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

232 michael hallettthis: What has to be added to the usual system, with both the Parallel Axiomand the Archimedean Axiom, but only the restricted Triangle CongruenceAxiom, to enable the Isoceles Triangle Theorem (or indeed the usual TriangleCongruence Axiom) to be proved?Hilbert’s first answer to this question is given in the 1902 lectures, and thenin the 1902/1903 paper, namely: one can prove the ITT if one adopts alongsidethe Archimedean Axiom a second continuity axiom which Hilbert calls theAxiom der Nachbarschaft. This latter axiom (see Hilbert, ∗ 1902, p.84) statesthat, given any segment AB, there exists a triangle (‘oder Quadrat etc.’) inwhose interior there is no segment congruent to AB. Hilbert later gave otheranswers, focusing, in place of Nachbarschaft, on an axiom he calls the Axiom derEinlagerung (Axiom of Embedding), which says that if one polygon is embeddedin another (i.e. its boundary contains interior but no exterior points of thefirst), then it is not possible to split the two polygons into the same number ofpairwise congruent triangles. (Bernays later gave an essential simplification.)³⁴The bulk of Hilbert’s ‘purity’ investigation is now devoted to showingthat this axiom (or respectively Einlagerung) and the Archimedean Axiom areboth essential if the ITT (respectively the broader version of the TriangleCongruence Axiom) is to be proved in this way. What Hilbert shows is thatgeometries can be constructed in which all the plane axioms hold (with theweaker version of triangle congruence), and where Nachbarschaft (respectivelyEinlagerung) holds, but where the Archimedean Axiom and the ITT both fail,or similarly where the Archimedean Axiom holds, but where Nachbarschaft(respectively Einlagerung) andtheITT fail.In his 1902 lectures, Hilbert is very careful to set out some of the importantthings which can be reconstructed on the basis of the weaker congruenceaxiom³⁵, and among them is Hilbert’s equivalent of Euclid’s theory of linearproportion. On the other hand, Hilbert’s full theory of triangular area (‘surfacecontent’), a conscious reconstruction of Euclid’s, does not apparently gothrough. Euclid’s fundamental theorem concerning this (Elements, I,39) isthat triangles on the same base and with the same area must have the sameheight. In establishing his version of this theorem (that two triangles which areinhaltsgleich and on the same base have the same height), Hilbert defines the³⁴ This new ‘very intuitive’ requirement was first introduced by Hilbert in a section added to thefirst reprinting of his paper in the Second Edition of the Grundlagen, i.e. Hilbert (1903, 88–107).Schmidt’s revision of Appendix II for the Seventh Edition (1930) omitted the consideration of theEinlagerungsaxiom. It was revived by Bernays in Supplements to later editions, where he points out thesimplification.³⁵ See §A of the 1902 lectures, which occupies pp. 26–32 of Hilbert ( ∗ 1902), to be found in Hallettand Majer (2004, 553–556).