"Famously, Goedel’s incompleteness theorems refuted (naive) logicism, the view that mathematical truth is just provability.
But one doesn’t need all of the technical machinery of the incompleteness theorems to refute that. All one needs is Goedel’s simple but powerful insight that proofs are themselves mathematical objects—sequence of symbols (an insight emphasized by Goedel numbering). For once we see that, then the logicist view is that what makes a mathematical proposition true is that a certain kind of mathematical object—a proof—exists. But the latter claim is itself a mathematical claim, and so we are off on a vicious regress."
However I want to add that the idea of David Hilbert was to get to the basic axioms that Mathematics and Physics. Not that he was saying that those axioms could be proved. Leonard Nelson applied this idea to philosophy also. That is the point that Dr. Kelley Ross makes that to avoid a regress of reason one needs to start with immediate non-intuitive knowledge. However Dr Michael Huemer has a way of getting out of this problem by means of the idea that reason is just a faculty that recognizes universals. Not that reason is infallible. And the way it recognizes universals in by probability--not infallibility. [See his treatment of these issue.]
However I want to add that the idea of David Hilbert was to get to the basic axioms that Mathematics and Physics. Not that he was saying that those axioms could be proved. Leonard Nelson applied this idea to philosophy also. That is the point that Dr. Kelley Ross makes that to avoid a regress of reason one needs to start with immediate non-intuitive knowledge. However Dr Michael Huemer has a way of getting out of this problem by means of the idea that reason is just a faculty that recognizes universals. Not that reason is infallible. And the way it recognizes universals in by probability--not infallibility. [See his treatment of these issue.]
