If you can't do math at a fairly advanced level - at least having competence with information theory, probability, statistics, and calculus - you can't understand the world beyond what's visible on its (metaphorical) surface.

While as a mathematician I find that claim touching, I can't really agree with it. To use the example that was one of the starting points of this conversation, how much math do you need to understand evolution? Sure, if you want to really understand the modern synthesis in detail you need math. And if you want to make specific predictions about what will happen to allele frequencies you'll need math. But in those cases it is very basic probability and maybe a tiny bit of calculus (and even then, more often than not you can use the formulas without actually knowing why they work beyond a very rough idea).

Similar remarks apply to other areas. I don't need a deep understanding of any of those subjects to have a basic idea about atoms, although again I will need some of them if I want to actually make useful predictions (say for Brownian motion).

Similarly, I don't need any of those subjects to understand the Keplerian model of orbits, and I'll only need one of those four (calculus) if I want to make more precise estimations for orbits (using Newtonian laws).

The amount of actual math needed to understand the physical world is pretty minimal unless one is doing hard core physics or chemistry.