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...
Furthermore, you could argue against the point of Riemannian geometry also by pointing out that Gauss was fond of measuring distances on the surface of the Earth. He too, thus, was compelled by rather practical considerations.
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
Not to argue here --- this is after all a question of taste --- but this remark is rather curious. I never thought that tensors, to the extent that Einstein needed them, were any less "basic" than integration. Should we no judge how "basic" these notions were at the time of their invention?
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.. Thanks for the clarification on this. Interestingly, this is exactly what people said about the work of Lobachevsky, Gauss, and Riemann in mid 1800s: what can this geometry possibly do with the real world?