alt Hacker NewsName a single physics phenomenon that was discovered purely with way 2. I can only think of one, the positron.
Newtonian physics was not kicked off by math "advances". Approximately speaking it was the other way, Newton created the math to explain p^2 ~ r^3, which was a surprising observation.
Even theory of relativity wasn't really a math advance, the math was already mostly worked out by mach, lorenz, and minkowski. Einstein put it together into a coherent story (v. Important)
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Bose-Einstein statistics, which originated purely from finding a derivation of an equation, without immediately understanding of the true physical meaning of the derivation.
And then Bose-Einstein condensates; a thought-experiment consequence of Bose-Einstein statistics.
Radio waves etc., from formulating Maxwell's equations.
The Higgs.