The problem I've always had with real numbers is that the vast majority (i.e. uncountably many) of them are not computable, and only countably many of them are computable.
That means that almost every real number, all but a vanishingly small subset, cannot be represented by any means whatsoever. No formulas, no algorithms, nothing. On top of which, they're surprisingly complicated to construct. There's several different ways of doing it, and they're all complicated.
At some point you've got to ask yourself, "are they really even there?" (Of course, those who already have non-Platonic leanings will find that question amusing.) I start to think that maybe we'd be better served by dropping all the non-computable numbers and start doing almost everything in the field of computables.
That means that almost every real number, all but a vanishingly small subset, cannot be represented by any means whatsoever. No formulas, no algorithms, nothing. On top of which, they're surprisingly complicated to construct. There's several different ways of doing it, and they're all complicated.
At some point you've got to ask yourself, "are they really even there?" (Of course, those who already have non-Platonic leanings will find that question amusing.) I start to think that maybe we'd be better served by dropping all the non-computable numbers and start doing almost everything in the field of computables.