The problem with using computable numbers in mathematics is that if you define two numbers x and y according to two different computations, then the question of

The problem with using computable numbers in mathematics is that if you define two numbers x and y according to two different computations, then the question of whether x equals y is itself uncomputable. That doesn’t inherently make math with computable numbers impossible, but it does mean that even when restricting yourself to computable numbers, you will inevitably start working with some uncomputable functions, so the whole thing doesn’t quite work cleanly.