We can put the point even more sharply. The Austrian-born mathematician Kurt Gödel taught that, given any mathematical question that hasn’t been answered yet (except for questions that boil down to a finite calculation, such as whether White has a winning move in a chess game), it’s possible that the answer is unprovable from the usual axioms of mathematics. And yet, 85 years after Gödel uncovered this gremlin at the centre of mathematics, the fact is that it’s remained mostly dormant. Gödel’s Incompleteness Theorem rears its head only in specialised situations: ...No, this is not correct. There was never a possibility that Fermat's Last Theorem might be demonstrably independent of the axioms of set theory.
It didn’t need to be that way (or, maybe it did, but it’s not obvious why). A priori, Fermat’s Last Theorem, the Poincaré Conjecture, and pretty much every other statement of mathematical interest could have been neither provable nor disprovable: if true, then totally disconnected from all the other interesting truths, an island unto itself, with the only question (a question of taste!) being whether we should add it on as a new axiom. But it didn’t turn out like that.
Any counterexample to FLT could be verified by a finite computation. Therefore the only possibilities were that FLT is provably true, true but not provable, and provably false. Any proof that FLT is independent of axioms would also be a proof that it is true about the integers.
While is it true that independence results are exceptional, the continuum hypothesis was certainly a mainstream math problem. It was the first of Hilbert's famous 1900 list of 23 open problems.
Lumo is flattered to be attaked in this essay, and disagrees with comments about the possibility of P=NP.
Aaronson gives the impression that mathematicians are solving the big questions, but his field of complexity became a real field over 40 years ago when everyone realized that P=NP was such an important question. A lot of really smart mathematicians and computer scientists have worked on it, without success.
And he specializes in the subfield of quantum computer complexity, and there is no consensus on whether a quantum computer is even possible.