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- His second "proof" of uncountability is very poorly explained. Strictly speaking it is false, since all his reasoning applies equally to the rationals. What he shows is that a countable set would have zero measure. You have to also show that (say) the real interval [0,1] has measure 1 (or at least positive measure) to get a contradiction. That requires some more work. You have to be using some property of the reals in order to prove uncountability, as of course the Cantor diagonal argument does.by francisdavey
- Norman Wildberger is a required mention on this topic.
Here's a great discussion on Curt Jaimungal's podcast:
https://www.youtube.com/watch?v=l7LvgvunVCM
And a good debate on the topic with Daniel Rubin, who takes the more orthodox position:
https://www.youtube.com/watch?v=edh5bbgSKqo
Wildberger has tons more on this topic on his own channel. His arguments are thought-provoking, even if you don't agree with them.
by jonahx - Also Nicolas Gisin on Curt's podcast:
Nicolas Gisin: Time, Superdeterminism, & Quantum Gravity
- Norman Wildberger's YouTube channel, Insight into Mathematics - https://www.youtube.com/@njwildbergerby lioeters
- Similar fun at this Baez blog post from a decade ago: Surprises in Logic: https://math.ucr.edu/home/baez/surprises.htmlby zyklu5
- I don’t understand constructivism at all.
No numbers are real. They are all an entirely abstract construction, like lines and planes and open sets and closed balls and metric spaces and everything else.
If I construct a number by saying it’s the limit of the sequence sqrt(2)-1/n as n->infinity that’s just as real as the number 1 or 1/2.
It just seems really arbitrary to privilege one kind of construction over another. We want a complete ordered field, so we built the reals. Saying they don’t exist or they aren’t real or whatever seems just to be completely beside the point. They are real enough to do the thing we want them to do.
They are just as real as anything else in maths.
by seanhunter - The numbers you mentioned are computable numbers. Constructivists and intuitionists have no problem with them, generally. The problem is that there is only a countable number of computable reals, so what do we do about all the other reals? The ones that nobody will give an example of because it is simply not possible to do so, which is to say, the vast majority of reals?
Does it exist if it is impossible to show an example? Only if "exist" is interpreted to mean "you cannot deduce a contradiction from assuming them", which is a logically consistent position. But if you mean "they are in some sense actually there", in some describable way, then it's muddier. And you can build a logically consistent position from declaring "they are not there" as well. Precisely because no counterexample can be produced.
And then there's ultrafinitists, and yeah, they are a bit bonkers.
by futune - 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.
by andrewla - > 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.
by LPisGood - >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.
by dmfdmf - 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.
by qsort - > 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.
- 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.
- > 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.
by BeetleB - > 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?
by xelxebar - >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?
by fasterik - We should also mention Nicolas Gisin.
One of his mantras is Time is real; Real numbers are not. He conceives of real numbers resulting from processes (approximations, relaxations, computable calculations) that unfold over time. So there is a Heisenbergish uncertainty principle of observable precision and elapsed time.
Selected papers:
Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real?
https://arxiv.org/pdf/1803.06824
Real Numbers are the Hidden Variables of Classical Mechanics
https://philarchive.org/rec/GISRNA
Time Really Passes, Science Can’t Deny That
https://arxiv.org/pdf/1602.01497v1
Popular articles:
Real numbers don’t cut it in the real world, this physicist argues
https://www.sciencenews.org/article/real-numbers-physics-fre...
- Related. Others?
How real are real numbers? (2004) - https://news.ycombinator.com/item?id=24029791 - Aug 2020 (112 comments)
How real are real numbers? (2004) - https://news.ycombinator.com/item?id=14080024 - April 2017 (265 comments)
by dang - I’ve had this paper downloaded for about a decade and haven’t gotten around to reading it, but thinking about it, especially if space and time are quantized (an undetermined question last I checked and almost certainly still so), there would exist numbers in ℝ that cannot be expressed as physical quantities, even with an infinite universe. It’s possible that even the algebraic numbers include numbers that are non-physical (although it might be a larger subset of numbers than the constructible numbers depending on the structure of space-time’s quantization).by dhosek
- What numbers in R would not be possible to express in an infinite universe?by d4ng
- Reality probably isn't quantized in the "on a grid" sense, but rather the "ability to resolve" sense. The more computation you put in the higher accuracy you can get.by CuriouslyC
- If reality is quantized and there is a smallest number that is physically relevant, you don't need the reals to break it. Take that smallest number and divide it by two, and now you have a physically meaningless number using only the rationals.
This isn't fair for what quantization means in reality, but I'm just pointing out that you don't have to introduce the real numbers to get physically meaningless quantities.
by andrewla - This doesn't seem like a very good point to make, sure reals are uncountable and any set of them with labels is countable and of measure zero. That doesn't say anything about physics at all. In QM things are only discrete in certain ways, like energy levels, not positions. A wave function over space can take on any real valued value in its range. Probabilities are real numbers(norms of the wave functions), and there is no reason to believe they would be discrete. Take cosine squared, given any angle it takes on all values between zero and one at some point.
- A more interesting, if not totally pointless question to me, is, are all the constants of nature computable?
If any are a "randomly" chosen real number, the answer would almost certainly be no. But a test of sufficient precision would of course be impossible.
by russdill - It is too bad we don't have a pithy familiar term for the computable / definable / "nameable" / constructible reals. The most general class of undisputed numbers, consistent with the forms we actually use to represent quantities and perform numerical operations.
I view "real" numbers, in the context of uncomputable, unnameable numbers, to be as unfortunately named as "imaginary" numbers.
by Nevermark - I don't think it makes sense to say anything except "computable real" - the computable / uncomputable distinction seems totally immaterial for the purposes of most real analysis, or even "pathological" topology and set theory involving R (except for puzzles directly involving computability). And the "interesting" transcendental computable reals are a bit of a grabbag.
The mean value theorem isn't true for the computable reals, differentiation of computable function isn't always computable, sequences tend to behave poorly, etc. There's still a lot you can say: https://en.wikipedia.org/wiki/Computable_analysis but in general calculus doesn't care about computability, that's a human problem.
by Diogenesian - > I view "real" numbers, in the context of uncomputable, unnameable numbers, to be as unfortunately named as "imaginary" numbers.
They're worse. Just having another dimension is significantly more relevant to reality.
And the reals also ruin the word "normal".
by Dylan16807 - I've long admired Chaitin for his original thinking and especially his ability to clearly convey his ideas about foundations, complexity and information in concise and digestible short proofs.
I'm a little surprised, however, to see him here proselytizing for a particular side in the constructivism debate. I associate him more with what he has described as a "quasi empirical" approach to mathematics[0] where the adoption of new axioms (such as for example the axiom of choice) is justified by their resulting in new, interesting mathematics.
But here, it seems his goal is to arrive at a somewhat Platonic conclusion, that either the reals are valid numbers or (seemingly he prefers) not.
My lay and naive take would be: if you adopt these rules (this Formal Axiomatic System) then you can have Big Fun in the playground of ever more esoteric and complex infinite cardinals, or if you adopt this other FAS you get to discover which results can and can't be obtained under a strict constructivist regime, and whichever FAS you choose it's just the same process of choosing axioms and applying valid deductive steps to arrive at a result you find interesting, with no more "existence" implied than the thoroughly non-Platonic existence of a solution to a problem, which can be demonstrated by solving it: if you take such-and-such steps then such-and-such result will follow.
I suppose the point being made is that avoiding axioms which imply the "existence" of the reals is more useful for doing physics, but that seems non-obvious in a field which for the last 200 years has seemingly sought to progressively make more and more phenomena intelligible by means of differential equations from infinitesimal calculus!
0: see, for example: https://arxiv.org/pdf/math/0303352, 1.9 Is Mathematics Quasi-Empirical
by ttctciyf - > According to Pythagoras everything is number, and God is a mathemati- cian. This point of view has worked pretty well throughout the development of modern science. However now a neo-Pythagorian doctrine is emerging, according to which everything is 0/1 bits, and the world is built entirely out of digital information. In other words, now everything is software, God is a computer programmer, not a mathematician, and the world is a giant information-processing system, a giant computer [Fredkin, 2004, Wolfram, 2002, Chaitin, 2005]
¯\_(ツ)_/¯ For some reason I get the same vibe from this as people referring to LLM inference using gendered pronouns instead of "it".
by exac - I think footnote 16 on pg 12 clarifies his view.by Diogenesian