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,
Hilbert's claim that the axioms of his geometry are definitions.
Hilbert's method of proving the consistency and independence of his set of geometric axioms
- 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.