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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 "

• 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 ;)
        • function_seven 8y ago
          Well, almost...
          • filmor 8y ago
            Nope, exactly. The measure of the interval [0, 1] excluding any countable subset (for example all rational numbers) is still exactly 1, that's usually what mathematicians mean by "almost every", here "almost every real number in [0, 1] is irrational" :)
            • function_seven 8y ago
              I was joking, but to a point. The problem is that "100%" =/= "All". But most of the time, it is considered to be equivalent. The "almost all" phrase is used instead of "100%" precisely because dividing by infinity doesn't yield a useful result in some contexts. In more... "conventional" cases, if I tell you that 100% of the items are blue, you can conclude that none of them are red. With the intermingling of countable and uncountable sets, this statement fails. You have a situation where 100% of the items in the superset are blue, and there are also some red ones. So it's "almost 100%" in the same way it's "almost all". I'm using the word "almost" to mean "infinitesimally close to"
      • resource0x 8y ago
        Really, exactly 100%.
        • function_seven 8y ago
          Yeah. But “100%”, and “all” or “every”, are considered synonyms most of the time. Not in this case though.
    • ouid 8y ago
      nice catch
  • windows_tips 8y ago
    How could you exclude "almost every real number", if such a thing should not even be considered?
    • amptorn 8y ago
      You just leave the axiom of infinity out.
    • gowld 8y ago
      You simply choose not to invent any of them.
• drvd 8y ago
  > 0.5 is a real number Yes, and it is one of the few we have no problem with (7 is an other example of this class): The rationals (or the algebraic). Unfortunately the vast majority of the reals are not that catchy. E.g. we still do not know how many reals there are. The reals are a beast if watched from a fundamental perspective.
  • OskarS 8y ago
    > E.g. we still do not know how many reals there are. Umm... yeah we do, the cardinality of the continuum is the same as the cardinality of the power set of natural numbers. We've known this for 140 years. Reading between the lines, I think the author is really talking about the "non-computable numbers" (i.e. those real numbers who can't be calculated to an arbitrary precision by any Turing machine), but if that's what the author is referring to, he should just say "non-computable numbers", not "real numbers", which is a much broader class.
    • danharaj 8y ago
      What is the cardinality of the powerset of the naturals?
      • OskarS 8y ago
        It's usually referred to as "c". I'm not sure what you're asking? It's like saying "3 is the number that follows 2", and then asking "yeah, but what really IS the number 3, man?". The answer is: it's the cardinality of the sets which can be in a bijection with the power set of the natural numbers. If you're referring to the continuum hypothesis, that is a distinct question from "what is the cardinality of the real numbers". Also: a solved one, the contiuum hypothesis is independent of ZFC, so saying "we don't even know what the cardinality is" is still wrong.
        • sykh 8y ago
          The cardinality of the continuum is a cardinal number. It’s one of the alephs. It is not known which aleph it is. So it’s not known what the cardinality of the powerset of the naturals is. It’s just known that it is the same cardinal number of the reals. Basically, we have two jars of marbles that contain the same number of marbles but it’s not known how many marbles that is.
• mathgladiator 8y ago
  So, I think the author meant that a real number is actually a sequence of approximations towards an ideal. So, 0.5 has infinite sequences like 5, 3, 2, 1, 3/4, 1/4, 3/5, 2/5, 1/2
• ssivark 8y ago
  I would phrase the crux as follows: Any physics theory is basically a map from one set of measurements (past) to another set of measurements (future). Any physical measurement can only have finite precision, due to fundamental constraints from physics and information theory. The set of finite precision numbers has vanishing measure in the set of all real numbers (since finite precision is a subset of rationals). So are real numbers really necessary? As a physicist, while this question is definitely interesting, it’s not clear what will come out of this line of questioning (aka research).
  • jwatte 8y ago
    I think you're suggesting that "real" numbers are an abstraction that is convenient to model with, but aren't actually fully representable in the universe, because of Planck and Heisenberg and such? I would agree with that view.
    • ssivark 8y ago
      It's not even necessarily about quantum stuff. The point is that any practical measurement system (even classical ones) can only deal in finite precision numbers. And since that set is of vanishing measure on the real line, there isn't an obvious notion of continuity/smoothness (my knowledge here is limited). That can lead to some surprises. For the more advanced readers: this result reminds me of how Haag's theorem shows (rigorously) that relativistic interacting QFTs can't exist (assuming operators sit at points indexed by real numbers), while in a Wilsonian sense QFTs are perfectly fine, because there's a UV cutoff, and all measurements have a finite resolution.
  • contravariant 8y ago
    If we're going with the theory that things with a wavelength shorter than the Planck length don't exist then the sampling theorem should allow us to describe the universe with a discrete set of points. General relativity complicates things (as usual) but the sampling theorem can be generalised to non-uniform samples so it should probably still work somehow. In both cases this discrete theory is equivalent to a theory based on real numbers though, so in that sense there's not really any way to say which is better.
• johnsonjo 8y ago
  I don’t know 0.5 still could be random. Relevant XKCD https://www.xkcd.com/221/ https://www.xkcd.com/221/. </sarcasm>
• candiodari 8y ago
  Although technically you're right, one can reasonably assume that when talking about random numbers, people mean non-rational random numbers. Like sqrt(2), e, pi, ...
  • OskarS 8y ago
    But none of those are really random either. The only sensible definition of randomness here is Kolmogorov complexity, and all of those numbers you just mentioned have incredibly low Kolmogorov complexity. I can write a tiny computer program that prints them out to billions of digits. What the author is probably referring to is "non-computable numbers", which are numbers that cannot be computed to arbitrary precision by ANY Turing machine. But it's incredibly confusing and wrong to use the term "real numbers" to mean "non-computable numbers".
    • sykh 8y ago
      A normal number can be computable. It is not known if sqrt(2) is normal.
      • OskarS 8y ago
        At no point was anybody talking about normal numbers. "Normal numbers" doesn't mean "numbers with random digits", it has a much narrower meaning that. There are non-computable numbers which are normal, there are non-computable numbers which aren't. There are computable numbers that are normal that don't look random in the least: https://en.wikipedia.org/wiki/Champernowne_constant https://en.wikipedia.org/wiki/Champernowne_constant
        • sykh 8y ago
          The author's paper is about normal numbers (numbers that contain infinite amount of information). If a normal number isn't considered to have random digits then what notion would you use?
          • 8y ago
• sykh 8y ago
  The complement of the set of normal numbers has measure zero. A normal number is what the author of the paper is talking about when referring to random numbers. I think most mathematicians, if they had to bet, would bet that sqrt(2) is normal. It is not known though. If it is normal then it’s digits are random. I haven’t read the paper but I think the author is arguing that most real numbers don’t make sense physically. If sqrt(2) were shown to be normal and since it’s the hypotenuse of a right triangle of legs with length 1, I wonder what the author's response to this would be. Perhaps he’d say that such a triangle doesn’t exist physically.
  • btilly 8y ago
    Perhaps he’d say that such a triangle doesn’t exist physically. I am certain that this is the case given that he's referring to the maximum information density density of space as one of the reasons why exact numbers make no physical sense. See https://en.wikipedia.org/wiki/Bekenstein_bound https://en.wikipedia.org/wiki/Bekenstein_bound if you don't know about that limit. More directly, the inability to represent exact lengths falls out of the Heisenberg Uncertainty principle. As an abstract mathematical concept, numbers could be anything. As a physically relevant concept, there are limits to how much precision can matter.
  • empath75 8y ago
    What does it mean to produce a physically realized triangle with length exactly 1? Some specific number of atoms in a specific place? How does that square with the uncertainty principle?
• plopilop 8y ago
  Well, computable numbers are countable, so almost all real numbers are non-computable (also they're dense in R). So while your objection is valid, in practice any truly randomly selected real number is non-computable.
  • sykh 8y ago
    A computable number can be a normal number. http://www.glyc.dc.uba.ar/santiago/papers/absnor.pdf http://www.glyc.dc.uba.ar/santiago/papers/absnor.pdf
  • OskarS 8y ago
    I guess I'm saying: such a basic mistake in terminology in the abstract doesn't bode well for the soundness of the paper in general. Being able to properly tell apart the terms "real number" and "non-computable number" is a fairly basic qualification.
    • gowld 8y ago
      It's not a "qualification", it's just some using a slightly different set of definitions for convenience. Anyone "qualified" to read the paper understands the meaning.
    • davidgay 8y ago
      A quick check on the author leads to https://en.wikipedia.org/wiki/Nicolas_Gisin https://en.wikipedia.org/wiki/Nicolas_Gisin. I think it's going to be rather safe to conclude that this author is well-aware of the difference between the computable and non-computable parts of R.
  • ajross 8y ago
    To nitpick: to say that "in practice" you won't select a rational is understating the case. Being uncountable, the set of irrationals is infinitely larger. The chance of selecting a rational "randomly" is zero as far (heh) as zero has any practical meaning.
  • zuminator 8y ago
• timwaagh 8y ago
  mathematicians like to skip steps, generalize and the like. 'randomness' for numbers is not really a well defined mathematical property. what i like about the summary is that you can understand the gist of what he is trying to do and why.
  • strainer 8y ago
    If we could read the coordinates of any physical object to infinite precision, we could only influence the higher digits by moving the object very carefully, beyond the digits which we have some influence over, the infinite tail of the real numbers digits must be completely random.
  • throwawaymath 8y ago
    > 'randomness' for numbers is not really a well defined mathematical property. Randomness is a well-defined property. We say that numbers are random, or "normal", if an infinite sequence of their digits forms a uniform distribution. The reason we call them "normal" and not "random" is because they are the norm - almost all real numbers are normal.
• meuk 8y ago
  You are talking about rationals and algebraic numbers, which are just a tiny subset of the real numbers. Your point about computable vs non-computable numbers is valid, but I think the author meant that you will never encounter a real number that is non-computable, since those are impossible to express in the first place. Real numbers can contain an infinite amount of information.
  • btilly 8y ago
    Actually there are non-computable numbers known that you can encounter. See https://en.wikipedia.org/wiki/Chaitin%27s_constant https://en.wikipedia.org/wiki/Chaitin%27s_constant.
    • chriswarbo 8y ago
      Yes, there's actually another distinction we can make: between incomputable numbers and indescribable numbers. The latter is very slippery, since it can lead to things like the Berry paradox https://en.wikipedia.org/wiki/Berry_paradox https://en.wikipedia.org/wiki/Berry_paradox For example, we would need some way to avoid expressions like "the smallest indescribable number", since that itself is a description ;)
• ttctciyf 8y ago
  The author alludes to the fact that the overwhelming majority of reals have an infinite number of decimal places that constitute an algorithmically random sequence[1] Sure, plenty of real numbers have finite (0.1, 0.01, ...) or non-random (0.111..., 0.0111..., ...) sequences of decimal places, but there are only a countable infinity of these, whereas there is an uncountable infinity of the other, "random," kind. Usually we almost exclusively deal in the only-countably-infinite subset of the reals which are not random and it's counterintuitive that these comprise a tiny tiny archipelago of order in a huge engulfing ocean of randomness, but such is the case. Gregory Chaitin's "How real are real numbers?"[2] has some relatively straightforward further reading on this if you're interested. 1: https://en.wikipedia.org/wiki/Algorithmically_random_sequence https://en.wikipedia.org/wiki/Algorithmically_random_sequenc... 2: https://arxiv.org/abs/math/0411418 https://arxiv.org/abs/math/0411418
  • tgb 8y ago
    The author's writing has a lot of background behind it that is true and yet the author's argument that they should be called "random numbers" is poor, at best. It is not typically helpful to think of real numbers that way - at the very least you're "missing" a tiny but hugely important subset that aren't random in any sense. But more importantly, it doesn't match up with any of the usual definitions of real numbers since those involve no such restrictions of randomness of digit expansions (and generally avoid digit expansions entirely). We might as well argue that the real numbers aught to be called the non-zero numbers since almost every one of them is non-zero. I don't think the author seems particularly aware of the existence of such arguments in the past (I assume he comes from a physics background and is new to this - though so am I). Glancing through the works cited I don't see references to past treatments of this idea like Chaitin's. If the author is not a crank, he is doing a poor job of convincing us of it.
    • ttctciyf 8y ago
      I don't know if it's still the case, but arxiv papers without a subject classifier used to be more likely to be "cranky" (contrast the /math/ in the Chaitin paper url.) So I don't know about the paper here - I only read the abstract - but I wouldn't get too hung up on terminology. "Random" is just an easier way of saying "uncomputable" I guess. But the broad topic itself is quite respectable and of interest to non-cranks, for example, see John Baez' "Struggles with the Continuum"[1]: > Is spacetime really a continuum, with points being described — at least locally — by lists of real numbers? Or is this description, though immensely successful so far, just an approximation that breaks down at short distances? Rather than trying to answer this hard question, let us look back at the struggles with the continuum that mathematicians and physicists have had so far.... 1: http://math.ucr.edu/home/baez/continuum.pdf http://math.ucr.edu/home/baez/continuum.pdf EDIT: The author seems like a perfectly respectable, non-crank physicist (https://en.wikipedia.org/wiki/Nicolas_Gisin https://en.wikipedia.org/wiki/Nicolas_Gisin) as per another comment here.
• chriswarbo 8y ago
  > 0.5 is a real number. Its series of bits is not random at all Would you say that an (arbitrarily long) sequence of coin tosses that all come up heads "is not random at all"? I'd say it's vanishingly unlikely to occur (if the coin is actually 50/50), but just as unlikely as any other particular sequence of the same length. It's like if a lottery gave out [1, 2, 3, 4, 5, 6]: it's a remarkable pattern, but no more or less likely than any other particular result. The same goes for the never-ending decimals of 0.5 and sqrt(2), and it's misleading to draw general conclusions from such specifically-chosen examples.
  • ethagnawl 8y ago
    > It's like if a lottery gave out [1, 2, 3, 4, 5, 6]: it's a remarkable pattern, but no more or less likely than any other particular result. Do you know if there's a name for the fallacy you've alluded to? (Assuming that an ordered sequence is less likely than any other.)
    • gowld 8y ago
      Interestingly, the fallacy is actually good reasoning. Even though every permutation is equally likely, the permutations fall into equivalence classes of "notoriety", and the class of ordered sequences is far smaller (== less likely) than the class of disordered sequences (which are all equivalently noninteresting). (This is entropy, as it applies to both information and the physical world.)
  • tzs 8y ago
    As of the end of April 2018, we've had 400 consecutive months where the world's temperature was hotter than average for that month, yet a whole lot of politician say that's just due to random variation. I bet they could take a long run of heads in stride without thinking that maybe they have a biased coin.
• GregBuchholz 8y ago
  A good book to read to get aquainted with these notions is "Meta Math" by Gregory Chaitin. The impatient can jump to chapter 5. https://arxiv.org/abs/math/0404335 https://arxiv.org/abs/math/0404335
• Tloewald 8y ago
  The question is whether the real numbers (as a set) are real, not whether each specific real number is a thing. Transcendental — the numbers that make the real numbers distinct from the algebraic numbers — are not necessary for anything we do, and all the weirdest corner cases in math seem to come from treating them as a thing. sqrt(2) and 0.5 aren’t transcendental. 0.5 is a rational. Sqrt(2) is an algebraic. You can express them in physical terms (as roots of finite polynomials) Most reals don’t need to exist for any purpose. All we need is some kind of vector space of constants (e.g. planck’s constant, electro-permeability, the speed of light, the mass of an electron) over the unbounded algebraics. It’s entirely possible that some of these constants would in fact be transcendental, but such a space is much much smaller than the reals. These are the numbers we can construct.
  • ducttapecrown 8y ago
    pi and e are very important numbers!
    • roywiggins 8y ago
      They also belong to the much smaller set of computable numbers. The set of all reals is almost entirely full of absolutely useless numbers- ones that can't be compressed or expressed by any algorithm or notation system with a finite amount of space.
      • gowld 8y ago
        I've never seen one of these supposedly useless numbers. Every one I've seen can be expressed. ;-)
    • Tloewald 8y ago
      Right, they might well be basis vectors.
• roenxi 8y ago
  Pretty much everything in mathematics was named on discovery, which was usually before they understood what the concept they were discovering actually was. And, somethings even the name that was given is meaningless in English because it was given in a foreign language. Real Numbers aren't somehow special here. There are a lot of elements in mathematics that arguably could be named better. Examples include (and obviously this is opinion, not a mathematician, in all humility, etc): + Negative numbers, real numbers, imaginary numbers, quaternion, etc (which should really all be named in a more systematic way) + sin, cos, tan (I doubt any link remains between the function and the concept in modern teaching) + Anything named after a person instead of named for what it is (eg, Euler's constant e; should have a more informative name) + plus, multiply, exponentiation, Knuth's up-arrow (a progression of the same concept, should be named systematically) + The assuredly long list of examples I don't know about The gains don't justify the costs, but one suspects if you could somehow give an alien modern math theory without any of the name or symbol conventions they would invent a much more systemic system than hat we have grown.