i.e. whether it is a logical consequenceof the restricted version. This investigation comes to the same thing as showingwhether or not the equality of the base angles in an isoceles triangle is provable on thebasis of the restricted version of the congruence axiom. The question has a closeconnection with that of the validity of the theorem that the sum of two sides ofa triangle is always greater than the third. (Hilbert, ∗ 1905, pp. 86–87)This question is the beginning of Hilbert’s investigations, and the consequencesare fascinating, both for the abstract mathematical structure, for what the investigationstell us about ‘geometrical intuition’, and because of the connectionHilbert draws with the property of triangles he states, which we will refer tohere as the Euclidean Triangle Property.Hilbert first shows that the restricted congruence axiom together with theITT itself implies the normal Triangle Congruence Axiom (see Hilbert, ∗ 1902,p. 32).³³ The key question, in effect first raised in Hilbert ( ∗ 1902), is then³³ In a note added to Appendix in the Sixth Edition of the Grundlagen, Hilbert points out that this isin fact only the case if one adds a further congruence axiom guaranteeing the commutativity of angleaddition. See Hilbert (1923, 259–262); a suitable further axiom due to Zabel is stated on p. 259. Thisnote does not appear in subsequent editions, but Schmidt’s revised version of Appendix II adopts asimilar additional congruence axiom, a weaker one attributed to Bernays. See Hilbert (1930, 134).