People do, it's called computable analysis. https://en.m.wikipedia.org/wiki/Computable_analysis

The first thing on that page are the computable real numbers. They have this annoying property that the set itself is not computable enumerable: there a real numbers that, if we only knew which ones they were, we could know all their digits; but there is no algorithm that will list all such numbers. https://en.m.wikipedia.org/wiki/Computable_number