37 ms·

Are real numbers really real?

147 points by d3ckard 8y ago

myWindoonn 8y ago
There exists a notion of "computable analysis", based on the delightful fact that Turing machines (or other Turing-complete enumerable computers) can be viewed as computing real numbers. These so-called computable reals are a little limited, but still usable for most tasks [0]. [0] https://en.wikipedia.org/wiki/Computable_analysis https://en.wikipedia.org/wiki/Computable_analysis
- chx 8y ago
  It's a real PITA to work with computable numbers because even their equivalence is not computable. Also, the field is really young. It was only in 2000 that Weihrauch has proven that every computable real function is continuous.
  - myWindoonn 8y ago
    Their equivalence can be given by a Turing machine, though, so the non-computability comes directly from the Halting Problem. This is worth mentioning because some folks get the incorrect impression that it's not possible to write a program that, say, outputs whether two computable reals are equal to each other. It is possible! But the program might not halt. I shouldn't trivialize the difficulty. It is a PITA. I remember my first foray into this; I wrote a program which computed sqrt(8), which worked fine, and then another program which computed sqrt(8)^2, which didn't halt. The edges are tricky.
  - voxl 8y ago
    As opposed to classical reals which have a computable equivalence? (Hint: they don't)
    - chx 8y ago
      But when working with reals we normally don't bother with computability. Whereas with computable numbers you don't even have anything else really.
- aeneasmackenzie 8y ago
  They are usable for all tasks. What are you going to do with a number you can't compute? Nothing, because you can't do anything with it.
- no_identd 8y ago
  Originally I wanted to make a point here about Interactive Computation, the strong Church-Turing hypothesis and persistent Turing Machines, however, while checking on the latest research in that area before doing so, I found the following paper: https://arxiv.org/abs/1702.06000 https://arxiv.org/abs/1702.06000 TE Raptis - 'Viral' Turing Machines, Computation from Noise and Combinatorial Hierarchies Which gets cited by the following paper by the same author, which cites the OP paper(!), AND elaborates on it: https://arxiv.org/abs/1805.06301 https://arxiv.org/abs/1805.06301 TE Raptis - Finite Information Numbers through the Inductive Combinatorial Hierarchy Which itself gets yet another follow on by the same author: https://arxiv.org/abs/1806.01637 https://arxiv.org/abs/1806.01637 - Encoding discrete quantum algebras in a hierarchy of binary words Highly interesting.
recursive 8y ago
"Real" numbers are mathematical abstractions. Perhaps it's unfortunate that "real" and "imaginary" were used instead of more abstract sounding terms that came with less attached connotation.
- jordigh 8y ago
  Just wait until cranks discover the hairy ball theorem and try to prove how wrong hairy balls are...
- danharaj 8y ago
  The argument of the author is not that the real numbers are poorly named, it's that the mathematical abstraction supposes physically impossible properties of the model. Therefore physics ought to be based on an abstraction that has more physically plausible properties instead of the real numbers. Not that I necessarily agree with such an argument.
- otakucode 8y ago
  There is value in comparing mathematics to reality even though mathematics is defined entirely independent to reality. We are able to use mathematics to model certain aspects of reality with great effectiveness.... why? Physical reality does not seem to resemble mathematical notions at all, yet shows some of the same results. Understanding that gap would be of monumental significance.
kavalec 8y ago
"Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic." This sounds like "hidden variable" and is not true for quantum physics.
- Retric 8y ago
  It is, but you need to give up more and allow information to travel faster than light speed. It's local hidden variables that are a problem.
  - kgwgk 8y ago
    Non-locality != faster-than-light communication
    - Retric 8y ago
      As I understand it 'Spooky action at a distance' does not allow communication, but in for hidden variable theory to work a particle needs access to another particles hidden variables. Unless you want to suggest another model?
      - kgwgk 8y ago
        Why is non-locality a problem for hidden variables theories and not for the "standard" interpretation of quantum mechanics?
        Retric 8y ago
        Quantom mechanics is based on non-locality, that's one of the things people have problems with. Quantum key distribution for example is seems to be based on this https://en.wikipedia.org/wiki/Quantum_key_distribution https://en.wikipedia.org/wiki/Quantum_key_distribution. As to the truth well ¯\_(ツ)_/¯
        8y ago
OskarS 8y ago
Ok, I'm not a mathematician (or a physicist), but this seems... dubious... Like... what?: > Moreover, a better terminology for the so-called real numbers is "random numbers", as their series of bits are truly random 0.5 is a real number. Its series of bits is not random at all. sqrt(2) is also a real number, and its digits looks sort-of random, but they're not really: they're exactly the sequence of digits that forms a number that, when squared, equals 2. You can easily develop very simple formulas or programs to generate them, so in a "Kolmogorov complexity" sense, it's not random at all. Point is: the concept of "having random digits" is in NO way intrinsic to "being a real number". I guess what he meant to say was "non-computable numbers"? So he's saying "in a finite volume, there can't be non-computable numbers"... or something? Again: not a mathematician or a physicist, but this comes off as pretty crank:y to me. I would love to be corrected.
- tgb 8y ago
  I'm a mathematician by training (though this is thoroughly outside the scope of a mathematician's job) and completely agree. That said, there's a real topic in the philosophy of mathematics called "finitism" that argues we should only consider finite mathematical objects and would exclude almost every real number. [1] [1] https://en.wikipedia.org/wiki/Finitism https://en.wikipedia.org/wiki/Finitism
  - fhood 8y ago
    Almost every is a very understated way to put that.
    - function_seven 8y ago
      “Almost every” is a term of art in mathematics. It means basically “100% wherever you look, but not really 100%” https://en.m.wikipedia.org/wiki/Almost_everywhere https://en.m.wikipedia.org/wiki/Almost_everywhere
      - meuk 8y ago
        Expressed as a percentage, it would be 100%, though ;)
jimhefferon 8y ago
"Since a finite volume of space can't contain more than a finite amount of information" Is that perfectly clear? Not so, to me.
- pjz 8y ago
  This stuck out to me too. Fractals teach us that infinite length can be fitted into finite area, so I'm dubious about this claim. Though I guess if we're in the realm of physics, the Planck constant determines a kind of minimum length, so the original statement may be true?
  - JoeAltmaier 8y ago
    Pendant here: the Planck constant is simply a convenient length for all things subatomic, like a hectare or a fortnight. No minimum whatsoever implied there. Many, many things are smaller than the Planck constant.
    - Koshkin 8y ago
      > the Planck constant is simply a convenient length m^2 * kg / s
    - deleted 8y ago
      [deleted]
    - jerf 8y ago
      "Many, many things are smaller than the Planck constant." Do we have a proof of that? Honest question. I know strings are often postulated to be smaller (by many orders of magnitude in some cases) but we are not yet particularly confident they exist. I'm not aware of us being reasonably confident that anything is smaller than Planck's size. ("Smaller than Planck's size" isn't even all that easy to define.)
      - JoeAltmaier 8y ago
        Event horizon of a black hole is many times smaller for instance.
noobermin 8y ago
The author makes statements with qualification of "almost all" even though "almost all" physically allowed systems (that is, solutions to the PDEs under question) are not physically relevant or beneficial. I can only see this applying to chaotic systems like the weather, which is a valid point. The key argument that finite volume contains finite information seems potentially true but suspect. Case in point, take a unit square, it will have a diagonal who's digits have a long "infinite" string of digits but sqrt(2) isn't "infinite information", it's very finite (ie., one number!) This desperately needs to be peer reviewed by people who study the foundations of physics.
- sykh 8y ago
  If sqrt(2) is a normal number then its decimal expansion would contain infinite information.
  - chriswarbo 8y ago
    It depends on how you define information, but I'd argue that it doesn't. In particular, those digits are not random variables which are free to take on any value; they're fixed by the definition of sqrt(2). In terms of Shannon information, the information is a measure of uncertainty in a message. If our protocol is that I'll send you digits of sqrt(2), then there's no information being transferred at all: since you could have worked them out for yourself. Alternatively, if our protocol is that I'll send you digits with uniform probability, then sending sqrt(2) would be "infinite information", but only because we're having to narrow it down out of infinitely many possibilities. I don't think this is the most useful definition though, since we can arrange for any amount of information we like: if the protocol is I will either send "5" or sqrt(2), then it contains 1 bit; and so on. Alternatively we could use algorithmic information theory, where the information content of a message is the length of the smallest tape for a universal turing machine which outputs that message. sqrt(2) can be calculated by a very small program (cranking out digits forever), so it contains very little information. Yet even here, since it's constant, we could define our turing machine such that it emits sqrt(2) when given an empty tape, and hence it again contains no information.
carapace 8y ago
Which real numbers? All of them? That's a kind of mental infinitude that I don't think exists, although Kurt Gödel did. Anyway no numbers exist unless you're some kind of mystic or dualist. This numeral "2" isn't the number two, nor are these two dots: . . Nor is the word "two". There are many twonesses, but they're not in the world, they are in our heads. Like "tuesday" doesn't exist. The words exists, and if you ask somebody what day is today they'll use that word, but there is no tuesdayness in the real physical world. You can prove this my going to the a pole (North or South) and wandering around. Now "tuesday" is stuck in gimbal lock. If numbers are "real" it is in some "other" Universe, some realm of archetypes and dream-stuff.
- teilo 8y ago
  Except that's exactly what Plato said. If one were to grant the possibility of some other realm where numbers are real or archetypes, then one is not an nominalist at all, as the author of this paper seems to be (although he may actually be a conceptualist). A nominalist would never grant that such a realm exists or he would be a realist/platonist.
- andrewla 8y ago
  > Which real numbers? All of them? That's a kind of mental infinitude that I don't think exists, although Kurt Gödel did. I read the article as saying that real numbers as a mathematical concept is not a useful physical concept, and that considering classical physics with that as a first principle yields non-determinism even in classical models. Which is to say that the number "5.78" (exactly) has no physical meaning, even though the number "5.78" (measured) is useful. If you put something 5.7800 cm from another object, and run an experiment that relies on something on the seventh decimal point of that distance, the result will be random. From a classical point of view, the argument would be that measurement error accrues, and the randomness is extrinsic. The author argues that if you consider the randomness to be intrinsic instead, then you potentially gain some insight into the bridge between classical and quantum mechanics. I'm not as convinced as the author, because quantifying where exactly the intrinsic error is introduced is very closely related to the foundations of quantum mechanics. There's no reason classically that you can't make an arbitrarily sensitive measuring device to resolve the "randomness".
NelsonMinar 8y ago
See also https://en.wikipedia.org/wiki/Constructivism_(mathematics) https://en.wikipedia.org/wiki/Constructivism_(mathematics)
zhte415 8y ago
Since a finite amount of space - my stomach and associated crevices - can't contain a finite amount of coffee, nor could the keyboard; I argue that 200ml on the cup isn't relevant, and I can order more without deterministic effect, perhaps. Clever. Nice. Very unexpected.
codewiz 8y ago
Wow. The author of this paper should be awarded the Ig Nobel Prize for both math and physics :-)
ajross 8y ago
I'm frustrated that everyone here seems to be hung up over an argument about the definition of computable numbers and some nattering about Cantor, when the real meat of the abstract seems to be this: > This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory. Wait, what? What does "a way similar to quantum theory" mean? Is he able to derive quantum postulates from information theory here, or is this just handwaving becuase "uncertainty" emerges like it did for Heisenberg? Not expert enough to read the paper. Anyone?
- wizeman 8y ago
  I'm very far from being an expert, but as far as I understood the reasoning, it seems that the author is arguing that as time passes, the current conditions of chaotic physical systems are going to start depending on more and more information regarding (or "binary digits" of) the variables defining the initial conditions. Since the deterministic equations of classical physics are defined over the real numbers but physical variables cannot contain infinite information, at some point some of the information/binary digits of the initial conditions that (partially) defines current conditions must be a random digit -- although this conclusion is only reached after arguing that almost all of the real numbers are numbers that contain random digits (as random as they can be, because they have no structure and can't be named/characterized/defined). As a layman, I'm not really convinced by this reasoning (if indeed this is what the author argues), but this is the gist of what I understood.
- chriswarbo 8y ago
  Not an answer to your question, but it reminded me of something one of my quantum mechanics professors said during undergraduate physics. Paraphrasing: "Quantum physics may seem hard, but it's much easier than classical physics. We can describe a quantum system using a finite set of integer (or half-integer) quantum numbers. In classical physics the properties are continuous, and describing the deeper decimal places would require knowing the entire history of all the objects involved. We could also never repeat an experiment exactly, due to these infinite decimals." Of course, we're not totally screwed in classical systems; the far decimals are only a problem in chaotic systems. Still, might be related to the author's claim.
Strilanc 8y ago
> almost all real numbers contain an infinite amount of information. Since a finite volume of space can't contain more than a finite amount of information [..] The Kolmogorov complexity can't be higher than the state of the initial system, the dynamics, and a time parameter. Even if the system's current state involves real numbers, you would have a finite description of it as long as the time values, dynamics, and initial conditions are computable. I also think it's subtle to apply the Bekenstein bound here. Even if the numbers going in have infinite precision, that doesn't mean the relevant information quantity is infinite. For example, a qubit's density matrix has three real-number parameters and yet its von Neumann entropy is never more than 1 bit.
- danharaj 8y ago
  This corresponds to type II computability of the reals whereas the paper, as I skimmed it, is about type I computability.
tntn 8y ago
Loads of questions here that seem to be based on only reading the abstract. Most of these are addressed in the paper.
geebee 8y ago
I'm curious about something, figured I'd ask since there are some math people on this thread (I was an undergrad math major but wouldn't include myself in that group, for the purposes of this question). The abstract contains the statement: "moreover, a better terminology for the so-called real numbers is "random numbers", as their series of bits are truly random." Has this been proved? I suppose another way to ask it is, if the sequence never repeats, does that mean it is random? For instance, if the sequence was .1121231234123451234561234567... isn't random, but does it resolve to a rational (can this sequence bet represented as the ratio of two integers)? It would be a never repeating sequence that has an upper and lower bound that approaches zero as the sequence continues, right? If my example turns out to be a rational, is it possible to construct a non-random sequence that would still never resolve to a rational? I might try to look this up and read about it a little, it's been a while.
- ouid 8y ago
  If the sequence doesn't repeat, and goes on forever, it cannot be a rational number. https://math.stackexchange.com/questions/61937/how-can-i-prove-that-all-rational-numbers-are-either-terminating-decimal-or-repe?utm_medium=organic&utm_source=google_rich_qa&utm_campaign=google_rich_qa https://math.stackexchange.com/questions/61937/how-can-i-pro... This does not "prove" the statement given in the abstract, however, as the statement isn't a mathematical statement. It cannot be proven.
  - geebee 8y ago
    Here's the question I have: Are the bits of a real, non-rational number random. I suppose another way to think of it would be Are the bits of a real, non-rational number expressed as a decimal random integers between 0 and 9? I'm out of my depth here, but it seems that is a statement that could be proven.
    - ouid 8y ago
      No, of course not. You just gave an example of a number which was real, not rational, and easily described with finite information 0.001012012301234...
gbacon 8y ago
“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.” — Leopold Kronecker[0] [0]: https://en.wikipedia.org/wiki/Leopold_Kronecker https://en.wikipedia.org/wiki/Leopold_Kronecker
madmax96 8y ago
>I do not make any metaphysical claims about space, time nor numbers, >but notice that the mathematics used in practice is always finite >...I argue that the so-called real numbers are not really real. >More precisely, I argue that the mathematical real numbers >are not physically real, by which I mean that they do not represent anything physical Well, isn't the statement 'only physical objects are real' a metaphysical claim?
- ckaygusu 8y ago
  I read it as "I do not make any new metaphysical claims". The statement you have put indeed is a metaphysical assertion, but you have to have a set of metaphysical "axioms" before you can proceed with physics, hence it's impossible to talk about physics without making implicit metaphysical assertions.
  - madmax96 8y ago
    That's certainly true. But one doesn't require the axiom "only physical objects exist" to do physics, and that principal is not universally accepted among physicists. This paper -- although I'm an outsider to physics, and I'm way more comfortable evaluating mathematics and computer science research -- seems like it has an interesting and novel approach to a theoretical problem in physics. I don't understand why the author felt the need to couple this with the given title and stress this point so much. It almost felt like click-bait, until I understood generally what was being done.
    - ckaygusu 8y ago
      You are right, just saying "he does not make new assumptions" does not explain the author's act quite well enough. On further deliberation, I have come to the following conclusion: there is a difference between assuming "only physical objects exist, and they are finite" and making an argument of that. The author is trying to say he is not making an argument for the validity of the axiom but implicitly says he just assumes it, to guide the focus of the reader to the main argument he is making. I acknowledge that you definitely do not need to make that assumption, but as I said, you have to start somewhere. I'd say the metaphysical assumption captures a substantial subset of all physics, hence his ideas are worthy of consideration.
acjohnson55 8y ago
I don't believe in the physical reality of real numbers, but that doesn't detract from the their extreme usefulness as an abstraction. For that matter the negative numbers don't really exist. Nor do fractions or zero. They're all just convenient because they do a great job modeling situations we care about in a consistent way. Or at least consistently enough for everyday needs. It was a bit of a mind bender initially coming to this conclusion, but shrug.
- cup-of-tea 8y ago
  Roger Penrose gives a great explanation in The Road to Reality. We know some subset of the natural numbers are real because we can count things. And we therefore know some subset of the rational numbers are real because we can cut things into pieces and count the pieces. But the reals? Well they are invented purely so we can take the square root of two. And the complex numbers? Invented so we can take the square root of minus one. Mathematics is a tool we've developed to help us understand reality. But it itself is not reality.
  - chriswarbo 8y ago
    > But the reals? Well they are invented purely so we can take the square root of two. I agree with what you say, but I also think that we overly conflate "importance" with "the order we found/defined them in". It's a big leap (conceptually, arithmetically, etc.) to go from the rationals to the reals; sure that's what the ancients actually did, but we don't need to follow the same path when e.g. generalising some result, teaching mathematics in school, defining a computer mathematics tool, etc. There's no reason to "skip past" algebraic numbers, computable numbers, etc. especially since they're most often the sets we're dealing with; e.g. in high school most problems were something like 'find the real number x such that something-involving-x = 0'. PS: You've inspired me to have another go at reading the road to reality; it's been gathering dust on my shelf for years!
  - SilasX 8y ago
    I'm not following the logic there: >We know some subset of the natural numbers are real because we can count things. So he's saying: 1) X is real = there exists a one-to-one mapping between X and a consistent physical process. 2) Natural numbers map to counting. Okay, but then: >Well they are invented purely so we can take the square root of two. And the complex numbers? Invented so we can take the square root of minus one. Not true: complex numbers also map to real-world dynamics, like oscillatory behavior. From a search for the physical significance of complex numbers: https://www.quora.com/What-is-the-physical-significance-of-complex-numbers https://www.quora.com/What-is-the-physical-significance-of-c... I would add that real numbers have physical significance in terms of having to arbitrarily operate on some fraction of another value. It's true that the universe might prohibit arbitrary precision, but when you don't know the depths of the permitted halvings, and you're permitted arbitrarily large units, it amounts to the same thing.
frankohn 8y ago
Not serious, just bla bla, no equations are given and the author does not suggest any finite representation to replace the real numbers. This kind of article is like the old philosophers way, just words, words and arguments, nothing exact or observable that can be verified or falsified. Normally this was over when physics began: we need hard fact, experiments and equations.
DArcMattr 8y ago
Keep in mind, this is using terminology that comes from courses appropriate for advanced undergrad/beginning graduate level math courses.
ah27182 8y ago
Norman Wilberger has some interesting/controversial views on this matter. Here is a link to his YouTube channel, where he has posted many lectures on subjects through his viewpoint: https://www.youtube.com/user/njwildberger https://www.youtube.com/user/njwildberger I will say, that he is viewed as a pretty notorious crank. Especially due to his views on finitsm. But I feel like his lectures on rational trigonometry are really interesting nonetheless.
- gowld 8y ago
  "crank" is the wrong word. Mathematicians don't like his attitude, because he is smug/brash and he rejects unphysical mathematics (which many mathematicians find offensive or irrelevant), but they don't dispute his math.
kruhft 8y ago
"God (sic) created the integers, the rest, was left to people." Can't remember. "If it cannot be done with integers, it cannot be done" Von Neumann
- shmerl 8y ago
  Letters in Sefer Yetzira.
  - kruhft 8y ago
    ty
  - kruhft 8y ago
    What book is that?
fizixer 8y ago
Unless your new classical mechanics predicts the outcome of an experiment that other existing theories cannot predict, and that prediction turns out to be true if such an experiment is conducted, your new idea is just another hypothesis. Furthermore, all the real physical quantities come from the fact that we assume space and time to be continuous (i.e., similar to real numbers). So your hypothesis boils down to the claim that space and time are discrete. And that is not new. Plenty of people have proposed that hypothesis, and the idea of planck length and planck time are conjectured to be the smallest units of space and time. But the two values are of the order of 10^-35 m, and 10^-44 s, respectively. This is way below our ability to prob smaller length and time scales. And to the extent that we have been able to prob length and time scales (roughly 10^-15), space and time appears to be continuous. Finally, when you're giving up continuity in any sense, you're essentially 'quantizing a theory'. So it's no longer a classical theory but a quantum theory. We have already quantized energy (quantum mechanics), as well as physical fields (quantum field theory), and associated phenomena. But quantum (i.e., discrete) space and time are only a hypothesis at the moment.
shmerl 8y ago
Terminology is rather artificial. I.e. mathematicians just had to come up with some names for various abstractions. So real, irrational, imaginary and etc. aren't supposed to be taken literally.
deleted 8y ago
[deleted]
satyajeet23 8y ago
Is anything real?

147 points by d3ckard 8y ago

There exists a notion of "computable analysis", based on the delightful fact that Turing machines (or other Turing-complete enumerable computers) can be viewed as computing real numbers. These so-called computable reals are a little limited, but still usable for most tasks [0]. [0] https://en.wikipedia.org/wiki/Computable_analysis https://en.wikipedia.org/wiki/Computable_analysis

chx 8y ago
It's a real PITA to work with computable numbers because even their equivalence is not computable. Also, the field is really young. It was only in 2000 that Weihrauch has proven that every computable real function is continuous.
- myWindoonn 8y ago
  Their equivalence can be given by a Turing machine, though, so the non-computability comes directly from the Halting Problem. This is worth mentioning because some folks get the incorrect impression that it's not possible to write a program that, say, outputs whether two computable reals are equal to each other. It is possible! But the program might not halt. I shouldn't trivialize the difficulty. It is a PITA. I remember my first foray into this; I wrote a program which computed sqrt(8), which worked fine, and then another program which computed sqrt(8)^2, which didn't halt. The edges are tricky.
- voxl 8y ago
  As opposed to classical reals which have a computable equivalence? (Hint: they don't)
  - chx 8y ago
    But when working with reals we normally don't bother with computability. Whereas with computable numbers you don't even have anything else really.
aeneasmackenzie 8y ago
They are usable for all tasks. What are you going to do with a number you can't compute? Nothing, because you can't do anything with it.
no_identd 8y ago
Originally I wanted to make a point here about Interactive Computation, the strong Church-Turing hypothesis and persistent Turing Machines, however, while checking on the latest research in that area before doing so, I found the following paper: https://arxiv.org/abs/1702.06000 https://arxiv.org/abs/1702.06000 TE Raptis - 'Viral' Turing Machines, Computation from Noise and Combinatorial Hierarchies Which gets cited by the following paper by the same author, which cites the OP paper(!), AND elaborates on it: https://arxiv.org/abs/1805.06301 https://arxiv.org/abs/1805.06301 TE Raptis - Finite Information Numbers through the Inductive Combinatorial Hierarchy Which itself gets yet another follow on by the same author: https://arxiv.org/abs/1806.01637 https://arxiv.org/abs/1806.01637 - Encoding discrete quantum algebras in a hierarchy of binary words Highly interesting.

recursive 8y ago

"Real" numbers are mathematical abstractions. Perhaps it's unfortunate that "real" and "imaginary" were used instead of more abstract sounding terms that came with less attached connotation.

jordigh 8y ago
Just wait until cranks discover the hairy ball theorem and try to prove how wrong hairy balls are...
danharaj 8y ago
The argument of the author is not that the real numbers are poorly named, it's that the mathematical abstraction supposes physically impossible properties of the model. Therefore physics ought to be based on an abstraction that has more physically plausible properties instead of the real numbers. Not that I necessarily agree with such an argument.
otakucode 8y ago
There is value in comparing mathematics to reality even though mathematics is defined entirely independent to reality. We are able to use mathematics to model certain aspects of reality with great effectiveness.... why? Physical reality does not seem to resemble mathematical notions at all, yet shows some of the same results. Understanding that gap would be of monumental significance.

kavalec 8y ago

"Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic." This sounds like "hidden variable" and is not true for quantum physics.

Retric 8y ago
It is, but you need to give up more and allow information to travel faster than light speed. It's local hidden variables that are a problem.
- kgwgk 8y ago
  Non-locality != faster-than-light communication
  - Retric 8y ago
    As I understand it 'Spooky action at a distance' does not allow communication, but in for hidden variable theory to work a particle needs access to another particles hidden variables. Unless you want to suggest another model?
    - kgwgk 8y ago
      Why is non-locality a problem for hidden variables theories and not for the "standard" interpretation of quantum mechanics?
      - Retric 8y ago
        Quantom mechanics is based on non-locality, that's one of the things people have problems with. Quantum key distribution for example is seems to be based on this https://en.wikipedia.org/wiki/Quantum_key_distribution https://en.wikipedia.org/wiki/Quantum_key_distribution. As to the truth well ¯\_(ツ)_/¯
        8y ago

OskarS 8y ago

Ok, I'm not a mathematician (or a physicist), but this seems... dubious... Like... what?: > Moreover, a better terminology for the so-called real numbers is "random numbers", as their series of bits are truly random 0.5 is a real number. Its series of bits is not random at all. sqrt(2) is also a real number, and its digits looks sort-of random, but they're not really: they're exactly the sequence of digits that forms a number that, when squared, equals 2. You can easily develop very simple formulas or programs to generate them, so in a "Kolmogorov complexity" sense, it's not random at all. Point is: the concept of "having random digits" is in NO way intrinsic to "being a real number". I guess what he meant to say was "non-computable numbers"? So he's saying "in a finite volume, there can't be non-computable numbers"... or something? Again: not a mathematician or a physicist, but this comes off as pretty crank:y to me. I would love to be corrected.

tgb 8y ago
I'm a mathematician by training (though this is thoroughly outside the scope of a mathematician's job) and completely agree. That said, there's a real topic in the philosophy of mathematics called "finitism" that argues we should only consider finite mathematical objects and would exclude almost every real number. [1] [1] https://en.wikipedia.org/wiki/Finitism https://en.wikipedia.org/wiki/Finitism
- fhood 8y ago
  Almost every is a very understated way to put that.
  - function_seven 8y ago
    “Almost every” is a term of art in mathematics. It means basically “100% wherever you look, but not really 100%” https://en.m.wikipedia.org/wiki/Almost_everywhere https://en.m.wikipedia.org/wiki/Almost_everywhere
    - meuk 8y ago
      Expressed as a percentage, it would be 100%, though ;)

jimhefferon 8y ago

"Since a finite volume of space can't contain more than a finite amount of information" Is that perfectly clear? Not so, to me.

pjz 8y ago
This stuck out to me too. Fractals teach us that infinite length can be fitted into finite area, so I'm dubious about this claim. Though I guess if we're in the realm of physics, the Planck constant determines a kind of minimum length, so the original statement may be true?
- JoeAltmaier 8y ago
  Pendant here: the Planck constant is simply a convenient length for all things subatomic, like a hectare or a fortnight. No minimum whatsoever implied there. Many, many things are smaller than the Planck constant.
  - Koshkin 8y ago
    > the Planck constant is simply a convenient length m^2 * kg / s
  - deleted 8y ago
    [deleted]
  - jerf 8y ago
    "Many, many things are smaller than the Planck constant." Do we have a proof of that? Honest question. I know strings are often postulated to be smaller (by many orders of magnitude in some cases) but we are not yet particularly confident they exist. I'm not aware of us being reasonably confident that anything is smaller than Planck's size. ("Smaller than Planck's size" isn't even all that easy to define.)
    - JoeAltmaier 8y ago
      Event horizon of a black hole is many times smaller for instance.

noobermin 8y ago

The author makes statements with qualification of "almost all" even though "almost all" physically allowed systems (that is, solutions to the PDEs under question) are not physically relevant or beneficial. I can only see this applying to chaotic systems like the weather, which is a valid point. The key argument that finite volume contains finite information seems potentially true but suspect. Case in point, take a unit square, it will have a diagonal who's digits have a long "infinite" string of digits but sqrt(2) isn't "infinite information", it's very finite (ie., one number!) This desperately needs to be peer reviewed by people who study the foundations of physics.

sykh 8y ago
If sqrt(2) is a normal number then its decimal expansion would contain infinite information.
- chriswarbo 8y ago
  It depends on how you define information, but I'd argue that it doesn't. In particular, those digits are not random variables which are free to take on any value; they're fixed by the definition of sqrt(2). In terms of Shannon information, the information is a measure of uncertainty in a message. If our protocol is that I'll send you digits of sqrt(2), then there's no information being transferred at all: since you could have worked them out for yourself. Alternatively, if our protocol is that I'll send you digits with uniform probability, then sending sqrt(2) would be "infinite information", but only because we're having to narrow it down out of infinitely many possibilities. I don't think this is the most useful definition though, since we can arrange for any amount of information we like: if the protocol is I will either send "5" or sqrt(2), then it contains 1 bit; and so on. Alternatively we could use algorithmic information theory, where the information content of a message is the length of the smallest tape for a universal turing machine which outputs that message. sqrt(2) can be calculated by a very small program (cranking out digits forever), so it contains very little information. Yet even here, since it's constant, we could define our turing machine such that it emits sqrt(2) when given an empty tape, and hence it again contains no information.

carapace 8y ago

Which real numbers? All of them? That's a kind of mental infinitude that I don't think exists, although Kurt Gödel did. Anyway no numbers exist unless you're some kind of mystic or dualist. This numeral "2" isn't the number two, nor are these two dots: . . Nor is the word "two". There are many twonesses, but they're not in the world, they are in our heads. Like "tuesday" doesn't exist. The words exists, and if you ask somebody what day is today they'll use that word, but there is no tuesdayness in the real physical world. You can prove this my going to the a pole (North or South) and wandering around. Now "tuesday" is stuck in gimbal lock. If numbers are "real" it is in some "other" Universe, some realm of archetypes and dream-stuff.

teilo 8y ago
Except that's exactly what Plato said. If one were to grant the possibility of some other realm where numbers are real or archetypes, then one is not an nominalist at all, as the author of this paper seems to be (although he may actually be a conceptualist). A nominalist would never grant that such a realm exists or he would be a realist/platonist.
andrewla 8y ago
> Which real numbers? All of them? That's a kind of mental infinitude that I don't think exists, although Kurt Gödel did. I read the article as saying that real numbers as a mathematical concept is not a useful physical concept, and that considering classical physics with that as a first principle yields non-determinism even in classical models. Which is to say that the number "5.78" (exactly) has no physical meaning, even though the number "5.78" (measured) is useful. If you put something 5.7800 cm from another object, and run an experiment that relies on something on the seventh decimal point of that distance, the result will be random. From a classical point of view, the argument would be that measurement error accrues, and the randomness is extrinsic. The author argues that if you consider the randomness to be intrinsic instead, then you potentially gain some insight into the bridge between classical and quantum mechanics. I'm not as convinced as the author, because quantifying where exactly the intrinsic error is introduced is very closely related to the foundations of quantum mechanics. There's no reason classically that you can't make an arbitrarily sensitive measuring device to resolve the "randomness".

NelsonMinar 8y ago

See also https://en.wikipedia.org/wiki/Constructivism_(mathematics) https://en.wikipedia.org/wiki/Constructivism_(mathematics)

zhte415 8y ago

Since a finite amount of space - my stomach and associated crevices - can't contain a finite amount of coffee, nor could the keyboard; I argue that 200ml on the cup isn't relevant, and I can order more without deterministic effect, perhaps. Clever. Nice. Very unexpected.

codewiz 8y ago

Wow. The author of this paper should be awarded the Ig Nobel Prize for both math and physics :-)

ajross 8y ago

I'm frustrated that everyone here seems to be hung up over an argument about the definition of computable numbers and some nattering about Cantor, when the real meat of the abstract seems to be this: > This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory. Wait, what? What does "a way similar to quantum theory" mean? Is he able to derive quantum postulates from information theory here, or is this just handwaving becuase "uncertainty" emerges like it did for Heisenberg? Not expert enough to read the paper. Anyone?

wizeman 8y ago
I'm very far from being an expert, but as far as I understood the reasoning, it seems that the author is arguing that as time passes, the current conditions of chaotic physical systems are going to start depending on more and more information regarding (or "binary digits" of) the variables defining the initial conditions. Since the deterministic equations of classical physics are defined over the real numbers but physical variables cannot contain infinite information, at some point some of the information/binary digits of the initial conditions that (partially) defines current conditions must be a random digit -- although this conclusion is only reached after arguing that almost all of the real numbers are numbers that contain random digits (as random as they can be, because they have no structure and can't be named/characterized/defined). As a layman, I'm not really convinced by this reasoning (if indeed this is what the author argues), but this is the gist of what I understood.
chriswarbo 8y ago
Not an answer to your question, but it reminded me of something one of my quantum mechanics professors said during undergraduate physics. Paraphrasing: "Quantum physics may seem hard, but it's much easier than classical physics. We can describe a quantum system using a finite set of integer (or half-integer) quantum numbers. In classical physics the properties are continuous, and describing the deeper decimal places would require knowing the entire history of all the objects involved. We could also never repeat an experiment exactly, due to these infinite decimals." Of course, we're not totally screwed in classical systems; the far decimals are only a problem in chaotic systems. Still, might be related to the author's claim.

Strilanc 8y ago

> almost all real numbers contain an infinite amount of information. Since a finite volume of space can't contain more than a finite amount of information [..] The Kolmogorov complexity can't be higher than the state of the initial system, the dynamics, and a time parameter. Even if the system's current state involves real numbers, you would have a finite description of it as long as the time values, dynamics, and initial conditions are computable. I also think it's subtle to apply the Bekenstein bound here. Even if the numbers going in have infinite precision, that doesn't mean the relevant information quantity is infinite. For example, a qubit's density matrix has three real-number parameters and yet its von Neumann entropy is never more than 1 bit.

danharaj 8y ago
This corresponds to type II computability of the reals whereas the paper, as I skimmed it, is about type I computability.

tntn 8y ago

Loads of questions here that seem to be based on only reading the abstract. Most of these are addressed in the paper.

geebee 8y ago

I'm curious about something, figured I'd ask since there are some math people on this thread (I was an undergrad math major but wouldn't include myself in that group, for the purposes of this question). The abstract contains the statement: "moreover, a better terminology for the so-called real numbers is "random numbers", as their series of bits are truly random." Has this been proved? I suppose another way to ask it is, if the sequence never repeats, does that mean it is random? For instance, if the sequence was .1121231234123451234561234567... isn't random, but does it resolve to a rational (can this sequence bet represented as the ratio of two integers)? It would be a never repeating sequence that has an upper and lower bound that approaches zero as the sequence continues, right? If my example turns out to be a rational, is it possible to construct a non-random sequence that would still never resolve to a rational? I might try to look this up and read about it a little, it's been a while.

ouid 8y ago
If the sequence doesn't repeat, and goes on forever, it cannot be a rational number. https://math.stackexchange.com/questions/61937/how-can-i-prove-that-all-rational-numbers-are-either-terminating-decimal-or-repe?utm_medium=organic&utm_source=google_rich_qa&utm_campaign=google_rich_qa https://math.stackexchange.com/questions/61937/how-can-i-pro... This does not "prove" the statement given in the abstract, however, as the statement isn't a mathematical statement. It cannot be proven.
- geebee 8y ago
  Here's the question I have: Are the bits of a real, non-rational number random. I suppose another way to think of it would be Are the bits of a real, non-rational number expressed as a decimal random integers between 0 and 9? I'm out of my depth here, but it seems that is a statement that could be proven.
  - ouid 8y ago
    No, of course not. You just gave an example of a number which was real, not rational, and easily described with finite information 0.001012012301234...

gbacon 8y ago

“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.” — Leopold Kronecker[0] [0]: https://en.wikipedia.org/wiki/Leopold_Kronecker https://en.wikipedia.org/wiki/Leopold_Kronecker

madmax96 8y ago

>I do not make any metaphysical claims about space, time nor numbers, >but notice that the mathematics used in practice is always finite >...I argue that the so-called real numbers are not really real. >More precisely, I argue that the mathematical real numbers >are not physically real, by which I mean that they do not represent anything physical Well, isn't the statement 'only physical objects are real' a metaphysical claim?

ckaygusu 8y ago
I read it as "I do not make any new metaphysical claims". The statement you have put indeed is a metaphysical assertion, but you have to have a set of metaphysical "axioms" before you can proceed with physics, hence it's impossible to talk about physics without making implicit metaphysical assertions.
- madmax96 8y ago
  That's certainly true. But one doesn't require the axiom "only physical objects exist" to do physics, and that principal is not universally accepted among physicists. This paper -- although I'm an outsider to physics, and I'm way more comfortable evaluating mathematics and computer science research -- seems like it has an interesting and novel approach to a theoretical problem in physics. I don't understand why the author felt the need to couple this with the given title and stress this point so much. It almost felt like click-bait, until I understood generally what was being done.
  - ckaygusu 8y ago
    You are right, just saying "he does not make new assumptions" does not explain the author's act quite well enough. On further deliberation, I have come to the following conclusion: there is a difference between assuming "only physical objects exist, and they are finite" and making an argument of that. The author is trying to say he is not making an argument for the validity of the axiom but implicitly says he just assumes it, to guide the focus of the reader to the main argument he is making. I acknowledge that you definitely do not need to make that assumption, but as I said, you have to start somewhere. I'd say the metaphysical assumption captures a substantial subset of all physics, hence his ideas are worthy of consideration.

acjohnson55 8y ago

I don't believe in the physical reality of real numbers, but that doesn't detract from the their extreme usefulness as an abstraction. For that matter the negative numbers don't really exist. Nor do fractions or zero. They're all just convenient because they do a great job modeling situations we care about in a consistent way. Or at least consistently enough for everyday needs. It was a bit of a mind bender initially coming to this conclusion, but shrug.

cup-of-tea 8y ago
Roger Penrose gives a great explanation in The Road to Reality. We know some subset of the natural numbers are real because we can count things. And we therefore know some subset of the rational numbers are real because we can cut things into pieces and count the pieces. But the reals? Well they are invented purely so we can take the square root of two. And the complex numbers? Invented so we can take the square root of minus one. Mathematics is a tool we've developed to help us understand reality. But it itself is not reality.
- chriswarbo 8y ago
  > But the reals? Well they are invented purely so we can take the square root of two. I agree with what you say, but I also think that we overly conflate "importance" with "the order we found/defined them in". It's a big leap (conceptually, arithmetically, etc.) to go from the rationals to the reals; sure that's what the ancients actually did, but we don't need to follow the same path when e.g. generalising some result, teaching mathematics in school, defining a computer mathematics tool, etc. There's no reason to "skip past" algebraic numbers, computable numbers, etc. especially since they're most often the sets we're dealing with; e.g. in high school most problems were something like 'find the real number x such that something-involving-x = 0'. PS: You've inspired me to have another go at reading the road to reality; it's been gathering dust on my shelf for years!
- SilasX 8y ago
  I'm not following the logic there: >We know some subset of the natural numbers are real because we can count things. So he's saying: 1) X is real = there exists a one-to-one mapping between X and a consistent physical process. 2) Natural numbers map to counting. Okay, but then: >Well they are invented purely so we can take the square root of two. And the complex numbers? Invented so we can take the square root of minus one. Not true: complex numbers also map to real-world dynamics, like oscillatory behavior. From a search for the physical significance of complex numbers: https://www.quora.com/What-is-the-physical-significance-of-complex-numbers https://www.quora.com/What-is-the-physical-significance-of-c... I would add that real numbers have physical significance in terms of having to arbitrarily operate on some fraction of another value. It's true that the universe might prohibit arbitrary precision, but when you don't know the depths of the permitted halvings, and you're permitted arbitrarily large units, it amounts to the same thing.

frankohn 8y ago

Not serious, just bla bla, no equations are given and the author does not suggest any finite representation to replace the real numbers. This kind of article is like the old philosophers way, just words, words and arguments, nothing exact or observable that can be verified or falsified. Normally this was over when physics began: we need hard fact, experiments and equations.

DArcMattr 8y ago

Keep in mind, this is using terminology that comes from courses appropriate for advanced undergrad/beginning graduate level math courses.

ah27182 8y ago

Norman Wilberger has some interesting/controversial views on this matter. Here is a link to his YouTube channel, where he has posted many lectures on subjects through his viewpoint: https://www.youtube.com/user/njwildberger https://www.youtube.com/user/njwildberger I will say, that he is viewed as a pretty notorious crank. Especially due to his views on finitsm. But I feel like his lectures on rational trigonometry are really interesting nonetheless.

gowld 8y ago
"crank" is the wrong word. Mathematicians don't like his attitude, because he is smug/brash and he rejects unphysical mathematics (which many mathematicians find offensive or irrelevant), but they don't dispute his math.

kruhft 8y ago

"God (sic) created the integers, the rest, was left to people." Can't remember. "If it cannot be done with integers, it cannot be done" Von Neumann

shmerl 8y ago
Letters in Sefer Yetzira.
- kruhft 8y ago
  ty
- kruhft 8y ago
  What book is that?

fizixer 8y ago

Unless your new classical mechanics predicts the outcome of an experiment that other existing theories cannot predict, and that prediction turns out to be true if such an experiment is conducted, your new idea is just another hypothesis. Furthermore, all the real physical quantities come from the fact that we assume space and time to be continuous (i.e., similar to real numbers). So your hypothesis boils down to the claim that space and time are discrete. And that is not new. Plenty of people have proposed that hypothesis, and the idea of planck length and planck time are conjectured to be the smallest units of space and time. But the two values are of the order of 10^-35 m, and 10^-44 s, respectively. This is way below our ability to prob smaller length and time scales. And to the extent that we have been able to prob length and time scales (roughly 10^-15), space and time appears to be continuous. Finally, when you're giving up continuity in any sense, you're essentially 'quantizing a theory'. So it's no longer a classical theory but a quantum theory. We have already quantized energy (quantum mechanics), as well as physical fields (quantum field theory), and associated phenomena. But quantum (i.e., discrete) space and time are only a hypothesis at the moment.

shmerl 8y ago

Terminology is rather artificial. I.e. mathematicians just had to come up with some names for various abstractions. So real, irrational, imaginary and etc. aren't supposed to be taken literally.

deleted 8y ago

[deleted]

satyajeet23 8y ago

Is anything real?