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

214 michael hallett1898/1899 lectures (Hallett and Majer, 2004, p.283). And he even states theapproximate nature of the physical interpretations of geometry as a reason whywe require the logical development of geometry separately:In physics and nature generally, and even in practical geometry, the axioms all holdonly approximately (perhaps even the Archimedean Axiom). One must howevertake the axioms precisely, and then draw the precise logical consequences, becauseotherwise one would obtain absolutely no logical overview. Necessarily finitenumber of axioms, because of the finitude of our thought. (Note on the frontcover of Hilbert’s copy of the Ausarbeitung. Date unknown, but after 1899, andvery probably before 1902; see Appendix to Chapter 4 in Hallett and Majer(2004, p.401), remark [15].)Two things are immediately clear from these passages. (1) Many interpretationsare possible, and even desirable, even where we are concerned apparentlywith only one general area of application (the application of the same theory tothe same spatial world). (2) There is an implicit assumption that the (unspecified)logical apparatus is sound, i.e. if the axioms hold under an interpretation, thenso do the theorems. (See also Hilbert’s letter to Frege of 29.xii.1899.) Mostimportantly, this latter means that the internal (logical) workings of thetheory, in short (‘pure’ or ‘free’) mathematics, can proceed independently ofany particular application, and this is the case even when one might firmlybelieve that application is the primary purpose. This is stressed by Hilbertin the continuation of the passage from the 1921/1922 lectures quoted onpage 296:Nevertheless this framework of concepts has a meaning for knowledge of theactual world, because it represents a ‘possible form in which things are actuallyconnected’. It is the task of mathematics to develop such conceptual frameworksin a logical way, be it that one is led to them by experience or by systematicspeculation. (Hilbert ( ∗ 1921/1922, p.3) inEwaldandSieg(2008))In short, interpretation (quite possibly manifold interpretations) can establishvarious connections to the world, and the more the better.¹⁷ Thus, themathematical theory is not determined by Wirklichkeit, it does not necessarilyextend our knowledge of it (it might or might not), and is in the endnot responsible to it. Mathematics can learn from intuition, observation, andempirical investigation more generally, but is not to be their slave, evenwhen they have played a major part in the establishment of the domain of‘facts’, and therefore in the axiomatization itself. A prime example is the¹⁷ See again Hilbert’s letter to Frege of 29.xii.1899. Inhisownletterof27.xii.1899, towhichHilbert’s is a reply, Frege had objected to the very idea of considering different interpretations forgeometry.