[menloparkbum]

"The world owes a great debt to Einstein and Godel, for relativity and incompleteness, but it is widely recognized that without them the same results would still have been achieved only a few years later. "

This is hardly widely recognized.

[bluishgreen]

1.The mathematics that sort of fell into and formed the structure of Einsteins work was about 50 years old when he used them. I am talking about the attempt to prove Euclids 5th axiom using the first 4 and the subsequent evolution of hyperbolic geometry.

2.Godel was working on the problem that was proposed by Hilbert (Google for Hilbert's List Of Problems). At the time when Godel was working on it, it was actively pursued by prominent math dudes of the era;

One connection that I find in both these cases is this: They were both trying to prove something is not possible. And whenever that happened in Science - it usually causes a huge splash. Another example is Group Theory: fell out of the proof for the fact that a 5th degree equation cannot be solved using algebra.

[bkovitz]

Actually, I believe group theory emerged very gradually from many simultaneous developments. Other sources: permutations, symmetries of crystals and linear transformations, quaternions and matrices (though theory of equations was indeed the main driver). I believe it wasn't until after a bunch of these had reached a high level of development (around 1880) that the abstract group concept began attracting attention as a field of research, because then people could see its use for unifying all those others. Abel's proof for 5th-degree polynomials came out in 1824, and Galois' general proof was published in 1846, so this was slow going, even by nineteenth-century standards.

BTW, I thought I read that Gödel worked pretty much as a lone wolf on his stuff, as most mathematicians weren't into it. And the trend has continued: foundations of mathematics has gotten research attention from surprisingly few people. I'm not 100% sure of this, though.

But anyway, yeah, proving that something can't be done is definitely a big deal. Interesting observation. I will percolate on that.