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

purity of method in hilbert’s grundlagen 225OFig. 8.2. Model for the failure of Desargues’s Theorem; diagram taken from p.31 ofHilbert’s 1898/1899 lecture notes on the foundations of Euclidean geometry.the upper part will appear as an ordinary straight line, while the lower partwill be a non-straight curved line. It might be said that what Hilbert has donehere is to take a simple example like this and to choose a particularly simplecurve.) However, while the direct perception (‘intuition’) of such objects isperhaps familiar, the analytic treatment is essential, and not merely a convenientway of proceeding. For one thing, the use of the algebraic manipulation isindispensable; it has to be shown that the model satisfies the plane axioms of Iand II, and this is by no means trivial. For instance, in considering Axiom I1,it has to be shown that any two points determine a straight line in the newsense, including the case where one point lies above the x-axis (has positivey-coordinate), and one point lies below the x-axis (has negative y-coordinate);in other words, it has to be shown that there is always a circle passing through0 and the upper point and which cuts the interval [−∞, 0) in a point below0 such that the tangent to the circle at that point is a straight line which passesthrough the given point in the lower half-plane. This, however, is correct, ascareful calculation shows.²⁸ It is hard to see how this could be accomplishedwithout the use of calculation. The point is that the use of the analytic planeand the accompanying algebra gives Hilbert extremely fine control over thepieces and how to glue them together in the right way, even though theresult (when transferred back to the intuitive level) is fairly easily understoodvisually.Hilbert’s model specifies the first example of a non-Desarguesian geometry.Hilbert’s treatment of the Desargues’s Theorem in the 1898/1899 notes is²⁸ Hilbert does not explicitly adddress this case, and I am grateful to Helmut Karzel for bringing itto my attention. It is more fully explained in my n. 46 to the text of Hilbert ( ∗ 1898/1899) in Hallettand Majer (2004, 237).