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

purity of method in hilbert’s grundlagen 221we have more axioms, then our logical analysis can be more refined. A niceexample is provided by the maintenance of the Archimedean Axiom as anindependent axiom while seeking an additional axiom which leads to pointcompleteness.A natural axiom would have been some version of the limitaxiom which Hilbert had used in earlier axiomatizations (e.g. in his 1893/1894lectures, in Hilbert (1895), and in his 1898 Ferienkurs), and which is evenpresented as a possibility in the 1898/1899 lectures. But this axiom would‘hide’ the Archimedean Axiom because it would imply it. Instead, Hilbertretains the Archimedean Axiom and seeks an axiom which will complementit and boost its power to full completeness. The axiom he chooses is his own‘Vollständigkeitsaxiom’.²⁴ Many of Hilbert’s metamathematical investigationsconcern the precise role of the Archimedean Axiom in the proofs of centraltheorems, typically whether, when used, its use is necessary or not. Oneexample is given by Dehn’s work on the Legendre Theorems and the Angle-Sum Theorem (Dehn, 1900), another concerns the involvement of the axiomin the theories of plane area and three-dimensional volume. The axiom andthe requisite analyses are also clearly important in the development of non-Archimedean geometries; and its presence without the Completeness Axiomis very important in the search for countable models. Yet another exampleis provided by Hilbert’s own work on the Isoceles Triangle Theorem to beconsidered below in Section 8.4.2.The point about the definitions, and this point about not reducing theconceptual level, are, of course, closely related. The creative purpose behinddefining, as Frege recognized, is that with properly chosen definitions, unprovablepropositions become provable, and therefore there is no need to takethem (and attendant propositions) as axioms. Examples abound, but a strikingone is furnished, of course, by Frege’s own work. With the Frege definitionof number, we can prove what is now known as Hume’s Principle (HP),²⁵ theprinciple in effect from which Frege arithmetic is derived. Without such aproof, we would have to take the principle as a primitive axiom, and the terms‘NxFx’ (i.e. the numbers) as primitive terms, contrary to Frege’s intention.However, it might be argued that the really revealing thing about Frege’swork is not that, with the Frege definition, HP is provable, but rather thatthe central arithmetic principles can be proved from HP alone, given Frege’sother definitions.²⁴ See my Introduction to Chapter 5 of Hallett and Majer (2004, §5, 426–435).²⁵ That is, the second-order proposition: ‘∀F, G[NxF = NxG ↔ F ≈ G]’, where ‘F’, ‘G’ arevariables for concepts, ‘NxF’ is a singular term standing for ‘the number of things falling under theconcept F’, and where ‘≈’ stands for the (definable) relation of equinumerosity between concepts. SeeFrege (1884, §63).