>>13172114Regardless of what his views actually were, both of the nature of mathematics and of what he discovered, he showed that he understood better than any of his contemporaries the limitations of mathematical theories as written on paper, and always took everything to the most bitter logical conclusion, with no personal fantasies attached.
As far as I can gather, he was a mathematical platonist himself. Then again, he was also theist, so perhaps his idea of mathematics was some sort of unattainable Logos, which we can only approximate with our mathematical theories. Nevertheless, he understood the fundamental importance of interpretation of symbols in mathematical theories in a way that even today mathematical practice doesn't give enough weight.
Meanwhile, his theories are a deadly blow to the naive mathematical platonists, who have coped with it by pretty much ignoring them, postulating ZFC = reality and misunderstanding model theory by speaking of standard and nonstandard models.
When Russell thought he could give self-serving foundations to mathematics, Godel showed with his incompleteness theorems that there will always be more mathematics, that a mathematical system cannot look at itself, and that interpretation is fundamental and necessary.
When the intuitionists/constructivists split from the "classical logician" platonists, each party calling the other's work incoherent schizo ramblings, Godel showed with the negative translation that they were two sides of the same coin, as they both interpreted one another.
When Einstein gave General Relativity, thinking that its mathematical beauty by itself was testimony of some unique physical correctness, Godel showed with his rotating universe that mathematics is not the end of all physics after all.
Again, I don't know and actively doubt that he meant to do all of this. I genuinely think that he liked to tinker a lot and stress-test everything. But he was, inadvertitely, absolutely based.
