8 ms·

How real are real numbers? (2004)

96 points by surprisetalk 3mo ago

dhosek 3mo ago
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).
- andrewla 3mo ago
  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.
  - thaumasiotes 3mo ago
    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.)
    - tromp 3mo ago
      > 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.
      - thaumasiotes 3mo ago
        Have you ever seen a map with a scale indicator?
    - testaccount28 3mo ago
      "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."
      - dhosek 3mo ago
        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.
        dhosek 3mo ago
  - wongarsu 3mo ago
    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
    - lanstin 3mo ago
      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”)?
- vee-kay 3mo ago
  [dead]
- CuriouslyC 3mo ago
  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.
- d4ng 3mo ago
  What numbers in R would not be possible to express in an infinite universe?
  - jojomodding 3mo ago
    Chaitin's constant. Or rather, you can never see that it has been laid out in your infinite universe.
andrewla 3mo ago
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.
- mathgradthrow 3mo ago
  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.
- BeetleB 3mo ago
  > 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.
zyklu5 3mo ago
Similar fun at this Baez blog post from a decade ago: Surprises in Logic: https://math.ucr.edu/home/baez/surprises.html https://math.ucr.edu/home/baez/surprises.html
amai 3mo ago
How complex are complex numbers?
- DarkNova6 3mo ago
  How integral are integers?
  - Ekaros 3mo ago
    How natural are natural numbers?
    - z4lis 3mo ago
      How 'P' are 'P' numbers?
deleted 3mo ago
[deleted]
protocolture 3mo ago
5 real
jonahx 3mo ago
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 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 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.
- lioeters 3mo ago
  Norman Wildberger's YouTube channel, Insight into Mathematics - https://www.youtube.com/@njwildberger https://www.youtube.com/@njwildberger
- mikhailfranco 3mo ago
  Also Nicolas Gisin on Curt's podcast: Nicolas Gisin: Time, Superdeterminism, & Quantum Gravity https://www.youtube.com/watch?v=jcHzgy0I6gk https://www.youtube.com/watch?v=jcHzgy0I6gk
ttctciyf 3mo ago
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 https://arxiv.org/pdf/math/0303352, 1.9 Is Mathematics Quasi-Empirical
- Diogenesian 3mo ago
  I think footnote 16 on pg 12 clarifies his view.
  - lioeters 3mo ago
    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 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 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.
reindeer2 3mo ago
[flagged]
Nevermark 3mo ago
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.
- Dylan16807 3mo ago
  > 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".
  - tialaramex 3mo ago
    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.
    - Dylan16807 3mo ago
      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.
      - tialaramex 3mo ago
        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.
gregdaniels421 3mo ago
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.
- russdill 3mo ago
  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.
twothreeone 3mo ago
> 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.
- cgio 3mo ago
  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.
dang 3mo ago
Related. Others? How real are real numbers? (2004) - https://news.ycombinator.com/item?id=24029791 https://news.ycombinator.com/item?id=24029791 - Aug 2020 (112 comments) How real are real numbers? (2004) - https://news.ycombinator.com/item?id=14080024 https://news.ycombinator.com/item?id=14080024 - April 2017 (265 comments)
francisdavey 3mo ago
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.
- darig 3mo ago
  [dead]
cwillu 3mo ago
https://ar5iv.labs.arxiv.org/html/math/0411418 https://ar5iv.labs.arxiv.org/html/math/0411418 for the in-browser html rendering
CodesInChaos 3mo ago
All real numbers are real, but some real numbers are more real than others.
- fuzzfactor 3mo ago
  Yes, as real as they can be. Or sometimes as real as you can fathom :)
- mikhailfranco 3mo ago
  All integers are happy; each real is unhappy in its own way.
seanhunter 3mo ago
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.
- futune 3mo ago
  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.
  - seanhunter 3mo ago
    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.
    - xscott 3mo ago
      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? :-)
singularity2001 3mo ago
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.
mikhailfranco 3mo ago
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 https://arxiv.org/pdf/1803.06824 Real Numbers are the Hidden Variables of Classical Mechanics https://philarchive.org/rec/GISRNA https://philarchive.org/rec/GISRNA Time Really Passes, Science Can’t Deny That https://arxiv.org/pdf/1602.01497v1 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-free-will https://www.sciencenews.org/article/real-numbers-physics-fre...

96 points by surprisetalk 3mo ago

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).

andrewla 3mo ago
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.
- thaumasiotes 3mo ago
  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.)
  - tromp 3mo ago
    > 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.
    - thaumasiotes 3mo ago
      Have you ever seen a map with a scale indicator?
  - testaccount28 3mo ago
    "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."
    - dhosek 3mo ago
      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.
      - dhosek 3mo ago
- wongarsu 3mo ago
  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
  - lanstin 3mo ago
    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”)?
vee-kay 3mo ago
[dead]
CuriouslyC 3mo ago
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.
d4ng 3mo ago
What numbers in R would not be possible to express in an infinite universe?
- jojomodding 3mo ago
  Chaitin's constant. Or rather, you can never see that it has been laid out in your infinite universe.

andrewla 3mo ago

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.

mathgradthrow 3mo ago
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.
BeetleB 3mo ago
> 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.

zyklu5 3mo ago

Similar fun at this Baez blog post from a decade ago: Surprises in Logic: https://math.ucr.edu/home/baez/surprises.html https://math.ucr.edu/home/baez/surprises.html

amai 3mo ago

How complex are complex numbers?

DarkNova6 3mo ago
How integral are integers?
- Ekaros 3mo ago
  How natural are natural numbers?
  - z4lis 3mo ago
    How 'P' are 'P' numbers?

deleted 3mo ago

[deleted]

protocolture 3mo ago

5 real

jonahx 3mo ago

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 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 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.

lioeters 3mo ago
Norman Wildberger's YouTube channel, Insight into Mathematics - https://www.youtube.com/@njwildberger https://www.youtube.com/@njwildberger
mikhailfranco 3mo ago
Also Nicolas Gisin on Curt's podcast: Nicolas Gisin: Time, Superdeterminism, & Quantum Gravity https://www.youtube.com/watch?v=jcHzgy0I6gk https://www.youtube.com/watch?v=jcHzgy0I6gk

ttctciyf 3mo ago

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 https://arxiv.org/pdf/math/0303352, 1.9 Is Mathematics Quasi-Empirical

Diogenesian 3mo ago
I think footnote 16 on pg 12 clarifies his view.
- lioeters 3mo ago
  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 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 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.

reindeer2 3mo ago

[flagged]

Nevermark 3mo ago

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.

Dylan16807 3mo ago
> 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".
- tialaramex 3mo ago
  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.
  - Dylan16807 3mo ago
    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.
    - tialaramex 3mo ago
      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.

gregdaniels421 3mo ago

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.

russdill 3mo ago
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.

twothreeone 3mo ago

> 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.

cgio 3mo ago
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.

dang 3mo ago

Related. Others? How real are real numbers? (2004) - https://news.ycombinator.com/item?id=24029791 https://news.ycombinator.com/item?id=24029791 - Aug 2020 (112 comments) How real are real numbers? (2004) - https://news.ycombinator.com/item?id=14080024 https://news.ycombinator.com/item?id=14080024 - April 2017 (265 comments)

francisdavey 3mo ago

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.

darig 3mo ago
[dead]

cwillu 3mo ago

https://ar5iv.labs.arxiv.org/html/math/0411418 https://ar5iv.labs.arxiv.org/html/math/0411418 for the in-browser html rendering

CodesInChaos 3mo ago

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

fuzzfactor 3mo ago
Yes, as real as they can be. Or sometimes as real as you can fathom :)
mikhailfranco 3mo ago
All integers are happy; each real is unhappy in its own way.

seanhunter 3mo ago

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.

futune 3mo ago
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.
- seanhunter 3mo ago
  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.
  - xscott 3mo ago
    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? :-)

singularity2001 3mo ago

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.

mikhailfranco 3mo ago

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 https://arxiv.org/pdf/1803.06824 Real Numbers are the Hidden Variables of Classical Mechanics https://philarchive.org/rec/GISRNA https://philarchive.org/rec/GISRNA Time Really Passes, Science Can’t Deny That https://arxiv.org/pdf/1602.01497v1 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-free-will https://www.sciencenews.org/article/real-numbers-physics-fre...