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What makes the proof to Fermat's Last Theorem difficult? Because the proof uses vocabulary that is rich in information?
The theorem itself seems so intuitively correct, that one could almost believe it solvable with logical calculus! More generally: what makes a math problem difficult?
It's an interesting question to which I don't have a strongly philosophical answer to offer. What I can suggest is a concept I encountered elsewhere called a "wicked problem". I don't think wicked and difficult are perfectly synonymous here, but I would offer that all wicked problems are difficult problems.
Wikipedia will tell you:
While I don't think that perfectly reflects what a difficult problem is, I think we could possibly derive a definition that a problem is difficult if it requires great effort to solve or nearly solve. A problem is subjectively difficult if it requires of the person in question great effort to solve or nearly solve. And a problem is objectively difficult if it requires specialist knowledge that is contemporary to solve or nearly solve.
But I have never seen difficult problems seen as a specifically philosophical problem.
It seems intuitively correct to you? That's strange. Take the subcase n = 3: There are no positive integers a, b, c such that a^3 + b^3 = c^3. Then modify it slightly: For which (positive or negative) integers k is it true that there are no positive integers a, b, and c such that a^3 + b^3 = c^3 + k? The case n = 3 of Fermat's Last Theorem is identical to this problem with k = 0.
It turns out that if k = 4 (modulo 9) or k = 5 (modulo 9) then it is very simple to show that there is no solution. For any other values of k except k = 0, it is believed that there will be an infinite number of solutions of which even one is often very, very, hard to find. But for the case k = 0, there is actually a very deep proof that there is not solution. I wouldn't call this in any way intuitive. It is actually quite counter-intuitive and surprising that FLT is true for n = 3. The same is true for n = 4, intuitively it seems reasonably likely that there would be solutions, but it turns out there is a proof that there are none. And after all, for n = 2 there is an infinite number of solutions. If n = 2 has an infinite number of solutions, what makes it intuitive to you that n = 3 would have none?
I seem to remember that the state of the art before Wiles' proof was that there are no solutions with n <= 125,000. It was indeed very, very unlikely at that point that solutions with n > 125,000 would be found, but by no means impossible. And by no means was it ever "intuitive" that this would be impossible.
It's usually a process of attrition. The first question to ask is what makes this question interesting? It was a question asked by a famous mathematician — Fermat who died in 1665 and all of whose outstanding question apart from this one were solved by the early 1800s; then in 1816 the French Academy established a prize to solve this one; by then it was recognised that established techniques weren't enough; and it is this that makes the problem interesting — what are these new techniques? this turned out to be the development of new number systems and their geometrisation.
Mathematicians learn the art of asking the possible, and the important; sometimes it's difficult to discern where the boundary lies; that again comes with time; Fermat's last theorem is an obvious generalisation of Pythagora's theorem (a much more immediately important theorem — it proved its importance in antiquity, and then in the development of vector and metric manifolds important to both special and general relativity). Fermat probably believed it to be a possible question — it turned out to be an important one.
Again through the process of time, the percolation and maturing of these new techniques it's just maybe possible that Fermat's Last Theorem will be an exercise set for university students, rather than being one that exercised the best mathematical minds of the last three centuries. After all it took Galileo to start measuring just hw long it took a pendulum to swing, whereas any schoolboy interested in physics and bothered enough to do the experiment can actually do this.
It's worth quoting a comment by Cartier on this:
These two problems, Fermat and Riemann, are in some sense rather futile: Fermat’s problem concerned a very particular equation, and the Riemann hypothesis can be interpreted in terms of very subtle regularities in the apparently random distribution of prime numbers. In itself, a counterexample to the Riemann hypothesis, given the present state of our knowledge, would have very small “practical” consequences and would certainly not be a catastrophe.
A note on the solution: Fermat's last theorem is solved by an appeal to the Modularity theorem, which is a special case of what is currently called the Langland's program, and which appears to be the correct higher and non-abelian generalisation of class field theory - which basically tells you how badly factorisation fails for these new number systems (rings). It's this context that appears to be its natural home.
