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

216 michael hallettThe things with which mathematics is concerned are defined through axioms,brought into life.The axioms can be taken quite arbitrarily. However, if these axioms contradicteach other, then no logical consequences can be drawn from them; the systemdefined then does not exist for the mathematician. (Hilbert ( ∗ 1902, p.47) orHallett and Majer (2004, p.563))This complex of views is a very important foundation for Hilbert’s theory ofwhat he calls ‘ideal objects’, where the concepts, freed from the constraints oftheactual,are‘completed’.Inhis1898/1899 lectures, Hilbert notes that whilethe Archimedean Axiom as sole continuity axiom shows that to every point,there corresponds a real number, the converse will not generally hold:That to every real number there corresponds a point of the straight line doesnot follow from our axioms. We can achieve this, however, by the introductionof ideal (irrational) points (Cantor’s Axiom). It can be shown that these ideal pointssatisfy all the axioms I–V; it is therefore a matter of indifference whether weintroduce them first here or at an earlier place. The question whether these idealpoints actually ‘exist’ is for the reason specified completely idle [völlig müssig]. Asfar as our knowledge of the spatial properties of things based on experience isconcerned the irrational points are not necessary. Their use is purely a matter ofmethod: first with their help is it possible to develop analytic geometry to its fullest extent.(Hilbert ( ∗ 1899, pp. 166–167); Hallett and Majer (2004, p.391))Thus, while one might start from the idea that the points in geometrycorrespond to points whose existence in actual space can be shown, perhapsthrough construction, this idea is left behind; as Hilbert says, it is ‘idle’ toconsider the question of whether the new points ‘actually exist’.¹⁸ It is preciselyin this context that Hilbert first introduces his new notion of mathematicalexistence; thus, full Euclidean geometry ‘exists’ because we can furnish amodel for the theory by using analysis. Put more abstractly, the procedureis this. We begin with a Euclidean geometrical system without continuityaxioms, and where ‘real’ points are instantiated by geometrical constructions.These constructions correspond to various algebraic fields over the rationals(depending on what is permissible in construction). These number fields areseen to have a maximal extension in the reals; therefore, we postulate that thereare points corresponding to all real numbers. These new objects are then ‘ideal’with respect to the original, real points. The geometrical system corresponding¹⁸ This view, that we can decide to extend the field of objects without being constrained by what‘really exists’, is very much in evidence in Dedekind’s 1872 memoir on continuity and irrationalnumbers; see Dedekind (1872, 11). The most notable difference compared with Hilbert is in theidea that objects are ‘created’ to fill the ‘gaps’; Hilbert’s reliance on consistency avoids this notion of‘creation’.