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

212 michael hallettintrinsic to which is the view that in general no one interpretation of anaxiom system is privileged above others, despite what might seem like theoverwhelming weight of the interpretation underlying the ‘facts’ as originallygiven: thus in the case of geometry the weight of the ‘intuitive’ or ‘empirical’origins. Thus, we see a radical departure from the kind of enterprise Frege andPasch (and even Euclid) were engaged in, enterprises part of whose very pointwas to explain (and thereby delimit) the primitives. Hilbert’s axiomatic methodabandons the direct concern with the kind of knowledge the individualpropositions represent because they are about the primitives they are, andconcentrates instead on what he calls ‘the logical relationships’ between thepropositions in a theory.Hilbert’s fundamental supposition of foundational investigation, going backto 1894 and stated repeatedly thereafter, is that a theory is only ‘a schema ofconcepts’ which can be variously ‘filled with material’. He says:In general one must say: Our theory furnishes only the schema of concepts, whichare connected to one another through the unalterable laws of logic. It is left tothe human understanding how it applies this to appearances, how it fills it withmaterial [Stoff]. This can happen in a great many ways. (Hilbert ( ∗ 1893/1894,p. 60), or Hallett and Majer (2004, p.104))In the 1921/1922 lectures already referred to, Hilbert calls axiomatization ofthis kind a ‘projection into the conceptual sphere’:According to this point of view, the method of the axiomatic construction of atheory presents itself as the procedure of the mapping [Abbildung] ofadomainofknowledge onto a framework of concepts, which is carried out in such a way thatto the objects of the domain of knowledge there now correspond the concepts,and to statements about the objects there correspond the logical relations betweenthe concepts.Through this mapping, the investigation is completely severed from concretereality [Wirklichkeit]. The theory has now absolutely nothing more to do withthe real subject matter or with the intuitive content of knowledge; it is a pureGedankengebilde [construct of thought] about which one cannot say that it is trueor it is false. (Hilbert ( ∗ 1921/1922, p.3), forthcoming in Ewald and Sieg (2008))It is not that a mathematical theory in this sense has nothing to do with reality;indeed it may have more to do with it, for the connections might be established‘in a great many ways’, to use Hilbert’s phrase from 1894.It is important to see how this view fits with the insistence in the 1890s,outlined above, that the root of geometry is empirical, and that geometry is,as Hilbert frequently put it, the ‘most perfect natural science’. For Hilbert,geometry is a natural science primarily because it can be applied to natureto furnish a more or less accurate description of it. The term ‘description’