Put another way: Can every mathematical proof be reformulated to be about some class of Turing Machines?
Any proof of the existence of infinite prime numbers is equivalent to the statement: "Any Turing Machine whose instructions amount to 'print the highest prime number' does not halt."
I say they are equivalent because any proof of the existence of infinite primes seems to verify the statement about the Turing Machine, and vice versa.
I feel like I can certainly do this with any algebraic, geometric, arithmetic, question that I am aware of.
I am considering putting such a remark into a paper I'm writing, but I'm not sure if such a remark is:
a) Obvious b) True, but requires the articulation of sophisticated ideas. In this case, I would love a reference. c) Subjective or at least philosophically contentious d) False, but requires the articulation of sophisticated ideas. e) Obviously False