[tphyahoo2]

"In constructive mathematics, proof by contradiction, while not universally rejected, is treated with caution and often replaced with direct or constructive proofs."

  (gemini llm answer to google query: constructive math contradiction)

"Wiles proved the modularity theorem for semistable elliptic curves, from which Fermat’s last theorem follows using proof by contradiction."

  https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem

So, will the Lean formalization of FLT involve translation to a direct or constructive proof? It seems not, I gather the proof will rely on classical not constructive logic.

"3. Proof by Contradiction: The core of the formal proof involves assuming ¬Fermat_Last_Theorem and deriving a contradiction. This contradiction usually arises from building a mathematical structure (like an elliptic curve) based on the assumed solution and then demonstrating that this structure must possess contradictory properties, violating established theorems. 4. Formalizing Contradiction: The contradiction is formalized in Lean by deriving two conflicting statements, often denoted as Q and ¬Q, within the context of the assumed ¬Fermat_Last_Theorem. Since Lean adheres to classical logic, the existence of these conflicting statements implies that the initial assumption (¬Fermat_Last_Theorem) must be false."

(gemini llm answer to google query: Lean formalization of fermat's last theorem "proof by contradiction")

[semolinapudding]

FLT is a negative statement ("there are no nonzero integers x, y, z such that..."), and proofs by contradiction are constructively valid for proving negative statements.

[tphyahoo2]

Thanks for articulating this.

https://math.andrej.com/2010/03/29/proof-of-negation-and-pro...

also

"It’s fine to use a proof by contradiction to show something doesn’t exist. When the assumption that it does exist leads to a contradiction, then that shows it can’t exist.

It’s not so fine to use a proof by contradiction to show something does exist. Here’s the situation. The assumption that it does not exist leads to a contradiction. What can you conclude from that? You would like say “therefore it exists”. But you haven’t got any idea what it is. You may know it’s out there somewhere, but you have no idea how to find it. It would be better to have a proof that tells you what it is.

That’s a difference between what’s called “classical logic” and “intuitionistic logic”. In classical logic, proof by contradiction is perfectly accepted as a method of deductive logic. In intuitionistic logic, proof by contradiction is accepted to show something doesn’t exist, but is not accepted to show something does exist."

David Joyce, https://www.quora.com/In-math-are-there-any-proofs-that-can-...

[bananaflag]

A purely universal statement, to be more clear.

[enricozb]

As far as I understand, The lead of this project Kevin Buzzard is a mathematician first. And the majority of mathematicians are untroubled by non-constructive proofs. I would imagine that proof directions that result in the most interesting additions to Mathlib would be chosen.

[CJefferson]

I have no idea why Gemini is saying that. Proof my contradiction is totally fine. Sure, many people prefer a more direct proof as they are nicer to read, but proof by contradiction is totally fine and sometimes the only way to prove important results.

[tphyahoo2]

I'm not disagreeing (I'm on the fence. Also a bit of a nube.). I thought this was a good read and on topic.

https://www.quora.com/In-math-are-there-any-proofs-that-can-...