Comment by elromulous
1 month ago
(disclaimer: I haven't watched this yet)
Mathematicians / folks knowledgeable on the subject - do you think Fermat had an error in his (lost) proof? Or that there exists a solution that perhaps is more straightforward than Wiles's?
My understanding is that Wiles had to use a ton of math (and invent some new math?) that had not yet been invented in Fermat's time.
One of the arguments I have heard that he did not have a proof is the following.
He wrote his note in his copy of Arithmetica around 1637.
He most likely wrote his proof for the case of n=4 in the 1640s.
He sent letters to other mathematicians in 1640, 1657 where he talks about the case of n=3 but writes in such a way that it seems like he does not have the answer.
Why would he write n=4 after? If he had a generalized proof?
Why would he tease other mathematicians with a special case in 1657 if he already had a generalized proof?
Consensus is that there's no way he had a correct, general proof.
My personal guess is he figured out later that his proof was broken or limited and never got around to fixing it.
The funny thing is that whether or not he had a proof, it was only definitively proved because he claimed he had a proof, so in a roundabout sense, he's still responsible for the theorem being proved. It's kind of crazy to think about your words carrying so much weight that somebody from hundreds of years in the future will dedicate (a significant portion of) their life to them.
> he's still responsible for the theorem being proved.
I think you have an odd definition of "responsible". Many (most?) theorems start out as conjectures, and I would strongly disagree that just because someone first formulated a problem that they're "responsible" for the eventual solution.
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I think he had a hunch, and maybe in his mathematic mind he got to the right answer by the wrong path
But yes had him not pointed it out maybe it would have been relegated to a mathematical curiosity or something
I think there's a deeper (but simpler) reason for it, besides the way Wiles proved it
We haven’t yet reached the era in which we’re allowed to know about the technique. The simple, intuitive proof will become available to us when the time is right.
[dead]
Quite literally, WTF is that supposed to mean?
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Pretty sure Fermat didn’t have a proof.
For one thing, Fermat wasn’t actually a “professional” mathematician. He was a lawyer and judge and was known for having remarkable intuition but not really troubling himself too much with details of proofs etc. For example Descartes famously called him a “deficient mathematician”[1] in an angry exchange of letters over a technique that Fermat had discovered to geometrically construct a tangent to a particularly problematic curve called Descartes’ Folia using a technique we would now recognise as being the definition of the derivative as the limit of the difference quotient.
[1] Explained entertainingly here https://youtu.be/xKfEmbWBgvM?is=qLnPvPXGDqgJCDC3
Many accomplished mathematicians were not professional mathematicians. Surprisingly enough, many were in fact lawyers.
Descartes was himself a lawyer (by training) then there were Leibnitz, Cayley, Viete. If you include Physics you will get a bigger list.
Descartes and Fermat ran a mutual admiration society, don't read to much into it.
Both had independently come up with coordinate geometry, linking algebra and geometry
I agree Fermat probably had a wrong proof but curb your condescension towards lawyers making contributions in mathematics.
You misunderstand. I have a tremendous respect for Fermat. I just meant he didn’t feel compelled to be totally rigorous. There are examples of contemporaries complaining about him skipping steps and hand waving etc. That was just how he did things.
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I don't know that much about math but I've read that one possibility is he had found a valid proof for n=4 and assumed that it generalized. Hopefully someone else who knows more about the subject chimes in!
> do you think Fermat had an error
Yes, with million-to-one odds. Though I'd say "error or equivalent shortcoming, for the general case".
> that perhaps is more straightforward than Wiles's?
I don't recall a mathematical definition of "straightforward", but yes. Ignoring minor improvements, I'd guess there's a much better proof...though that might require a century of new developments in related parts of mathematics, before we'll have the necessary tools to write it down.
When I was at uni my number theory lecturer said that if Fermat had a solution, it was only valid for regular primes.