alt.hn

7/27/2026 at 3:40:17 PM

How real are real numbers? (2004)

 https://arxiv.org/abs/math/0411418

by surprisetalk

7/29/2026 at 1:32:49 PM

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...

by mikhailfranco

7/27/2026 at 10:54:41 PM

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

7/28/2026 at 10:36:14 AM

> 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

7/28/2026 at 10:55:46 AM

Arguably starts with Wheeler and “it from bit”. Zuse also, who predates Wolfram. For me it’s a little bit like calling the egg you hold in your hand the centre of the Universe while rolling on a skateboard.

by cgio

7/28/2026 at 12:10:44 PM

> 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

Can this be true? DNA is the information system of living creatures and as far as I know, it is not coded with 0s and 1s. So, how can we justify that "the world is built entirely out of digital information"?

by nyc111

7/28/2026 at 12:34:46 PM

For all I, a Victorian everyman, know the world is built from small pistons, gears and pulleys.

Every age has their technology which they will project onto the world. A century ago one may have started to talks about everything being electrical wires and switches.

by atoav

7/28/2026 at 1:22:22 PM

It doesn't matter whether the model assumes bits, pulleys, elves or whatever, as long as it does a better job at describing physical reality than whatever exists at the time anyway. People will try and very probably succeed in providing alternative formalism anyway.

All that matters is whether it facilitates reasoning towards the goal.

by d_tr

7/28/2026 at 12:19:24 AM

I think footnote 16 on pg 12 clarifies his view.

by Diogenesian

7/28/2026 at 1:47:04 AM

From page 12:

> Why should we believe in real numbers, if most of them, it turns out,[^15] are maximally unknowable like Ω? [^16]

The footnotes:

> [^15]: See the chapter entitled The Labyrinth of the Continuum in [Chaitin, 2005]

> [^16]: In spite of the fact that most individual real numbers will forever escape us, the notion of an arbitrary real has beautiful mathematical properties and is a concept that helps us to organize and understand the real world. Individual concepts in a theory do not need to have concrete meaning on their own; it is enough if the theory as a whole can be compared with the results of experiments.

---

The reference [Chaitin, 2005] in footnote 15 links to..

Meta Math! The Quest for Omega - http://arxiv.org/abs/math/0404335

> This book presents a personal account of the mathematics and metamathematics of the 20th century leading up to the discovery of the halting probability Omega. The emphasis is on history of ideas and philosophical implications.

Irreducible Complexity in Pure Mathematics - http://arxiv.org/abs/math/0411091

> By using ideas on complexity and randomness originally suggested by the mathematician-philosopher Gottfried Leibniz in 1686, the modern theory of algorithmic information is able to show that there can never be a "theory of everything" for all of mathematics.

by lioeters

7/27/2026 at 11:15:14 PM

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

7/28/2026 at 12:05:22 AM

> 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

7/28/2026 at 11:18:46 AM

I actually think both "real" and "normal" are helpful in building the correct intuition about how complicated the world we inhabit is rather than how simple we want it to be.

by tialaramex

7/28/2026 at 1:08:44 PM

As far as we can tell, there is nothing resembling a number with infinite digits in the real world.

It's complicated in a completely different way from how the real world is complicated. I don't think that gives you useful intuition.

by Dylan16807

7/28/2026 at 1:43:50 PM

Of course "having infinite digits" isn't from the real world because it's about an artifice, our choosing to represent numbers with digits. But the distinction between the Rationals and the Reals isn't about those digits. The discovery that there's some fixed ratio between the diameter of a circle and its circumference is fascinating and yet though we can't (AFAIK) prove it's normal that ratio sure looks normal and across mathematics we find this ratio again, and again, and again, it's something fundamental but it clearly isn't rational.

Likewise for the square root of 2 and for Euler's Number. These numbers are ever so real and yet they sure fucking look normal to me. If you assure me they are not normal, but you can't prove it, I shall not believe you.

by tialaramex

7/28/2026 at 1:49:02 PM

The real world doesn't have any of those numbers, only approximate matches.

My point isn't about digits, it's about precision. You can do math by hand with more precision than actually exists in the real world. And once you add any slack at all, even one part per googol, your numbers stop being normal and they can all be computed and represented in rational form.

by Dylan16807

7/28/2026 at 2:22:43 PM

> You can do math by hand with more precision than actually exists in the real world.

This claim can't mean anything because it disappears up its own backside. If you, here in the real world, can "do math by hand" that math is in the real world.

by tialaramex

7/28/2026 at 6:56:07 PM

All you have in the real world is scribbles on paper.

I'm sorry, what did you mean by "how complicated the world we inhabit is" because I thought you were talking about physical interactions of matter and energy.

If you're including all math as "real world", then I think the claim that math teaches you something useful about the complexity of the real world is what actually disappears up its own backside.

by Dylan16807

7/28/2026 at 6:51:34 PM

what he probably meant is that if you write down pi with 200 digits there's nowhere in the universe where you can find pi with that precision

by singularity2001

7/28/2026 at 8:16:53 PM

Are you sure? Naively sure, you won't find sufficiently enormous circles and even if you could such a huge circle won't have the Pi ratio here, that's a property of Euclidean space and we don't live in a Euclidean space, ours is slightly off IIRC.

But Pi shows up in other places and I'm not at all sure you can show there are no such places which distinguish some arbitrary approximation from the actual ratio.

by tialaramex

7/28/2026 at 11:47:29 PM

The most precise situations I can think of involve impossibly perfect measurements of volume. And even there, okay a cubic meter is 10^105 cubic planck units and the visible universe is 10^186. Finite math can easily throw a million digits at any problem. How do you reach a point where you need reals to describe actual things?

by Dylan16807

7/29/2026 at 6:48:38 AM

Pi, i and e show up with apparent perfect "precision" in all kinds of physics.

Waves are pervasive and described by relationships involving those numbers. They show up in other relationships. With important properties such as conservation of energy that any partial precision wouldn't be able to achieve.

Numbers are not just evident by single value measurement, but even more powerfully when they govern a system, where any problem with the definition would result in an easily recognizable failure of an entire theory.

I think "precision" is the wrong way to look at what can mean something or not.

I think the boundary between numbers that "make sense", relative those that don't is better found by looking at the progression of numbers.

From naturals, to integers, to rationals, to algebraic (both non-rational roots, and roots of negatives), all the way to limits and series. (Note that the infinite computation associated with expanding digits is not a definition problem. Even 1/3 requires infinite digits in decimal, but the relationship between 1 and 3 is clear.)

What is true about all these numbers is not precision, but that they emerge from a finite number of relationships.

They can be written exactly, defined perfectly, with finite numbers of symbols. (Meaning, abstracting away notation, with a finite number of relationships.)

And all those types of numbers do show up exactly (for all appearances), in waves, and other relationships. The relationships themselves make predictions more powerful than the practical precisions we might have in measuring single values.

So pi really exists. All kinds of physics would fail if it didn't. That doesn't mean we can make a perfect pi circle with plan length, since that would be an arbitrary test, and if the medium is discrete units, one chosen to a priori fail.

Contrast with: The uncomputable, undefinable numbers, which we can't define, can't measure, etc., and are introduced via shaky (relative to the general body of mathematics) means. They require infinite information to define exactly. Not just measure, but even to define. Which is a remarkable postulation, and is not needed to solve any problems they don't themselves introduce.

by Nevermark

7/29/2026 at 1:29:20 PM

> With important properties such as conservation of energy that any partial precision wouldn't be able to achieve.

If your measurement of energy is 150 +/- 2, you only need a handful of digits to do calculations involving that value that preserve it just fine. Insisting that that billionth digit and more still match is no longer working with the real world.

> The relationships themselves make predictions more powerful than the practical precisions we might have in measuring single values.

You can make a prediction to infinite precision but it's not falsifiable. The infinite precision isn't real any more than aether theory is real.

Any powerful results that impact the real world don't need all that precision.

> if the medium is discrete units, one chosen to a priori fail

I choose discrete units because that's what the universe is, as far as we can measure.

If there's something more subtle than Planck, we can't measure it.

It's possible the universe does round at some point. We can't tell.

Can you describe any theoretical experiment that could tell the difference between perfect pi and thousand digit pi?

> Contrast with: The uncomputable, undefinable numbers,

I agree that there's a stark difference there. But I don't think computability is the specific point where it detaches from reality, it's just where the disconnect gets the most obvious.

by Dylan16807

7/29/2026 at 8:48:51 PM

> If your measurement of energy is 150 +/- 2, you only need a handful of digits to do calculations involving that value that preserve it just fine. Insisting that that billionth digit and more still match is no longer working with the real world.

You are repeating the misunderstanding.

Direct magnitude measurements do not limit our ability to test accuracy.

Numbers like pi are not just magnitudes, but form critical relationships. And relationship tests offer (unimaginable) orders of magnitude more stringent testing.

"Weak" relationship test example: The 3-body problem. There are stable modes, but even small discrepancies results in an unstable system falling apart. Accuracy rapidly compounds over observation or reconstructible time.

Strong example: If wave equations were not exact to pi, the discrepancy would be obvious in a nanosecond, much less thousands, millions or 14 billion years.

Pi isn't just a magnitude, it is a very special magnitude, where any offset completely destroys its properties. Properties that have held for billions of years of plank time intervals, themselves distributed over non-linear space time and all the other disturbances of the universe's complexities.

Try to come up with a non-pi number that can do that. The maximum discrepancy you can come up with would be infinitesimal, and shrinking faster and faster every Plank unit of time since the Big Bang, and also relative to the increasing volume, in Plank lengths, of observable space ever since the Big Bang.

There is no measured magnitude accuracy that means anything against that test.

We know it is impossible to create a circle made up of discrete lengths (Plank or not) in flat space, due to basic geometry. So using that as a test, when no theory predicts or depends on that kind of "perfect" circle, is a red herring. Test the exactness of naturally appearing pi where it actually appears to be, not where we know it isn't.

by Nevermark

7/28/2026 at 11:25:15 PM

I agree but with a slightly different definition.

There are no numbers in the real world, that require and infinitely long definition.

1/3 has infinitely many digits in decimal, but has a finite definition. So its good.

I would write down the opposite, a definition of an uncomputable, unnameable real, but there are not enough atoms in the universe (multi-verse, ...) to being to do that.

by Nevermark

7/28/2026 at 2:38:35 PM

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

7/28/2026 at 11:29:03 PM

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.)

by Nevermark

7/28/2026 at 12:17:22 AM

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.

by gregdaniels421

7/28/2026 at 12:30:42 AM

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

7/27/2026 at 8:02:15 PM

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

7/27/2026 at 8:26:51 PM

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

7/27/2026 at 8:37:54 PM

You're taking an anomalously narrow view of the parent comment. Say the minimum distance is one inch.

You want to say that the concept of half an inch lacks physical representation, but that isn't true. You can easily demonstrate it as the ratio between one inch and two feet, compared to the reference ratio between one inch and one foot.

dhosek is saying that in a quantized space, there are reals that cannot be demonstrated this way, and he is right, but the same thing is untrue of rationals.

("In a quantized space", by the way, just means that all measured quantities are necessarily integers. That causes all kinds of problems, but "lacking examples of arbitrary rational numbers" isn't one of them.)

by thaumasiotes

7/27/2026 at 8:46:02 PM

> You can easily demonstrate it [half an inch] as the ratio between one inch and two feet, compared to the reference ratio between one inch and one foot.

You seem to have your units confused. Half an inch is a distance, while ratios, or comparisons of ratios, are all dimensionless scalars.

by tromp

7/27/2026 at 8:54:16 PM

Have you ever seen a map with a scale indicator?

by thaumasiotes

7/27/2026 at 10:48:00 PM

"suppose that the minimum distance is one inch. well, that's one of something. so now imagine half of that! there you go: one half. a physically unrealizable number."

by testaccount28

7/27/2026 at 10:58:02 PM

Except that you’re assuming that 1 must necessarily correspond to that minimum distance. Keeping that situation, we can just say that 1 corresponds to two inches and then realize 1/2 as that number.

Classical geometry (a la Euclid) allows for constructing a lot of numbers. We get natural numbers pretty cheaply, negative integers through adding in a concept of directionality (zero is a bit of an imaginative leap which is why it was absent from Western mathematics for so long). Constructing arbitrary ratios is possible through similar triangles and square roots through right triangles, but some basic algebraic numbers like cube roots cannot be constructed with a ruler and straight edge (which raises the question of whether, in a quantized universe, whether irrational cube roots actually exist). Of course there’s no guarantee that the quantization is going to be uniform and we also have the ɣ factor of special relativity (1/sqrt(1-v²/c²)) which gives us a non-Euclidean space to complicate things, but it’s not clear that if you can find a value for 1 that allows you to get a measurement for every irrational number.

by dhosek

7/27/2026 at 11:02:19 PM

One could make the argument that the only numbers that actually “exist” are the natural numbers, but the question ultimately is can you model any real number in the physical universe. Modeling ½ is simply a question of picking a unit to be 1 and finding its midpoint (or for that matter, declaring two apples to be “1” and thus a single apple would be “½”, although it’s a bit of a challenge to use apples to model (2-√3)/5

by dhosek

7/28/2026 at 12:11:31 AM

Make a line of 10 apples and declare it to be 2 units long, then make a square that's 15 applies diagonal, finally measure how much longer the 10 apples are than the side of the square.

It's a pretty linear increase in complexity between the math and the apples.

by Dylan16807

7/28/2026 at 1:35:00 AM

in what way does drawing a diagram have anything to do with physically realizing a number?

by testaccount28

7/28/2026 at 2:20:20 AM

I didn't say a diagram? I'm talking primarily about physical objects.

And I'm just explaining how to use the same methods as the comment I replied to.

by Dylan16807

7/28/2026 at 6:03:27 AM

"Make a line of 10 apples and declare it to be 2 units long" <- what number does this physically realize?

by testaccount28

7/28/2026 at 6:19:11 AM

Both ten and two. And it lets you model fifths.

But don't ask me about that. That part wasn't my idea at all. Ask dhosek about that way to do fractions. My contribution was the square root and the subtraction.

by Dylan16807

7/28/2026 at 8:10:09 AM

Take the smallest number corresponding to a physically meaningful distance in meters, divide it by two, and that number is still a physical meaningful distance if you switch the unit to decameters or kilometers.

Any rational number has some meaning if you just add the right unit, even if the units become increasingly ridiculous. But for reals that trick does not work

by wongarsu

7/28/2026 at 2:08:06 PM

Sqrt(2) “any unit that is an integral multiple of the shortest interval”.

But it would get complicated, for any given allowable velocity and allowable length, you’d get more lengths from Lorentzian contraction.

There are really a couple of different ideas being combined: are there an infinite number of quantum states for the universe, are space and time continuous, is the forward direction of time resolved by computable processes.

And even bigger ones lurk: are space and time emergent properties from quantum waveforms that lack an inherent idea of space and time (but things that are highly correlated give rise to a notion of being near each other in “space time”)?

by lanstin

7/28/2026 at 1:00:44 AM

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

7/28/2026 at 9:08:46 AM

What numbers in R would not be possible to express in an infinite universe?

by d4ng

7/28/2026 at 9:54:44 AM

Chaitin's constant. Or rather, you can never see that it has been laid out in your infinite universe.

by jojomodding

7/27/2026 at 10:12:08 PM

[dead]

by vee-kay

7/27/2026 at 8:15:05 PM

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

7/27/2026 at 9:57:42 PM

>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

7/28/2026 at 2:34:30 AM

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.

by btilly

7/28/2026 at 5:58:34 AM

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.

by fasterik

7/27/2026 at 9:42:50 PM

> 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

7/27/2026 at 10:53:24 PM

> 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.

by testaccount28

7/28/2026 at 8:30:54 AM

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.

by xelxebar

7/28/2026 at 2:32:05 AM

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.

by btilly

7/27/2026 at 8:37:33 PM

> 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

7/27/2026 at 8:51:42 PM

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).

by andrewla

7/28/2026 at 2:19:33 PM

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.

by lanstin

7/27/2026 at 8:43:45 PM

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.

by mathgradthrow

7/27/2026 at 11:11:50 PM

> 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.

by TheOtherHobbes

7/27/2026 at 9:07:45 PM

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

7/27/2026 at 9:16:05 PM

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.

by andrewla

7/27/2026 at 10:04:27 PM

> 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.

by qsort

7/27/2026 at 11:32:44 PM

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

by mathgradthrow

7/27/2026 at 8:48:07 PM

>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

7/27/2026 at 9:02:27 PM

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.

by andrewla

7/27/2026 at 9:28:22 PM

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.

by xscott

7/28/2026 at 5:30:38 AM

> 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

by discarded1023

7/28/2026 at 8:28:57 AM

Thanks! That was a nice read.

by poetaster

7/27/2026 at 11:20:52 PM

> 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

7/28/2026 at 3:09:06 PM

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

7/28/2026 at 3:57:51 PM

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

7/28/2026 at 4:04:20 PM

They are in a very meaningful sense actually there. If I draw a curve I want the line not to have holes in it, and they have to be there for that to be true.

More importantly, a function is its graph so if I want my functions to be continuous I need them to be there.

by seanhunter

7/28/2026 at 4:36:15 PM

No matter how small you go, between every two real numbers is a computable number, and between every two computable numbers is a real number that's not computable. If you restricted yourself to computable numbers, are you sure there are any holes in your graphs and functions? Can you point to or name one?

:-)

by xscott

7/28/2026 at 5:00:46 PM

I understand the point about density and I'm fine with the concept of functions being continuous over a restricted domain (eg the rationals even) but if I draw a line and label one end 0 and one end 1 then I have plotted the set of all numbers in that interval, not just the ones we find computationally convenient.

The historical context about the constructivist movement is that it was a religiously-inspired objection to the work of Cantor, who some random bishop said was challenging God with his work on transfinite numbers because God owned infinity. I just find it weird that now people try to pretend that it's somehow more rigorous when really it's just an alternative axiomatic perspective that started in this shonky way and has grown to a point where it's just about respectable.

by seanhunter

7/28/2026 at 5:37:58 PM

I can't speak to the religious bits or history. That's certainly not my motivation for thinking about this stuff. It's not about computational convenience either. Both of those seem like strawmen, but maybe they're relevant to other people.

The problem to me is that the Reals which aren't computable are absurd. We've never used any of them in all history. We can only put names on a very special few, those are countable, and we don't even know their value very well.

And the main way we prove that the Reals are uncountable is to use a proof by contradiction. It would take too long to spell it out, but they aren't really just contradicting "Reals are Countable". It's "All that other stuff we think is true AND Reals are Countable" that gets contradicted.

Once you accept the Reals, the Axiom of Choice is not simply obvious any more. And if you go down that path you get things like the Banach-Tarski paradox. To me THAT ought to be a proof by contradiction that we've made a mistake somewhere.

More interesting than that though: If the universe we live in requires non-computable Reals to describe it carefully, then it says something very weird about determinism. In order to compute a future state of a system, we need to use numbers we can't compute?!?

by xscott

7/28/2026 at 7:55:07 PM

> We've never used any of them in all history.

You just used them yourself in your previous post to make your argument that computable numbers are dense in incomputable numbers.[1] So presumably that makes you the first person in all of maths history to use them. Congratulations I guess? The other possibility is they get used a lot and we just don’t make a fuss about it because most of the time it’s exactly like you used them - to express an argument where it doesn’t matter whether they are computable or not.

I have no idea why you think they are absurd but as I say it’s an alternative axiomatic perspective. It just makes a huge amount of maths very inconvenient without really yielding (as far as I can see) much of anything.

I don’t have a perspective on your questions about the universe. Non-computable numbers don’t really trouble my world view. We invented the calculus and the language of continuous functions which gave rise to the rigorous construction of the real numbers precisely to better describe the universe, so it seems that they are pretty useful in that context.

[1] Which is interesting right, because this is a trivial argument in the sense that in standard first year analysis you learn that the rationals are dense in the irrationals and rationals are obviously computable so it must be that rationals are dense in the non-computable numbers. But this is a very basic argument that we can only make because we permit the Reals to be complete. It’s not possible to express the argument in the restriction to computable numbers because there is no ambient space outside the computable numbers for them to be dense into.

by seanhunter

7/28/2026 at 8:24:40 PM

> And the main way we prove that the Reals are uncountable is to use a proof by contradiction. It would take too long to spell it out, but they aren't really just contradicting "Reals are Countable". It's "All that other stuff we think is true AND Reals are Countable" that gets contradicted.

Btw that's not true. Cantor's diagonal argument isn't a proof by contradiction, it's a purely constructive argument. The way he made it in his original paper is a bit more technical than this but this is the "modern" version that's a bit easier to put into layman's terms.

Say you say you can construct a (countably finite) list containing all the real numbers. I say I don't care how you made your list I can give you a number that's not on it, and in fact cut your list down to just numbers between 0 and 1. If you have all the real numbers you must have all the numbers between 0 and 1, but even if you make a countably infinite list of numbers between 0 and 1 I'll give you a procedure that will construct a number that's not on your list no matter how you made it.

1) Read the first number on your list. If it has 1 in the first decimal place, make the first decimal place of my number a two otherwise make it a 1. 2) Read the second number on your list. If it has 1 in the second decimal place, make the second decimal place of my number a two otherwise make it a 1. ....

Proceed in that manner.

At the n-th step I read the n-th number on your list. If it has 1 in the n-th decimal place make the n-th decimal place of my number a 2 otherwise make it a 1.

Now: My number is clearly nowhere on your list as it differs at in least one decimal place from every number on your list.

Therefore it is not possible to construct a countably infinite list of real numbers.

by seanhunter

7/28/2026 at 4:37:06 AM

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

7/28/2026 at 4:58:16 AM

[dead]

by darig

7/27/2026 at 10:50:08 PM

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

7/28/2026 at 6:58:21 PM

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.

by singularity2001

7/28/2026 at 9:48:39 AM

All real numbers are real, but some real numbers are more real than others.

by CodesInChaos

7/29/2026 at 1:15:37 PM

All integers are happy; each real is unhappy in its own way.

by mikhailfranco

7/28/2026 at 2:45:16 PM

Yes, as real as they can be.

Or sometimes as real as you can fathom :)

by fuzzfactor

7/28/2026 at 1:11:53 AM

> 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.

by twothreeone

7/28/2026 at 10:48:07 AM

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.

by cgio

7/27/2026 at 9:07:45 PM

How complex are complex numbers?

by amai

7/27/2026 at 9:09:40 PM

How integral are integers?

by DarkNova6

7/27/2026 at 10:53:22 PM

How natural are natural numbers?

by Ekaros

7/27/2026 at 11:34:01 PM

How 'P' are 'P' numbers?

by z4lis

7/27/2026 at 11:01:03 PM

[flagged]

by reindeer2

7/27/2026 at 10:44:02 PM

5 real

by protocolture