It seems like you're suggesting that mathematicians replace the reals with the computables. This is a reasonable thing to try, and is likely of particular interest to constructivists. There's even this whole field:

https://en.wikipedia.org/wiki/Computable_analysis

> It seems like you're suggesting that mathematicians replace the reals with the computables.

No, not any more. :-)

I liked the idea a year or two ago, but I've come to believe that even the Integers are too bizarre to worry about. For now, I'm content just playing with fixed and floating point, maybe with arbitrary precision. Stuff I can reason about.

I just think people are a little too casual thinking they are really using the Reals. It might be like Feynman's quote about saying you understand QM.