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

224 michael hallettfor the first time we subject the means of carrying out a proof to a critical analysis.(Hilbert ( ∗ 1898/1899, p.30), p. 236 in Hallett and Majer (2004))Then follow the general remarks about ‘purity of method’ which were quotedearlier (see p. 282). What this passage raises is the question of the appropriatenessof a given proof, a ‘purity’ question par excellence, for here we have to do witha theorem, the Planar Desargues’s Theorem, which is proved by means whichat first sight might be thought inappropriate, owing to the proof’s appeal tospatial axioms.What we are faced with first is an independence question, and Hilbert setsout to show that the proof of the Planar Desargues’s Theorem is not possiblewithout appeal to spatial incidence axioms. As he puts it:We will rather show that Desargues’s Theorem is unprovable by means of I 1–2,II 1–5. One will thus be spared the trouble of looking for a proof in the plane.For us, this is the first, simplest example for the proof of unprovability. Indeed:Tosatisfy us, it is necessary either to find a proof which operates just in the plane, or toshow that there is no such proof. Prove so, that we specify a system of things = pointsand things = planes for which axioms I 1–2, II1–5 hold, but the DesarguesTheorem does not, i.e., that a plane geometry with the axioms I, II is possiblewithout the Desargues Theorem. (Hilbert ( ∗ 1898/1899, p.30), pp. 236–237 inHallett and Majer (2004))To show this, Hilbert constructs an analytic plane of real numbers, with theclosed interval [0, +∞] removed. The key thing is that some of the newstraight lines are gerrymandered compositions: below the x-axis, the line is anordinary straight (half-)line; above the x-axis, the line is in fact an arc of a circleuniquely determined by the conditions (a) that it goes ‘through’ the origin,O (it is open-ended here, since O is not itself in the model), and (b) that thegiven straight line below the x-axis is a tangent to it. Hilbert’s figure (Fig. 8.2)illustrates the essentials of the model.²⁷The first thing to note about this is that we are here operating at one removefrom the intuition ordinarily thought to underlie projective geometry. For onething, the model produces highly unintuitive ‘straight lines’, even though theyare pieced together using intuitable objects. The gerrymandered ‘straight lines’are again ‘intuitable’ in the sense that one can easily visualize them (witnessFig. 8.2). Moreover, there is a sense in which lines made up of two distinctpieces are extremely familiar to us; think of a straight stick (a line segment) halfimmersed in still, clear water and then viewed from above looking down intothe water. (Indeed, if it is viewed when the water is not still, then at any instant²⁷ For full details, see Hilbert ( ∗ 1898/1899, 31), or the Ausarbeitung, p.28; these are pp. 237 and 316respectively in Hallett and Majer (2004).