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

purity of method in hilbert’s grundlagen 251Hilbert remarks that there are in principle three different ways in whichone might provide the basis of the theory of number: ‘genetically’ as wascommoninthe19th century; through axiomatization, Hilbert’s preferredmethod whenever possible; or geometrically. With respect to the latter, aftersketching how such a reduction might in principle proceed, Hilbert says thefollowing:The objectionable and troublesome [mißliche] thing in this can be seen immediately:it consists in the essential use of geometrical intuitions and geometricalpropositions, while geometry and its foundation are nevertheless less simple thanarithmetic and its foundations. One must also note that to lay out a foundation forgeometry, we already frequently use the numbers. Thus, here the simpler wouldbe reduced to the more complicated, or in any case to more than is necessary forthe foundation. (Hilbert, ∗ 1905, p.9)But while the foundational investigation of geometry presupposes arithmetic(and analysis), there was at this time no similar foundational investigation ofarithmetic, and no investigation of the conceptual connection between moreelementary parts of arithmetic and higher arithmetic and analysis.⁵³ In particular,this complex of theory was not subject to axiomatic analysis and indeed noteven axiomatized. In the course of Hilbert’s work on geometry, he doesaxiomatize an important part of it, namely the theory of ordered fields, mainlyfor the purpose of revealing certain analytic structure in the geometry ofsegments in the analysis of Desargues’s Theorem, giving rise to the systemof complete, ordered fields published in Hilbert (1900a). Nevertheless, thetheory of natural numbers was not treated axiomatically by Hilbert untilvery much later. And the important extensions of Archimedean and non-Archimedean analysis involving (say) complex function theory were nevertreated by Hilbert as axiomatic theories. Of course, what is in question inthe work examined here are certain aspects of the foundations of geometry.Nevertheless, Hilbert was well aware that the results garnered are in a certainstrong sense relative, and that the foundational investigation of geometry mustbe part of a wider foundational programme. Indeed, Hilbert’s famous lectureon mathematical problems from 1900 sets out (as Problem 2) preciselytheproblem of investigating the axiom system for the real numbers, in particularshowing the mathematical existence of the real numbers, where there is norecourse to a natural theoretical companion such as is possessed by syntheticgeometry. Thus, a limited foundational investigation gives birth to anothermore general one.⁵³ In the 1920s, Hilbert stated decisively his rejection of the Dirichlet thesis, though it is not clearwhen he abandoned it.