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

purity of method in hilbert’s grundlagen 239by using very sophisticated mathematical analysis: complex analysis over anon-Archimedean field. In other words, the higher mathematical/logicalanalysis based on complex numerical structures instructs and informs lowerlevelgeometrical intuition. This, note, is just such a case where ‘functiontheory’ is used to show something about elementary geometry. (See thequotation from Hilbert on p. 201 above.)In short, this example illustrates Hilbert’s view that one is guided bygeometrical intuition, one asks questions suggested by intuition, but in the endit is higher mathematics which instructs intuition, not the other way around.Thus, while Hilbert does carry out a kind of ‘purity of method’ investigation,it is much more focused, as he puts it, on the ‘analysis of intuition’. One ofthe reasons why Hilbert thinks that intuition requires analysis is that it is not,for him, a certain source of geometrical knowledge, and certainly not a finalsource. Thus the analysis, which is designed to throw light on the question:what is one committed to exactly when one adopts certain principles, amongthem principles suggested by intuition?8.4.3 The Three Chord TheoremThe third example considered here concerns another fairly elementary geometricaltheorem, which says that the three chords generated by three mutuallyintersecting circles (lying in the same plane) always meet at a common point.Call this the Three Chord Theorem (TCT). (See Fig. 8.5.)The theorem is not an ancient one, but was apparently first discovered byMonge in the middle of the 19th century. It has an interesting generalization,much studied by 19th century geometers, concerning the lines of equal powerFig. 8.5. Diagram of the Three Chord Theorem, adapted from the Ausarbeitung ofHilbert’s 1898/1899 lectures, p.61, (Hallet and Majer (2004, 335)).