Here’s research that suggests otherwise: http://reports-archive.adm.cs.cmu.edu/anon/isr2012/CMU-ISR-12-113.pdf

And the tools referenced in this academic work have already begun adding in words as component pieces.

]]>Actually knowing or thinking a bit about the history of Wiles’s proof may convince you of this. I myself do not understand well the argument but even the faint picture I have of it gives me confidence in FLT, more than, say, a proof of the n=3 or 4 cases.

So those historical details I am thinking about: many people have contributed to the only currently accepted proof. (Though stay tuned for another proof if Mochizuki’s work settles ABC, which implies FLT for large exponents, and perhaps all exponents with some tuning.) Actually what motivated Wiles to start working on FLT was not really working on FLT itself, but on the Shimura-Taniyama-Weil modularity conjecture. Because the biggest insight into FLT had been obtained before, Hellegouarch and Frey (and others) sought to prove that Fermat triples for any n>2 yielded elliptic curves with strange properties. It would contradict, via a naturally associated elliptic curve, a conjecture of Szpiro related to the ABC conjecture. And Frey also looked at a similar contradiction supposing that curve was modular. He was helped (or rescued) by Serre and Ribet. And once this connection to modularity was well understood Wiles was quite confident and “only” had to prove modularity, the STW conjecture. In fact he did not have himself the insight that FLT was reachable, but when he understood the work of the above-mentioned people he was convinced, enough to work hard for 7 years on STW. So that tells you that definitely he got insight that FLT was true from those partial results toward it, and he completed the work, and certainly he got surer and surer that FLT was true as he fleshed out his arguments and uncovered key parts of his eventual proof.

Similarly, all the people involved in that (Nicholas Katz, Ribet, Sarnak, grad students like Buzzard) felt more and more sure as they learned and thought, each at their own pace, about the different parts of the proof, some old (the theory of elliptic curves, or even algebraic number theory) some new (Galois representations in GL(2,F_p)).

If you invested much time in studying that you would also gain insight into FLT, become comfortable with it.

Unfriendly computer proofs are just hard to fit to our relatively flexible neural constructs but they can be grasped. Now the question of whether it’s more efficient to look for proofs by ourselves rather than try to make computers good at that and then understand them is a very interesting question.

But in the case of Wiles’s proof it definitely does shed light on FLT. I guess though it is still a matter of taste: you may say I am satisfied with checking 3 exponents up to a,b<1000 on the computer to be convinced of FLT, or you may be satisfied with Kummer's proof for regular primes.

The questions seems to be "How much certainty am I looking for?", "How much work am I willing to put?", "What is the best tradeoff?". From this point of view we can question complicated proofs, but if they are full and fully understood proofs compared to heuristics, they do provide more insight.

We can also think of the interesting situation where a proof is so complicated that we do not have the ressources to understand it, and we do not trust expert that much. Then a heuristic may be more reliable than the proof, though not fully reliable. And we can argue the definition of insightful, it may be a product "information x speed to get it" = "information / time to get it". But I still like to say Wiles et al's proof provides insight beyond partial results.

Well, this was much rambling, I hope at least it will not bother you too much.

I have to read more of your work to understand it but I should already take the opportunity to thank you for it.

]]>But some questions arrised when I was reading it.

PCE claims -as you say- then, that complexity is achieved soon or later because it has a specific limit. Then is almost logical to think that this Principle include the thought system.

This make me think that to achieve current “thinking” human level, we have had crossed around a long road of evolution (is this correct accordint with your comment?).

If the last is a correct asuption derivated from your comments, how then could we name “evolution” to a progress that is pushing us (the human race) to the destruction (as Carl Sagan claims)? Is that evolution or the opposite process?

You know what? When I was reading your blog, I remembered an interesting (in my view point) comment by Rupert Sheldrake (author of the morphic fields) who said that (according with his theory) about thinking and imagination (I understand the mind in general) there hasn’t been such a process: the evolution.

The comment came from an original question: in prehistoric ages of humans, there were a Motzart? or a Ban Gogh? an Einstein or maybe a Wolfram? (this last reference to Wolfram is mine) The response is “Yes” according with Rupert. This could then explain the supposed alien aztec technology, even the alien existence, the art in caves, sophisticated weapons and so on.

Under this particular vision… What about PCE? it involves always an evolution trying to achieve the last limit of complexity? If the evolution really dones’t exists… Wolfram’s PCE still being true? Even more deeply… I understand that evolution is implicit in the Wolfram’s PCE, then it seems a contradiction -for me- to think on an evolutional universe with simple fixed rules causing the birth of the most amazing complexity, Is not true (by maths) that, if a system is in constant evolution, then the rules that define it have to evolve also? ]]>