I think we're in a circle: I'm objecting that we teach people to map everything onto the real line and you're saying that it's essential to quantitatively under

I think we're in a circle: I'm objecting that we teach people to map everything onto the real line and you're saying that it's essential to quantitatively understanding the world. But isn't that what "quantitatively" means? My point is that in the zoo of mathematics, the reals are just one exhibit. Equally valid is to map phenomena onto topological spaces, inner product spaces, sets, groups, rings, fields, lattices, topoi, etc... People have been standing on Newton's shoulders for so long that all they can see from there is ground well worn by their colleagues who stood on the same shoulders. I think we'd be much better off if you had to specialize in some part of math, but that different people specialized in different parts of it without necessarily taking a major in it. This would maximize the sort of happy accidents that lead to discovery because for any given phenomena you now have a wider variety of perspectives on it, rather than just a classroom full of analysts. I'm against the reals in particular because I think they're especially suited to zero sum games, and I wish we played fewer of those.