[h-cobordism]

> It's silly to think that the sqaure root of negative one is something real to be discovered.

As you said, the complex numbers are the unique (up to isomorphism) algebraic closure of R. Given that they arise as the unique solution to (in my opinion) a pretty fundamental question about the real numbers, I think it's fair to say that they've been "discovered", not "invented".

[jeffdavis]

Real numbers might be less real than you think. Once you get beyond the number of possible quantum states in the universe, what do higher numbers really mean?

[h-cobordism]

1. Isn't an infinite universe consistent with observations?

2. Yes, real numbers, much like complex numbers, were "invented". But complex numbers lay hidden, waiting to be discovered, as the algebraic closure of the reals, and similarly, the reals can be discovered from the "simpler" ideas of ordered field and Dedekind-completeness.

3. That's the weird thing about mathematics --- when you invent things, you leave a world of discoveries for others to make, and sometimes those discoveries are that your invention has inside it a perfect mirror image of another invention.

4. Once you leave classical logic, you suddenly have a lot more room for invention, because there are several competing definitions of "real number", and none is definitively better.