I would prefer that mathematicians routinely perceived "the reals" as a peculiar construction, and instead of throwing it in routinely when working on concepts in geometry or symmetry as the standard tool to modeling positions and distances, thought about what properties they actually need to get the job they're doing done.

Why is it that mathematicians so love the idea of doing their work blindfolded and with their hands tied behind their backs? Someone invented the reals. They're awesome things. And people invented all sorts of techniques you can use the reals for. Make the most of it! Leave proving stuff about when reals are useful to and how such a peculiar construction can be derived and angsting about how deep and convoluted the basis must be to specialists in angsting about how deep and convoluted the basis for using reals is.

I know of one mathematician who thinks the real numbers are a peculiar construction in the context of topology because of the pathological things you can do with them — continuous nowhere-differentiable curves, space-filling curves, and so on. That's why she studies motivic/A1 homotopy theory instead of classical homotopy theory; only polynomial functions are allowed.

So you would prefer that, instead of having one all-purpose number system that we can embed just about any kind of number we like into (not to mention do all kinds of other things with), we had a collection of distinct number systems, each used for some different ad-hoc purpose? How would this be an improvement?

You might consider the fact that, once upon a time, people actually started with the natural numbers -- and then, over the ages, gradually felt the need to expand the system of numbers further and further until they ended up with the standard objects of modern mathematics, such as the real number system.

This was not a historical accident. Each new kind of number corresponds to a new kind of operation people wanted to do, that they couldn't do with existing numbers. If you want to do subtraction, you need negative numbers; if you want to do division, you need rationals; and if you want to take limits of Cauchy sequences, then you need real numbers.

I don't understand why this should cause computer-programming types any anxiety. A real number is not some kind of mysterious magical entity; it is just the result of applying an operation, call it "lim", to an object called a "sequence" (a_n).

Real numbers are used because people want to be able to take limits (the usefulness of doing which was established decisively starting in the 17th century). So long as you allow the taking of limits, you are going to be working with the real numbers, or something equivalent. Yes, you could try to examine every particular limit that anyone has ever taken, and put them all into a special class (or several special classes), but that would be ad-hoc, ugly, and completely unnecessary.

The set of all definable objects is countable, for obvious reasons.

With this definition, your diagonal trick actually doesn't work (which is good, because otherwise we'd have a paradox): Definability isn't a notion expressible in the theory itself, only in the metatheory. Hence if you attempt to "define" something in terms of the set of all definable numbers, the result is only, uh, "metadefinable".

We could similarly argue that the definable objects should be thought of as "meta-countable" rather than countable, right? The reals-implied-by-a-theory would always be uncountable-in-the-theory. (I'm tempted to imagine a world in which this ended the argument between constructivists and classicists, but realistically, one side or the other would end up feeling uneasy about such a compromise... more likely, both.)