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

purity of method in hilbert’s grundlagen 213is important. In the Vorrede to his lectures on mechanics published in 1877(Kirchhoff, 1877), Kirchhoff insists that physics should only set itself therestricted aim of describing the phenomena, and not that of trying to getat the underlying ‘causes’, since these frequently suffer from deep-seated‘conceptual unclarity’. In his 1894 lectures on the foundations of geometry,Hilbert refers directly to Kirchhoff’s aim of ‘describing’ only, and states thatcorrespondingly the aim of geometry is to ‘describe’ the ‘geometrical facts’.See Hilbert ( ∗ 1893/1894, p.7) or Hallett and Majer (2004, p.72). And thebasis of application in this sense is interpretation (or, as Hilbert says, ‘Deutung’),and this is essentially inexact. In these 1894 lectures, Hilbert says:With the axioms given hitherto, the existence [incidence] and position [order]axioms, we can already describe a large collection of geometrical facts andphenomena. We require only to take bodies for points, straight lines and planes,for the relation of passing through, touching, for being definite, immovable orfixed (perhaps, in the unrefined sense, when nudged by the hand). The bodies weshould think of as finite in number, and such that the axioms are satisfied underthis interpretation [Deutung] (for which, as one recognises, it will be necessaryto have for the bodies taken in place of points, straight lines, planes, somethinglike grains, rods or stretched threads or wire, cardboard) and indeed precisely.Then we know that all the propositions set up so far are also satisfied, and indeedprecisely satisfied.The direct continuation of this passage also makes it clear that applicationsonly hold approximately, and this is stated in one breath with the claim thattheories are only schemata of concepts, represented here by the ellipsis:If one finds that, with an application, the propositions are not satisfied (or notprecisely satisfied), this arises because an inappropriate application has been taken,i.e. the bodies, movement, touching do not agree with our scheme of axioms. Inthis case it will be necessary to replace the things: bodies, movable, touching, byothers, perhaps by smaller grains, blots [Klexe (sic)], tips, thinner wires, thinnercardboard, touching with firmer contact, movability [of the bodies] even whenwe blow on them [Anpusten], in such a way that the axioms are satisfied. Then weknow that the propositions also hold (precisely). ... But always when the axiomsare satisfied, the propositions also hold. The easier and more far reaching theapplication, then so much better ∗ )thetheory.∗ ) All systems of units and axioms which describe the phenomena are equallyjustified. Show nevertheless that the system given here is in a certain respectthe uniquely possible one. (Hilbert ( ∗ 1893/1894, pp. 60–60A), pp. 104–105 inHallett and Majer (2004))Hilbert repeatedly stresses the inexactness of application, for example on p. 92of the 1893/1894 lectures (Hallett and Majer, 2004, p.122), or p. 106 of the