Quote:  "Without the reals you can't assume that all limits exist."  Oh, I agree completely.  I can't assume that all limits exist.<p>Modded you up because I agree with your premises, but I disagree with your conclusion.  Both constructivism and "regular" mathematics are systems, but rather than views computable numbers as approximations of reals, I prefer to think of real valued calculus as a degenerate case numerical analysis where delta goes to zero.<p>Here's the thing:  most of mathematics was developed prior to computers, and I question how much of what's in mainstream mathematics is influenced by the requirement that ultimately the math had to be tractable to humans without the aid of a computer.