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Since Wiles proof is in essence proof by contradiction, it relies on the law of excluded middle. Which as I understand intuitionists / constructivists do not accept as an axiom. So what is their view of the Wiles proof? Do they still consider Fermat theorem not proved?

Rentgen Rukogama
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  • I am not familiar with the details of the proof, so I do not know whether it is actually a contradiction proof. If it is, I guess they do not accept it. – Peter Dec 05 '20 at 13:37
  • I think , we have to sacrifice many important results if we omit this very important , useful and moreover utterly obvious principle. – Peter Dec 05 '20 at 13:39

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Wiles' proof is not, in essence, a proof by contradiction.

Wiles has shown that every elliptic curve over $\mathbb{Q}$ (well, a large enough subset of them) satisfies the following theorem. At the time, this was a conjecture, and a big one, and Wiles had begun his work on FLT after a connection was established between FLT and this conjecture.

A positive integer solution to FLT could be manipulated to construct an elliptic curve which isn't modular, but this kind of connection is only an indication that the modularity conjecture would imply FLT. So FLT would not be proven by assuming it's true and seeing it's impossible, but rather it was already established that if FLT is true, then something really strange happens, and Wiles had undertaken to prove that it was impossible.

Wiles, in fact, uses the word contradiction only 3 times throughout his amazing paper "Modular elliptic curves and Fermat's last theorem.". FLT isn't even stated directly after modularity is proven, since the connection was already known,

As for the proof itself, it was accepted as correct (not without some obstacles), and still is accepted, and will remain so. Wiles was awarded, among many other honors, a special plaque from the ICM (I heard someone refer to it as a "quantized" Fields medal), and in 2016 he was awarded the Abel prize.

  • I am aware it was accepted by mainstream mathematicians. Constructivists is a separate matter, no? As for contradiction - in the very beginning of his proof he starts with "let's assume there is a solution", and then shows that then it would be possible to construct a specific elliptic curve. And then he shows that it is impossible to construct such a curve. – Rentgen Rukogama Dec 05 '20 at 14:51
  • You're right, and there is still no inherent contradiction. The construction of a nonmodular elliptic curve from a sol to Fermat's equation is quite easy, and the fact that FLT is true - IF the modularity conjecture is true, was known. This is a matter of rigor, so when Wiles does prove that it's impossible to build a nonmodular curve, his proof is really a constructive proof for the fact that every semistable EC over $\mathbb{Q}$ is modular. FLT follows. –  Dec 05 '20 at 15:01
  • Now, about proofs by contradiction: it can be shown that there exist statements that are provable by contradiction, but impossible to prove otherwise. What I've tried to emphasize so far is that there is no inherent contradiction in Wiles' proof, but let's put that aside - most mathematicians, at least most of those I meet - don't really take to constructivism for the above reason(and others), hence most simple don't really consider its viewpoint on FLT's proof to be important. –  Dec 05 '20 at 15:11
  • https://math.stackexchange.com/questions/243770/can-every-proof-by-contradiction-also-be-shown-without-contradiction –  Dec 05 '20 at 15:11
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    Thanks, this is interesting. I'm not a professional mathematician, but dabble in basic Category Theory and Type Theory, having come there from the FP side, so eventually ended up with Homotopy Type Theory book, which builds the math on TT foundation in constructive way. It really is beautiful, but the whole deal with the law of the excluded middle being, uhm, excluded, bothers me as well - so it's interesting to know what mainstream math thinks of that. – Rentgen Rukogama Dec 05 '20 at 15:22
  • HoTT is compatible with excluded middle. It can be added as an axiom. The problem, among others, is that it in general it will break canonicity. So for terms using excluded middle (say, the definition of a natural number), it may not be possible to calculate a meaningful value (numeral). This is why some people were concerned about determining such a computational meaning for univalence. A principle being 'obvious' doesn't do actual work. The HoTT book does use EM in some cases, but perhaps it's instructive how few of its results require it. – Dan Doel Dec 06 '20 at 17:08
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There is a very subtle difference between proof of negation and proof by contradiction in constructive mathematics. If you assume some $A$ and derive a contradiction then you conclude $\lnot A$. In fact, this is how we define negation. Now on the other hand a proof by contradiction assumes for some $A$ that $\lnot A$ holds and derives a contradiction. By the definition of negation they have shown $\lnot \lnot A$ but with the double negation law (equivalent to Excluded middle) they can then conclude $A$.

The wiles proof assumes that there is a solution and shows a contradiction. This is proof of negation. So we may conclude $\lnot (\exists x, \phi(x))$. Where $\phi$ is some encoding of fermats last theorem. Now it is constructively valid to then conclude $\forall x, \lnot \phi(x)$. I’ll try to link a good reference that discusses the demorgan identities that do hold constructively (not all of them do.)

ToucanIan
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