"WM" <email@example.com> wrote in message news:firstname.lastname@example.org... Matheology § 202
<snip altered cut and paste, + insert orgional text, thank you WM, for bringing this to my attention>
4. The Deeper Disagreement
The bulk of Frege's critique of Hilbert consists of criticizing Hilbert's lack of terminological clarity, particularly as this applies to the differences between sentences and various collections of thoughts. He takes Hilbert to task for misleadingly using the same sentences to express different thoughts, and points out repeatedly that Hilbert's use of axioms as definitions needs considerably more-careful treatment than Hilbert affords it. The more-substantial criticism flows naturally from this terminological critique: Frege takes it that once one disentangles Hilbert's terminology, it becomes clear that he is simply not talking about the axioms of geometry at all, since the sets of thoughts he actually deals with are the misleadingly-expressed thoughts about e.g. real numbers. And, adds Frege, one cannot infer the consistency of the geometric axioms proper from that of the thoughts Hilbert treats.
Frege's complaints against Hilbert essentially end here. Having pointed out what he takes to be the illegitimate shift in subject-matter from geometric thoughts to non-geometric ones, and noted that Hilbert's reinterpretation strategy will always introduce such an illegitimate shift, he takes himself to have discredited that strategy. The interesting philosophical question which receives considerably less emphasis from Frege is that of why, exactly, the shift is illegitimate. Why is it that the consistency of AXG doesn't follow from that of the structurally-similar AXR, particularly when each of these sets is expressible via the same set AX of sentences?
We should note, to begin with, that from Frege's point of view the burden of argument is squarely with Hilbert: if Hilbert thinks that the consistency of AXG follows from either the consistency of AXR or from the truth of AXR's members, then it is up to Hilbert to show this. Frege does not go out of his way to demonstrate that the crucial inference is invalid, but seems to take his point to have been essentially made once he has pointed out the need for a justification here.
From Hilbert's point of view, of course, there is no need for such a justification. The differences that Frege insists on over and over again between the sets of sentences (AX) and the different sets of thoughts (AXG, AXR etc.) are entirely inconsequential from Hilbert's standpoint. Because consistency as Hilbert understands it applies to the "scaffolding" of concepts and relations defined by AX when its geometric terms are taken as place-holders, the consistency he has in mind holds (to put it in terms of thoughts) of AXG iff it holds of AXR, since both sets of thoughts are instantiations of the same "scaffolding." The same point can be put in terms of sentences: Frege insists that the consistency-question that arises for the sentences under their geometric interpretation is a different issue from the one that arises for those sentences under their real-number interpretation; for Hilbert on the other hand, there is just one question, and it is answered in the affirmative if there is any interpretation under which the sentences express truths. Hence while Frege takes it that Hilbert owes an explanation of the inference from the consistency of AXR to that of AXG, for Hilbert there is simply no inference.
We turn now to the more substantial issue, namely, why the inference from the consistency of AXR to that of AXG is in fact fallacious from Frege's point of view. Frege clearly takes it that the consistency of the set of thoughts expressed by a set ? of sentences is sensitive not just to the overall structure of those sentences, but also to the meanings of the non-logical (here, geometrical) terms that appear in the members of ?. What we need to understand, in order to see why this should be the case for him, is how Frege understands the relationship between the meanings of terms and the logical implications that hold between thoughts expressed using those terms. This relationship comes out most clearly when we turn to Frege's method of demonstrating that a given thought follows logically from other thoughts. In general, for Frege, we can show that a given thought ? follows logically from a set T of thoughts via a two-step procedure in which we (i) subject ? and/or the members of T to conceptual analysis, bringing out previously-unrecognized conceptual complexity in those thoughts, and (ii) prove the thus-analyzed version of ? from the thus-analyzed members of T. The clearest examples of this procedure appear in Frege's work on arithmetic. Frege holds for example that the thought expressed by
The sum of two multiples of a number is a multiple of that number
follows logically from the thoughts expressed by
(?m)(?n)(?p)((m+n)+p = m+(n+p)) and by
(?n)(n = n+0).
He demonstrates this by providing a careful analysis of the notion of "multiple of" in terms of addition, giving us in place of (i) a more-complex (i') which is then derived from (ii) and (iii). Similarly, a significant part of Frege's logicist project consists of the careful analysis of such arithmetical notions as zero and successor, analysis which brings out previously-unnoticed complexity, and facilitates the proof of arithmetical truths. (For a discussion of the logicist project, see Frege.)