The Wikipedia article on Euler's totient function lists, in the Growth rate section, the following: $$\varphi(n)<\frac{n}{e^\gamma\log\log n}\quad\text{ for infinitely many }n$$ and says: "The method of proof is interesting, in that the inequality is shown first under the assumption that the Riemann hypothesis is true, secondly under the contrary assumption".
If I'm understanding the intuitionist thesis, propositional formulae in intuitionistic logic are not assigned a definite truth value and are only considered "true" when we have direct evidence, hence proof (from Wikipedia).
We don't have proofs that the Riemann hypothesis is true or that it's false so I'm inlcined to think that intuitionist logic rejects this "interesting method of proof" but, in any case (RH is true or RH is false) we do have an explicit construction that ensures the truth of the proposition.
What is the "status" of this type of proof under an intuitionist/constructivist viewpoint?? (other philosophies are of course also welcome for completeness)
Maybe this question belongs in the Philosophy Stack Exchange and if so, make me know please
Thanks!
That kind of proof doesn't work in intuitionistic logic; given $p \implies q$ and $(\neg p) \implies q$, we can't conclude $q$.
If this kind of proof were valid, we could prove the law of excluded middle, $r \lor \neg r$, by setting $p = r$ and $q = r \lor \neg r$.