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(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 th
by elromulous 1mo 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.
- jerkstate 1mo agoI 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!
- kadoban 1mo agoConsensus 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.
- odyssey7 1mo agoWe 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.
- lmntryflt 1mo ago[dead]
- hn_throwaway_99 1mo agoQuite literally, WTF is that supposed to mean?
- kjs3 1mo agoIt's the internet. Don't feed the trolls. Or the loons.
- applfanboysbgon 1mo agoThe 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.
- hn_throwaway_99 1mo ago> 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.
- applfanboysbgon 1mo agoIt's not just that he formulated the problem, though. It was specifically the fact that he claimed he had a proof which led to the proof, and it was only because people credibly believed he had one that they spent so much effort trying to (supposedly, re-)discover it.
- hn_throwaway_99 1mo ago> and it was only because people credibly believed he had one that they spent so much effort trying to (supposedly, re-)discover it. That's not true. Even early on, most mathematicians doubted he had a proof. But I'd check out the history of the problem as described on Wikipedia. There was considerable research into the problem before Fermat, and general research into Diophantine equations had gone on for over a thousand years before Fermat. I have no doubt there would have been significant interest into solving the problem even if Fermat had never written his "I have a proof but it's too big to fit in the margin" note.
- duskwuff 1mo agoI have to agree. Even if it hadn't been designated as Fermat's Last Theorem, the problem would likely have ended up in some comparable list of interesting conjectures (e.g. Hilbert's problems, the Erdős problems, the Millennium Prize problems, etc).
- hackthemack 1mo agoOne 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?
- alichapman 1mo agoWhen I was at uni my number theory lecturer said that if Fermat had a solution, it was only valid for regular primes.
- seanhunter 1mo agoPretty 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 https://youtu.be/xKfEmbWBgvM?is=qLnPvPXGDqgJCDC3
- srean 1mo agoMany 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.
- seanhunter 1mo agoYou 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.
- srean 1mo agoAgreed. Modern rigour is, well, quite modern.
- bell-cot 1mo ago> 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.