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

200 michael hallettall these questions on an objective footing requires gathering certain kinds ofinformation. For example, we require answers to questions of the followingsort: Can two different ways of developing a theory match each other inthe kinds of theorem they can prove? Can a certain theorem only be provedby using the extended theoretical means? Are the connections so establishedaccidental, or can one find a deeper theoretical reason for them?² To answersuch questions as these, as Hilbert suggests, is just to pursue the ‘basic principle’set out in the Grundlagen der Geometrie. Certainly this is what Hilbert doessay with respect to the principle’s application in the case of his work ongeometry:In fact, the geometric investigation carried out here seeks in general to cast lighton the question of which axioms, assumptions or auxilliary means are necessary inthe proof of a given elementary geometrical truth, and it is left up to discretionaryjudgement [Ermessen] in each individual case which method of proof is to bepreferred, depending on the standpoint adopted. (Loc. cit., pp. 89–90, p.526 inHallett and Majer (2004))In other words, in geometry, no general decision is to be made as to what isto be preferred and what not. The purpose of the foundational investigation,neutrally stated as it is in the Grundlagen der Geometrie, is therefore to assesswhat we might call the ‘logical weight’ of the axioms and central theorems.This, in any case, is what there is in abundance in Hilbert’s work on geometry.We will come to some examples in due course.The Grundlagen itself was immediately preceded by a long series of lecturesheld in 1898/1899, which Hilbert entitled ‘Elemente der EuklidischenGeometrie’. These lectures contain most of what is novel in the Grundlagen,but they contain also many more philosophical and informal remarks, and arevery differently arranged from the presentation in the Grundlagen der Geometrie.The notes for the lectures exist in two different forms: 110 pages of Hilbert’sown notes, and then a beautifully executed protocol of the lectures followingthe notes very carefully (in German, an ‘Ausarbeitung’), which Hilbert hadcommissioned from his first doctoral student at Göttingen, Hans von Schaper,whose own field of research was analytic number theory, specifically the PrimeNumber Theorem.³ Towards the end of these notes, there are several passageswhich are clearly the origin of the citations from the Grundlagen given above.² Recall the emphasis in Hilbert’s paper of 1918 on the ‘Tieferlegung der Fundamente’. See Hilbert(1918).³ For a more detailed description of the relationship between the 1898/1899 lectures and theGrundlagen of 1899, as well as that between the notes and the Ausarbeitung, see my Introduction toChapter 4 in Hallett and Majer (2004).