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

230 michael hallettcongruent by the side-angle-side criterion; consequently, the base angles mustbe the same. The proof appears to be, thus, little more than trivial.Hilbert’s proof can be traced to Pappus; Euclid’s is different and a bit moreinvolved. In his edition of the Elements, Heath notes of Pappus’s proof:This will no doubt be recognised as the foundation of the alternative prooffrequently given by modern editors [of the Elements], though they do not refer toPappus. But they state the proof in a different form, the common method beingto suppose the triangle to be taken up, turned over, and placed again upon itself,after which the same considerations of congruence as those used by Euclid in I, 4are used over again. (Heath (1925, Volume 1, p.254))But Heath points out that Pappus himself avoids the assumptions about ‘lifting’and ‘turning’, simply describing the same triangle in two ways. Hilbert thusfollows Pappus, not those ‘modern editors’ Heath mentions.Heath notes, though, that even Pappus’s proof depends indirectly on asuperposition argument, since it rests on Euclid’s I, 4. Hilbert’s proof doesnot, simply because the side-angle-side congruence criterion depends on theTriangle Congruence Axiom, not on any special proof method. It seems thatmatters have been shifted to the ‘conceptual sphere’, and no longer directlyconcern intuitions of space and movement in space.Nevertheless, a ‘purity of method’ question can still be posed at the intuitivelevel: is ‘flipping’ and turning of the triangles, and thus a spatial dimension,essential to the argument for ITT? In some work first presented in his 1902lecture course on the foundations of geometry, and then more fully in a paperpublished later in that year, Hilbert transposed this question at the intuitivelevel to a correlative question at the level of his axiomatization.³² At the centreof concern is the Triangle Congruence Axiom. In its usual form, this involvesno reference to what intuitively might be called the ‘orientation’ of thetriangles involved; two triangles with matching side-angle-side combinationswill be determined as congruent regardless of whether their orientation isthe same or not. To mix idioms, accepting the quasi-physical interpretationof the concepts and axioms, and with the notion of congruence formulated³² The lecture notes are Hilbert ( ∗ 1902), in Chapter 6 of Hallett and Majer (2004); Hilbert’s paperis Hilbert (1902/1903). This paper was reprinted as Anhang II to the editions of the Grundlagen derGeometrie from the Second on. An Appendix (mentioned below) was made to the first reprinting inHilbert (1903), and then a note (also mentioned below) was added to the Sixth Edition publishedin 1923, i.e. Hilbert (1923). The Appendix was radically revised by Arnold Schmidt for the SeventhEdition (Hilbert 1930) oftheGrundlagen, as Hilbert makes clear in the Preface, this being the lastedition of the monograph to be published in Hilbert’s lifetime. We will actually concentrate here onHilbert’s presentation in his 1905 lectures on the ‘Logische Principien der Mathematik’, since of all theversions, this is the one where Hilbert is most philosophically expansive about the results and whatthey show. These lectures will appear in Ewald, Hallett and Sieg (Forthcoming).