Why didn't Fermat provide proofs of his theorems?

Question

Apparently Fermat stated but didn't provide proofs of various theorems named after him, including Fermat's little theorem, Fermat's theorem on sums of two squares, Fertmat's polygonal number theorem, and most famously, of course, Fermat's Last Theorem. Why is this?

Answer 1

Much of the formal reasoning that modern mathematicians today consider to constitute formal "proof" was not known to Fermat or his contemporaries. Even Euler did not always furnish proofs of his claims to an extent that would be regarded as satisfactory today. The efforts of the Bourbaki group of mathematicians could be regarded as the foundation of a modern axiomatic approach to mathematics.

That said, we can infer from Fermat's writings that he does demonstrate some intuition supporting his theorems, and this is really how much of mathematics progressed during his time. Even as late as Poincaré's and Ramanujan's time did we see this sort of emphasis on results at the expense of rigor. But at some point in the history of mathematics, there was a realization that what was regarded as sufficient proof was in fact not so firm, and the critical difference is whether one was aware that their arguments lacked sufficient rigor.

Answer 2

I heard a talk by Fernando Villegas, in which he mentioned that Fermat was a lawyer by profession, and most of his mathematical activity was in the form of annotations he made on his copy of Diophantus's Arithmetica, the one with the famously narrow margins :) So, while he was familiar with the notion of a proof (although his notion was not a modern one), he does seem to have considered mathematical "revelations" more important than proofs.

Answer 3

While Fermat did not provide proof of the FLT he certainly provided proofs of many seminal results. Thus he applied his technique of adequality to give a variational proof of Snell's law. He similarly solved the cycloid problem. In many cases his reasoning is diagrammatic in the tradition of the Greeks, in the sense that the figure is an essential part of the proof rather than being merely illustrative, as R. Netz argued. For details see this recent article.

Answer 4

In Fermat's era, algebra was not a complete branch. With complete, I don't mean "finished" of course, but well defined. Although algebra includes the proof methods, during Fermat's time, the proof methods were not as clear as they are now. Even though Fermat had proved some of his theorems, we would not accept that proof as they were not well formed. Math was being studied like physics back then. This means, it was not completely analytic, but also hypothetical.

Most of the theorems of Fermat are proved after him, because he was only guessing and saying "It seems like true as I couldn't find any counter examples" or "Its proof is trivial". Also Euler and Gauss didn't prove most of their theorems. I don't remember which one, but Euler used 1000 x 1000 prime number table to prove a hypothesis about a binary operation on prime numbers, as he couldn't prove his hypothesis analytically. He just wanted to make sure that it works to feel free for using it.

Of course Fermat is the worst example of all, as he is not even a mathematician but a lawyer who is really into math. However, after Cantor's, Frege's, Zermelo's and Russell's hard work, math and algebra are completely analytic now. We know if something is provable or not, even though we can't prove it.