> There's no separating physics and math.There is. Even though most physics research is extremely mathematical and abstract these days, it's still ostensibly grounded in empirical science. Math is not science, it just provides useful tools and insights for studying science. Unlike physics, the disciplinary imperative of math is not to provide us with truths about this world or any other world. Its imperative is to tell us what must follow as a consequence from a given set of assumptions and definitions. This is a very important philosophical dichotomy because it means that even the most lackadaisical, abstract problems in physics (such as moonshine in high energy physics) are still grounded in something "real." Math need not be grounded to anything real; it can be decoupled from what is real or even possible entirely.
> Keeping math "pure" really means keeping it "purely abstract", so it resists any kind of practical application.
I'm not one to be elitist with regards to pure versus abstract mathematics so I sympathize with your point here. That being said, purely abstract mathematics can be extremely useful even if it doesn't ultimately relate to the real world. Consider what G.H. Hardy wrote nearly a century ago in A Mathematician's Apology:
"...both Gauss and lesser mathematicians may be justified in rejoicing that there is [number theory] at any rate...whose very remoteness from ordinary human activities should keep it gentle and clean."
If only Hardy had lived long enough to see his pure and beautiful number theory sullied with the applications to error-correcting codes and cryptography.