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Really? From Euclidean geometry to Rimannian. How about the classical and variational calculi?

I am not sure why you refer to Banach-Tarski. These things, as far as I understand stem from the Axiom of Choice and should not be THAT surprising: once you accept it, you are stuck, for instance, with non-measurable sets.

So how come we first "invent" an area of mathematics and only subsequently "discover" that it describers Nature?

You are correct sir, e.g., tensor analysis for general relativity, and functional analysis for quantum mechanics.

Hmmm. I take the point about Euclidean and Reimannian geometry. Calculus? I am not entirely sure why Leibniz invented/discovered/needed calculus, but Newton was certainly trying to describe physical phenomena. On the other hand, these are relatively basic mathematical constructs to begin with. I suspect that the area under, or the slope of, a curve is not exactly tensor calculus (which I guess Einstein had to learn before he could formalize general relativity, which is more your point than mine).

I'm not sure about any point with B-T. It was an example, (in your favor) of a well-established mathematical fact which you would never expect to describe a physical phenomenon, but, in fact may. And, yes, the sets involved are not Lebesgue-measureable, but if (and that's a big if), there is a physical process that is modeled this way, well that's pretty funky if you ask me...