For 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.
It 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.
Without knowing too much about about the subject, I've vaguely wondered about this idea for a long time, now, but I figured it likely an un-respectable position. I think it's too bad that beginners are often shielded from controversies in foundations.
Within the past few years I ran across an alternate approach to calculus, which if I recall correctly, achieves the same basic results, but without the same notion of infinitely small slices and so on... now I can't find it to link.
Yep, and as you can see in that Wikipedia article, both Brouwer and Hilbert were finitists. It is not that it is an unrespectable opinion as much as mathemticians these days -- from what little I know -- are not generally expected to hold any dogmatic opinion on philosophical foundations, but maybe that attitude will shift again.
> I think it's too bad that beginners are often shielded from controversies in foundations.
Mathematicians generally shouldn't worry about foundations, as it is the intention of foundations to stay hidden -- except in cases where the foundation requires a rewrite of much of math, as in the case of Brouwer's intuitionism, but constructive math is pretty advanced anyway. Foundations are usually the concern of logicians, but from what little I've seen in online discussions, it seems like many logicians aren't interested in philosophy, either, and that's a real shame.
> I ran across an alternate approach to calculus, which if I recall correctly, achieves the same basic results, but without the same notion of infinitely small slices and so on
There's constructive analysis[1], which recreates analysis within the framework of constructive math (by finite means etc.), and there's also non-standard calculus[2] (which I know nothing about) that makes treat infinitesimals as actual numbers, which may be the opposite of what you meant, but maybe not -- I saw something about there being constructive versions of non-standard analysis, too.
I think you're right on the money bringing Turing into this.
I don't have a real, formal horse in this race, but the way that I have things arranged in my mind is that these are all disagreements on the notion two things in logical foundations: the need to consider time/resources and the conception of logical systems as closed or open.
Brouwer and Hilbert accept that time is a factor, but handle it in different implicit ways. Hilbert differentiates between finite (time accessible) and infinite (not finite) and then makes a plea that this divide doesn't much matter. His Program sought to prove it and then Turing and Godel and the like shattered that dream. Brouwer philosophically treated proof as communication and thus Intuitionism handles time and resource naturally (though not explicitly like a Kripke model).
Brouwer was forced by this choice to leave his mathematical foundations open ended as well. At any given level of effort there is a finite set of things we can prove/talk about/know and the rest must be left unknown, but likewise at any point in time we can incrementally increase our knowledge. Intuitionism is complex in order to handle that openness.
Bringing Turing in makes openness and expense/time immediately obvious. Turing machines are a model for an open world of knowledge: once you have a partial halting oracle G I can build a new program which your oracle fails on. Godel took it to the next level: your own formal system cannot prove its own consistency, but using it I can create a larger universe where it can be proven consistent.
In each case, the result arises from wanting a formal closure of your foundation and both use self-reference to create a (co-)inductive means of incrementally enlarging your system forever.
So I think Brouwer beats them to the punch in a certain way by taking that continuous finitary expansion as foundational itself.
Free choice can't be modeled formally by "Turing machines" because we possess a means for writing them all down. On the other hand, if I give you the trace of a Turing machine you have no (finite) means to predict it so "Turing machine traces" do provide the largeness you need. I don't really know how to handle the fact that someone might want to form a bijection between the two. To me it makes me think immediately of information hiding behind an abstract interface: even though you could eventually know an implementation of the interface, you spend most of your time working ignorant to it and often have more power for doing so.
I'm definitely no expert here, but I think it's all really fascinating. Definitely agree that Brouwer didn't produce anything of much use to day-to-day mathematicians, but that's sort of Hilbert's (and Turing's) point: practical common sense doesn't seem too impeded by this.
I think as computer scientists we're essentially carrying out a certain mathematical/philosophical program in this debate, fwiw. Turing showed that if you take working computationally very seriously that you will run into open-endedness and time-as-resource constraints very quickly. Programming is a long exercise in discovering just have very seriously you have to take both of those things (interfaces, abstraction on one side, resources, human patience on the other).
Brouwer and intuitionism are subsequently having a bit of a resurgence in mechanized proving. The non-computability of reals shows its ugly head every time someone working in computational optimization has to write `compare_espilon` instead of just `equals`. Or, perhaps even better, every time someone gets bitten by thinking IEEE reals are "mathematical reals".
I also really like to imagine this because I love this idea that each programmer is unknowingly taking very tiny steps along a grand philosophical program. Unromantically it's obviously true, but there's something fun in trying to imagine just how profoundly computers may yet still impact our understanding of the world.
> I think you're right on the money bringing Turing into this.
Well, his most famous paper, the one in he was first discovered the essence of what computation is (rather than merely asking what functions can be computed by a "definite procedure"), was named On Computable Numbers, and he uses the very problem of the real numbers as a segue into the more general topic of computation.
> Free choice can't be modeled formally by "Turing machines" because we possess a means for writing them all down. On the other hand, if I give you the trace of a Turing machine you have no (finite) means to predict it so "Turing machine traces" do provide the largeness you need. I don't really know how to handle the fact that someone might want to form a bijection between the two.
I don't think there is any such bijection (if only for the cardinality argument you mention), but that it turns out (thanks to halting), that lawful and lawless sequences are equally opaque: it is as impossible to determine (using finite means) whether two lawful sequences are equal as it is to determine whether two lawless sequences are, which is all that's required (I think) for the theory. Therefore the same idea that Brouwer had in mind does not really require lawless sequences (which, I think, he wouldn't have introduced if he'd known about halting).
> Turing showed that if you take working computationally very seriously that you will run into open-endedness and time-as-resource constraints very quickly. Programming is a long exercise in discovering just have very seriously you have to take both of those things (interfaces, abstraction on one side, resources, human patience on the other).
It's a bit more complicated, as Turing himself -- in his published conversations with Wittgenstein -- opposed intuitionism as a foundation, believing that math neither has nor requires one true formal foundation. After all, Formalism is also finitistic, and you can just as well use a program to prove a theorem about (classical) real numbers using finite means; there's no need to construct -- i.e. give specific meaning to -- each real number. That was Hilbert's point: it is enough to reason about mathematical proposition in a finitary manner, and there's no actual need to construct the objects in their domain of discourse with finitary means.
> Brouwer and intuitionism are subsequently having a bit of a resurgence in mechanized proving.
Yes, but I think the relationship is more complicated. Again, a mechanical prover can prove classical propositions -- by using deduction rules -- just as it can prove intuitionistic propositions. The resurgence with respect to mechanical provers, as far as I understand, has two reasons: 1. it is easier to create a tiny universal core for a mechanical prover based on intuitionsitic logic, and classical logic can then be built on top of that. 2. much of the research on the theory of functional programming rests on the desire (explicitly stated by Martin-Löf) to unify programming and constructive math. I personally find this approach misguided for various reasons (I'm not a mathematician or a logician, so I see this from a practicing programmer's perspective), but I can understand the appeal.
> there's something fun in trying to imagine just how profoundly computers may yet still impact our understanding of the world.
Oh, absolutely. Once Turing realized that computation is in a very strong sense a fundamental natural phenomenon, he understood the many implications (on logic, AI, biology and physics).
> it is as impossible to determine whether two lawful sequences are equal as it is to determine whether two lawless sequences are
I also think that's enough.
I think when you mention that it's easier to build a mechanistic theorem prover on an intuitionistic foundation that it's a bit more "foundational" a problem than it seems. Theorem provers need (some kind of partial) equality on propositions and on proofs in order to function, but propositions are too rich to support finite equality. This backs you into a more intuitionistic approach.
> but propositions are too rich to support finite equality
Why do you say that? The finitary deduction rules of all logic systems provide both equivalence (<=>) and partial order (=> or |-) relations. Classical logic gives rise to a boolean algebra, while intuirionistic logic forms a Heyting algebra, of which boolean algebra is a special case. The latter is more general, but both are bounded lattices, and both are perfectly fine.
All numbers are ultimately "probabilistic" in calculations.
Sure you can state them as a "fact", but that undermines quantum theory itself and the Universe.
If you want to do the math right you really need to consider terms as peak probabilities and work from there. (See Chaos theory)
IMHO
Of course doing math with hard constraints is perfectly adequate for most macroscopic calculations.
That said, errors do add up given sufficient entropy and time.
It's quite simple to add fuzziness into the equations but quite a pain in the ass to compute with, plus you will in most cases get the same answer anyway.
As far as real numbers, they are no different from integers, just shift the decimal.
"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!"
only 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.
Doesn'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?
You are right, this reasoning depends on only allowing formal systems with finitely many symbols. Turing himself actually gave justification for this in a beautiful footnote in his 1936 paper [1]:
page 249:
> I shall also suppose that the number of symbols which may be printed is finite. If we were to allow an infinity of symbols, then there would be symbols differing to an arbitrarily small extent.
And then in the footnote:
> If we regard a symbol as literally printed on a square we may suppose that the square is 0 < x < 1, 0 < y < 1. The symbol is defined as a set of points in this square, viz. the set occupied by printer's ink. If these sets are restricted to be measurable, we can define the "distance" between two symbols as the cost of transforming one symbol into the other if the cost of moving unit area of printer's ink unit distance is unity, and there is an infinite supply of ink at x = 2. y = 0. With this topology the symbols form a conditionally compact space.
Doesn'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.
Physics would have something to say about infinite precision.
But yes, with those assumptions, it does not hold. It is not as much that the argument is circular, but that the only tech we know how to use (symbols) limits our expressiveness.
I think you (and some other commenters) are responding to something different - potential vs actual infinity - not being able to write down all natural numbers, but we can potentially write any number(with some assumptions like an infinite universe).
Whereas the point being made is different and needs some math background which is the work of Cantor. Countable can include all natural numbers that you count until infinity, and the reals are uncountable in that they exceed even this.
The proof of this fact(Reals are uncountable), has the same idea involved in Turing machines halting problem and also Godel's theorem. The general version is the power set of a set(set of subsets of a set) cant be put in one to one correspondence with original set. If you assume such a correspondence, then (<guess the clever trick>) leads to a contradiction.
Reals are sequences of naturals which include sequences of 1's and 0's which can be interpreted as subsets of naturals(a sequence represents the subset of indexes which get assigned the value 1). So reals are bigger than the power set of naturals.
Also, the infinite countable union of countable sets is countable. The number of grid points(integer coordinates) on a plane is the same as the number of grid points on a line. You can set up a zig-zag correspondence. A salesman can visit all grid points on a plane in infinte time. This is used to prove rationals are countable. So even countably infinitely different symbolic systems doesnt help.
The proof of the reals being uncountable depends on the idea that one can build a number that depends on being able to make an infinite number of choices, each of which depends on the absolute truth or falseness of a statement.
But what happens if we open it up to have statements be true, false, or currently unknown? That is we develop a system of mathematics that could be in principle done inside of a Turing machine?
Then Cantor's diagonalization argument falls apart because of all of those "currently unknown" options. See https://news.ycombinator.com/item?id=13843725 for a previous explanation that I gave of this.
Mathematician here. If you mean to say that you cannot constructively prove "the real numbers are not countable", then you're wrong. As a rule of thumb, you can usually prove negative statements constructively as you would prove them classically.
A constructivist would probably state the result more positive (and stronger, constructively): To every countable set M of real numbers, there is a real number not contained in M.
A constructivist would state the result in a variety of ways. But none of them would involve a potentially self-referential construction based on the absolute truth of an infinite number of statements. Which really does rule out Cantor's argument.
I don't understand this. For a constructivist, "¬A" means "from A, falsity is derivable". You can derive falsity from the statement "there is a countable enumeration of the real numbers". A diagonal function given such an enumeration is constructively definable, and it is also constructively true that it is not equal to any of the enumerated ones (because 0 != 1 is true constructively). Thus, the diagonal is not enumerated all real numbers, but it also is by assumption, thus falsity.
What does make real numbers weird when working constructively is that you cannot conclude |x| > 0 from x != 0. This means that 1 / x need not exist even if x != 0, so the real numbers do not form a field in the classical sense.
> The interpretation of Cantor's result will depend upon one's view of mathematics. To constructivists, the argument shows no more than that there is no bijection between the natural numbers and T.
A surjection seems silly: doesn't that imply that some reals would be indistinguishable? Or can you indeed not prove that more than the natural number of reals are distinguishable in constructive mathematics?
Honestly, the result hinted at in the wikipedia article caught me by surprise as well: There are some flavors of constructive mathematics (notably CZF), that are still consistent if you also add the statement "There is a subset A of the natural numbers such that there is a surjection from A onto the real numbers". Note that this does not imply that you can prove this in CZF, it only means that you cannot disprove it. See http://www1.maths.leeds.ac.uk/~rathjen/acend.pdf prop. 8.2 if you're interested.
Wikipedia says that the Cantor argument shows "no more" than that there is no bijection between the natural numbers and the real numbers. As far as I can tell, it also proves that there is no surjection from the natural numbers onto the reals in every system of constructive mathematics I know of, so I think this is slightly misleading.
What do you mean by "distinguishable"? In constructivism, a set A such that x = y or x != y for all x, y in A is sometimes called decidable. And yes, it is indeed not true that the real numbers are decidable. This would imply that there is an algorithm that decides whether two infinite sequences of ones and zeroes will differ at some point or agree indefinitely, which is pretty much equivalent to solving the halting problem.
On the other hand, the natural numbers, integers, rational numbers and every finite algebraic extension of the rational numbers (for example Q[i], "rational imaginary numbers") is decidable.
Furthermore if we limit all mathematics to things that can in principle be done on a Turing machine, then this statement is trivially true because all possible Turing machines is a countable set.
Moving from Turing machines to a Turing-complete programming language, we could define a real number as an equivalence class of functions from positive integers to fractions such that for each function |f(n) - f(m)| <= 1/n + 1/m, and two such functions are equivalent if |f(n) - g(n)| <= 2/n for all positive n.
The question of whether a given function is actually a real number is in general undecidable. The question of whether two functions represent the same real number is also undecidable. However it is easy to create a set of functions that will definitely include all real numbers under this definition (and a few things that are not), but some of those real numbers will be included multiple times.
And the really important point about this is that suddenly "uncountable" doesn't mean "more". It just means that there is an undecidable question or three creating complexity in the way of generating the mapping.
>A constructivist would state the result in a variety of ways. But none of them would involve a potentially self-referential construction based on the absolute truth of an infinite number of statements. Which really does rule out Cantor's argument.
But in any case you misunderstand Cantor's argument and constructive mathematics.
mbid is correct; Cantor's diagonalization argument constructively proves the uncountability of the real numbers, see e.g. Bishop-Bridges' CONSTRUCTIVE ANALYSIS, Theorem 2.19, page 29.
I am not a mathematician (physicist). I think the concept of infinity is a con that mathematicians have pulled on us (as there isn't an easy reality to map on to). I can understand arbitrarily big set; however, I never managed to make the jump from arbitrarily finite to infinity. Mathematicians made that jump and glossed over, then continue to show the difference between countable infinity and infinity beyond. Since I never could make that jump, it all sounds nonsense to me.
I have a similar (and I believe to be equivalent) problems with infinitesimal as well. How does arbitrarily small but non-zero become infinitesimal? Since I have problem with infinitesimal, I find the differentiation of real numbers and rational numbers equally non-sensical.
So mathematician, take a pause, could you explain what is countable infinity? Since you never can finish counting (all the natural numbers), how does it make the set countable? What do we mean exactly by countable here?
Wikipedia refers to the idea of one-to-one correspondence. But since you can never exhaust the correspondence, what do we mean by one-to-one? Give me any unique real, I'll give you a unique natural number, and we can go on forever, so how does that not count as one-to-one correspondence?
Unlike mathematicians, physicists are fine with unresolved :)
PS: I guess countable can be defined as there is a definite way of ordering the set, which is true for natural numbers but questionable for real numbers. I still don't see how the ordering connects to the size of infinity and one to one correspondence. Even for the set of real numbers, I can have an algorithm continuously generate random numbers (discarding re-occurring ones so it will be a unique sequence) and prove there is an order of the set (non-exhaustively defined, same as the set of natural number). The ordering may not be describable though. But non-describable ordering is still an ordering, right? Just as a real number that cannot be exhaustively described is still a number. I don't have to describe it, I can hand-wave it just as the way mathematicians hand-waved the infinity.
A set being "countably infinite" only means that you can write a function that maps each distinct entry in the set to exactly one natural number (0, 1, 2, etc.) without duplicates. That's it.
So for example, the set of natural numbers is countably infinite and we know this because we can write a function that maps each natural number to exactly one natural number: the id function.
We can extend this and say that the set of even natural numbers is countably infinite because it has a mapping function of x => x / 2.
The same is true for all integers (natural numbers + negative numbers): x => if (x < 0) { x * -1 * 2 } else if (x == 0) { 0 } else { x * 2 + 1 }, i.e. if it's negative map it to an even number and if it is positive map it to an odd number, if it's zero map it to itself.
You can even write a function that maps all rational numbers to the natural numbers, since each rational number can be written as a fraction of two integers. (Figuring out the function is a fun exercise but it is also easy to google)
However, you can't write a function that maps any real number to a natural number. The easiest to understand proof of this is Cantor's Diagonal Argument[0], which is a proof by contradiction that shows that any attempted function must exclude some real numbers. Therefore, the real numbers are not countably infinite, and we call them uncountably infinite.
EDIT: In response to your edit, Cantor's Diagonal Argument basically shows that for any given function (and you have to define the function completely ahead of time - that's key) I can give you a real number that is not included in the domain of your function.
Why a pre-definable function matters here? We have allowed a real number (that cannot be exhaustively described) in, so why a real function (that you cannot exhaustively define) has to be excluded? Isn't it unfair that I am only allowed to use a finitely definable function while you can choose a non-pre-finitely-describable real number? Isn't that a loop proof -- that the set of real numbers is uncountable simply because it is defined to be uncountable, and it is larger infinity than that of natural numbers simply it is defined to be bigger (as what bigger means in the context of infinity is otherwise undefined).
Here is how you do it. Have a function p(r) which evaluates to the previous real number. Then your mapping function is:
f(r) = if (r == 0) { 0 } { else f(p(r)) + 1 }
If your objection is "You can't determine what the previous real number is." Then my counter-objection is "Please prove that you can't." Which I don't think is possible without first assuming reals are uncountable.
What is your definition of "previous real number"? If you define p(r) such that p(r) < r and there does not exist any real number x such that p(r) < x < r, then it is easy to show that p(r) cannot exist using a proof by contradiction.
Given real number r, assume there exists a real number q such that q < r and there does not exist a real number x such that q < x < r. Let real number y = (q + r) / 2. It is trivial to show that q < y < r. Therefore we have a contradiction, and therefore there does not exist a real number q that meets our conditions for a "previous real number".
If you take issue with this, then I suggest you read up on the standard construction of the number systems from the naturals up to the reals. This is all very rigorously defined in terms of ZFC set theory.
Your definition of f is circular: to calculate f(r) we need to know p(r), which in turn depends on f(r).
Cantor's diagonal argument shows that any mapping from the natural numbers to the real numbers must necessarily miss some real numbers out. It takes some time to get your head around if you aren't used to mathematical proofs, but it's definitely worth looking it up and trying to work through it if you're interested in this subject.
Suppose there exists an r' such that for some real number r, p(r)=r', where p(r) gives the real number that precedes r.
What does that mean? Does it mean there are no numbers between r and r'? Because that's what I think predecessor means, even though it's trivial to prove that there are numbers between r' and r. For example, (r'+r)/2, the average of r and r', is between the two numbers. And there are also numbers between r and (r'+r)/2, and between r' and (r'+r)/2.
So there can't possibly be a real that is the predecessor to another real, because there are always more real numbers between any two reals.
Reals are not equivalently as unrealizable or dubious as infinitesimals. Infintesimals can be constructively specified as nilpotent/nilsquare entities, numerical entities which when squared equal 0 but where those entities themselves aren't reducible to 0. All of this can be done in a constructive manner avoiding any use of infinity or classical logic that depends upon indirect proofs (ie excluded middle). John Bell's 'Primer of Infinitesimal Analysis' has good details.
The computational techniques from Automatic Differentiation use these types of entities to calculate derivatives exactly without approximating infinite (limiting) processes.
Calculus can be done constructively without infinite limiting processes purely algebraically using these nilpotents. And from a geometric interpretation there is nothing nonsensical about a tangent line to a curve.
Also you don't differentiate numbers (real or rational), you can only differentiate functions.
Also the idea of a actual infinity is a poetic mathematical one. It doesn't have to fit reality. The issue is whether it is useful and to what extent.
Seems I agree with you (or you agree with mine) :)
I don't have problem with calculus -- or I wouldn't be able to do physics. I am having problem with calculus based on infinity.
> Also the idea of an actual infinity is a poetic mathematical one. It doesn't have to fit reality. The issue is whether it is useful and to what extent.
Well said.
> Also you don't differentiate numbers (real or rational), you can only differentiate functions.
Not what countable means. The meaning meant is the one about having a map from the set to the integers, where no two things of the set map to the same integer.
The correspondence is meant as either a full thing, or at least a well defined specification which could be applied to any of the things, if you want to get philosophical about ontology or something.
Also, the specification can't depend on things like, what reals have been given as input "previously".
Say that in this game, you have two obligations:
1: for any n,m I give, you must tell me what the mth binary digit of the real number associated with n is, in the mapping you are considering (or, if there is no real number associated with that natural number.).
2: For any finite collection of the binary digits of the real number I am choosing, you must tell me the first natural number such that the corresponding real number matches all the digits I specified.
With these rules, I can choose my real number (specifying more and more digits) such that for any natural number, I will be able to show that my real number doesn't correspond to that natural number, or any lower one.
Therefore, there is no natural number that corresponds to my number in the matching system you are providing.
(To win, I repeat this procedure:
Ask what the first natural not ruled out already by my specification of my number is. (Call that n)
Ask what the nth digits of the real associated with n is.
Inform you that the nth digit of my number turns out to have the other value for its nth digit, so no natural less than or equal to n corresponds to my number.
Repeat.
This will only ever tell you more about my number, and will not result in me changing my mind about any of the digits of my number, yet there is no natural number which won't ever be ruled out as potentially corresponding to my number.
Therefore, no natural corresponds to my number in your system.
Why should you get to choose your real number -- specifying more and more digits -- implies an un-exhaustive process, while I am not allowed to do the same? An unfair game will have unfair winners, how would it mean anything? If we both are allowed an un-exhaustive process of specifying what we have, this goes back to the counting game. As we never can finish, how does it make countable (or not)?
The reason that the real number being defined can be defined in terms of the function is because, that is the situation being considered in the statement.
For any way of doing the mapping, there is a real number that the mapping misses. Alternative statement: "there is no such mapping that doesn't miss any of the reals".
This is shown because, given a mapping, I can find a real that the mapping misses.
When one says "for all x, there exists a y such that P(x,y)", the y is allowed to depend on the x.
That's what this is.
Why wouldn't your objection apply to the proof that the halting problem is uncomputable ? The program that the halting checker can't check is defined in terms of the halting checker. Why is that allowed? Because that is what the statement is saying. For any purported halting checker, there exists a program it doesn't decide the halting of.
Similarly here, for any purported bijection between the integers and the reals, there is a real that the purported bijection misses.
I don't know if you are using the word "countable" in the standard way, so I don't know what you mean by that last sentence.
Given any finitely described real numbers, there will be a finitely described mapping map it to a unique natural number. But if you use real numbers that can never finish describing, we have to use a mapping that cannot be finished in describing. That is not the same as "there is no such mapping that doesn't miss any of the reals". If all mappings that cannot be finitely described are excluded, what will be the reason? And why that reason cannot be applied to exclude some real numbers (that cannot be finitely described) as numbers? I understand that is what it is, but being what it is seems meaningless.
The general halting problem is uncomputable. It can only be simulated. However, any program is still assumed to be finitely described. Any discussion in finite domain (including arbitrarily big finite) cannot lead to conclusions or insight toward infinite.
> I don't know if you are using the word "countable" in the standard way, so I don't know what you mean by that last sentence.
In the context of infinity, words such as countable, bigger, order, etc. all lose its standard meaning. We don't really know what it means if we don't really know what infinity is. Mathematicians simply made a definition to countable here -- a finitely described mapping -- that is fine on its own, but completely useless. Since we can't draw any parallels from infinity to finite (including arbitrarily big), we can't really relate any definitions over the infinity to "standard" meaning of those words to the domain of finite.
At future timelike infinity you'll reach infinity.
Now for the reals. Your goal is to step from 0 to 1, by way of 0.1, 0.01, and so forth.
0
0.0........
At future timelike infinity you still haven't stopped adding in zeroes to the right of the decimal point.
In the first case, at any finite time before \breve{i}^+ you will have counted out some finite natural number. In the second case, you will not yet have counted out your first nonzero real.
This survives across changes of positional counting systems, and almost certainly survives arbitrary choices of non-lossy notation, as long as you start with a finite representation of 0.
> At future timelike infinity you'll reach infinity.
> Now for the reals. Your goal is to step from 0 to 1, by way of 0.1, 0.01, and so forth.
> 0 0.0........
> At future timelike infinity you still haven't stopped adding in zeroes to the right of the decimal point.
> In the first case, at any finite time before \breve{i}^+ you will have counted out some finite natural number. In the second case, you will not yet have counted out your first nonzero real.
The same reasoning applies for rationals, yet they can still be counted.
The definition of «can be counted» means there is a bijection between your set and the naturals. Such kind of bijection can easily be created for the rationals[1] and Cantor's diagonal argument[2] shows that you can't create such bijection for reals.
There is nothing really intuitive about this concept, but fortunately the proofs are pretty straigtforward which give a kind of «intuition» around this.
> but fortunately the proofs are pretty straigtforward which give a kind of «intuition» around this.
It is only straightforward once you accepted infinity and all other definitions/description based on infinity.
If you, like me, cannot accept infinity, then all the proofs/descriptions that contain infinity become apparent non-sensical.
>> Start counting the naturals: 1, 2, 3, ...
>> At future timelike infinity you'll reach infinity.
This highlights the flaw. You reach infinity with infinity. Nothing is really being said about infinity. But somehow if you accepted the understanding infinity here, the rest of thesis such as cantor's diagonal argument may seem to be natural, except you forget that you really didn't know what infinity is.
All property of infinity cannot be finitely described. So anything about infinity is built on top of infinity. Turtle all the way down (or up), and we don't really know what it is.
The mathematical concept of infinity generally derives from the axiom of infinity in Zermelo-Fraenkel set theory, which asserts the existence of the inductive set (which is the basis for the construction of the natural numbers). Specifically, it states that there exists a set N such that the empty set is in N, and for all x in N, x union {x} is also in N. This is an axiom, so it cannot be proven to be right or wrong. You can either accept this axiom, or attempt to create your own system of mathematics that does not require the axiom of infinity.
Infinite defined as not finite is meaningless. For example, infinity is not really a number in a conventional sense. What it is is not clear other than it is not finite.
A real number with infinite precision is equally meaningless.
(Above are not directly related to your comments. I simply like to summarize my thought).
Now to your comment. You can start counting from 0 by 2s, you'll never hit 1, but that doesn't show 1 is not countable. It only shows that 1 is not countable in this particular counting scheme. Yes, you can devise a counting scheme that never hits some numbers, doesn't really contribute to either proof or insight.
I assume you are familiar with the counting scheme of rational numbers, and in that scheme, it can hit any number within any (finite) precisions. Just as infinity, a number with infinite precision is unclear. You certainly can define it, but the definition will have infinite built-in, and it is not clear what meaning does such definition adds.
Counting integers from 0 by 2s and never hitting 1 (or 3 or 5 ...) still results in a finite number of natural numbers counted in finite time, skipping a finite number of natural numbers at each step. How many reals do you have to skip between 0.0 and 2.0 and between 2.0 and 4.0 ?
Returning to my previous attempt, you could think of instead a successor function; for any finite natural number the immediately adjacent natural number can be found in finite time. For any real number, the immediately adjacent real number cannot be found in finite time because the step from one real number to the next is infinitesimally small.
All of these examples are "de-generalizations" of the mapping argument. Counting integers from 0 by 2s maps bijectively onto the natural numbers.
The naturals map injectively and surjectively onto the reals; you exhaust all the naturals counting between 0.0 and 1.0, or 1.0 and 2.0, or even between 0.01 and 0.011.
"A real number with infinite precision is equally meaningless": uhm, integration of infinitesimals (dS, dV, ...) ?
> How many reals do you have to skip between 0.0 and 2.0 and between 2.0 and 4.0 ?
Here you sneaked the concept of reals in. Remember reals are defined on top of infinity. You can't have reals if we are still debating what infinity is. There are infinite amount of numbers between 2.0 and 4.0, in the same sense there are infinite amount of numbers in the natural set.
Your successor function defines any finite natural number, it does not define infinity. In the rational counting scheme, we can reach any number within any finite precision. A real number that is defined on the base of infinity precision requires infinity time to reach with the same counting scheme -- the same way infinity requires infinity time to reach by 1, 2, 3, ... So if you allow infinity time, the same way you allowed infinity in your definition of real, then all real numbers can be reached (including infinity time) by counting -- not that provide any meaning.
Calculus is based on taking limit -- that is assuming a finite precision, albeit arbitrary. Infinitesimals are still finite, not infinite. Otherwise, you cannot divide them.
I find it helpful to think of countability in terms of the following game: You have a set of items in mind. You propose a (non-terminating) scheme for listing all the items in the set. An adversary attempts to name any item X, hoping your scheme misses it. However, you then show your scheme does, in fact, get to X after a _finite_ amount of time. The set is said to be countable if you prove that your adversary cannot win this game.
Yes, the counting process is non-terminating. But every item gets counted after only finite time.
Since it is non-terminating, you didn't really prove every item gets counted after finite time, did you?
Refer to my other reply, you asserted a requirement of predefined (describable) counting scheme here. Why that requirement has any relevance here (in the context of infinity)?
If I start at 0 and successively add 1, do you agree that I eventually hit any positive integer you could pick after a finite number of steps? Does that not prove to you that I hit every positive integers? Which one do I not hit?
You will only eventually hit any given integer after finite steps. It does not prove you will hit every positive integer. You'll miss those that one never can finish giving you -- for example, I'll start with digit 1, and I'll infinitely adding 1 behind it (never finishing). It is an infinite natural number that you can't hit within any finite number of steps.
The game itself is certainly not the proof that is required. The proof in question is a proof that the game cannot be won by the adversary. By no means do you need to play all possible rounds of the game to conclude this.
The discussion really helps me understand the problem. Real numbers are infinity in disguise. Because infinity is not really defined, real numbers are not really defined. Just saying infinity is not finite does not make much sense (in the sense of adding anything helpful). Any real number that you can finitely describe can be included in a finitely described counting scheme. Let's use the counting schemes of rational numbers, I'll hit arbitrary numbers to arbitrary precisions. Taking a limit, it is not clear that I miss any numbers (including pi). To defeat this scheme, the adversary has to keep adding digits to his real number, as well as keep shrinking his allowed precision infinitely. Now both the number (that is being infinitely being described) and my counting process is infinite, we didn't and can't prove anything. There is simply no conclusions to be drawn.
> you didn't really prove every item gets counted in finite time
The proof that the adversary cannot name such an item is logically the same as a proof that every item gets counted. This is a fundamental logical truth, namely de Morgan's law for universal/existential quantifiers: (not exists x such that P(x)) is the same as (forall x, not P(x))
> Why that requirement has any relevance here
'A counting scheme' is an intuitive way of providing a 'one-to-one correspondence to the natural numbers,' which is used in the technical definition of countability.
That sounds like, if you use ternary logic (true, false and CU), then the relationship between sizes of N and R is different. Intuitively, that sounds kinda wrong - why sizes of N and R and their relationship would depend on which kind of logic model you're using to reason about them? If you could build a 1-1 mapping between N and R somehow employing your logic system (which is equivalent to saying they have the same size then shouldn't it be possible to build the same without using your special logic? At least intuitively it sounds so, what am I missing?
Yes, when you restrict to computable numbers, then there are countably many reals. Countable in classical sense. You wont have an computable enumeration, again of because of diagonalization.
The formulation I prefer: a countable union of counted sets is countable.
(Any countable set has a well-ordering, because a bijection with the natural number gives you one in an obvious way. The trouble is that you need well-orderings for all of them together. The unusual term "counted" emphasizes that we need the actual "countings" to do the job, whereas for me "well-ordered" is sufficiently commonplace that it doesn't shove in my face the requirement that each set come along with a specific choice of well-ordering.)
> 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
Yes it does: there are countably many numbers we can define, and the reals are uncountable, therefore there are uncountably many numbers that cannot be defined.
Whether or not it does is a matter of which philosophy you choose.
A Platonist may believe that those numbers exist in an abstract space we cannot reach. A Formalist simply defines "existence" in such a way that this is true without worrying about whether they really exist. And a Constructivist denies the existence of things that cannot actually be written down, at least in theory.
This may be clearer to the parent commenter if expressed as "there are only countably many strings drawn from the Unicode alphabet, so if we fix a coding scheme such that a string expresses at most one number in that scheme, then there are only countably many numbers we could have defined. If you try and get around this by saying that the coding scheme is arbitrary so the symbol L can be made to stand for any real, then you have dodged the question of how you specify that coding scheme; there are only countably many coding schemes which can be expressed by strings of Unicode once a coding scheme is fixed, and in particular there are only countably many coding schemes expressible in English".
"Real numbers that cannot be defined with language" is not a well-defined set though. Just like "integers that cannot be described in less than 100 words" is not well-defined (I could reach a contradiction by pointing to the "smallest integer that cannot be described in less than 100 words").
Wait, you're agreeing with me that the set in question is ill-defined, but still somehow comparable to other sets? I am not sure I follow. My statement is that you cannot meaningfully talk about "all the numbers that cannot be described with language". If you could, I'd ask you if this set intersected with [0, 1] has a lower bound / 'inf' that's contained in it, and if so, did I just describe that lower bound with language? I'm sure some sort of paradox similar to the one with integers can be constructed here...
In my view, every real number is well-defined and there's nothing controversial about the set of real numbers. If the infinite aspect of it causes some researchers to call it a "mathematical fantasy", so be it, so is literally every other mathematical model we use in our lives.
> I'd ask you if this set intersected with [0, 1] has a lower bound / 'inf' that's contained in it, and if so, did I just describe that lower bound with language?
The "indescribable numbers" are dense in [0,1], and so (if the set exists) the inf of the set of indescribable numbers which are between 0 and 1 is 0. Perfectly describable.
My example is incomplete, not faulty. I left it as a question (does the inf belong to the set?). If the answer is yes, we reached a contradiction. If the answer is no, we have to continue further zooming in to this interval (or some other construction along those lines).
See, I claim that this set is ill-defined, so I can't know its properties like whether or not it's dense, open, closed, Borel-measurable, etc. etc.
You have to tell me what its properties are, and I will come up with a concrete proof that the set in question is ill-defined.
EDIT: After I RTFA'd, this is actually the paradox in section 2.3 of the linked article
Then we mean different things by define. I am saying the set R (with all its elements) is an uncontroversial, well-defined construction within ZFC.
I am leaving out any linguistic or Turing-computability aspects out of this, and people try to bring it back in, mixing computability with definability.
Defining the set of real numbers is very different from defining all real numbers. Yes, Chaitin's constant is defined(with a computable system as a parameter).
But that's the point - we cant produce such a definition for almost all reals.
> Defining the set of real numbers is very different from defining all real numbers.
I'm saying that ^ sentence makes no sense to me, I don't know how to parse it formally. If you start talking about the set of "definable" numbers (not computable, but specifically "definable"), I believe you're gonna run into paradoxes as it's an ill-defined concept, similar (in spirit) to "all integers described under 100 words". In fact, the linked article actually talks about it in 2.3.
> For any given language, like for instance ZFC, we can say that definable numbers are a countable subset. Hence measure zero.
If I can describe a set of objects, then we're all set as far as I'm concerned (mathematically speaking). Being able to efficiently construct individual elements of this set using Turing machines or other computational devices is an orthogonal problem.
Also, I don't think having only countable number of utterances in ZFC precludes you from having well-defined uncountable sets described in that system (quite obviously, for any set S take 2^S which is very well-defined).
The point is straightforward - the fact that you have defined a country on a map, doesnt mean you have defined all its cities and towns. Especially if the number of markers you have are less than the number of towns.
Also, we can talk about definable numbers as long as we choose some specific system which we assume is consistent. So we are talking about numbers which are definable by predicates using the language of ZFC or Peano Axioms.
There is no need to invoke computability, just definability is sufficient. There are lots of definable numbers which are not computable(like Chaitins constant or the real number whose digits encode information about halting of Turing Machines).
But even with this more relaxed constraint, we still dont have enough definable numbers.
If you can describe a set of objects thats fine. But by itself that would be nearly useless. The problem is that ZFC and the like add an axiom that you can identify an element of any described set, and use that to prove further theorems. That makes no sense; its an elimination rule with no corresponding introduction, materializing members of a set from nothing.
There are more people on earth than, say, kings. That's true, even though I can't enumerate all non-kings, and even if the set of non-kings is somehow ill-defined.
A sentence which fails to specify a real uniquely… fails to specify a real uniquely. Borel's talking about numbers which can be defined uniquely: that is, picked out, identified. Your Berry-paradox description doesn't identify an integer.
A rather more technical – but very interesting – take on this question is [0], which opens:
> One occasionally hears the argument — let us call it the math-tea argument, for perhaps it is heard at a good math tea — that there must be real numbers that we cannot describe or define, because there are are only countably many definitions, but uncountably many reals. Does it withstand scrutiny?
then explains in detail why it does not withstand scrutiny.
Consider the set of all real numbers which you can actually specify IN ANY FASHION AT ALL, so that the person writing a paper about it and the person reading the paper are talking about the same number. This includes easy ones like "3" or "Square root of Pi". It includes oddballs like Chaitin's constant -- the probability that a randomly constructed program will not get stuck in an infinite loop (for some particular choice of computer and programming language) -- which is so ornery a number that we cannot (even in THEORY) figure out a single digit of it (other than its being between 0 and 1).
Consider that great big set. It's still countable. The thing that makes "real numbers" bigger than integers, the "extra" that real numbers have must be very, very peculiar.
>Chaitin's constant -- <snip> -- which is so ornery a number that we cannot (even in THEORY) figure out a single digit of it (other than its being between 0 and 1).
"A Chaitin Omega number is the halting probability of a universal Chaitin (self-delimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random(its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly noncomputable. The aim of this paper is to describe a procedure, that combines Java programming and mathematical proofs, to compute the exact values of the first 64 bits of a Chaitin Omega: 0000001000000100000110001000011010001111110010111011101000010000"
Unfortunately, that is not a Chaitin Omega, since the notion of program length in that paper is flawed. In particular, it counts a 0 or a 1 provided as binary data in the program as adding 7 bits of length to the program. Later papers by Calude fix this problem, while reducing the number of computed bits to 43.
> Consider that great big set. It's still countable.
You didn't prove that statement, and actually Cantor's diagonal arguments [1] proves you wrong:
Consider the set of
«all real numbers which you can actually specify IN ANY FASHION AT ALL». If you can count it, you can order them in a certain fashion: n0, n1, n2, n3 …
It's easy to specify a number X which is not part of this set (which is in contradiction with the definition of this set, and demonstrates ad absurdum that this set is not countable). To build such X, consider the n-th decimal of the n-th number: if it's zero, the n-th decimal of X is 1, otherwise the n-th decimal is 0. Then by construction, X is different from any number of this set, which mean X is not a part of the set.
□
How do you guarantee that you can compute the n-th decimal of the n-th number in the list? In fact, from what I understand, this paper[1] shows how to specify exactly a number can't be computed like that.
Oh, you're right. For that I would need to explicit the construction of the sequence n0, n1, … but that's impossible : if I can find such sequence, my proof holds but at the same time such sequence shows that the set is countable => contradiction, hence there is no such sequence.
But well, now that we've proven that such sequence doesn't exist, we've proven that the given set is not countable !
Not the most elegant proof ever, but I think it works.
Actually it doesn't: what if the given sequence exists but cannot be expressed with words ? (This is exactly the same thing than «a real number that can't ne expressed with words» which is what I'm trying to demonstrate not to exist. I'm going nowhere)
What about phrases that specify some numbers on Thursdays, and another numbers when Moon is in second quarter?
What about phrases that some people agree specifies a number, while other people think it's another number, and yet another people just aren't sure?
What about phrases that specify one and the same number for a specific person, but once that person reached age of 40, then he/she is not sure anymore?
I just making counter-examples to Jules Richard paradox formulation, showing that it's not properly formalized, but rather expressed in vague language, so cannot be considered strict mathematics.
> What about phrases that specify some numbers on Thursdays, and another numbers when Moon is in second quarter?
Would you mind producing such a phrase?
> What about phrases that some people agree specifies a number, while other people think it's another number, and yet another people just aren't sure?
> What about phrases that specify one and the same number for a specific person, but once that person reached age of 40, then he/she is not sure anymore?
You may as well ask whether someone understanding a theorem has any bearing on its truth value. A well-formed sentence in a formal language has one and only one meaning regardless of whether a specific person understands it. That's the whole point of using a formal language. If meaning is actually somehow bound to its interpretation then communication of even the simplest of mathematics is totally impossible.
> These sentences describe functions, not numbers.
Yes, and if we think deeper about it, we'll find out that in mathematics we very often use functions in place for numbers, like it was the same thing. Simple example: √2. Even it form suggests that we apply function "square root" to number 2.
Less obvious example: π. It's also a relation between diameter and circumference.
If we go even deeper, trying to understand what, for example, number "2" means, we'll come up to their definition through functions (or classes) of equivalence of sets cardinalities (cardinal numbers), or order (ordinal numbers), where sets are defined using ZFC, for example.
I met this discrepancy while naively trying to program mathematical logic in college. In mathematics you just "take" a number, while in programming you "create" or "instantiate" a number, and it has a ton of consequences.
>> "One, if P=NP, zero otherwise".
> This value is a constant. It doesn't change through time. We simply don't know what it is.
Not necessarily a constant, as it's not proven to be a constant yet. As an example of how it may turn out to be non-constant, check out Continuum hypothesis proof (https://en.wikipedia.org/wiki/Continuum_hypothesis) : solution is independent of our axiom set. In other words -- we can state that ℵ1=c, or we can say ℵ1!=c, and both statements will not contradict anything else. We'll just get two different models, with one more axiom each.
> Yes, and if we think deeper about it, we'll find out that in mathematics we very often use functions in place for numbers, like it was the same thing. Simple example: √2
√ is a function. √2 is the result of applying √ to 2, which is a number. Since "how Giants scored last season" is a function, it can be applied to a value such as "2017-04-17", therefore "how Giants scored last season, and today is 2017-04-17" would be a number.
I don't follow your P=NP argument. Either P=NP or P!=NP. They can't both be simultaneously true. I also don't follow how the continuum hypothesis is relevant. = is not comparing the cardinalities of P and NP, it's asking whether they're the same set (i.e. all elements of P are in NP and vice versa). Again, either they're the same set or they're not. Even if there was a cardinality between that of N and that of R, I don't see how that would change how we compare sets for identity.
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
In all seriousness, I don't know that we know for sure that the former doesn't in fact exactly mean the latter.
Numbers are not physical things, they are symbols that are part of a system that has changed significantly over years. We can make a metaphorical link between a number and a measurement of our universe, but that does not mean that numbers are part of the physical world.
Think about a real number with digits: 0.a_1 a_2 a_3 ... in which a_i encodes whether Turing Machine with index i (or program i) halts or not. Since Halting problem is not computable, we will never be able to compute the digits of this real number. Never in our universe.
In the usual setting in which we do mathematics, this is a well-defined real number. That it's hard to determine its digits (even impossible when restricted to use an algorithm) is of no relevance.
However, you might be interested in the effective topos. This is an alternate mathematical universe in which one can't define construct this real number. It has a number of curious properties:
* The statement "1 + 1 = 2" is true in that topos, just like it is in the ordinary topos.
* The statement "any number is either prime or not prime" is true in that topos, just like it is in the ordinary topos, however its truth is not trivial.
* The statement "any real number is either equal to zero or not equal to zero" is false in that topos.
* The statement "any function is computable" is true in that topos (and wildly false in the standard topos).
* The statement "any function is continuous" is true in that topos (and wildly false in the standard topos).
Yes but you defined the number just fine, right?
"Compute" is a separate issue from "define", you can't compute a lot of things. Is this going in the intuitionistic logic direction?
Oh it's definitely more real, as in it belongs to R and not to any of its extensions. You guys realize that the definition of R is non-controversial in modern math, right? There are fringe theories like constructivist logic and other groups that reject all infinite constructions, but this is not the consensus view among practicing mathematicians...
The way you defined that number makes it a perfectly valid element of the set R, as described by, say, the axiomatic definition here:
Whether it's easy or hard or computationally intractable to compare it to other numbers, that's a totally different question unrelated to its definition.
Plus, you can actually empirically compute a finite set of initial digits (a specific Turing machine can be analyzed to see if it terminates or not), so you can compare this number with one that's constructed by flipping its digits, or with pi, etc.
In constructive mathematics, we don't reject "all infinite constructions". The only axiom which we don't generally assume is the axiom which says "any statement is either true or not true". (Note that we also do not use the counterfactual axiom "there is a statement which is neither true nor false". In fact, we're just agnostic on some truth values.)
In constructive mathematics, there is a perfectly well-defined set of real numbers. The usual diagonalization proof that this set is not a countable set applies.
I said "and other groups that reject all infinite constructions". Some schools of thought within that general intuitionist/constructivist/etc branch of mathematical logic do reject all infinite constructions:
https://en.wikipedia.org/wiki/Finitism
Either way, my point above was that this entire branch is not "mainstream math" by any means, AFAIK
I totally agree that mainstream mathematics doesn't have any problem whatsoever with infinite constructions and in fact embraces them.
I just wanted to clarify that intuitionistic and constructive mathematics don't have any problems with infinite constructions either. Finitism and ultrafinitism do, but they're not what's usually called "constructive mathematics".
There are at least three orthogonal axes which you can classify mathematical schools of thought in:
* Is the law of excluded middle accepted? ("Any statement is either true or not true.")
* Are infinite sets accepted? (They are not in finitism, but they are in constructive mathematics and of course in ordinary mathematics.)
* Can constructions implicitly refer to the result of what is being constructed? Is the powerclass of a set again a set? (Yes in ordinary mathematics and in constructive mathematics, no in predicative mathematics.)
> Plus, you can actually empirically compute a finite set of initial digits (a specific Turing machine can be analyzed to see if it terminates or not).
Well, the fact that the number is not computable means that there will exist an index i, for which you will not be able to compute a_i (no matter how hard you try). In other words, you will not be able to analyze the Turing Machine i, i.e. it will not be possible to prove the termination or non-termination of the Turing Machine i.
So this specific digit will be a mystery forever, and you would not be able to compare it to anything.
>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
It depends, as they say, on what the definition of is is. What does it mean for a number to exist if it cannot be described? What does it mean for a construction to exist if it cannot be constructed?
>if you confuse extant with useful you might end up believing that some random large integers aren't "there!"
The question isn't whether it's useful to claim the existence of uncomputable reals; the question is whether it's meaningful. A construction needn't be useful to be meaningful, but surely it must be meaningful to be useful.
The standard delta-epsilon type of reasoning crucially depends on existence of arbitrary reals. Many geometry proofs / lines of reasoning crucially depend on the ability to position a point on a line at an arbitrary distance from another point.
All numbers ever written are rationals and thus countable. But those endless irrational numbers make a lot of ways of reasoning simpler, or possible at all.
>How does analysis require the existence of uncomputable reals?
I don't know what the other person was referring to, but from many important theorems of analysis you can prove the existence of uncomputable reals. For example, from the theorem that a continuous function on a bounded interval is uniformly continuous you can prove that there exist uncomputable reals. If we think that this theorem and many more like it are necessary for analysis, we must conclude that analysis requires the existence of uncomputable reals.
The above fact comes from the area of mathematical logic known as reverse mathematics. The idea is in the name: rather than proving theorems from axioms, we go in reverse. We take some theorem and we see what axioms we can derive from it (using a weak background theory). This gives us a sense of the logical strength of a theorem; the stronger the axioms we can prove from it the stronger the theorem.
An amazing fact is that most classical theorems of mathematics have their logical strength captured by one of five sets of axioms. The weakest are those which are outright provable from the weak background theory (briefly, the background theory, called RCA_0, says that you're allowed to do computable processes). The next is the theory WKL_0, formed from RCA_0 by adding an additional axiom, which goes by the name weak Kőnig's lemma. This axiom states that any infinite binary tree has a cofinal branch (it's a lemma in 'ordinary' mathematics but in reverse mathematics gets treated as an axiom). This is the theory which captures the logical strength of the theorem I mentioned in the first paragraph. From weak Kőnig's lemma we can prove the existence of uncomputable reals. We can construct a computable infinite binary tree so that any branch through it must be uncomputable. From such a branch one can build an uncomputable real, say by insisting that the nth bit in its binary expansion is 1 if and only if the branch went left at level n.
(The easiest way to build such a tree is to go through the halting problem. The basic idea is to construct a tree of attempts to solve the halting problem. It turns out this can be done in a computable way, and any branch through the tree will allow us to solve the halting problem and thus must be uncomputable. The key fact used is that while there's no computable procedure to check whether a Turing machine halts at some time, you just want to know whether it halts in n steps (for fixed n), then you can check that with a computer. The program is easy: simply simulate running the Turing machine for n steps and see whether it stops at some point.
To build our tree: list the Turing machine programs as p_0, p_1, p_2, ... (This listing can be done in a computable way.) We will associate level n+1 on the tree with p_n. Start with a single node for the root of the tree. Then, to build the (n+1)-st level of the tree, look at each node t on the nth level. Check, in a computable way, whether p_0, p_1, ..., p_(n-1) halt within n steps. If none of them halt, then t has a left and right child at the next level. If p_k does halt within n steps, then look below t to see whether you took the left or right path at level k to get to the node t. If you took the right path, then t gets no child nodes. Otherwise, if you took the left path whenever p_k halts within n steps, then t gets two child nodes. Intuitively speaking, at stage n we keep our options open and don't decide whether p_n halts. But if we later see that it does halt, then we retroactively fix any mistake by not going any further along paths where we guessed wrongly before.
For example, it could be that p_0 halts but it takes 1000 steps for this to happen. So while building the tree when you get to level 1000 you'll suddenly stop building any further the entire right half of the tree, since you finally learned that p_0 halts and that you wanted to go left at the root node all along.
This process for generating the tree is computable, so by definition it's a computable tree. It's infinite, because you can keep building the tree upward if you happened to have guessed correctly whether each Turing machine so far halts. However, no branch through this tree can be computable. This is because a branch goes right at level n if and only if p_n didn't halt within k steps for any k. That is, the branch goes right at level n if and only if p_n doesn't halt, so from a branch we can decide the halting problem.)
----
Probably the best reference for reverse mathematics is Stephen Simpson's book Subsystems of Second-order Arithmetic. The first chapter is freely available on his website. It nicely lays out the philosophy behind the project and the major results while deferring (most of) the technical details to the rest of the book. http://www.personal.psu.edu/t20/sosoa/chapter1.pdf
You might be reading too much into the phrase "mathematical fantasies". This is not saying Borel claims they literally do not exist. More that they are not "accessible" or "usable".
"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely."
If that were so, might it not offer a way to an "explanation" for the Banach-Tarski paradox that even the likes of I could imagine I understood? - that duplicate volume you constructed is made from the same reals, specified differently!
Not that that would be an argument for the proposition quoted.
I was going to say that I am not a mathematician, but that would be redundant.
Banach-Tarski isn't a paradox. It's just what happens when you build geometric "shapes" out of nonmeasurable sets. In actual, real-world geometry, we almost always assume the set of points inside the shape/construction we care about is measurable, and that any geometric operations we perform must preserve measure.
Thanks - I have seen it mentioned in many places (often with "paradox" appended, e.g. [1]), but never with such a succinct commentary as you have given here.
>Besides, if you confuse extant with useful you might end up believing that some random large integers aren't "there!"
In addition to the points already made about the "all at once" claim, every integer (every rational number, even) can be given a finite description (just write it down).
The 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.
> The complex numbers come about by simply adding one dimension
Complex numbers are also best thought of (in my opinion) as an abstract completion of the reals under the operation of taking roots of polynomials.
Because otherwise, what would be the difference between the real plane and the complex numbers? They are topologically identical, after all. There has to be something more substantial than simply adding a dimension.
If by real plane you just mean the standard vector space R^2, the answer is algebraic rather than topological. R^2 as a vector space has an addition operation, but no multiplication operation. The complex plane is obtained by simply picking the appropriate multiplication operator.
In general, completing the rationals into the reals is more complex than constructing the complex plane from the real numbers. For the latter, you just need to adjoin a single element (sqrt(-1)), enforce existing arithmetic rules, and the rest falls into place. For the former, you can't just adjoin a single new element like sqrt(2). Doing so will get you the ring (actually field) Q[sqrt(2)], but not R.
If you take R and adjoin two special new elements (sqrt(-1) and the point at infinity), you do obtain a topologically different result: the Riemann sphere. This sphere is in many ways the more natural domain for complex analysis than the complex plane.
Complex numbers are (isomorphic to) a certain subset of linear operators on a 2-dimensional Euclidean vector space which rotate and/or scale the vectors in the plane without skewing or anisotropically stretching them. (We usually also include a zero operator here; depending on use case we sometimes omit it (the “punctured plane”), or sometimes add a point at infinity (the “Riemann sphere”).)
If you want you can write them down as matrices acting on vectors in an orthonormal basis by matrix multiplication:
[a -b]
[b a]
Or if you prefer you can consider i to be a unit bivector in the plane, with a complex a + bi acting on vectors by Clifford’s geometric product.
Or if you want you can write them using a length and an angle measure, and use high school trigonometry to figure out how they apply to vectors.
The difference between the real plane and the complex numbers is that for two vectors in the plane, the product is not a vector. (Indeed, if you use Clifford’s geometric product, then the product of two vectors is a complex number (scalar + bivector).)
Agreed. The jump from real to complex numbers is about as difficult to explain as the jump from natural numbers to integers. Integers are the numbers you need for "subtracting numbers gives you a number". Complex numbers are the numbers you need for "factoring polynomials with numeric coefficients gives you numeric roots". There's a similar argument for real numbers that you hinted at, but I don't understand it well enough to give a simple explanation for it, so I pretty much always hand-wave over it.
For reals, it's "Real numbers are the numbers you need to ensure every convergent sequence of rational numbers has a terminating point"
Convergence is determined in the "Cauchy" sense by having a vanishing distance between subsequent sequence entries, so as not to rely on the (potentially nonexistent) limit.
The continuity of real numbers provides a clean theoretical basis for continuity of functions.
Exactly. Real numbers are an invention to make mathematics simpler and cleaner. If you don't have continuous reals, it takes extensive case analysis to prove relatively simple things for, say, a binary floating point representation. It's possible to do that; Boyer and Moore did work like that to formalize floating point and check its correctness. (That work was funded by AMD, after the famous Pentium divide bug.)
Newtonian mechanics assumed that the physical universe is described by real numbers. But today, it seems that time, length, and mass are all quantized. There's thus not a physical basis for the existence of reals.
Maybe reals should be viewed merely as a useful convenience for analysis, not some fundamental part of mathematics.
"God created the integers. All the rest is the work of Man." - Kronecker
> But today, it seems that time, length, and mass are all quantized. There's thus not a physical basis for the existence of reals.
This is actually not the case. Space and time are not regarded as quantized by the majority of physicists and there's also absolutely no evidence suggesting that. To the contrary, such quantization would lead to inconsistencies – e.g. with relativity – pretty quickly.
An interesting idea that follows from this: what other kinds of "numbers" might we come up with if we relax our logical blinders?
I have this concept of "materialization" and wonder if there is a formal mathematical term for it. Complex numbers are actual, in the sense that they can be used in calculations that finally would give us a number we can make sense of (materialization), even if we cannot actually imagine a complex quantity.
In the same way, what if we invent a new class of real numbers that are quantized (such that there exists a smallest quantity x)?
What if we propose the existence an "imaginary" algorithm (call it omega, if you like) that can decide the halting problem?
These mathematical objects do not have to "exist" for them to be useful. I think there is a whole lot of new and interesting math that would be unlocked by dispensing with our logical blinders.
> What if we propose the existence an "imaginary" algorithm (call it omega, if you like) that can decide the halting problem?
Although our mathematical systems rely on the existence of computable procedures (that halt) to e.g indicate whether a formula is well formed or verify a proof, they have no trouble talking about uncomputable functions. There actually ends up being a hierarchy, as once you allow "solve the halting problem for this Turing machine" as an operation the expanded set of computable functions is now unable to solve its own halting problem: https://en.wikipedia.org/wiki/Arithmetical_hierarchy
> These mathematical objects do not have to "exist" for them to be useful. I think there is a whole lot of new and interesting math that would be unlocked by dispensing with our logical blinders.
These kind of things are actually pretty well studied. Some interesting examples are:
* The quaternions and octonions, which along with complex numbers are the only finite-dimensional algebras over the real numbers that can have "division" and "absolute value" in the usual sense: https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(compositi...
In general, mathematicians are really good at investigating things of the form "Okay we have structure X with properties A, B, C. What if we get rid of C? How about B? How about B and a weaker version of A? ...".
> What if we propose the existence an "imaginary" algorithm (call it omega, if you like) that can decide the halting problem?
This is called an oracle (https://en.wikipedia.org/wiki/Oracle_machine). We can posit oracles for solving computable problems in constant time (e.g. factoring the product of two arbitrarily large primes) as well as for solving uncomputable problems (e.g. halting problems).
Oracles are a great tool for studying complexity and computability, since oracle machines have their own complexity and computability limits; an oracle machine for the halting problem can determine whether a simple Turing machine will halt, but cannot determine whether it itself will halt (this is called a Turing jump). Thus oracle machines for halting problems form a class hierarchy, which Post's theorem shows is precisely the arithmetic hierarchy.
> An interesting idea that follows from this: what other kinds of "numbers" might we come up with if we relax our logical blinders?
It depends on exactly what you mean by this, but I would argue that we do this absolutely everywhere. For example, matrices have similar properties to numbers. You can add/subtract them, you can multiply them, you can (sometimes) divide them. Depending upon how you restrict your set of matrices, ab might equal ba or it might be the case that ab makes sense while ba does not (i.e. not only is it not commutative, but it doesn't even make sense to multiply the other way).
> In the same way, what if we invent a new class of real numbers that are quantized (such that there exists a smallest quantity x)?
Basically you're adding in a number smaller than everything else that squares to 0.
> What if we propose the existence an "imaginary" algorithm (call it omega, if you like) that can decide the halting problem?
I'm no logician, but I believe that they would refer to this sort of thing as "model theory". I.e. you're basically taking proofs (sequences of logical statements) and studying different logical systems in which these statements make sense. For example, you could extend your theory/model to take as an axiom that there exists an algorithm to solve the halting problem (though this would be stupid since you can already prove within most theorems that the halting problem is impossible...i.e. you would have a self-contradictory axiomatic system).
> These mathematical object do not have to "exist" for them to be useful. I think there is a whole lot of new and interesting math that would be unlocked by dispensing with our logical blinders.
I 100% agree. Math is a tool that we invent to better understand the world and our thoughts (as well as do more). However, you'll probably have to track down a true logician if you were to go down that rabbit hole...
This 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
Infinite values (e.g. the density of a black hole, the size of the universe) and infinitesimals (e.g. continuous space-time) are assumed to be real, rather than inaccurate but useful approximations.
Not a physicist but I thought the Planck length specifically denies infinitesimality? And one of the proposed solutions to the black hole information problem is that due to local relativistic effects, they never actually reach singularity in finite time.
The Planck length isn't a minimum length or size, and doesn't have a whole lot of real significance other than as it results from a particular choice of units.
Assumed to be real by whom? Not all physicists believe the same thing.
Also why is this unfortunate? Why does it matter if they do or do not "believe" it? Are they able to make useful predictions with their models? Are they able to better understand physics? If so, why worry about their personal beliefs?
If we want to put physics on the faith table, I'm cool with whatever anyone wants to believe, and more power to them. But you can't be objectively right on the faith table - that's the price of admission.
If we want to be "true" and fully rational then we need to try to accurately represent our degree of knowledge about the world. Thus we shouldn't be making strong statements about things with an absence of evidence.
I wouldn't say assuming infinite density of black holes involves an absence of evidence. It is a hypothesis made in advance of evidence, and it leads to specific predictions that should be falsifiable, and in that sense it's considered scientifically sound.
As for implications that can't be falsified, for example ones that (to butcher Douglas Adams) "rather involve being on the other side of the event horizon," scientists can and do feel free to disregard those. Contradictory assumptions by scientists do not imply that one of them is "wrong" unless their theories imply contradictory observable phenomena. In which case there is probably a fruitful experiment to be done.
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.).
>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.
One 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.
The 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!)
> "the halting probability Ω, which is irreducibly complex (algorithmically random), maximally unknowable, and dramatically illustrates the limits of reason"
"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."
This 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?
Free 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.
I 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.
Thats 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.
All 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.
"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.
Richard'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.
If one takes natural language as a means to definite sets, you quickly get a plethora of paradoxes, for example Russel's paradox("Takes the sets of all sets that don't contain themselves. Does that contain itself?"). So if one takes "natural language" as one's system of defining set, one has to assume it's inconsistent and any statement is provably true and false. Thus "Richard's Paradox" is in the same boat as all statements in our "system of natural language".
That said, I think the proof in the text falls apart in another way - it neglects the distinction made in Skolem's Paradox. The Löwenheim–Skolem [2] theorem show that any system definable with a finite alphabet has a countable model. This is only an apparent paradox because this model is only countable when "viewed" from outside the model, within the model, it is possible to have ostensibly uncountable sets. So ones could certainly haves a countable model of the real numbers while the real numbers themselves remained uncountable as "countable" and "uncountable" were defined in the countable model.
"If one takes natural language as a means to definite sets, you quickly get a plethora of paradoxes"
Which is why I think if you resort to natural language to refute the consequences of a more rigorous treatment of the concept of number then there will be trouble. The whole paper attempts to refute things like Cantor through a weird recourse to French.
However I think it is possible to reduce real numbers as being a sort of subset of French purely through the same construct that Mr Cantor describes because that's the way descriptions work. If you define a real in some way in some form of symbolic language - I recall from GED that SSS might embody "three" and so does "trois". So I don't see why French can't encompass reals SSS can be considered exactly equivalent to trois.
I suspect I need to know and understand the formal, rigorous definition of "real" before I really give it some.
I don't know what the rigorous definition of a real number is nor even what the rigorous definition of French (texts) is but I'm pretty sure it would possible to define "reals" in terms of "French texts" in such a way that all the results hold.
Funnily enough: 0,1,2,3,4,5,6,7,8,9 are French words along with . and ,
Not 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.
"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.
I 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.
But 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.
I'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.
I 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.
Let'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.
We 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.
General 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.
Do 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?
It 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.
I 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.)
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.
I'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.
Yes - 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.
Does 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.
What if the function takes input, and for each input, the program halts, but for every instruction, there is an input such that the program when run on that input will execute that instruction? Would that imply a contradiction? It doesn't seem like it to me. Yeah, no, that should be fine.
Example: consider a language where 0 means "if the input register is at most 0, halt with the answer "no", otherwise, decrement the input register and continue to the next instruction" and 1 means the same thing except that it gives the answer "yes" instead.
Then the fractional part of any real, expressed in binary, would be a program that takes an integer as input, and answers whether that digit of the number is 1. This seems like it would count as a program to me.
So, this would allow there to be uncountably many "programs", only, almost all of them would be impossible for us to refer to specifically.
Hey, this way you could define a uniform measure over programs. Hah.
Wait, is that how Solomonoff induction stuff works? (Of course, with a richer language than what I described)
Doesn'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?
> "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.
Is 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.
A 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.
I've taken a minor personal interest in some of Ramanujan's more quickly converging sequences for pi. Giving myself a crash course in computable analysis/computable reals. Wish there was more out there about them (particularly limitations vis a vis traditionally defined reals).
Agreed, I think that one of the consequences of the article is that computable numbers are generally a substitute for how most people think about the reals (e.g. the fundamental theorem of algebra is true for computable numbers extended with the square root of -1, not just complex numbers).
> "pi" and "e" - among the uncountably infinite transendentals - are probably reasonable responses to this article in it's entirety
pi and e are computable; we even know a bunch of programs which will compute them (to arbitrary precision; Google for "spigot algorithms"). Computable numbers are countable, so we can identify them with the natural numbers, and there's no need for "reals". For example, we might think about the tapes of a binary universal turing machine: we can enumerate them (empty; 0; 1; 00; 01; etc.), and hence the tapes/programs are isomorphic to the natural numbers; the computable numbers are the results of running those programs (this is easier to imagine with a separate "monotone" tape for writing "output").
What about the rest of the "uncountably infinite transcendentals"? I would claim that such an argument is too hand-wavey in the form you've presented it. Can you write down a uniquely determined transcendental number which is uncomputable?
Such numbers can be defined; Chaitin's "omega" is an example. However, it's difficult to decide whether such numbers "exist" or not: we can assume that the physical world "exists", by definition; things which can't exist physically, but can be explored mathematically, like pi and e, can be said to exist, although it's a leap of faith; uncomputable numbers can't really be explored mathematically, or in any way that we know. Can we still say they "exist"?
Consider that the digits of omega: whatever formal system you use to study it, the digits you "discover" are nothing more than a scrambled representation of your formal system's axioms; since the axioms are assumed rather than proven, the "discovery" is actually a circular argument.
> Whitehead & Russell's Principia Mathematica which (and please correct me) used logic to construct the integers
It's easy to construct the integers "using logic". For example, Peano constructed the naturals like this (more or less):
The symbol '0' is a natural
The symbol 'S', followed by a natural, is a natural
We can count through Peano's naturals as '0' (0), 'S0' (1), 'SS0' (2), 'SSS0' (3), etc.
We can likewise construct the integers like this:
A natural, followed by the symbol ':', followed by a natural, is an integer
We can interpret the first natural as positive, the second as negative, and the resulting integer as their sum. Hence '0:0' is 0 (0 + 0), and so is 'SSS0:SSS0' (+3 + -3); 'SSS0:0' is 3, '0:SSS0' is -3, and so on.
It's not too hard to construct something logically. The problem is what assumptions you're willing to make. Whitehead and Russell wanted to prove everything using only the assumptions of set theory; that's why it took them hundreds of pages to prove seemingly trivial theorems about integers.
Goedel's incompleteness theorem(s) showed that no set of assumptions is complete and consistent; hence proving some result using set theory is no better than proving it with some other system. In particular, if it's easier to construct a theory of integers (like the one above) than it is to derive one from set theory (like Whitehead and Russell), we might as well take the easy path :)
Note 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.
That's a much less profound statement, though, because you're talking about an implementation detail of "modern computers".
First of all, the set of values in the range [0,1] that can be represented by a computer is not at all the same as the set of rationals. If you're talking about fixed-point or floating-point data types, then there are only a finite number of bit patterns (corresponding to dyadic rationals), so obviously the probability of hitting one of them precisely is zero.
You may think I'm just being pedantic here, but I'm really not. The theorem you described is only interesting because there are an infinite number of rationals between zero and one, and yet the set still has measure zero. But if you're willing to say that any rational number is theoretically "representable" in a computer, then you're already assuming that we can go beyond the natively-available data types, and come up with our own representations -- e.g., a tuple of numerator and denominator. And in that case, we can also come up with representations of irrational numbers; for example, you could have a datatype that can finitely specify any algebraic number, such as sqrt(2) or the golden ratio. So the fact that a typical CPU happens to only be able to natively deal with rationals is irrelevant.
The beauty of the diagonalization argument described in this paper is that it applies to all possible discrete representations, no matter how weird or complex, without having to make any assumptions.
Is concept of "infinity" viable at all considering that number of states of our Universe is huge but finite. And so is any vocabulary for number forms - it is finite.
"probability of randomly picking a rational number from [0,1] is exactly 0."
If to consider that as a lemma only ...
You cannot and will never be able to prove that, right?
This is a mind-blowing. So rational numbers are an invention of humans. That is to say, natural numbers simply don't exist unless and until they are explicitly defined. Am I understanding correctly?
The answer to your question kinda gets into question-begging about what you mean about "invention". But, nominally, all numbers you know are inventions of humans. Exactly how well they correspond to the real universe is still an open problem. Obviously in many cases the answer is "well enough that you need not worry about the deviations very often", but things like "Is the universe continuous or discrete at the core, or some hybrid?" are open questions, which has direct implications on the question of how "real" any of the numbers are. Even the naturals may be quite artificial.
Someone else responded to me in a recent Hacker News discussion that really clarified this in my head: A real number essentially has an infinite number of digits after the decimal place - the difference between a rational and an irrational number is that the digits end up in a repeating pattern in a rational number (you can think of rationals that terminate as really having an infinite number of zeros, e.g. 1.5 as 1.5000000...). Thus, to randomly pick a real number, you randomly select digits an infinite number of times after the decimal. Clearly there is no way a random process will produce an infinite number of repeating digits, thus the probability of picking a rational would be 0.
Wouldn't it be simpler to explain it this way: there are infinitely many reals between any two different reals, and thus the theoretical probability of picking any number between them at random is 1/infinity, which we think if as 0.
But in that case, why is is more probable to pick an irrational number?
I don't think it's simpler that way at all, mainly because the ways in which some "infinities" are larger than others, which explains why it's more probable to pick an irrational number. The rational numbers are countable, while the real numbers are not.
I'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.
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.
Epsilon is a variable, not a number, so that wouldn't make sense. The probability is less than epsilon for all epsilon > 0. The only non-negative real (and we know the probability must be some non-negative real) satisfying that is 0.
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.
Richard'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.
Just 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.
A number is also a concept, and there are number systems that include infinities (e.g. 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.
Are 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.
It 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).
i 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
I 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.
Unrelated, 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.
This 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.
1. 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.
1-3: 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.
The 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.
This 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
Mathematical 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.
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-Gu...