18 ms·
How real are real numbers? (2004)
- jfaucett 9y agoThis was a light and interesting read. Here is the direct link: https://arxiv.org/pdf/math/0411418.pdf https://arxiv.org/pdf/math/0411418.pdf "Indeed, the most important thing in understanding a complex system is to understand how it processes information. This viewpoint regards physical systems as information processors, as performing computations. This approach also sheds new light on microscopic quantum systems, as is demonstrated in the highly developed field of quantum information and quantum computation. An extreme version of this doctrine would attempt to build the world entirely out of discrete digital information, out of 0 and 1 bits." It would indeed be quite a blast to discover we are to a high degree of probability in a simulation.
- klodolph 9y agoI don't see the connection between figuring out that the universe is discrete and learning that the universe is a simulation. Not even in our universe are all simulations done using digital computers, some are done using analog computers.
- Smaug123 9y agoBut in our universe, if it's discrete on a macroscopic level, then it's either inherently a natural number (it can be counted), or it's intelligently created.
- maverick_iceman 9y agoOur best theories, General Relativity and the Standard Model, say that the world is a continuum.
- Smaug123 9y agoThen my statement is vacuously true. (This was intentional, but perhaps a bit obscure.)
- Smaug123 9y agoI'm a bit confused why this got downvoted. I can understand my parent post being downvoted, but the explanation for my parent post? Is it false?
- klodolph 9y agoI simply can't make heads nor tails of the parent comment. It was probably downvoted because people think it is nonsensical. Here is my explanation: 1. What does it mean to be "discrete on a macroscopic level"? (If a universe were discrete, would it not be discrete at any scale? When we use the term "discrete", are we talking about state space, or discrete spacetime, or what?) 2. What does it mean for a universe to "inherently [be] a natural number"? (Universes are not numbers, right? Is some sort of claim that the universe's state space is finite?) 3. What does have to do with whether a universe was "intelligently created"? (How can it possibly make sense that "intelligent creation" is an alternative to a discrete universe? The claims seem entirely unrelated.) The "explanation" comment doesn't attempt to explain anything so I'm not sure why you call it an explanation. (Well, perhaps it explains that a conditional with a false antecedent is logically true, but most people here already know that, so pointing it out is not a great way to contribute to the discussion.)
- pacman128 9y agoDo they really say this? They assume this just as Newtonian mechanics does. And like Newtonian mechanics give reasonable answers in the domains they describe. How would they change if there was a cutoff at some infinitesimal scale?
- klodolph 9y agoI'm a little uncomfortable with the language that the theories "say that the world is" X. General Relativity and the Standard Model both model the world using real numbers, but they're both known to be wrong, and the fact that they are continuous is not a great reason to claim that the universe is continuous. On the other hand, observations about Lorentz symmetry holding at distances on the order of the Planck scale put a wrench in a bunch of discrete spacetime theories. I don't really understand the math, though. All this is somewhat tangential to the issue of real numbers. Real numbers are not necessary for continuity.
- ska 9y agoAll models are wrong. Some models are useful. Some of them extremely useful :)
- maverick_iceman 9y ago>Real numbers are not necessary for continuity. You know of any continuum that doesn't include the real numbers? That will contradict the continuum hypothesis.
- klodolph 9y agoLet's avoid equivocating here: "the continuum" is sometimes used to refer to the real numbers, but "continuity" in this context is a property of functions between metric spaces (or possibly topological spaces). "The continuum hypothesis" and "continuous functions" are actually from completely different branches of mathematics. This happens fairly often in mathematics, where similar-sounding terms are used to describe completely different concepts, or the same term sometimes means different things in context, or sometimes an Adjective Noun is neither described by Adjective nor by Noun.
- mikhailfranco 9y agoI think it is easy to get around the problems of Lorentz invariance in discrete models, as long as it is not the spacetime that is explicitly discretized on the lattice. The lattice must be some other combinatorial graph structured algebra, with non-local 'propagation' of fields. Quantum Mechanics says that space is only defined relationally on the intervals between field interactions: a 'particle' is only localized as a particle when it interacts (position or momentum observable). So discrete space and time appear as values on some subset of lattice nodes in response to some propagating fields ('particles' only at the interaction event). The slogan for this is 'spooky distance at an action', because it is the (inter)action that defines the space(time). QM (and QFT) assume a background time, it is not an observable, even though it appears to commute with energy (e.g. energy is momentum in the time direction). So it's more tricky to understand how time emerges in a discrete Quantum Gravity, but I suspect there is an intrinsic proper time, defined by interactions with the 'vacuum' (minimal field states on the underlying lattice), which bootstraps a relational time defined over intervals between interactions.
- AnimalMuppet 9y agoI'm not sure that I'd agree with you on the Standard Model. To me, it's view of the universe is pretty discrete. (That is, presuming that you're speaking of the particle physics Standard Model, not something in cosmology.)
- klodolph 9y agoThe Standard Model has discrete energy levels but continuous spacetime.
- AnimalMuppet 9y agoELI5: Where does the Standard Model say anything about spacetime being continuous? I thought it was only about what particles exist.
- klodolph 9y agoIt is definitely far, far more than just what particles exist. The Standard Model describes how those particles interact with via the three forces other than gravity. The particles themselves are modeled using quantum fields and the properties of particles arise from operators on those fields. Those operators don't work unless the field is continuous. The operators will have a spectrum which describes how the corresponding observable is quantized. That's not really ELI5, but remember learning derivatives in calculus? Just like you can't take the derivative of a function which isn't continuous, you can't use the Standard Model if spacetime isn't continuous.
- enugu 9y agoWe have the holographic principle from physics which says that the amount of information in a finite volume of space is discrete. If true, then we can think of the continuity just as a convenient interface. All experimental questions can be answered by a simulation with a discrete state space.
- CuriouslyC 9y agoGeneral relativity and the standard model also produce infinite values, which to me is a pretty sure sign your model is broken. Viewing black holes as infinitely dense hasn't yet caused us problems in terms of predictions (but they're still a bit wild west area), and they fixed the standard model with renormalization (which seems like a bit of a hack). They're good enough to be useful but I'm pretty sure everyone expects them to be radically reformulated at some point.
- jawarner 9y agoI never understood this argument. If we are in a simulation, what manifold do the simulators live in?
- whatshisface 9y ago"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely." (Paraphrasing, because there are only countably many possible math papers that might describe a number.) I think Borel has confused names with things. The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never write expressions for - only that in a single symbolic system we can't have expressions for all of them at once. Besides, if you confuse extant with useful you might end up believing that some random large integers aren't "there!"
- xyzzyz 9y agoonly that in a single symbolic system we can't have expressions for all of them at once. I don't think that's true. There are only countably many different symbolic systems[1], and as we can only express countably many numbers in each, we don't leave the realm of countable. [1] - A "symbolic system" must at least come with a procedure to tell whether a sequence is a part of it, and, unless you disbelieve the Church-Turing thesis, there are only countably many procedures.
- j2kun 9y agoDoesn't this (and by extension, Curch-Turing) rely on the assumption that the universe is discrete? Provided it were not, and one were able to harness infinite precision, one could presumably make a new symbolic system based on it. Not saying I believe it, just teasing out assumptions. If one is arguing whether the universe is continuous and using the Church-Turing thesis as justification for something, there's a danger of circular reasoning. Also, I think the parent commenter is getting more at naming vs existence rather than naming versus "could be named in the future." Is the argument boiling down to that something does not exist (is not "real") if it cannot be named?
- xyzzyz 9y agoDoesn't this (and by extension, Curch-Turing) rely on the assumption that the universe is discrete? Provided it were not, and one were able to harness infinite precision, one could presumably make a new symbolic system based on it. That's a valid point -- if you read it in a certain way, the Church-Turing hypothesis indeed states that continuous models are no more powerful than the discrete ones. In fact, we have every reason to believe it's true. See [1], and references [BCGH07], [GCB08] in that paper. [1] - http://perso.ens-lyon.fr/yassine.hamoudi/wp-content/uploads/2014/08/pnpgpac.pdf http://perso.ens-lyon.fr/yassine.hamoudi/wp-content/uploads/...
- DanBlake 9y agoUnrelated, but I read a article a while back which said something similar, but it was based on the fact that our entire mathematical system is designed around "base 10" and as such is only relevant in many constructs to our specific human 'ten fingered', interpretation of math.
- Smaug123 9y agoThis is false. As soon as you get to first-year undergraduate maths, one discards log-to-base-10, all but permanently. Nearly all the rest of integer arithmetic is performed in an arbitrary base (if it's being considered "additively" in some sense), or much more commonly, using prime factorisation, which is independent of any base. Areas of maths which are not number theory basically never mention the number which is twice five at all, and could be done happily without ever knowing that there was some privileged base in which we count the naturals.
- unit91 9y agoIronically the publisher of that article used a variety of radices to make the claim that 10 is the only radix! 2 - to enter and transmit data on the machine 10 - radix in question 16 - color description on the page etc.
- gertef 9y agoOf course 10 is the only radix http://cowbirdsinlove.com/43 http://cowbirdsinlove.com/43
- Jason-Andrade 9y agoThis entire subject is very academic and theoretical, and will never impact anything in the real world. Using Big Fractions instead of floating-point numbers is a far more concrete argument with definite real-world impact! https://news.ycombinator.com/item?id=13855198 https://news.ycombinator.com/item?id=13855198
- Retra 9y agoMathematical structure can -- and has -- impacted the world with far reaching and unparalleled effectiveness without ever even having to have had introduce the concept of a number. Math really isn't about numbers.
- dbcurtis 9y ago1. Given any two real numbers on the real number line, you can find another real number between those two points. 2. The Planck length is the smallest unit of distance with any meaning. 3. The universe has finite diameter. Discuss. 4. For extra credit: Given 2 and 3, above, it follows that both the diameter and circumference of the universe can be expressed in Planck lengths as integers with a finite number of digits. Discuss the concept that Pi is a ratio of two finite integers.
- AnimalMuppet 9y ago1-3: You can define a (mathematical) real number that cannot be interpreted as a position in the universe that can be physically realized, taking the Planck Length into account. 4: At the precision of the Planck Length, you have two integers that are the closest physically-meaningful values whose ratio approximates pi. In both cases, you're confusing mathematical abstractions with what is physically realizable in a discrete system. You can't (meaningfully) do that.
- tossaway322 9y agoThe article unfortunately does not address the question of whether current mathematical analysis is an appropriate framework for a description of space/time. We are, in the end, dealing with elements "smaller than" (if that's the right phrase!) the Planck length. Of course, you can choose to ignore that and use what's already been provided, but to do so is "whistling past the graveyard". This has ramifications for all of string theory, quantum gravity et al.
- kazagistar 9y ago/me goes to wikipedia > Theoretical significance > There is currently no proven physical significance of the Planck length.
- Sharlin 9y agoThe author of this paper is Gregory Chaitin of Chaitin's constant fame, among other things (I didn't know that Kolmogorov complexity is also known as Chaitin-Kolmogorov complexity!)
- FabHK 9y agoI like the idea of encoding answers to all questions, or for that matter all books written so far (or both, while we're at it), in one real number between 0 and 1. My favourite number, really.
- gertef 9y agoThe encoding of all books written so far (and will ever be written in finite time), is a rational number. Don't need Reals
- kazagistar 9y agoThats kinda the whole point of the article, in fact. All possible encodings of all possible thoughts, books, formal systems, and whatever, fit into the rationals, and the reals are categorically outside that.
- FabHK 9y agoAll books written and will be written (in finite time) - yes, rational. All possible questions (infinitely many) - no, that would be a non-terminating non-periodic binary, right. From the article: > 2.4 Borel’s know-it-all number > The idea of being able to list or enumerate all possible texts in a language is an extremely powerful one, and it was exploited by Borel in 1927 [Tasi ́c, 2001, Borel, 1950] in order to define a real number that can answer every possible yes/no question! > You simply write this real in binary, and use the nth bit of its binary expansion to answer the nth question in French. > Borel speaks about this real number ironically. He insinuates that it’s illegitimate, unnatural, artificial, and that it’s an “unreal” real number, one that there is no reason to believe in.
- mrcactu5 9y agoi think it's no coincidence Godel's proof of uncountable reals, comes around the same time as Lebesgue integration. As mathematicians started exploring what the serious use of Fourier series
- ska 9y agoI think you are mixing up Godel (incompleteness of axiomatic systems) and Cantor (uncountable reals). Timing wise Cantors proof was done the year before Lebesque was born; Cantor was a generation before Lebesgue, and Fourier a generation before that iirc. Godel's is a generation younger than Lebesgue. Lesbesgue and Borel were working at the same time - Godel was very young when he published his incompleteness theorem in the 1930s.
- mrcactu5 9y agoi think it's no coincidence Godel's proof of uncountable reals, comes around the same time as Lebesgue integration. As mathematicians started exploring what the serious use of Fourier series
- CogitoCogito 9y agoThe continuity of real numbers provides a clean theoretical basis for continuity of functions. I personally view it as more of a theoretical tool that seems to do pretty well and instead steer clear of the philosophical questions. One thing that I think is important to note though is that the jump from real numbers to complex numbers is nothing compared to the jump from integers/rational numbers/etc. to real numbers. The complex numbers come about by simply adding one dimension whereas the real numbers come about from an abstract "completion" of (say) the rational numbers in a very specific mathematical sense. My point is that deriding "imaginary numbers" (as many do) is total nonsense if you accept real numbers.
- hammock 9y agoThis makes sense. It feels analogous to economics (and economic models) - something fake we make up to be able to create functions and theories and so on
- gertef 9y agoRationals are conintuous, but not a continuum. you don't need real numbers to have continuous functions. http://math.stackexchange.com/a/672151 http://math.stackexchange.com/a/672151
- CogitoCogito 9y ago> Rationals are conintuous, but not a continuum. No. In fact, your claim is contradicted by your own link. As stated by Asaf Karagila in the comments: > Continuity is a property of functions. You seem to ask why the rational numbers are not connected (or path connected). But you are correct that you don't need real numbers to describe continuous functions. As you link points out, one is a property of spaces (or domains or whatever you want to call it) and the other is a property of function. However, talking about continuous functions between complete spaces (basically "complete" is what makes the real numbers "real") is extremely natural and basically goes hand in hand with continuous functions. It really ties together a lot of the theory if you're talking about metric continuity. Regardless, you also don't _lose_ anything by talking about real numbers. You can of course use it as a tool to develop a theory and then choose to apply the results to the rational numbers (or algebraic, etc.).
- scythe 9y ago>In addition to this mathematical soul-searching regarding real numbers, some physicists are beginning to suspect that the physical universe is actually discrete [Smolin, 2000] and perhaps even a giant computer [Fredkin, 2004, Wolfram, 2002]. It will be interesting to see how far this so-called “digital philosophy,” “digital physics” viewpoint can be taken. Here is how far: Everything written in words about the physical universe is, by necessity, discrete. Thus all information that can be encoded in human languages is discrete. Any non-discrete behavior of the physical universe which causes a change in the discrete information available to us, must, by assumption, have a component which is orthogonal to all of the prior discrete information (otherwise it is fully discrete). Since this component is independent of all previously available information, it looks like randomness. In other words: from the viewpoint of a discrete (linguistic) observer, the behavior of a continuous universe looks identical to that of a discrete universe that contains random fluctuations. What is interesting, then, is that observationally, our discrete observable universe is full of random fluctuations. Speculation as to their true continuous underpinnings is, however, unfalsifiable, unless the randomness itself can be made to disappear. I usually turn the question around: is it inconceivable that there would be a continuous universe with inhabitants that used a discrete language? So: >According to these ideas the amount of information in any physical system is bounded "the amount of observable information in any physical system" -- any unobservable continuous information shows up as unpredictable changes in the observable information.
- mikhailfranco 9y agoOne of the few comments here that shows great insight. I think you should push the argument down a few layers, to talk about the discreteness of quantum observables, the underlying unobservable continuous wavefunction, and the randomness of the Born rule that maps between the two (Copenhagen collapse or forking of Many Worlds). Tegmark takes a similar, but rather extreme, MW approach in his book 'Our Mathematical Universe'. Personally, I struggle with Tegmark's use of Measure Theory, and proponents of various Anthropic Principles, because they seem to have a completely broken frequentist view of probability and inference.
- chairmanwow 9y ago> "the halting probability Ω, which is irreducibly complex (algorithmically random), maximally unknowable, and dramatically illustrates the limits of reason" I really enjoyed the beauty of this statement.
- nemo1618 9y agoI think you would also enjoy some of the statements here: http://www.mathrix.org/liquid/archives/the-history-of-the-chaitin-leibniz-medallion http://www.mathrix.org/liquid/archives/the-history-of-the-ch... "God has chosen that which is the most simple in hypotheses and the most rich in phenomena. But when a rule is extremely complex, that which conforms to it passes for random." "Everything can be summarized in one thing, but the thing itself cannot be reached." "Mathematical facts are true by chance." "To make all things from nothing, unity suffices."
- prmph 9y agoMy philosophical take: the halting problem illustratres how free will can exist in a deterministic universe
- datadata 9y agoThis is a really interesting comment, could you elaborate on it a bit more? Which part of a deterministic universe would the halting problem serve to enable free will? -The universe as a whole? -Any agent claiming to have free will? -Some physical process that couldn't be simulated faster by something else in the universe?
- Retra 9y agoFree will is just what it feels like to have a mind that can construct models of realities that are not fact. And you can model nondeterminism in a deterministic system just fine. So I would argue that it illustrates nothing of significance under either of those issues.
- prmph 9y agoHow do you model true non-determinism in a deterministic system? And if that can indeed be done, would that not support my original point?
- option 9y agoNote that the probability of randomly picking a rational number from [0,1] is exactly 0. That is, with P=1 you will end up with an irrational number, not representable in any modern computer.
- 7373737373 9y agoReally? Interesting. I would have expected it to be something like epsilon.
- jerf 9y agoI'm not sure if this is entirely mathematically valid, but it's at least intuitively valid. Imagine we're in decimal (for familiarity). Imagine we're generating our random number by drawing digits out of a bag containing the ten digits, then replacing and drawing again. This is of course a magical perfectly uniformly distributed bag. In order to draw a rational number, you must randomly select a number that has a repeating pattern in decimal. Imagine that for the sake of argument that we're in this "repeating pattern" and, amazingly, we've already drawn it 1000 times! What a roll we're on! In order for the number to be rational, how many more times must we draw this pattern? Infinitely many. What is the probability that in one of those infinite fair random draws, the repeating pattern gets broken? 1. It's just not possible for infinitely many fair draws to produce a repeating pattern; that would be proof that the process generating the number was in fact not random in the first place. (I can't be that sloppy if we're doing finitely many draws but I believe that's justifiable at infinity.) Imagining through this process of drawing a random number I think makes this more intuitively obvious than imagining being presented with a completed number and trying to figure out if it's rational.
- option 9y agoit is entirely mathematically valid. See here for some math https://www.quora.com/Is-there-a-point-where-statistical-improbability-transitions-to-functional-impossibility https://www.quora.com/Is-there-a-point-where-statistical-imp... The catch, here is "selecting a number randomly" - that isn't constructively defined. And to me this looks kind of similar to the notion of "measurement" in quantum theory - not precisely defined either.
- gertef 9y agoLawrence Spector (professor at CUNY, Manhattan) on this topic: http://www.themathpage.com/acalc/anumber.htm http://www.themathpage.com/acalc/anumber.htm
- SeanLuke 9y agoRichard's Paradox for some reason reminded me of the proof that there is an infinite number of interesting whole numbers. The proof goes like this. Assume instead that the number is in fact finite. Consider the first number higher than any of the interesting numbers, and thus bounding them. Now that's an interesting number! QED.
- AndrewOMartin 9y agoShut up and calculate.
- snarfy 9y agoI tend to think of real numbers as a composite made of whole numbers and an operator.
- Smaug123 9y agoWhich operator?
- snarfy 9y agoIt depends on the real. 0.5 would be 1/2 using division operator, while an irrational like square root of 2 is using the power and division operators (raising to the 1/2 power).
- ducttapecrown 9y agoThose are the algebraic numbers.
- gerdesj 9y agoRichard's Paradox seems a bit shaky to me (p4): "Since all possible texts in French can be listed or enumerated" Unless I have completely missed the point then he has simply stated a way to generate another member of the set of French texts which of course is part of that set and so on. You can easily squint hard enough to generalize to all texts in all languages, now, earlier and possible then allow that grammar, spelling and so can be pretty slack. Now translate that lot into numbers in some way (a bunch of IT bods should be able to manage that!) To be honest French on it's own is probably more than enough. "How very embarrassing! Here is a real number that is simultaneously nameable yet at the same time it cannot be named using any text in French." The very act of naming the number (in French) constructs the French text that adds to the set of possible French texts. I think that the set of possible French texts is exactly as large as the set of reals. So is the set of all language texts and that the "paradox" is merely trying to use the Cantor argument backwards.
- surement 9y ago> I think that the set of possible French texts is exactly as large as the set of reals. How? The set of French texts is countable and the set of reals is uncountable.
- gerdesj 9y ago0,1,2 etc are French words. All reals are French words 8)
- jawarner 9y agoThis is not true, no French word has an infinite length. Any set of finite strings is countable.
- Retra 9y agoNot all French texts need be finished either. Nobody his picked a date boundary to constrain all French text. Nor has anyone formalized which texts are French, since the French language is evolving.
- gerdesj 9y ago"According to Pythagoras everything is number, and God is a mathematician. This point of view has worked pretty well throughout the development of modern science. However now a neo-Pythagorian doctrine is emerging, according to which everything is 0/1 bits, ... , God is a computer programmer, not a mathematician, and the world is a ... a giant computer" [p13 of the pdf] If you only have one finger then zero and one are just as real (ahem) as numbers that arise naturally when you have 10 fingers. The 10 toes are a bonus. I doubt that we can really know what Pythagoras really thought but given some of the results attributed to him I think Chaitin does him a disservice. Getting wound up over whether French is a sophisticated enough language to describe numbers and some of the odder consequences of allowing construction to equate existence will probably only lead to a headache. As a civilian wandering on the outskirts of all this philosophical foot stamping, I believe there are a fair few pretty rigorous arguments out there that can't be denied by resorting to "it looks wrong, cos reasons" style illustrations in a 13 page pdf.
- throwaway5752 9y agoDoesn't this basically rehash stuff covered 100 years ago by Hilbert, Whitehead & Russell, and Godel? If it wasn't such an eminent author, I would give it a pretty solid eye-roll. As some other poster noted in a link they provided, "pi" and "e" - among the uncountably infinite transendentals - are probably reasonable responses to this article in it's entirety. And again, even the point about describing the world in 1s and 0s at the end seems to me to be repetitive of Whitehead & Russell's Principia Mathematica which (and please correct me) used logic to construct the integers?
- surement 9y ago> "pi" and "e" - among the uncountably infinite transendentals - are probably reasonable responses to this article in it's entirety. The point of the article is that most reals are useless in practice; pi and e might be transcendental and the transcendental numbers are uncountable, but they are both computable, and the set of computable numbers is countable.
- throwaway5752 9y agoIs pi computable? I would have thought it would be a good halting problem example. I am admittedly not strong in modern developments in computability. Was aware of Chaitin and some of his work prior to this, but that's about the limit. If his point is that the universe is finite and finite methods are a more correct basis for physical sciences, then I'm open to that even if I'm not particulary interested (theoretical math objects are perfectly interesting in their own right to me). But if there's more to it than that, I'd appreciate the help.
- hawkice 9y agoA computable number one where there is a finite length representation of the number, namely, there is the program that, given a number of digits, can output the number it describes accurate to that many digits. There are a tremendous number of ways to calculate pi and e.
- prmph 9y agoSo is there full agreement on what a number is, in the first place? Some would argue that PI is not an actual number; but that it is a concept, like infinity
- marplebot 9y agoA number is also a concept, and there are number systems that include infinities (e.g. https://en.wikipedia.org/wiki/Surreal_number https://en.wikipedia.org/wiki/Surreal_number). It all depends on your definitions/axioms. So in a sense there is not full agreement on what a number is, but several sets numbers are generally agreed upon: integers, rationals, computables, etc.
- plg 9y agoI love LaTeX in general, but man alive, the default choice of font style/size/face for headings & sub-headings looks god-awful
- kingkawn 9y agoMephistopheles from Goethe’s “Faust”: “Theory, my friend, is gray, but green is the eternal tree of life.”
- tel 9y agoFor those interested in constructive and intuitionistic approaches here Dummett's [0] Elements of Intuitionism is an extremely good read. Intuitionism is a form of a constructive foundation for mathematics which (a) notes that any attempt to deny the uncountability of reals leads to difficulties and (b) any attempt to internally define them violates constructivity. The resolution proposed is to posit the existence of "free choice sequences". These are essentially "unpredictable" sequences of potentially infinite binary choices. As there is no a priori reason to believe that they can be predicted they are able to be much larger than what is computable and thus can be used to give a characterization of the reals. Atop this you build a constructive understanding of free choice reals which behaves very nicely (at least foundationally... it points out all kinds of weirdnesses about what we assume classically to be the structure of reals). What's very nice about this solution is that it sidesteps the difficulty. Free choice is a weaker thing to ask for than finite/constructive reals, but finite/constructive reals could be transparently encoded into free choice sequences and all the math would just work. [0] https://www.amazon.com/Elements-Intuitionism-Oxford-Logic-Guides/dp/0198505248 https://www.amazon.com/Elements-Intuitionism-Oxford-Logic-Gu...
- pron 9y agoIt sidesteps the difficulty by making the math itself (at least for the time being) much more difficult, and that is the reason it was rejected by most mathematicians in the Hilbert/Brouwer debates. Because here's the question: suppose you can't philosophically justify the "existence" of the real numbers, yet they coincide perfectly with observation and result in math that is much simpler than constructive math. Should you reject them? Brouwer -- who was the first to recognize just how much ordinary math relies on non-constructive principles -- said yes, because non-constructive math is philosophically wrong, period. Hilbert -- who was a finitist -- instead suggested that the propositions of math come in two flavors: real propositions, those that are finitary and can be taken to say something about physical reality, and ideal propositions, that are not. He said that as long as the ideal propositions are consistent with the ideal ones, they should not be rejected on a priori philosophical grounds even if no finitary meaning can be assigned to them. I.e. they are philosophically justified after the fact by virtue of their consistency with the real propositions. This philosophical classification of mathematics into "real" and "ideal" is called formalism, because it does not require that the ideal propositions be assigned a finitary meaning beyond their formal statement (as a finite string of characters). Of course, most mathematicians are not finitist, so they require neither intuitionism nor formalism -- both essentially finitist philosophies -- and are Platonists, believing that even ideal objects that are beyond physical reality and computation have a "real existence" in some Platonic sense. BTW, I think that after Turing (who used Brouwer's choice sequences in his construction of computable numbers) it is no longer necessary to rely on "free choice" (or lawless) sequences because of the halting theorem, and both lawlike and lawless sequences can be unified, and Brouwer's "creating subject" identified with a Turing machine. But I'm not sure about that. Turing -- who was a mathematical philosopher himself -- rejected any dogmatic a priory philosophy of mathematics, except for common sense, as the one true foundation, and suggested that the value of a formal system be derived not from its a priori philolosphy but from its ad hoc utility.
- graycat 9y ago"The natural numbers were invented by God. All the rest are man made."
- D_Alex 9y agoInfinity is a weird thing, isn't it? Now: one of the proofs in the paper relied on an assumption that all possible computer programs are countable, which I think implies that they are finite in length. But it is fairly trivial to generate computer programs that are infinitely long, say by assigning characters or expressions in some language to the digits of transcendental numbers such as pi. It is also possible to generate infinitely many such programs, simply by using pi/2, pi/3... etc. Now, the proof as presented fails, since these programs cannot be ordered by size. Can the proof be modified to take account of this? I don't know... comments invited.
- bo1024 9y agoI'm not sure it's so easy to define or generate an infinitely-long program. Such a thing doesn't sound to me like it would be either possible in practice or equivalent to a Turing Machine in theory. For example, you suggest an assignment of expressions to digits of pi. Now how would you run such a program? Presumably by generating the digits of pi, interpreting them as expressions, and evaluating the expressions, etc. But the program you used to do that was finite. So are you running the infinite program? I think it's more fair to say you are running the finite one.
- D_Alex 9y agoYes - I was just thinking the same thing. I have not come to a conclusion one way or another on whether it is possible to de-couple the generating program from the generated programs for the purpose of analysing the proof. It seems that there should be a way to do it... unfortunately I cannot spend more time on this now.
- jawarner 9y agoDoes there have to be a generating program? The set of all finite programs is countable, so it could not possibly describe the uncountable set of real numbers - this would be a surjection from a countable set to an uncountable set. In particular only the subset of computable numbers [1] can be described by the countable set of finite computer programs. Also, any infinite program either passes through a finite number of instructions or never terminates. So any program which does not have a finite representation will never terminate. [1] https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf
- lngnmn 9y agoNumbers? Real? Neither molecular biology nor the sane part of physics has any of em. Btw, it is heuristic - if there are numbers involved then it is human made. Reality as it is has no such notion. Biology does not count. Numbers require an observer, which is a by-product of the processes in vastly complex brain structures of the cortex, and cannot be the basis of anything in the underlying universe. Any good (which means Eastern) philosophy arrived at these simple conclusions millennia ago.
- psyc 9y agoAre they real? Well, when a human says something about a thing, they can never be completely sure whether they're saying something about an actual thing with an independent existence, or whether they're only saying something about what they say.
- threepipeproblm 9y agoJust wanted to thank the OP, caustic, for this. I first thought it might be past my available background/resources for a casual read. But I found it accessible and rewarding.
- mikhailfranco 9y agoFor the parallel historical development of 'the continuum' in physics, I recommend this readable survey: Paper: https://arxiv.org/abs/1609.01421 https://arxiv.org/abs/1609.01421 https://math.ucr.edu/home/baez/continuum.pdf https://math.ucr.edu/home/baez/continuum.pdf Blog summary with discussion: https://johncarlosbaez.wordpress.com/2016/09/08/struggles-with-the-continuum-part-1/ https://johncarlosbaez.wordpress.com/2016/09/08/struggles-wi... https://johncarlosbaez.wordpress.com/2016/09/09/struggles-with-the-continuum-part-2/ https://johncarlosbaez.wordpress.com/2016/09/09/struggles-wi...