I've thought a little bit about this as a physicist. Given that the numbers we really need for science are countable and they are dense in the real numbers, why don't people imagine how math would look like restricted to this set?

My best example: epsilon delta proofs...well, usually when I run physics simulations, I have an explicit error I can't overcome, so I don't really need to know whether a limit converges for all epsilon, just some larger than the largest error in my problem.