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

purity of method in hilbert’s grundlagen 217to the completion is axiomatized; this new system is consistent, since it has amodel in the reals. As Hilbert says:Euclidean geometry exists, so long as we take over from arithmetic the proposition that thelaws of the ordinary real numbers lead to no contradiction. With this we have shown theexistence of all those other geometries which we have considered in the courseof this investigation. (Hilbert ( ∗ 1899, p.167); Hallett and Majer (2004, p.391).See also Hilbert’s own lecture notes, p. 104; in Hallett and Majer (2004, p.280).)From the point of view of this theory, any dispute then over the ‘real’ existenceof some points as opposed to others is ‘idle’; the theory exists, and it has theexistence of all these points as consequences; the real/ideal distinction wasalways only ever relative to the original domain. As Hilbert puts it later:The terminology of ideal elements thus properly speaking only has its justificationfrom the point of view of the system we start out from. In the new system we donot at all distinguish between actual and ideal elements. (Hilbert, 1919, p.149,book p. 91)¹⁹Furthermore, the core system of elementary geometry possesses many differentand incompatible ‘ideal extensions’, some of them giving rise to the ‘othergeometries’ Hilbert mentions. All ‘exist’ in so far as they are consistent.In short, advanced ‘pure’ mathematical knowledge is now knowledge aboutvarious conceptual schemes, and a significant part of this is logical knowledge,in the first instance, knowledge of the existence or non-existence of derivationsfrom certain starting points. It is the abandonment of the idea of fixed contentand the shift to the ‘conceptual sphere’ which allows Hilbert to introduce asophisticated, and relative, conception of the ‘real/ideal’ distinction, one whichis of great importance for his later foundational work.The arrival at this ‘pure thought form’ is, I think, the key to the correctexplanation for the progression from intuition to concept to ideal statedat the beginning of Hilbert’s Grundlagen in an epigram from Kant’s Kritikder reinen Verknunft.²⁰ And it is as a ‘pure thought form’, thus through theprojection into the ‘conceptual sphere’, that mathematics falls under the‘general law’ on the nature of the mind which Hilbert stated in his lectureon mathematical problems, namely that all questions ‘which it [the mind] asksmust be answerable’.Important in all this is the relationship between mathematical theories. Forone thing, the approximate (not to say coarse) nature of interpretation in¹⁹ See also p. 153 of the 1919 lecture notes (Hilbert, 1919, book p. 94).²⁰ The epigram is the following passage: ‘So fängt denn alle menschliche Erkenntnis mit Anschauungenan, geht von da zu Begriffen, und endigt mit Ideen.’ (See Kritik der reinen Vernunft, A702/B730.A very similar remark is also to be found at A298/B355.)