Well, a lot of math doesn't use continuous sets of numbers. Lots of algebra is done using countably infinite sets of numbers like the rationals and the algebraic completion of the rationals. It's about using the right tool for the job. And to that end, the computable numbers are more of a curiosity than something of practical use. Compared to the Reals, the computable numbers are difficult to define and have a lot of logical overhead for virtually no gain. Also, logically, you will never see most all numbers even from a countably infinite set, so a very similar qualm can be had with computable numbers being used in place of the reals (for wherever they can be substituted, which isn't all situations we encounter in math).