Comment by pron
17 hours ago
Oh, if you mean that most algorithm correctness proofs are finitary and therefore don't need to explicitly rely on the excluded middle, that may well be the case, but they certainly don't try to avoid it either. Look at any algorithm paper with a proof of correctness and see how many of them explicitly limit themselves to constructive logic. My point isn't that most algorithm/program proofs need the excluded middle, it's that they don't benefit from not having it, either.
> My point isn't that most algorithm/program proofs need the excluded middle, it's that they don't benefit from not having it, either.
Because if they benefited from it (in surfacing computational content, which is the whole point of constructive proof) they'd be comprised within the algorithm, not the proof.
> in surfacing computational content, which is the whole point of constructive proof
The point of a constructive proof is that the proof itself is in some way computational [1], not that the algorithm is. When I wrote formal proofs, I used either TLA+ or Isabelle/HOL, neither of which are constructive. It's easy to describe the notion of "constructive computation" in a non-constructive logic without any additional effort (that's because non-constructive logics are a superset of constructive logics; i.e. they strictly admit more theorems).
[1]: Disregarding computational complexity
> When I wrote formal proofs, I used either TLA+ or Isabelle/HOL, neither of which are constructive.
True, but this requires using different formal systems for the algorithm and the proof. Isabelle/HOL being non-constructive means you can't fully express proof-carrying code in that single system, without tacking on something else for the "purely computational" added content.
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