9 ms·

Are Real Numbers Real? (2017)

38 points by yamrzou 4y ago

Koshkin 4y ago
Is the integral a "thing"? - No, it, along with the reals, is the method (we call "analysis") that we humans invented to solve a certain class of problems in science and technology.
PaulHoule 4y ago
I find it ironic that the "real" numbers are so named because they are definitely less "real" than the integers or rational numbers since they cannot (with p=1) be individually named or specified.
- jjgreen 4y ago
  The name was introduced by Descartes in contrast to the disreputable "imaginary" numbers occuring in investigations of roots of polynomials. So not that ironic, more like rain on your wedding day ...
  - lupire 4y ago
    It's extremely ironic: imaginary numbers are real, and real numbers are imaginary.
    - Koshkin 4y ago
      The situation is complex indeed.
- martopix 4y ago
  Integers and rational numbers are also reals. What aren't reals are imaginary numbers.
  - deleted 4y ago
    [deleted]
  - psychoslave 4y ago
    You mean imagery encompasses reals, I guess :D
    - Aardwolf 4y ago
      nonzero imaginary and real are both disjoint sets, and complex encompasses them both
  - hackandthink 4y ago
    "Integers and rational numbers are also reals" Many mathematicians don't think so. Category Theory has a category of sets. Integers/Rationals/Reals are different Objects (sets) in this category. Type Theory is similar s/Category/Type.
    - voxl 4y ago
      Any mathematician, literally all of them, will tell you there is an injection from the set if rationals and reals, and that your foundational system is likely impoverished if this injection does not behave as the identity
      - klodolph 4y ago
        I think it’s worth talking about how the language works here. We say, “the natural numbers are a subset of the reals,” and this is a sensible thing to say. We also might say, “you construct each successor number in the natural numbers as n + 1 = n ∪ {n}”. And then we say, “a real number is a Cauchy sequence of rational numbers, or a Dedekind cut of rational numbers.” From a set-theoretic perspective, “the natural numbers are a subset of the reals” is obviously untrue with these definitions, and it’s worth spending a moment to think what the statement actually means, or how you would have to interpret it in order to understand the truth of the sentence. I might translate the sentence as “there is a left-cancellative morphism from natural numbers to real numbers,” but then I’d have to define what category I’m using, and what the morphisms are—which is usually implied. You end up having to stand on top of a surprisingly tall stack of proofs in order to say “the natural numbers are a subset of the reals” and actually explain what you mean by that, rigorously, from foundations. Or, put another way, it’s sometimes useful to understand what you mean by “behave as the identity”.
- nuncanada 4y ago
  Worse, if you take "Imaginary Numbers" to be only the necessary complement to make polynomials factoring complete... And "Imaginary Numbers" are definitely a necessity to describe reality... Meanwhile almost all "Real Numbers" are nothing but phantasmagorical numbers which we cannot name nor talk about them... It seems beyond the pale that we still are teaching these bizarre axioms instead of the Computable Reals which don't have all these phantom numbers...
yamrzou 4y ago
This is a response to Chaitin's "How real are real numbers?" (2004), which was discussed here twice: - (2017) https://news.ycombinator.com/item?id=14080024 https://news.ycombinator.com/item?id=14080024 - (2020) https://news.ycombinator.com/item?id=24029791 https://news.ycombinator.com/item?id=24029791
- crdrost 4y ago
  It is interesting that the source article from Chaitin it seems to regard its own title as mere wordplay, it does not want to get into the Morpheus Question “What is real? How do you define real?”... But the response seems to take it very seriously and get somewhat upset about it. Other than those aspects the discussion is kind of snore-fest...
  - lupire 4y ago
    What do you mean? Chaitin is the most well known author who intended and is received as arguing against the existence of the set of real numbers.
psychoslave 4y ago
>concludes that these difficulties undermine the common assumption that real numbers underlie physical reality, strongly suggesting that physical reality may in fact be discrete, digital and computational. Well, it should precise from what population this "common assumption" is supposed to be taken of. Arguably, most humans that ever existed had basic integer arithmetical notions at best. What humans take for facts are not given but constructed on social interactions, even when protocolled around scientific methods relying on falsifiable models and sound metrics. The best that science can tell us is how much sound our model is in regard of available data and how much resources we need to produce and test an other one. Note that philosophy of Karl Popper is not itself falsifiable. > This statement describes a faith, not a scientific principle. Ultimately, principles lie on some mental faith or an other. There is no irrefutable evidences that solipsism is less sound than believing in an "outside world".
kirkules 4y ago
Still reading the article, but upon reading > Despite these difficulties, the cognitive notion of a continuum is not at all difficult to grasp. The difficulties arise only when trying to communicate, for example by naming or describing all the real numbers. But one can understand without communicating. I feel it's important to point out that humans are very good at convincing themselves that they have done a great job of understanding things when they have not. Including brilliant minds. Since this is in "trust our intuitions" territory about, essentially, philosophy of mind, it's worth bringing along this awareness.
- kirkules 4y ago
  I also think it's notable that somehow DNA "encoding" our minds for mind-inheritance is ruled out using the channel capacity theorem. Shouldn't this same kind of argument rule out the transferral of minds from one moment to the next, unless minds do not actually arise from our bodies? In other words, does this criticism by Lee commit to mind-body dualism?
  - pencilguin 4y ago
    It demonstrates that the statement is nonsensical.
- bell-cot 4y ago
  >> [...] cognitive notion of a continuum is not at all difficult to grasp. The difficulties arise only when trying to communicate, for example by naming or describing all the real numbers. But one can understand [...] Notice that one can do a very similar analysis after replacing "continuum" with "rules of the game of chess" and "naming or describing all real numbers" with "name or describing all positions which could occur in a chess game". You don't need uncountable infinities, topological oddities of R, subtle but deep differences from how the real world works, etc. to get out to where humans aren't all that good at understanding things.
- zmgsabst 4y ago
  In fact, the formalization of calculus (and in particular, continuity and differentiability via limits) came about because people had different intuition about “continuous” functions. I would say the implications of a continuum versus a dense cloud of rationals is incredibly unintuitive.
hackandthink 4y ago
Some people think: If you can't compute it, it does not exist. or If you can't compute it, I don't care. Computing and real numbers are a confusing subject. Edit: https://ncatlab.org/nlab/show/computable+physics https://ncatlab.org/nlab/show/computable+physics There are nice discussions an Andrej Bauer's blog: https://math.andrej.com/ https://math.andrej.com/. Can't find them right now. "type II computability" is the concept. Look at the table in: https://ncatlab.org/nlab/show/Type+Two+Theory+of+Effectivity https://ncatlab.org/nlab/show/Type+Two+Theory+of+Effectivity
- bell-cot 4y ago
  Quip: Can the tuna salad sandwich I am eating for lunch be computed? Does it exist? Should I care?
  - pixl97 4y ago
    I think first you would have to define what it means to be computed in this context.
  - Koshkin 4y ago
    I am not even sure if one can say that tuna itself "exists," in this context.
  - hackandthink 4y ago
    Not a sandwich but an equilibrium. What is it good for if there's no mechanism to reach it? "What they’ve found is that some games have Nash equilibria that are impossible for a computer to determine and even more equilibria are computable only if one of the largest open problems in computer science, P = NP, is solved." https://blogs.cornell.edu/info2040/2015/09/21/the-computability-of-nash-equilibria/ https://blogs.cornell.edu/info2040/2015/09/21/the-computabil...
dragontamer 4y ago
What is "real"? The ancients may have understood negative numbers, but IIRC the concept of zero (additive identity) wasn't understood until Arabic Numbers / Algebra. So "zero" isn't real, at least not to the Romans or Greeks. Once Algebra is invented, we learn that not only is "zero" a useful concept (additive identity), but so is "one" (multiplicative identity). More importantly, the inverse of multiplication can be defined as "blah times number == 1". IIRC, the Greeks somehow understood this concept before the invention of Algebra, but its easier to see once you have Algebra. Once we have those, its not too difficult to extend into irrational numbers, such as sqrt(2). The Greeks almost discovered it, but were too stubborn. There's a lot of discussion in Greek Mathematics over sqrt(2)... they're so so close but they were too much of a fanboy of integers / rational numbers to actually reach the concept. Once we understand sqrt(2), its not too difficult to extend the principle to sqrt(-1) or sqrt(-2), reaching the concept of "i". ----------- sqrt(-1) occurs in "real" voltages / currents all the time today in Electrical Engineering (https://en.wikipedia.org/wiki/Phasor https://en.wikipedia.org/wiki/Phasor). Its as "real" as any other number. IE: We can truly measure real-world objects (ie: the voltage going through your power-lines) and assign them a value of "i", and use that to predict what happens to the electricity as it flows through your equipment and/or house. So all of these things are "real" numbers, that describe truly physical real-world objects. Its just an unfortunate naming convention of "Real" vs "Imaginary" vs "Rational" vs "Irrational", which comes down to history / order of discoveries, rather than usefulness.
- chriswarbo 4y ago
  > sqrt(-1) occurs in "real" voltages / currents all the time today in Electrical Engineering I'd say sqrt(-1) describes "real" voltages / currents; but we can find other models which don't involve sqrt(-1). Those models may be more complicated, and have fewer rich connections to pre-existing work, but they're just as valid. This is usually where appeals to Occam's Razor come in, and physicists start appealing to "elegance", "beauty", etc.
  - dragontamer 4y ago
    You're not going to be able to describe how voltages/currents work in an AC circuit unless you reach into 2-vectors / matrices. In Complex Arithmetic, we call it the "imaginary axis". But reach into Linear Algebra and you'd just call it a "2nd dimension". When things oscillate (and AC current is oscillating), there's just certain mathematical properties that occur. It doesn't matter if its electricity, a spring/mass system, a bridge, sine-waves, music theory, Fourier Series or anything else, its all going to be represented by complex-math extremely elegantly. Or, have a 2-vector linear-algebra representation, since its equivalent. EDIT: I guess there's the differential-equations representation, but I feel that reaches into linear-algebra very, very quickly... or complex numbers. (E^(pi * i) keeps coming up...)
adrian_b 4y ago
Both this article and that of Chaitin which is cited in it, fail to say clearly what they mean by "physical reality" and they also mention some things that are irrelevant to the title question, such as DNA. All the physical quantities that can be modeled as continuous, i.e. by real numbers, are derived in some way or another from the positions in space and time. All the physical quantities whose dimensional formulas (in natural units, not in SI, which obscures some of the relationships between physical quantities) do not include space/time quantities, e.g. electric charge, magnetic flux, angular momentum and a few others, are discrete. So the question whether the real numbers correspond to the physical reality must be reformulated less ambiguously as the question whether the space and time are continuous or discrete. The traditional model of the space and time as continuous matches successfully all known experiments. There have been a few attempts of creating models for various parts of the quantum physics by using a discrete space-time, i.e. where the points in space-time form a lattice (i.e. there are minimum distances between points that cannot be subdivided to find an intermediate point) and any physical process progresses by jumps between discrete points, instead of continuous movements. However, I am not aware of any useful result obtained with such models.
chriswarbo 4y ago
> If in fact the mind relies on a continuum for its cognitive functions, this would explain why cognitive functions cannot be inherited and why our minds can deal with real numbers, despite the paradoxes. I disagree with the assertion that "our minds can deal with real numbers". AFAIK there's no way to distinguish an uncomputable number from a computable one, using a finite set of computations (i.e. there's always some computable model which gives the same answers as our uncomputable number, even if it's highly convoluted); so we're still only learning about the countable, computable sub-set.
- Nomentatus 4y ago
  "Deal with" in detail, absolutely not, just as you say. Pardon my rephrasing you point (I think): You can show that some functions that give real numbers are computable. (Let's say, computing that number digit by digit. Binary digit or base-ten, you choose.) You can't show that all Turing machines halt, so you can't use the mere existence of a Turing machine as proof of a computable function. Therefore, we can't know or compute the set of all real numbers because we can't compute the set of irrational numbers; there will always be some Turing machine that, unbenownst to us, calculates a (likely irrational) function that spits out a real number. If we can't describe even a way to sort functions into two piles: "computes a real number (digit by digit)" and "doesn't compute a real"; then we can't be said to "deal with" the set of real numbers at all well. This rather presupposes the (now pretty much accepted) concept that numbers "exist" if they can be computed (whether they have been computed, or conceived of, or not.) "Intuitionism" in other words, IIRC.
mjburgess 4y ago
All of physics stands as an objection to "digital physics" which uses real numbers indispensably. Many things fall apart with discrete numbers (classical physics becomes non-deterministic, quantum mechanics becomes non-linear, etc.). The loons on the "digital bandwagon" have yet to produce a single plausible work showing the geometrical spaces hiterto essential to physics can be replaced with anything discrete. This is just another pipe-dream of idealist-Platonist-AI fanatics for whom the world is an abstract discrete number only because their favourite toy is so-described.
- lupire 4y ago
  An epithet is not an argument. I'm not saying you are wrong; I am saying you are not even wrong.
  - mjburgess 4y ago
    There is never a cabal of zealots who gather together to defend material reality, is there ? Who are the loons of the scientific method ? Which zealot says, "let's be conservative in our dreaming?" 'digital physics', constructive mathematics, AI, idealism -- etc. are all apparently respectable symptoms of an ideological project i reserve the right to call it as i see it, and name a purely ideological project for what it is the world is not a simulation, there is no digital physics, your consciousness will not be uploaded -- etc
    - DontchaKnowit 4y ago
      Preach
- ingleso 4y ago
  Is Planck a loon? > In 1900, Max Planck derived the correct form for the intensity spectral distribution function by making some strange (for the time) assumptions. In particular, Planck assumed that electromagnetic radiation can be emitted or absorbed only in discrete packets, called quanta, of energy https://en.m.wikipedia.org/wiki/Ultraviolet_catastrophe#Solution https://en.m.wikipedia.org/wiki/Ultraviolet_catastrophe#Solu... How do you explain the zenos paradox’s place in reality? https://en.m.wikipedia.org/wiki/Zeno's_paradoxes https://en.m.wikipedia.org/wiki/Zeno's_paradoxes What’s with the hate on Plato’s ideals? What is a “chair”? Draw me a “circle”.
GTP 4y ago
I'm still reading it, but I don't agree with the following: >In Plato and the Nerd, I review Claude Shannon's channel capacity theorem [Shannon, 1948], which states that any noisy observation of anything conveys only a finite number of bits of information. Borel's know-it-all number cannot be encoded with finite number of bits unless the list all possible yes-no questions is finite, which it is not. It is easy to construct an infinite sequence of valid yes-no questions. For example, let the first question be "Is one a whole number?" Let the second question be "Is the answer to the first question YES?" Let the third question be "Is the answer to the second question YES?" And so on. As a consequence, Borel's number cannot reveal itself to us unless we invent some noiseless way of observing a thing in itself. It is true that he showed an example of an infinite list of valid yes/no questions, which would lead to a number with an infinite amount of digits. But despite it being infinite, by the way it was constructed you already know all the digits (they will be all 1s), meaning that that number's entropy is remarkably low and that number can be expressed with a finite number of bits of information. So Shannon's theorem isn't a problem here.
- kirkules 4y ago
  The presentation in the quote is awkward and doesn't work, as you pointed out, but Lee could instead have just mentioned that there's such a construction with questions of the form "is the answer to the previous question YES [NO]?" which is not constant (not all 1s), and does have the property he wants, i.e. the are such sequences that cannot be specified finitely (corresponding to binary sequences with that property). I don't think you've shown that Shannon's theorem isn't a problem here so much as the argument presented did not successfully convey that it's a problem. Tangentially, I'm interested to see a formal statement/proof of the form of Shannon's theorem add used in this article, because I can sort of intuitively see why it makes sense but I'm not familiar enough with the technical details to guarantee that this is even an appropriate use of the theorem, assuming the yes/no question construction has all the properties Lee wants it to have.
  - pencilguin 4y ago
    For all i, "Is i a Collatz number?"
daveslash 4y ago
"God made the integers; all the rest is the work of Man." ~ Leopold Kronecker, when criticizing Cantor's work [0]. See also Stephen Hawking's Anthology by a similar name. [1] [0] https://en.wikipedia.org/wiki/Leopold_Kronecker https://en.wikipedia.org/wiki/Leopold_Kronecker [1] https://en.wikipedia.org/wiki/God_Created_the_Integers https://en.wikipedia.org/wiki/God_Created_the_Integers
- another_devy 4y ago
  God should have made uint. I don’t want to deal with all these negativity
- bediger4000 4y ago
  That's an argument from authority isn't it? Rational numbers are pretty easy to understand. There are easy to understand proofs that at least some irrational numbers (famously square root of 2) exist. That is, the square root of 2 cannot be represented as a rational number. I'm sure there are other such proofs. My take is that Kronecker and Hawking were either making an ironic statement or just expressing irony or exasperation.
  - constantcrying 4y ago
    I think you are missing the point. Kronecker does not deny that irrational numbers exist, he states his belief that the fundamental number system which is inherent to reality are the natural numbers. That (by divine authority) are the irreducable thing which other number system must take from. E.g. the rationals are pairs of natural numbers, the reals cauchy sequences of rational numbers etc. > I'm sure there are other such proofs. The reals are uncountable.
raoof 4y ago
if Reality!=0.111... : the Universe is computable ( the list of infinite binary strings between [0,1) is countable ) else: the Universe is non-computable ( the list of infinite binary strings between [0,1] is uncountable ) how do you solve this dilemma?
- Koshkin 4y ago
  Isn't the list of infinite binary strings between [0,1) also uncountable?
  - raoof 4y ago
    if you put 1 out then there is no reason to believe it is uncountable because you can sort the list such that the diagonal is all 0s (if you know an argument that shows [0,1) is uncountable I'd be happy to know)
    - Koshkin 4y ago
      Well, if you remove one element from an uncountable set, then...
      - raoof 4y ago
        you can't prove something just by assuming it that would be trivial, if we can build or convince ourselves how to build a wormhole or a time machine then we can believe in Real numbers
        d_tr 4y ago
        What Koshkin says is correct and has already been proven by others. You want to get a book that covers sets, maps and cardinal numbers, and read it carefully if you are interested in this sort of stuff.
        raoof 4y ago
pixl97 4y ago
Veritasium: Math's Fundamental Flaw https://www.youtube.com/watch?v=HeQX2HjkcNo https://www.youtube.com/watch?v=HeQX2HjkcNo
thwayunion 4y ago
If you want to understand the epistemic status of any formal object related to "infinity" in Mathematics, the best three starting points are Religion, Sociology, and the History of Mathematics. I think a good History of World Religions course plus some focused reading on notions of infinity is likely more useful than an Analysis course for really understanding questions like "Are Real Numbers Real?" I do not think it is possible to divorce even modern notions of infinity from the deep impact that superstitious thinking has had on the direction of Mathematical thinking -- and thinking in general about anything remotely metaphysical, really -- for the last few thousand years (with only very recent counter-examples).
- yamrzou 4y ago
  Any specific readings you'd recommend?
  - thwayunion 4y ago
    Yes, but unfortunately they require some digestion. This thesis isn't, for example, fleshed out in a single pop sci book or even dissertation or academic book afaict. It's more like: reading a lot about the history of philosophy, mathematics, and religion. (That said, I think most people in those three fields -- philosophy, history of mathematics, and religious studies at least as an academic field -- will agree in one way or another with the general sense that mathematical notions of infinity have definitely influenced and been influenced by religion and spirituality.) The closest single source that comes to mind, at least for one region of the world, might be Russell's history of western philosophy, which commits more time to both theology and mathematical history that is typical for that type of text. But, it's a big tome and only here and there provides nuggets of insight on this topic. Thinking about other sources, if you are interested in this thesis, I think you'd get a lot out of reading the works of people like Descarte, Leibniz, or even Godel. Also biographies about the same, because the writings by the men themselves are more like primary sources than summative/synthetic expositions. By which I mean, you have to study the text, its cultural context, and its impact on the history of mathematics (rather than just read and comprehend the words on the page). So reading the texts plus the biographies plus knowing enough about mathematics to understand how their ideas do and do not permeate the field will, I think, give you a decent sense for the thing I'm saying above. Sorry I don't have a single easily digestible podcast/book/audio book :(
kortex 4y ago
Of the three linguistic entities in this sentence, only one is well defined. Let's rephrase from a question to a statement and look at the veracity: "Real numbers are real". - "Real Numbers" - this is the one thing we can formally define, but there is implied baggage - does the concept of real numbers exist, or does at least one real number exist? The former is just ∃x(x ∈ ℝ), √2 proves that, the set ℝ must exist, no major problem there. Really this question asks "For ∃x(x ∈ ℝ), is x 'real, existing in reality'?" For that we need to look to the other two words. - "are" - the copula (is/to be) is a doozy and a perennial semantic footgun, put a pin in that - "real" - this is a nebulous concept, and where the weight of the discussion lies. Basically it is asking can something with infinite information be? And there's that pesky copula. Is/be is tricky, particularly in English, because its usage is heavily overloaded. It is used to express - Strict equality: "Two plus two is four" - Categorization: "A frog is an amphibian" (all frogs are amphibians, not all amphibians are frogs) - Description: Grass is green (grass has the property green, but green is not grass). - Predicates (in general): "It is raining" (in many languages, there is no copula, and the sentence is closer to "Raining." or "It rains." see Zero Copula [0]) (arguably, all the above are involved in predicates, definitions are hard) In "Real numbers are real", I would contend we are talking about the description usage or predicate. Real numbers have the property of "existence". This is where a messy problem gets ultra messy. But at the same time, I believe the problem more broadly collapses to "Does X Exist?" [1] for any X. Chairs are made up of components which do not individually have anything chair-like about them, but the arrangement of matter (the process of producing the chair) nonetheless has the description of chair-like-ness. Real numbers comprise a process which produce them (stating a mathematical formula and asking what the answer is), and thus the process defines their existence. Whether or not an object with infinite information exists is irrelevant if the process to produce that object is finite, and that object can be used to produce another answer (e.g. we can use √2 to define the dimensions of the A Series of paper, or broadly solve for the dimension of a 45° triangle with side length 1). Its existence can no more be denied than the existence of chairs, or heaps, or other intangible ideas. (I suppose this argument only applies to constructable/algebraic numbers, which means the original question is really only talking about transcendental reals with no "nice" expression. But that gets into Hilbert's seventh problem, and whether you can derive an expression of any arbitrary transcendental using two algebraics). "Reality" as a linguistic concept is also highly relative. Are dreams real? They are experienced, they can be measured with medical equipment, and they can have lasting psychological side-effects; they are "real". But the events of the dream do not themselves have the observed side-effects in "consensus reality"; they are not "real". tl;dr - language is nebulous, and nebulosity is challenging [2] [0] - https://en.wikipedia.org/wiki/Zero_copula https://en.wikipedia.org/wiki/Zero_copula [1] - Do chairs exist? - Vsauce - https://www.youtube.com/watch?v=fXW-QjBsruE https://www.youtube.com/watch?v=fXW-QjBsruE [2] - https://meaningness.com/nebulosity https://meaningness.com/nebulosity
zzo38computer 4y ago
DNA is digital, but life is made from its environment, not only from DNA. However, it does not answer the original question, anyways. I think Borel's number is not very well defined, and the examples given there make that easy to see, because an infinite sequence will not reach all possible questions. However, that is not the only problem with Borel's number, anyways. The numbers, people, etc are not "existence" except by relation of other things (e.g. you can define additions/multiplications of numbers, greater/less/equal of numbers, and physical objects moving by the laws of physics according to the numbers, physical objects collide with other objects and interfere with them by gravity and other forces (i.e. they are not in isolation), etc). The relations between everything is "real", rather than the objects themself. Due to this, it can occur by many ways, and they are ultimately related to "you", too. Essentially, all existence must be by all relations of everything by everything. There is problem of what is "is" and what is "reality"; they are not so clearly defined. "Real numbers" can be explained by mathematics, but the meaning of the words in English (or other languages) is not really as precise as it can really be. (And then, even with mathematics, there will be "abuse of notation", etc.) But, anyways, absolute Truth is inexpressible, so, due to this, you cannot possibly explain everything. You only explain certain concepts, which are descriptions of some things, but whatever is explained, there will be everything else which is not explained and cannot be explained. A real number might be given a finite description, but it is not necessarily the case. However, sometimes by a part of it, you can figure out some of the other relations, but not all of them, e.g. if you have 100 digits then you can figure out if it is less or greater than another number with 100 different digits, but if they are the same then you will not figure out. But, that is only one way. It is not the only way to make a description, finite or infinite or otherwise, even if there will be one. Even so, that does not mean that such a description necessarily "is" such a "real number", anyways (especially since, as I mentioned above, absolute Truth is inexpressable). By mathematics, you can prove mathematical things, but you will not make the "set of every possible things mathematically"; rather, it will have to be specifically the theories which are being handled/proven, which are going to be done by mathematics.

38 points by yamrzou 4y ago

Is the integral a "thing"? - No, it, along with the reals, is the method (we call "analysis") that we humans invented to solve a certain class of problems in science and technology.

PaulHoule 4y ago

I find it ironic that the "real" numbers are so named because they are definitely less "real" than the integers or rational numbers since they cannot (with p=1) be individually named or specified.

jjgreen 4y ago
The name was introduced by Descartes in contrast to the disreputable "imaginary" numbers occuring in investigations of roots of polynomials. So not that ironic, more like rain on your wedding day ...
- lupire 4y ago
  It's extremely ironic: imaginary numbers are real, and real numbers are imaginary.
  - Koshkin 4y ago
    The situation is complex indeed.
martopix 4y ago
Integers and rational numbers are also reals. What aren't reals are imaginary numbers.
- deleted 4y ago
  [deleted]
- psychoslave 4y ago
  You mean imagery encompasses reals, I guess :D
  - Aardwolf 4y ago
    nonzero imaginary and real are both disjoint sets, and complex encompasses them both
- hackandthink 4y ago
  "Integers and rational numbers are also reals" Many mathematicians don't think so. Category Theory has a category of sets. Integers/Rationals/Reals are different Objects (sets) in this category. Type Theory is similar s/Category/Type.
  - voxl 4y ago
    Any mathematician, literally all of them, will tell you there is an injection from the set if rationals and reals, and that your foundational system is likely impoverished if this injection does not behave as the identity
    - klodolph 4y ago
      I think it’s worth talking about how the language works here. We say, “the natural numbers are a subset of the reals,” and this is a sensible thing to say. We also might say, “you construct each successor number in the natural numbers as n + 1 = n ∪ {n}”. And then we say, “a real number is a Cauchy sequence of rational numbers, or a Dedekind cut of rational numbers.” From a set-theoretic perspective, “the natural numbers are a subset of the reals” is obviously untrue with these definitions, and it’s worth spending a moment to think what the statement actually means, or how you would have to interpret it in order to understand the truth of the sentence. I might translate the sentence as “there is a left-cancellative morphism from natural numbers to real numbers,” but then I’d have to define what category I’m using, and what the morphisms are—which is usually implied. You end up having to stand on top of a surprisingly tall stack of proofs in order to say “the natural numbers are a subset of the reals” and actually explain what you mean by that, rigorously, from foundations. Or, put another way, it’s sometimes useful to understand what you mean by “behave as the identity”.
nuncanada 4y ago
Worse, if you take "Imaginary Numbers" to be only the necessary complement to make polynomials factoring complete... And "Imaginary Numbers" are definitely a necessity to describe reality... Meanwhile almost all "Real Numbers" are nothing but phantasmagorical numbers which we cannot name nor talk about them... It seems beyond the pale that we still are teaching these bizarre axioms instead of the Computable Reals which don't have all these phantom numbers...

yamrzou 4y ago

This is a response to Chaitin's "How real are real numbers?" (2004), which was discussed here twice: - (2017) https://news.ycombinator.com/item?id=14080024 https://news.ycombinator.com/item?id=14080024 - (2020) https://news.ycombinator.com/item?id=24029791 https://news.ycombinator.com/item?id=24029791

crdrost 4y ago
It is interesting that the source article from Chaitin it seems to regard its own title as mere wordplay, it does not want to get into the Morpheus Question “What is real? How do you define real?”... But the response seems to take it very seriously and get somewhat upset about it. Other than those aspects the discussion is kind of snore-fest...
- lupire 4y ago
  What do you mean? Chaitin is the most well known author who intended and is received as arguing against the existence of the set of real numbers.

psychoslave 4y ago

>concludes that these difficulties undermine the common assumption that real numbers underlie physical reality, strongly suggesting that physical reality may in fact be discrete, digital and computational. Well, it should precise from what population this "common assumption" is supposed to be taken of. Arguably, most humans that ever existed had basic integer arithmetical notions at best. What humans take for facts are not given but constructed on social interactions, even when protocolled around scientific methods relying on falsifiable models and sound metrics. The best that science can tell us is how much sound our model is in regard of available data and how much resources we need to produce and test an other one. Note that philosophy of Karl Popper is not itself falsifiable. > This statement describes a faith, not a scientific principle. Ultimately, principles lie on some mental faith or an other. There is no irrefutable evidences that solipsism is less sound than believing in an "outside world".

kirkules 4y ago

Still reading the article, but upon reading > Despite these difficulties, the cognitive notion of a continuum is not at all difficult to grasp. The difficulties arise only when trying to communicate, for example by naming or describing all the real numbers. But one can understand without communicating. I feel it's important to point out that humans are very good at convincing themselves that they have done a great job of understanding things when they have not. Including brilliant minds. Since this is in "trust our intuitions" territory about, essentially, philosophy of mind, it's worth bringing along this awareness.

kirkules 4y ago
I also think it's notable that somehow DNA "encoding" our minds for mind-inheritance is ruled out using the channel capacity theorem. Shouldn't this same kind of argument rule out the transferral of minds from one moment to the next, unless minds do not actually arise from our bodies? In other words, does this criticism by Lee commit to mind-body dualism?
- pencilguin 4y ago
  It demonstrates that the statement is nonsensical.
bell-cot 4y ago
>> [...] cognitive notion of a continuum is not at all difficult to grasp. The difficulties arise only when trying to communicate, for example by naming or describing all the real numbers. But one can understand [...] Notice that one can do a very similar analysis after replacing "continuum" with "rules of the game of chess" and "naming or describing all real numbers" with "name or describing all positions which could occur in a chess game". You don't need uncountable infinities, topological oddities of R, subtle but deep differences from how the real world works, etc. to get out to where humans aren't all that good at understanding things.
zmgsabst 4y ago
In fact, the formalization of calculus (and in particular, continuity and differentiability via limits) came about because people had different intuition about “continuous” functions. I would say the implications of a continuum versus a dense cloud of rationals is incredibly unintuitive.

hackandthink 4y ago

Some people think: If you can't compute it, it does not exist. or If you can't compute it, I don't care. Computing and real numbers are a confusing subject. Edit: https://ncatlab.org/nlab/show/computable+physics https://ncatlab.org/nlab/show/computable+physics There are nice discussions an Andrej Bauer's blog: https://math.andrej.com/ https://math.andrej.com/. Can't find them right now. "type II computability" is the concept. Look at the table in: https://ncatlab.org/nlab/show/Type+Two+Theory+of+Effectivity https://ncatlab.org/nlab/show/Type+Two+Theory+of+Effectivity

bell-cot 4y ago
Quip: Can the tuna salad sandwich I am eating for lunch be computed? Does it exist? Should I care?
- pixl97 4y ago
  I think first you would have to define what it means to be computed in this context.
- Koshkin 4y ago
  I am not even sure if one can say that tuna itself "exists," in this context.
- hackandthink 4y ago
  Not a sandwich but an equilibrium. What is it good for if there's no mechanism to reach it? "What they’ve found is that some games have Nash equilibria that are impossible for a computer to determine and even more equilibria are computable only if one of the largest open problems in computer science, P = NP, is solved." https://blogs.cornell.edu/info2040/2015/09/21/the-computability-of-nash-equilibria/ https://blogs.cornell.edu/info2040/2015/09/21/the-computabil...

dragontamer 4y ago

What is "real"? The ancients may have understood negative numbers, but IIRC the concept of zero (additive identity) wasn't understood until Arabic Numbers / Algebra. So "zero" isn't real, at least not to the Romans or Greeks. Once Algebra is invented, we learn that not only is "zero" a useful concept (additive identity), but so is "one" (multiplicative identity). More importantly, the inverse of multiplication can be defined as "blah times number == 1". IIRC, the Greeks somehow understood this concept before the invention of Algebra, but its easier to see once you have Algebra. Once we have those, its not too difficult to extend into irrational numbers, such as sqrt(2). The Greeks almost discovered it, but were too stubborn. There's a lot of discussion in Greek Mathematics over sqrt(2)... they're so so close but they were too much of a fanboy of integers / rational numbers to actually reach the concept. Once we understand sqrt(2), its not too difficult to extend the principle to sqrt(-1) or sqrt(-2), reaching the concept of "i". ----------- sqrt(-1) occurs in "real" voltages / currents all the time today in Electrical Engineering (https://en.wikipedia.org/wiki/Phasor https://en.wikipedia.org/wiki/Phasor). Its as "real" as any other number. IE: We can truly measure real-world objects (ie: the voltage going through your power-lines) and assign them a value of "i", and use that to predict what happens to the electricity as it flows through your equipment and/or house. So all of these things are "real" numbers, that describe truly physical real-world objects. Its just an unfortunate naming convention of "Real" vs "Imaginary" vs "Rational" vs "Irrational", which comes down to history / order of discoveries, rather than usefulness.

chriswarbo 4y ago
> sqrt(-1) occurs in "real" voltages / currents all the time today in Electrical Engineering I'd say sqrt(-1) describes "real" voltages / currents; but we can find other models which don't involve sqrt(-1). Those models may be more complicated, and have fewer rich connections to pre-existing work, but they're just as valid. This is usually where appeals to Occam's Razor come in, and physicists start appealing to "elegance", "beauty", etc.
- dragontamer 4y ago
  You're not going to be able to describe how voltages/currents work in an AC circuit unless you reach into 2-vectors / matrices. In Complex Arithmetic, we call it the "imaginary axis". But reach into Linear Algebra and you'd just call it a "2nd dimension". When things oscillate (and AC current is oscillating), there's just certain mathematical properties that occur. It doesn't matter if its electricity, a spring/mass system, a bridge, sine-waves, music theory, Fourier Series or anything else, its all going to be represented by complex-math extremely elegantly. Or, have a 2-vector linear-algebra representation, since its equivalent. EDIT: I guess there's the differential-equations representation, but I feel that reaches into linear-algebra very, very quickly... or complex numbers. (E^(pi * i) keeps coming up...)

adrian_b 4y ago

Both this article and that of Chaitin which is cited in it, fail to say clearly what they mean by "physical reality" and they also mention some things that are irrelevant to the title question, such as DNA. All the physical quantities that can be modeled as continuous, i.e. by real numbers, are derived in some way or another from the positions in space and time. All the physical quantities whose dimensional formulas (in natural units, not in SI, which obscures some of the relationships between physical quantities) do not include space/time quantities, e.g. electric charge, magnetic flux, angular momentum and a few others, are discrete. So the question whether the real numbers correspond to the physical reality must be reformulated less ambiguously as the question whether the space and time are continuous or discrete. The traditional model of the space and time as continuous matches successfully all known experiments. There have been a few attempts of creating models for various parts of the quantum physics by using a discrete space-time, i.e. where the points in space-time form a lattice (i.e. there are minimum distances between points that cannot be subdivided to find an intermediate point) and any physical process progresses by jumps between discrete points, instead of continuous movements. However, I am not aware of any useful result obtained with such models.

chriswarbo 4y ago

> If in fact the mind relies on a continuum for its cognitive functions, this would explain why cognitive functions cannot be inherited and why our minds can deal with real numbers, despite the paradoxes. I disagree with the assertion that "our minds can deal with real numbers". AFAIK there's no way to distinguish an uncomputable number from a computable one, using a finite set of computations (i.e. there's always some computable model which gives the same answers as our uncomputable number, even if it's highly convoluted); so we're still only learning about the countable, computable sub-set.

Nomentatus 4y ago
"Deal with" in detail, absolutely not, just as you say. Pardon my rephrasing you point (I think): You can show that some functions that give real numbers are computable. (Let's say, computing that number digit by digit. Binary digit or base-ten, you choose.) You can't show that all Turing machines halt, so you can't use the mere existence of a Turing machine as proof of a computable function. Therefore, we can't know or compute the set of all real numbers because we can't compute the set of irrational numbers; there will always be some Turing machine that, unbenownst to us, calculates a (likely irrational) function that spits out a real number. If we can't describe even a way to sort functions into two piles: "computes a real number (digit by digit)" and "doesn't compute a real"; then we can't be said to "deal with" the set of real numbers at all well. This rather presupposes the (now pretty much accepted) concept that numbers "exist" if they can be computed (whether they have been computed, or conceived of, or not.) "Intuitionism" in other words, IIRC.

mjburgess 4y ago

All of physics stands as an objection to "digital physics" which uses real numbers indispensably. Many things fall apart with discrete numbers (classical physics becomes non-deterministic, quantum mechanics becomes non-linear, etc.). The loons on the "digital bandwagon" have yet to produce a single plausible work showing the geometrical spaces hiterto essential to physics can be replaced with anything discrete. This is just another pipe-dream of idealist-Platonist-AI fanatics for whom the world is an abstract discrete number only because their favourite toy is so-described.

lupire 4y ago
An epithet is not an argument. I'm not saying you are wrong; I am saying you are not even wrong.
- mjburgess 4y ago
  There is never a cabal of zealots who gather together to defend material reality, is there ? Who are the loons of the scientific method ? Which zealot says, "let's be conservative in our dreaming?" 'digital physics', constructive mathematics, AI, idealism -- etc. are all apparently respectable symptoms of an ideological project i reserve the right to call it as i see it, and name a purely ideological project for what it is the world is not a simulation, there is no digital physics, your consciousness will not be uploaded -- etc
  - DontchaKnowit 4y ago
    Preach
ingleso 4y ago
Is Planck a loon? > In 1900, Max Planck derived the correct form for the intensity spectral distribution function by making some strange (for the time) assumptions. In particular, Planck assumed that electromagnetic radiation can be emitted or absorbed only in discrete packets, called quanta, of energy https://en.m.wikipedia.org/wiki/Ultraviolet_catastrophe#Solution https://en.m.wikipedia.org/wiki/Ultraviolet_catastrophe#Solu... How do you explain the zenos paradox’s place in reality? https://en.m.wikipedia.org/wiki/Zeno's_paradoxes https://en.m.wikipedia.org/wiki/Zeno's_paradoxes What’s with the hate on Plato’s ideals? What is a “chair”? Draw me a “circle”.

GTP 4y ago

I'm still reading it, but I don't agree with the following: >In Plato and the Nerd, I review Claude Shannon's channel capacity theorem [Shannon, 1948], which states that any noisy observation of anything conveys only a finite number of bits of information. Borel's know-it-all number cannot be encoded with finite number of bits unless the list all possible yes-no questions is finite, which it is not. It is easy to construct an infinite sequence of valid yes-no questions. For example, let the first question be "Is one a whole number?" Let the second question be "Is the answer to the first question YES?" Let the third question be "Is the answer to the second question YES?" And so on. As a consequence, Borel's number cannot reveal itself to us unless we invent some noiseless way of observing a thing in itself. It is true that he showed an example of an infinite list of valid yes/no questions, which would lead to a number with an infinite amount of digits. But despite it being infinite, by the way it was constructed you already know all the digits (they will be all 1s), meaning that that number's entropy is remarkably low and that number can be expressed with a finite number of bits of information. So Shannon's theorem isn't a problem here.

kirkules 4y ago
The presentation in the quote is awkward and doesn't work, as you pointed out, but Lee could instead have just mentioned that there's such a construction with questions of the form "is the answer to the previous question YES [NO]?" which is not constant (not all 1s), and does have the property he wants, i.e. the are such sequences that cannot be specified finitely (corresponding to binary sequences with that property). I don't think you've shown that Shannon's theorem isn't a problem here so much as the argument presented did not successfully convey that it's a problem. Tangentially, I'm interested to see a formal statement/proof of the form of Shannon's theorem add used in this article, because I can sort of intuitively see why it makes sense but I'm not familiar enough with the technical details to guarantee that this is even an appropriate use of the theorem, assuming the yes/no question construction has all the properties Lee wants it to have.
- pencilguin 4y ago
  For all i, "Is i a Collatz number?"

daveslash 4y ago

"God made the integers; all the rest is the work of Man." ~ Leopold Kronecker, when criticizing Cantor's work [0]. See also Stephen Hawking's Anthology by a similar name. [1] [0] https://en.wikipedia.org/wiki/Leopold_Kronecker https://en.wikipedia.org/wiki/Leopold_Kronecker [1] https://en.wikipedia.org/wiki/God_Created_the_Integers https://en.wikipedia.org/wiki/God_Created_the_Integers

another_devy 4y ago
God should have made uint. I don’t want to deal with all these negativity
bediger4000 4y ago
That's an argument from authority isn't it? Rational numbers are pretty easy to understand. There are easy to understand proofs that at least some irrational numbers (famously square root of 2) exist. That is, the square root of 2 cannot be represented as a rational number. I'm sure there are other such proofs. My take is that Kronecker and Hawking were either making an ironic statement or just expressing irony or exasperation.
- constantcrying 4y ago
  I think you are missing the point. Kronecker does not deny that irrational numbers exist, he states his belief that the fundamental number system which is inherent to reality are the natural numbers. That (by divine authority) are the irreducable thing which other number system must take from. E.g. the rationals are pairs of natural numbers, the reals cauchy sequences of rational numbers etc. > I'm sure there are other such proofs. The reals are uncountable.

raoof 4y ago

if Reality!=0.111... : the Universe is computable ( the list of infinite binary strings between [0,1) is countable ) else: the Universe is non-computable ( the list of infinite binary strings between [0,1] is uncountable ) how do you solve this dilemma?

Koshkin 4y ago
Isn't the list of infinite binary strings between [0,1) also uncountable?
- raoof 4y ago
  if you put 1 out then there is no reason to believe it is uncountable because you can sort the list such that the diagonal is all 0s (if you know an argument that shows [0,1) is uncountable I'd be happy to know)
  - Koshkin 4y ago
    Well, if you remove one element from an uncountable set, then...
    - raoof 4y ago
      you can't prove something just by assuming it that would be trivial, if we can build or convince ourselves how to build a wormhole or a time machine then we can believe in Real numbers
      - d_tr 4y ago
        What Koshkin says is correct and has already been proven by others. You want to get a book that covers sets, maps and cardinal numbers, and read it carefully if you are interested in this sort of stuff.
        raoof 4y ago

pixl97 4y ago

Veritasium: Math's Fundamental Flaw https://www.youtube.com/watch?v=HeQX2HjkcNo https://www.youtube.com/watch?v=HeQX2HjkcNo

thwayunion 4y ago

If you want to understand the epistemic status of any formal object related to "infinity" in Mathematics, the best three starting points are Religion, Sociology, and the History of Mathematics. I think a good History of World Religions course plus some focused reading on notions of infinity is likely more useful than an Analysis course for really understanding questions like "Are Real Numbers Real?" I do not think it is possible to divorce even modern notions of infinity from the deep impact that superstitious thinking has had on the direction of Mathematical thinking -- and thinking in general about anything remotely metaphysical, really -- for the last few thousand years (with only very recent counter-examples).

yamrzou 4y ago
Any specific readings you'd recommend?
- thwayunion 4y ago
  Yes, but unfortunately they require some digestion. This thesis isn't, for example, fleshed out in a single pop sci book or even dissertation or academic book afaict. It's more like: reading a lot about the history of philosophy, mathematics, and religion. (That said, I think most people in those three fields -- philosophy, history of mathematics, and religious studies at least as an academic field -- will agree in one way or another with the general sense that mathematical notions of infinity have definitely influenced and been influenced by religion and spirituality.) The closest single source that comes to mind, at least for one region of the world, might be Russell's history of western philosophy, which commits more time to both theology and mathematical history that is typical for that type of text. But, it's a big tome and only here and there provides nuggets of insight on this topic. Thinking about other sources, if you are interested in this thesis, I think you'd get a lot out of reading the works of people like Descarte, Leibniz, or even Godel. Also biographies about the same, because the writings by the men themselves are more like primary sources than summative/synthetic expositions. By which I mean, you have to study the text, its cultural context, and its impact on the history of mathematics (rather than just read and comprehend the words on the page). So reading the texts plus the biographies plus knowing enough about mathematics to understand how their ideas do and do not permeate the field will, I think, give you a decent sense for the thing I'm saying above. Sorry I don't have a single easily digestible podcast/book/audio book :(

kortex 4y ago

Of the three linguistic entities in this sentence, only one is well defined. Let's rephrase from a question to a statement and look at the veracity: "Real numbers are real". - "Real Numbers" - this is the one thing we can formally define, but there is implied baggage - does the concept of real numbers exist, or does at least one real number exist? The former is just ∃x(x ∈ ℝ), √2 proves that, the set ℝ must exist, no major problem there. Really this question asks "For ∃x(x ∈ ℝ), is x 'real, existing in reality'?" For that we need to look to the other two words. - "are" - the copula (is/to be) is a doozy and a perennial semantic footgun, put a pin in that - "real" - this is a nebulous concept, and where the weight of the discussion lies. Basically it is asking can something with infinite information be? And there's that pesky copula. Is/be is tricky, particularly in English, because its usage is heavily overloaded. It is used to express - Strict equality: "Two plus two is four" - Categorization: "A frog is an amphibian" (all frogs are amphibians, not all amphibians are frogs) - Description: Grass is green (grass has the property green, but green is not grass). - Predicates (in general): "It is raining" (in many languages, there is no copula, and the sentence is closer to "Raining." or "It rains." see Zero Copula [0]) (arguably, all the above are involved in predicates, definitions are hard) In "Real numbers are real", I would contend we are talking about the description usage or predicate. Real numbers have the property of "existence". This is where a messy problem gets ultra messy. But at the same time, I believe the problem more broadly collapses to "Does X Exist?" [1] for any X. Chairs are made up of components which do not individually have anything chair-like about them, but the arrangement of matter (the process of producing the chair) nonetheless has the description of chair-like-ness. Real numbers comprise a process which produce them (stating a mathematical formula and asking what the answer is), and thus the process defines their existence. Whether or not an object with infinite information exists is irrelevant if the process to produce that object is finite, and that object can be used to produce another answer (e.g. we can use √2 to define the dimensions of the A Series of paper, or broadly solve for the dimension of a 45° triangle with side length 1). Its existence can no more be denied than the existence of chairs, or heaps, or other intangible ideas. (I suppose this argument only applies to constructable/algebraic numbers, which means the original question is really only talking about transcendental reals with no "nice" expression. But that gets into Hilbert's seventh problem, and whether you can derive an expression of any arbitrary transcendental using two algebraics). "Reality" as a linguistic concept is also highly relative. Are dreams real? They are experienced, they can be measured with medical equipment, and they can have lasting psychological side-effects; they are "real". But the events of the dream do not themselves have the observed side-effects in "consensus reality"; they are not "real". tl;dr - language is nebulous, and nebulosity is challenging [2] [0] - https://en.wikipedia.org/wiki/Zero_copula https://en.wikipedia.org/wiki/Zero_copula [1] - Do chairs exist? - Vsauce - https://www.youtube.com/watch?v=fXW-QjBsruE https://www.youtube.com/watch?v=fXW-QjBsruE [2] - https://meaningness.com/nebulosity https://meaningness.com/nebulosity

zzo38computer 4y ago

DNA is digital, but life is made from its environment, not only from DNA. However, it does not answer the original question, anyways. I think Borel's number is not very well defined, and the examples given there make that easy to see, because an infinite sequence will not reach all possible questions. However, that is not the only problem with Borel's number, anyways. The numbers, people, etc are not "existence" except by relation of other things (e.g. you can define additions/multiplications of numbers, greater/less/equal of numbers, and physical objects moving by the laws of physics according to the numbers, physical objects collide with other objects and interfere with them by gravity and other forces (i.e. they are not in isolation), etc). The relations between everything is "real", rather than the objects themself. Due to this, it can occur by many ways, and they are ultimately related to "you", too. Essentially, all existence must be by all relations of everything by everything. There is problem of what is "is" and what is "reality"; they are not so clearly defined. "Real numbers" can be explained by mathematics, but the meaning of the words in English (or other languages) is not really as precise as it can really be. (And then, even with mathematics, there will be "abuse of notation", etc.) But, anyways, absolute Truth is inexpressible, so, due to this, you cannot possibly explain everything. You only explain certain concepts, which are descriptions of some things, but whatever is explained, there will be everything else which is not explained and cannot be explained. A real number might be given a finite description, but it is not necessarily the case. However, sometimes by a part of it, you can figure out some of the other relations, but not all of them, e.g. if you have 100 digits then you can figure out if it is less or greater than another number with 100 different digits, but if they are the same then you will not figure out. But, that is only one way. It is not the only way to make a description, finite or infinite or otherwise, even if there will be one. Even so, that does not mean that such a description necessarily "is" such a "real number", anyways (especially since, as I mentioned above, absolute Truth is inexpressable). By mathematics, you can prove mathematical things, but you will not make the "set of every possible things mathematically"; rather, it will have to be specifically the theories which are being handled/proven, which are going to be done by mathematics.