On Math and Faith

Back in the 1630s a mathematician by the name of Fermat, writing on the margin of a book, posited a very simply stated theorem: a^n+b^n cannot equal c^n where n is a positive integer greater than 2. Next he wrote he had a proof for this theorem, but that he lacked the space to write it out.

Until the mid-1990s, no proof was ever found. When it finally was, it took the talent of several mathematicians and some very powerful computing tools. Given all this, does it seem reasonable that Fermat worked out a proof for his theorem? That is the poll question.

So let's think about it. On the one hand Fermat's Last Theorem stumped mathematicians for over three and a half centuries. On the other hand, he claims to ahve worked out a proof, pressumably working alone and without any computaional aides at all. Is it even possible.

Yes, it is. He might have had the necessary genius to see at once, or nearly so, where the proof lay. Or he may have had the luck to stumble upon the solution early on. Such things have happened before and since. Einstein worked out the theory of relativity largely in his own head. So it is possible.

Is it likely? I don't know. I'm not familiar with Fermat's other work and I suspect I wouldn't be able to judge it if I were. But even a plodder can have a moment of inspiration, and anyone can get lucky.

Of course Fermat could have avoided all the speculation by providing his proof elsewhere. Writing on the margin of a book seems bizarre, but I've no idea what the customs were at the time, or how easily was paper obtained. Modern, cheap, wood cellulose paper is a relatively new invention.

In the end I think he may have worked out his proof, but absent the proof itself I can't say that he did.

Was it his own book, or someone else's that he borrowed? If the former, was it one that he loaned out, intended to loan out, or specified an heir to in his will?

If he never intended his comment to be seen by anyone other than himself, he would have no impetus to lie about having worked it out. If there's even the slightest chance that he expected the book to be looked at by anyone other than himself, however, all bets are off.

Quote: heather

Was it his own book, or someone else's that he borrowed? If the former, was it one that he loaned out, intended to loan out, or specified an heir to in his will?

No clue.

Quote:

If he never intended his comment to be seen by anyone other than himself, he would have no impetus to lie about having worked it out. If there's even the slightest chance that he expected the book to be looked at by anyone other than himself, however, all bets are off.

That's the thing. Times were different. There were scientific publications, mostly in Latin, but not the kind we know today. No peer-reviewed journals, for example, nor science conferences or conventions. Newton himself took a long time publishing his work. Mendel never published his

I think we'd have to know something about Fermat himself. Was he a joker, an amiable sort who liked to kid? Or was he serious, someone who was scrupulous on his way through the world?

I've read a lot about this, and I read the general statement of the solution for lay people using the elliptic curves. My guess is that Fermat was joking, making it seem like there was a simple solution when he knew it was true, but complicated.

I voted for "Fermat did not work out the proof." I'm sure he thought he could when he made the remark in the margin, but it was probably a false epiphany. If he sat down to do it I think he would hit a brick wall somewhere and realize he was harder than he thought.

Quote:

a^n+b^n cannot equal c^n where n is a positive integer greater than 2

I say it *can* but I also probably won't publish my proof heh heh

The problem is mathematics has gotten so much abstract and complicated during the last 50 years that it is impossible for someone not trained in the exact specialized field to follow what is going on. Since Fermat's last theorem has caught the public attention, people want to know what the fuss is all about. Alas, they really cannot understand it or even appreciate the mathematics in general without strong background in number theory, Galois theory, elliptic curves, and so on. All books that try to cater to the layman have to decide where to draw the line. If they water down the mathematics, then they really cannot explain how the proof was got satisfactorily; and if they throw in one too many equations, it becomes incomprehensible to many--even mathematicians not in that specialized field. My advice to the general public is to watch the video on Fermat's last theorem. I think for the layman visual media is better than a book. Just Google the excellent UKTV documentary on Fermat's last theorem.

At the time books were not as often and freely loaned as now. Books would never be carried openly but always in a concealed pocket. Often a mild anathema would be inscribed to place a curse on a book thief. Writing in the margin was common and would not be a reliable indication of intent to disseminate.

It is likely that a passing reference might allude to a mathematical proof that was not yet fully formed or tested but which he did indeed have in mind at the time. Do not dismiss ancient knowledge as inadequately arrived at. Precision may have been lacking, computers may have been not readily available but often knowledge was acquired nevertheless.

The discovery of the ice-entombed caveman in the Italian alps showed us a bow and quiver of several components and each one was optimal for what materials were available. Homer gave us the best way for escaping a vortex and all our studies of fluids and vortexes has still not found a better way. Early navigation and commerce was far more extensive than we ever thought. Damascus steel was truly of high quality.

So did the proof exist? Its unknown and unknowable, but I'd put my money on "yes".

I suspect Fermat thought he had an elegant proof (I thought I had a simple one once, but it took a short time to work out it was nonsense) but further work would have shown he was wrong.

Thus I didn't vote cos none of these quite fit...

Quote: thecesspit

I suspect Fermat thought he had an elegant proof (I thought I had a simple one once, but it took a short time to work out it was nonsense) but further work would have shown he was wrong.

Thus I didn't vote cos none of these quite fit...

That is also my opinion. I thought "Fermat did not work out the proof" was the closest choice to my position on the matter.

Looks like he did prove the cases for x^4 + y^4 = z^4 has no integer solutions (and thus the exponent could never be even).

He set this an exercise for his correspondents.