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Recently I have come across the interesting debate of Frege and Hilbert regarding the Foundations of Geometry. It seems to me that the main concern of Frege was on the Logical Consistency of Hilbert's Geometry and in this sense I have found Frege's concept of "consistency" to be a very natural one (although "natural" doesn't necessarily carry exactly the same meaning to all). Moreover, it seemed to me that the three main targets of Frege's criticism all makes sense (at least intuitively). His criticisms mainly concentrated upon,

  1. Hilbert's claim that the axioms of his geometry are definitions.

  2. Hilbert's method of proving the consistency and independence of his set of geometric axioms

  3. Hilbert's doctrine that if a set of axioms is consistent, then the axioms are "true" and the things defined by the axioms exist.

My question is as follows,

Is there any solution to the second of Frege's criticisms without the application of Model Theory?


A Brief Explanation of Frege's Concept of "Consistency". For Frege a set of sentences is consistent if (roughly) it is not possible to deduce a contradiction from it using logic and the definitions of the terms in the sentences. Similarly, a sentence $\phi$ is independent of a set of sentences $\Gamma$ if (roughly) it is not possible to deduce either $\phi$ or the negation of $\neg\phi$ from $\Gamma$ using rules of logic and the definitions of the terms in $\phi$ and the members of $\Gamma$. In Frege's sense of the term 'consistent', then, to say that a set of sentences is consistent is to attribute senses in such a way that the truth of all these sentences is logically compatible with the meanings they have. Indeed, he said,

"If we take the words 'point' and 'straight line' in Hilbert's so-called Axiom II. 1 in the proper Euclidean sense, and similarly the words 'lie' and 'between', then we obtain a proposition that has a sense, and we can acknowledge the thought expressed therein as a real axiom. ... Now if one has acknowledged [II.1] as true, one has grasped the sense of the words 'point,' 'straight line,' 'lie,' 'between'; and from this the truth of [II.2] immediately follows, so that one will be unable to avoid acknowledging the latter as well. Thus one could call [II. 1] dependent upon [II.1]"

A Brief Explanation of Hilbert's Concept of "Consistency". On the other hand, Hilbert's proofs can be regarded as straightforwardly model-theoretic. Within the framework of first-order logic, a set of sentences is by definition "consistent" iff there is an interpretation (or structure) in which the set of sentences is true, and to prove the consistency of a set of axioms, one need only show that there is a model or structure in which the axioms are all true. Thus, Hilbert proved the consistency of his set of geometric axioms by constructing a model of the axioms from the real number system.


So, in particular it seems to me that (and I am not an expert) the point of dispute between Hilbert and Frege was their different philosophies regerading the axioms of Euclidean Geometry. For Hilbert, the axioms would characterize a kind of structure. Since the sentences of Hilbert's new geometry are uninterpreted sentences, the theorems of the geometry turn out to be not even true statements. But on the contrary, Frege viewed geometry as a theory of physical space.

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    $\begingroup$ For the sake of self-containment, could you tell us what Frege's concept of "consistency" was and what Frege and Hilbert disagreed about? (What I find somewhat interesting about Hilbert's axioms is that he was deliberately trying to avoid using any concept of real numbers in his axiomatization of geometry. Euclid used no such concept because he had none; yet he was able to do something that in modern terminology we describe as proving that the ratio of lengths of the diagonal of the square to the side is irrational.) ${}\qquad{}$ $\endgroup$ – Michael Hardy Oct 21 '15 at 5:22
  • $\begingroup$ I don't know what there is to "solve"; it sounds like they're just using the same words with different meanings. $\endgroup$ – Eric Wofsey Oct 21 '15 at 7:04
  • $\begingroup$ @EricWofsey: I don't understand. Can you elaborate a bit? $\endgroup$ – user 170039 Oct 21 '15 at 7:06
  • $\begingroup$ Frege is assuming that the terms used in Hilbert's axiomatization must be given "natural" meanings, and Hilbert is not. This is no more of a problem than the fact that Americans use the word "chips" with a different meaning than British people. They should just be clear which meaning they are using when they try to communicate with each other. $\endgroup$ – Eric Wofsey Oct 21 '15 at 7:24
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    $\begingroup$ Well, I don't know what Hilbert said at the time, but the modern response would be that the axioms come along with some rules of logical deduction which allow you to prove theorems from them. The difference between Hilbert and Frege's approach is that in Hilbert's we can write down these rules explicitly and precisely (though Hilbert himself may not have done so at the time), whereas Frege's interpretations seem to depend on some ill-defined intuition about what these words mean. $\endgroup$ – Eric Wofsey Oct 21 '15 at 9:10

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