Kurt Gödel
I don't think that Gödel proved that 2+2 <> 4. He proved that any system sufficiently complex could not be both complete and consistent. It used to be believed (as per the Shrödinger article) that if we knew the the initial conditions of a system and we knew all of the rules determining the system, that we could then perfectly predict everything about the system at any moment in the future. (This assumes of course that the system was deterministic.) We believed that the universe was understandable, that it was deterministic, that it was causal, that it was logical. But ... weird stuff was happening ... things that contradicted basic assumptions - some of them assumptions so basic that few, if any, had ever thought to question them.
Gauss, Bolyai, et. al., showed that despite supposed proof by Kant to the contary, that one could develop a consistent geometry replacing Euclid's fifth postulate with something inconsistent with it.
Cantor showed that not all infinities are equal, some infinities are bigger than others.
Michelson and Morley show that there is no ether ... but if there is no ether, through what do light waves travel? How can you have a wave without a medium?
Russell and Whitehead spent years in a thought-provoking, but failed attempt at showing that mathematics could be derived from logic.
A patent clerk publishes a paper in Annalen der Physik that in a very few paragraphs (and if I recall correctly not a single reference) questions our ideas about simultaneity and absoluteness. The speed of light does not depend on the speed of the source or the reciever. It's absolute (in a vacuum). [A common layman's MIS-understanding is that Einstein said "everthing is relative."] In any case, what he actually proposes violates our classical assumption that velocity vectors are additive.
Said clerk had previously published a paper in same journal explaining that in addition to obvious wave properties, light also exhibits some particle properties (waited for De Broglie, Germer, et al to 'reconcile').
Heisenberg states remarkable conclusion that we can't perfectly know position AND momentum of a particle, not because we have bad eyes, or faulty equipment, but because of a fundamental aspect of nature - when we look at something (measure it), we change it. We can know these things, but only within the constraints of the uncertainty relation.
Bohr and others advance the copenhagen interpretation (which I don't understand and I don't even remember what I read about it) and Schrodinger graciously volunteers his cat for an experiment to show the aburdity of this view.
While the feline is in the box, she's out of the bag. Mystics the world over use this knowledge as final proof of what they've always believed - that there is no objective reality. (No idea how they get there from here, and I don't understand this stuff well enough to argue the point - but I have a strong suspicion that these guys are misinterpreting things.)
Godel develops his incompleteness theorem. Even if you knew all the rules of a sufficiently complex system, you STILL couldn't prove that all the true statements are true. If you could magically find another true statement and at it to the list of axioms, you still would YET be true statements that couldn't be proved to be so.
It's not easy for me to recognize one thing or person that is more important than the others in the demolition of the classical view (although I, too, have a soft place for Godel.) Each of these is a pretty startling development - even now.
k
