"Prove Fermat's Last Theorem" an assignment with a mutating meaning?

[Boojatta]

Aren't there some assumptions/axioms that are generally accepted by mathematicians today, but that had not yet been thought of (and that therefore were not yet accepted) when Fermat put forward the conjecture?

Yes, the meaning of the conjecture has not changed. However, the rules of the game have changed, haven't they? We're not just talking about the creation of new techniques within the rules of the same game that Fermat played, are we?

Suppose something like Wiles' proof had been (miraculously) proposed while Fermat was still alive. What would the reaction have been?

Maybe this? "These are fascinating assumptions that you have put forward. It's very convenient that those assumptions allow you to prove the theorem that you wanted to prove. However, we'll have to wait and see whether or not we think that those assumptions are actually legitimate."

[Botany]

1. x + y = x + Y

This DU if you want to discuss string theory and or the proof of a mathematical
formula this is not the place.


Gibberish in = Gibberish out

Yes, the meaning of the conjecture has not changed. However, the rules of the game have changed, haven't they? We're not just talking about the creation of new techniques within the rules of the same game that Fermat played, are we?

[longship]

3. Have a gorilla!

with apologies to Peter Sellers, Spike Milligan, and Harry Secombe

[Boojatta]

5. This is DU, but this also happens to be the Science section of DU.

[longship]

2. I have a very elegant proof of this theorem.

But unfortunately, it doesn't fit in a DU post.
:evilgrin:

[bemildred]

4. Such concerns are not new.

The followers of Hilbert pursued one such project, and the concerns you point to here are under current discussion, the meaning and legitimacy of proofs of the sort of Wile's theorem. From my own studies, now long past, I think the roots of that discussion begin around 1900, in the great flowering at the beginning of the last century, and it is certainly true that the discussion has been radically re-framed since the time of Fermat. The results of Turing and Goedel and their successors seem relevant to this too, since one can view a deduction as a sort of computation, and then consider the computability and incompleteness results as they would apply to "proof". This is especially so with the rise of "proof by computer". It does not seem to me to be an area in which any well-understood consensus view exists though, and I doubt that you will see that sort of thing soon.

Your conjecture about the reaction of Fermat's contemporaries to Wile's effort seems reasonable. One suspects at least that it would have taken some time to digest the meaning of his work.

[eppur_se_muova]

6. Not really. Mathematicians have been striving to REDUCE the number of axioms

which must be assumed, without proof, to be true. The prime example is Euclid's "parallel postulate". In the 19th century, it was discovered that by dropping this axiom from geometry, it was possible to develop from the remaining axioms a number of "noneuclidean" geometries which were internally consistent and thus no less "true" than the geometry of Euclid.

"...creation of new techniques within the rules of the same game that Fermat played" is pretty much what happened. "Mere" technical progress, accumulated over a few centuries, creates an environment that differs not merely in degree, but in kind, with that previous, even if there is no sharp transition point. Wiles had tools to work with that Fermat did not.

If Wiles' proof had been presented while Fermat was still alive, there would have been a handful of savants who could have followed it, provided they were familiar with the previous theorems (not axioms) on which it was built. With a few months/years of study, it would have been accepted just as it was when it was actually presented -- but it would have involved a LOT more work, and probably revolutionized number theory at the time. As it was, Wiles' proof has had almost no follow-on results (AFAIK, big caveat).