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).
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.
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!
> 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".
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.
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.
In this case, the term "uncomputable" also means "undefinable with less than infinite symbols".
As in, not even computable in theory. It is isn't about normal "computable" concerns. It is "proven to never be characterizable".
Which is a class of numbers whose "existence", if that can term can even be applied coherently for undefinable things, is contested, in theory. In practice they certainly do not exist.
1/3 has infinite decimal digits, but is definable with a finite number of symbols, so it is a computable real. Even if we had no algorithm yet to compute those digits.
Try and define a specific number, that requires infinite symbols to define. As far as I am aware of, no part of calculus involves specific values that have no finite definition, except when the need for uncomputable/undefineable reals are asserted on a circular basis (i.e. they are needed to resolve problems with assumptions that already assume them.)
(Note that a number defined by interpreting the infinite digits of pi as mathematical relations, would still be considered a definable number, assuming some form of convergence could be proven. Because pi is finitely defined.)
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.
> To prove that Ω is computationally and therefore logically irreducible,
requires a theory of program-size complexity that I call algorithmic infor-
mation theory (AIT) [Chaitin, 2005]
Interesting, I think everyone else calls this Kolmogorov complexity.
Actually it’s Solomonoff. But it’s called Solomonoff-Kolmogorov-Chaitin. So Chaitin is among the few people entitled to call it something and AIT is real and less pretentious than using his name.
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.
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.
there is a practical reason to get rid of the fantasy reals and restrict oneself to normal reals or some other new invention:
Since Lean has become more popular as a proving system I've stumbled upon one very annoying feature of reals: they are not computably comparable. The system says you can never know whether two arbitrary real numbers are the same because you don't have enough time to compare them.
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?
26 comments
[ 0.27 ms ] story [ 21.1 ms ] threadBut 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.
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.
Nicolas Gisin: Time, Superdeterminism, & Quantum Gravity
https://www.youtube.com/watch?v=jcHzgy0I6gk
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
¯\_(ツ)_/¯ For some reason I get the same vibe from this as people referring to LLM inference using gendered pronouns instead of "it".
I view "real" numbers, in the context of uncomputable, unnameable numbers, to be as unfortunately named as "imaginary" numbers.
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.
As in, not even computable in theory. It is isn't about normal "computable" concerns. It is "proven to never be characterizable".
Which is a class of numbers whose "existence", if that can term can even be applied coherently for undefinable things, is contested, in theory. In practice they certainly do not exist.
1/3 has infinite decimal digits, but is definable with a finite number of symbols, so it is a computable real. Even if we had no algorithm yet to compute those digits.
Try and define a specific number, that requires infinite symbols to define. As far as I am aware of, no part of calculus involves specific values that have no finite definition, except when the need for uncomputable/undefineable reals are asserted on a circular basis (i.e. they are needed to resolve problems with assumptions that already assume them.)
(Note that a number defined by interpreting the infinite digits of pi as mathematical relations, would still be considered a definable number, assuming some form of convergence could be proven. Because pi is finitely defined.)
Interesting, I think everyone else calls this Kolmogorov complexity.
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)
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.
Since Lean has become more popular as a proving system I've stumbled upon one very annoying feature of reals: they are not computably comparable. The system says you can never know whether two arbitrary real numbers are the same because you don't have enough time to compare them.
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
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