[andrewla]

I think people overindex on the continuity problem with the reals. I'm personally a bit of real-number denier myself as a constructivist / intuitionalist.

But when we say things like "the rationals are discrete" or "the computable numbers are discrete" these are very specific claims in the domain of measure theory, a theory which yields almost nothing of value except endless paradoxes and naval-gazing nonsense. Similarly when people say "the rationals are countable" and "the computable numbers are countable" this is taking for granted the Cantor notion of measuring cardinality by bijective correspondence, once again, a theory that yields nothing of value except endless paradoxes and naval-gazing nonsense.

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. And any useful number can be approximated arbitrarily closely by rationals.

And for computable numbers there's even less of a gap. With rationals you can only approximate. But you can have a computable number that is exactly equal to the square root of 2, because a computable number is the algorithm by which you form arbitrarily close approximations. The square of that computable number is itself computable and is exactly equal to 2.

What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable? Basically nothing.

And if you're worried that sticking to the rationals and the computable numbers is too much of a concession to "physical reality", rest assured -- the rationals are just as unphysical as the real numbers because they are continuous already, and physics does not give us the power to measure the difference between two sufficiently precise rational numbers just as it barfs when you throw "real" numbers at it.

[fasterik]

>What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable?

I guess the naive answer is completeness. Every Cauchy sequence converges to a member of the space. For example, quantum mechanics relies on the formalism of Hilbert space, defined as a complete inner product space. This gives us nice things like the spectral theorem for unbounded operators, without which we wouldn't be able to define probability (the Born rule) or time evolution (the operator exponential e^-iHt).

Can you formalize quantum mechanics using computable numbers? I don't actually know, but let's say yes. I assume it's more work with more edge cases, so I would ask the same question: what do you get for the trouble of building a formalism around computable numbers?

[btilly]

Bad example. You can do all of this with constructivism. Any constructable Cauchy sequence converges to a constructable member of the space.

What you get for the formalism around computable numbers is this. Every mathematical object in the theory is something that can be, at least in principle, actually written down. When we say that it exists, this existence is of the most tangible form that any mathematical thing could have.

[fasterik]

Having constructible Cauchy sequences doesn't guarantee that we can construct unbounded operators. I'm no expert, but the little searching I've done suggests this is an open research question.

I don't see the benefit of being able to write something down "in principle." A number can only ever be computed to a finite number of digits in practice. If we're talking about finite approximations, then the standard approach using numerical solutions to the Schrödinger equation handles this just fine, no alternative mathematics needed. If we're talking about theories, then we should choose whatever abstraction is most convenient for expressing the theory.

Personally, I don't believe numbers "exist." The physical universe exists, and numbers are abstractions that we invent to describe it. In that sense, uncomputable numbers are just as "real" as computable ones.

[xelxebar]

> 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?

[testaccount28]

> equality is undecidable

equality is always undecidable until you see the light of intuition. consider the rational number whose numerator is 0 if $theorem is true, and 1 if it is false, and whose denominator is 1.

[xelxebar]

Okay, theorem=generalized-continuum hypothesis. If you use exotic axioms to give that a definite result, the go eat a Gödel.

We define computable numbers to be Turing machines, lambda reduction processes, or whatever your favorite model of computation happens to be. If you don't like this kind of definition, then we need to talk philosophy of computation.

To decide equality, we let your machines clunk along until they both produce a result, which we then compare (using another machine). Hello Mr. Halting Problem. Specific programs are fine, but comparing against arbitrary classes of program is the bugger. This is why discontinuous functions cannot exist in a hardline computable analysis theory.

[btilly]

That is not a number in constructivism.

But there are numbers in constructivism for which it is unknown whether they are zero. Some of which must remain unknown, if mathematics is consistent. This is a rather important and weird edge case.

[BeetleB]

> What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable? Basically nothing.

It buys you the rigor of doing calculus, which buys you a lot of results that, while could be computed without calculus, would also be very difficult without it.

[andrewla]

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).

[lanstin]

Like the old joke: The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?

But can you do stats without measure theory? Normal distribution in the limit and all that?

There is no linearly additive measure on rationals, and therefore no way to grab a rational uniformly from (0,1). Has to be skewed to some level of complexity in the denominator.

[mathgradthrow]

Navel gazing is a physical phenomenon, so if you would like to know everything about physical phenomena, you have to be able to predict the navel gazers.

[TheOtherHobbes]

> What do "real" numbers buy you?

They're a powerful abstraction - the base concept of a smooth continuous complete domain which encodes non-trivial relationships, and is a prototype for other analytic abstractions.

The reals are the philosophical base class for some very useful mathematical objects. Computability and physicality are both side issues.

[qsort]

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.

[andrewla]

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.

[qsort]

> The big Cantorian leap that we make is when we use the diagonal argument to argue that the rationals are countable.

Again, I don't think your position is indefensible, but this doesn't strike me as particularly convincing. The usual definition of R is that there exists a unique ordered complete Archimedean field up to isomorphism. We get the kitchen sink from the least upper bound property. As a constructivist you're gonna say that I don't get to define R like that, but you can't pretend it's done for no reason or that it buys nothing.

> I certainly learned about areas in geometry

And how were they defined? In elementary geometry we just sweep the question under the rug, usually...

If you get to say that being able to articulate why the measure of Q is 0 is unimportant and uninteresting, then I get to claim that the supposed problems with the usual definitions are also unimportant!

Saying that the non-constructive world leads to worse problems is a respectable position. Pretending the usual way of doing things is completely arbitrary isn't very honest.

[mathgradthrow]

The word you are looking for, probably, is "totally disconnected". Discrete always refers to the "discrete topology".

[discarded1023]

> Similarly when people say "the rationals are countable" and "the computable numbers are countable" this is taking for granted the Cantor notion of measuring cardinality by bijective correspondence, once again, a theory that yields nothing of value except endless paradoxes and naval-gazing nonsense.

You may enjoy a recent update on that story [0] that maybe avoids a few paradoxes and looks at things other than navels.

[0] https://nicholasdibella.com/cantor.pdf

[poetaster]

Thanks! That was a nice read.

[LPisGood]

> these are very specific claims in the domain of measure theory, a theory which yields almost nothing of value except endless paradoxes and naval-gazing nonsense

Measure theory is used for lots of practical things, for example probability theory.

[dmfdmf]

>With rationals you can only approximate.

Approximate relative to what? All actual measurement is implicitly or explicitly approximate such as L = x meters +/- epsilon. There is no infinite precision by which to discount rational measures as "approximate" and thus "invalid" in any way.

>What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable? Basically nothing.

You "buy" all of mathematics which operates on the assumption of "infinite" precision. It is an abstraction necessary to prove theorems and relationships of math. Abstracting from precision isn't a denial that it exists, it is the assumption that I can ignore it or leave it undefined. This is the assumption that distinguishes math from physics/engineering. Mathematicians deal with abstract e or pi but in the real world pi=3.14 if you are tiling your patio and 3.14159265... or whatever is necessary to get to the moon.

[andrewla]

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.

[xscott]

I'm on your side for most of what you say. This topic has been interesting to me for years. I've considered going back to school to build on my math degree, specifically because of this topic.

However, I thought things like Chaitin's Constants (you could make one per programming language) are real numbers you can name but not compute. I think you could do this from any undecidable problem.

Of course there only a countable number of those Reals. And they still don't seem useful for much more than naval gazing.