How real are real numbers? (2004)

Agree to disagree!

Doing calculus with computable numbers is totally possible and you get all the continuity you need. You need to drop the Lebesgue formulation of the integeral and either use a Reimann integral or the gauge integral (Henstock–Kurzweil) if you need a well-behaved integral in the face of very poorly-behaved functions, but in physical reality these don't exist and in abstract mathematics they are rarely of interest and the gauge integral is as robust as Lebesgue without all the measure theory nonsense.

Intuitionalist analysis and calculus are very well established; the only thing you can't do with them is nonsense like showing that integrating over the characteristic function of the rationals is zero (who cares) or showing that you can break a three dimensional sphere up into three pieces are reassemble them after translations and rotations into a larger sphere (obviously not true).

You are overestimating what real numbers buy you.

pi and e and sqrt(2) are real numbers and not rational, to be sure. But they are computable! Computable just means that they are arbitrarily approximable. "approximate relative to what" is that whatever criteria defines the number. You can't represent the "true" value of a non-rational number in the rationals, but you can prove that the error of an approximation is (rationally) bounded above and below, and you can have another approximation with a tighter bound.

Rational numbers are already infinitely precise relative to other representations -- finite decimals are another representation that is functionally equivalent to the rationals, but even a simple rational like 1/3 does not have a finite decimal value.

You can prove all the interesting theorems with computable numbers and rational/decimal numbers. You don't need the real numbers because you can't name a real number that exists and is not computable, BY DEFINITION! No mathematical construction can define a real number that is not constructible. These numbers are useless and there's no reason to continue even in abstract mathematics to pretend that they are useful because we have the formalisms to ignore them.

I don't think your position is silly, but this is not a great argument for it.

> But when we say things like "the rationals are discrete"

In the usual topology they are not?

> In the practical sense the rational numbers are quite continuous -- between any two rational numbers there are an infinite (unbounded) number of rational numbers -- there's no notion of a "leap" the way there is with the integers.

This characterization captures neither the intuitive nor the formal definition of continuity. You are effectively saying that Q is dense in R, but this is insufficient to prove, for example, the intermediate value theorem.

> measure theory, a theory which yields almost nothing of value except endless paradoxes

Come on now. The usual definition of concepts as basic as areas is tethered to measure theory. We say it's "obvious" that the integral is the area under the curve (and it is: e.g. the Riemann integral is trivially the Peano-Jordan measure) but this only works because we're appealing to it. You can route around it, but let's not pretend we're doing it for no reason.

I can see the elegance of a purely intuitionistic construction, but the "usual" real numbers are much closer to how we intuitively (no pun intended) work with numbers.

No, the rationals are not discrete in the usual topology. They end up being discrete when we consider continuous mappings from R->Q though. That is the "technical" sense that I refer to. The rationals, as you say, are dense in R but they are also dense in the computables.

The big Cantorian leap that we make is when we use the diagonal argument to argue that the rationals are countable. All the real construction techniques (Dedekind cuts or Cauchy sequences) effectively only yield the computable numbers, the real numbers outside of the computables are inherited from the diagonal argument rather than being foundational to the construction. I mean, this is trivially true because constructions are constructive.

I disagree that area is tethered to measure theory; I certainly learned about areas in geometry long before I ever heard of anything with measure theory. Measure theory exists to tie up some of the horrifying poorly behaved functions that increasingly wily mathematicians invented to break our notions of area and continuity. But we have better tools now for dealing with those that don't involve measure theory so there's no reason to ever hear the phrase "almost everywhere" or "subadditive" ever again.

To back it up to your closing and my main point -- the constructive numbers are way closer to the way we work with numbers because all numbers we ever deal with, even abstractly, fit this definition much better.

> What do "real" numbers buy you?

They're well-known and have a simpler implementation, and we are familiar with their quirks. There is a giant body of useful knowledge built up around standard real analysis. That doesn't really exist if you insist on using only computable numbers.

The computables are also more fiddly in many ways. Because equality is undecidable, you can't have discontinuous functions, you need to carry around error epsilons all over the place, and we lose useful tools like the Heine-Borel theorem, I think.

Try proving some results in PDE theory, and I think you might change your mind.

In general, I find clarity in thinking of numbers as the system that implements them, rather than as platonic objects with individual reality. What does using Old Boring tech buy you over using Shiny New Thing?