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

218 michael hallettthe observable world means that these interpretations are less than ideal asways of demonstrating ‘possibility’, and the same holds for investigations ofindependence. Secondly, the process of ideal extension as described aboveappears to rely heavily on the notion of embedding one mathematical theoryin another, and indeed Hilbert himself describes the procedure in just this wayin his 1919 lectures:Precisely stated, the method [of ideal elements] consists in this. One takes a systemwhich behaves in a complicated way with respect to certain questions which wewant to answer, and one transforms to a new system in which these problemstake on a simpler form, and which in addition has the property that it containsa sub-system isomorphic to the original system. ... The theorems concerning theobjects of the original system are now special cases of theorems of the new system,in so far as one takes account of the conditions which characterise the sub-systemin question. And the advantage of this is that the execution of the proofs takesplace in the new system where everything becomes much clearer and easier totake in [übersichtlich]. (Hilbert, 1919, pp. 148–149, pp. 90–91 of the book)This procedure can be brought out, too, by looking more closely at thegeneral form of argument in the independence and relative consistency proofs,a form first articulated by Poincaré in one of his explanations of the proof ofthe independence of the Parallel Postulate.²¹ The general procedure is this.Proving the independence of Q from P 1 , P 2 , ... , P n with respect to a givenconceptual scheme T 1 , that is, showing that there cannot be a derivation ofQ from P 1 , P 2 , ... , P n , involves showing that, if there were, there would thenalso be a derivation of τ(Q) from τ(P 1 ), τ(P 2 ), ... ,τ(P n ) with respect toanother conceptual scheme T 2 ,where:(a) we know that each of the τ(P i ) aretheorems of T 2 ;(b) we know that the conceptual scheme T 2 can already derive¬τ(Q); (c) we assume (or know) that the scheme T 2 is not in fact inconsistent.Here, crucially, τ(P) is some map from formulas to formulas which preserveslogical structure. (This point was first articulated in Frege (1906).) An obviousvariant of this procedure can be used to prove the relative consistency of theoryT 1 with respect to T 2 . Hilbert puts it this way: After specifying a minimalPythagorean (countable) field of real numbers which determines an analyticmodel of the axioms for Euclidean geometry, he continues:We conclude from this that any contradiction in the consequences drawn fromour axioms would also have to be recognizable in the domain . (Hilbert(1899,p. 21); p. 455 in Hallett and Majer (2004))Thus, crucial to the investigation are what we might call base theories, thosethrough which the propositions of the ‘home theory’ are interpreted.²¹ See Poincaré (1891).