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David Hilbert's Lectures on the Foundations of Geometry, 1891-1902 (English and German Edition)

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This book is of course invaluable for studying the development of Hilbert's Grundlagen. It also contains lectures that are not directly related to foundations, such as excerpts from a more conventional projective geometry course of 1891 (apparently taught to two students and "ein für Geometrie interessierter Mahler") and a few delightful Ferienkurs lectures (including for example three reasons why number theory is the queen of mathematics, pp. 154-156).

The foundations lectures naturally differ from the published Grundlagen in that they contain things that were already well known, such as discussions of non-Euclidean geometry. The first set of lecture notes on the foundations of geometry is from 1893/94. Most aspects of Hilbert's Grundlagen are here already. The story is still thoroughly enjoyable, however, and greatly enhanced by the editors meticulous attention to pointless details (including things like the colour of the ink Hilbert used).

Consider for example the case of the amphibiousness of Desargues's theorem. 3D Desargues is obvious, so Desargues must hold in any sensible 3D geometry. Could it be that Desargues is not only necessary but also sufficient to create a sensible 3D space, i.e., is it true that "if the Planar Desargues Theorem is added as a new postulate to the planar order and incidence axioms, then this will yield spatial consequences, so that, in particular, the full set of spatial incidence and order axioms will hold"? In his 1898/99 lectures Hilbert thinks the answer is probably yes: "Diese Frage ist wahrscheinlich zu bejahen". Apparently he did not have a proof at this stage but he soon found one; in fact, so soon that the proof appears later in the same course, prompting someone to go back to the Lesezimmer notes and cross out "wahrscheinlich" and underline "zu bejahen", "in rough hand, unlike the usual underlining of the Ausarbeitung, which is done carefully with a straightedge."

The fact that we used spatial methods to prove Desargues raises the question of the purity of method ("Reinheit der Methode"): Could Desargues be proved from plane axioms alone? We prove that the answer is no, using a contrived model later replaced by the Moulton plane. In this connection Hilbert's notes read: "It is fashionable to always guarantee the purity of method. In fact, this is appropriate: we are often not satisfied when a proof in number theory uses geometry or geometrical truths of function theory ... [But a detailed study may reveal] a deeper, legitimate basis and beautiful and fruitful connections, e.g. primes and the zeta function, potential theory and analytic functions, etc."

An appendix lists all Hilbert's lecture courses throughout his career. Hilbert was a real, classical professor. Unlike the disgraceful "professors" of today who whine about their teaching "load", he did not disrespect teaching and learning. He taught widely not only in every area of mathematics (including basic calculus courses several times as a full professor) but also in physics (mechanics, hydrodynamics, potential theory, elementary particles, electromagnetism, special and general relativity, quantum mechanics) and the philosophy of mathematics and science.