>>11383688Absolutely no one should have ever been surprised that mathematical truth cannot be equated with theoremhood in some finite axiomatic system.
An infinitude of mathematical truths are uninteresting trivia, with no obvious route to being proved. I think even Gauss knew this.
E.g check if the digits of square root of 47 is contained in the decimal expansion of pi^101. Well, we can just calculate out the sequences of digits and check. If it's true, you can find the digits that match, but no amount of just grinding out the digits and checking will ever prove it: there are always more digits to check. If it's not true, same problem, you cant disprove it. That is an unprovable mathematical fact. It is also a very, very, very uninteresting one.
All Gödel did was find a clever way to construct a provably unprovable mathematical fact, given any consistent and finite set of axioms to work with. The work is clever piece of technicality but in no way profound. It should have come as no surprise at all. And surely it never really impacts most of mathematics, except for some narrow branch.
At Hilbert's time it would have made more sense to have this sentiment that mathematics took some dark turn due to the incompleteness theorem, because it killed Hilbert's meme program. But it has been a century since then, it should've been clear that Hilbert's vision was too idealistic, and of course he was overconfident. Godel's theorem marks the boundary of Enlightenment, not its refutation (boundary that first rate mathematicians should've seen coming long before it was reached). Again I think philosophically, this stuff was known since Gauss who spent a lot of time staring at natural numbers, if we drop Godel's paper on Gauss's office, he would skim over it and wouldn't have cared much for it and continue sipping his tea and read his geodesic survey and cartography, which would actually impacts math and physics century later (differential geometry, General Relativity, etc)
