Name 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)