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

220 michael hallettso, whether one succeeds or fails. The point is that it is not the central goalof foundational investigation. This means that, to a large extent, Frege-styledefinitions can be dispensed with. They appear now rather as assignments forthe purpose of modelling the principles being investigated. The prime exampleis perhaps Hilbert’s use of the theory of real numbers (or rather a minimalPythagorean field drawn from them) to model the axioms of elementarygeometry by giving ‘temporary’ definitions of point, straight line, congruence,and so on; this is in place of the standard procedure behind analytic geometry,which effects a reduction of geometry to analysis by making these assignments asFrege-style definitions. Similarly, one could use the Dedekind-cut constructionor the Cauchy-sequence construction to model the axioms for the theory ofreal numbers, rather than to define them, and likewise with the ‘definitions’ ofthe integers as equivalence classes of ordered pairs of natural numbers, complexnumbers as ordered pairs of reals, and so on.²²There is a further important point to be made. Even if we seek a conceptualreduction, it is important to have available ‘local’ axioms, for instance for thenatural numbers or for the reals. To take an example, in seeking to show inhis 1898/1899 lectures (and later the Grundlagen) that line segments themselvesexhibit a field structure like that of the real numbers, it was necessary for Hilbertto have a detailed axiomatization of field structure, in order to say that the fieldsare alike in certain respects but differ in others. This is the origin of Hilbert’saxiomatisation of the real numbers in the lecture notes, published as axioms forwhat Hilbert calls ‘complex number systems’ in the Grundlagen of 1899, andthen completed in (Hilbert, 1900a). The point is simple: to show that a conceptual‘reduction’ (like Frege’s) has worked, one has to be able to derive theoremswhich say that the defined objects (e.g. the numbers) have the right properties.²³Thus, one has to have in effect an axiom system, and this is overriding.Frege sought a reduction to fewer and fewer principles, as in a sense didEuclid; Hilbert’s work on the other hand shows that, in some key cases if²² Hilbert’s later paper ‘Axiomatisches Denken’ makes the parallel point that the uncovering of ever‘deeper’ primitives is a constant theme in the development of mathematics and physics. The papereven seems to suggest that the use of the Axiomatic Method incorporates such a search; he uses thephrase ‘Tieferlegung der Fundamente’. The point to stress, though, is that there is never thought to be anultimate conceptual layer, and the ‘foundations’ for a theory might be given, and productively so, indifferent, even incompatible, ways. As Boolos observes with respect to the Frege analysis of numberand the attempt to reduce arithmetic to logic:Neither Frege nor Dedekind showed arithmetic to be part of logic. Nor did Russell. Nor didZermelo or von Neumann. Nor did the author of Tractatus 6.02 or his follower Church. Theymerely shed light on it. (Boolos, 1990, 216–218).²³ Frege seeks to establish just this in his (1884, §§78–83).