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ChristianKl comments on Rationality Quotes March 2012 - Less Wrong
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I feel quite confident saying that mathematics will never undergo paradigm shifts, to use the terminology of Kuhn.
The same is not true for empirical sciences. Paradigm shifts have happened, and I expect them to happen in the future.
The frequentist vs. baysian debate is a debate of computing mathematical paradigms. True mathematicians however shun statistics. They don't like the statistical pradigm ;)
Gödel's discovery ended a certain mathmatical pradigm of wanting to construct a complete mathematics from the ground up.
I could imagine a future paradigm shift way from the ideal of mathmatical proofs to more experimental math. Neural nets or quantum computers can give you answer to mathematical question that you ask that might be better than the answer s that axiom and proof based math provides.
Except, in practice mathematics still works this way.