[HN Gopher] How real are real numbers? (2004)
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How real are real numbers? (2004)
Author : MindGods
Score : 50 points
Date : 2020-08-02 16:53 UTC (6 hours ago)
(HTM) web link (arxiv.org)
(TXT) w3m dump (arxiv.org)
| karmakaze wrote:
| > We discuss mathematical and physical arguments against
| continuity and in favor of discreteness
|
| I can appreciate the physical arguments which makes me think of
| this as a question in physics. What might the mathematical ones
| be? Math is about defining a system and playing it out, what
| would make continuous numbers off-limits?
| karmakaze wrote:
| [I'll assume the downvote was a misclick and that math/physics
| hasn't also become political.]
| sebastialonso wrote:
| I don't understand. You didn't say anything remotely
| political
| ojnabieoot wrote:
| The article goes into significant detail, but the problem is
| that the _computable_ real numbers are a tiny slice of all real
| numbers, and in general even in principle mathematicians can
| only deal with a countable subset of the reals. So
| (metamathematically) if almost every single real can 't
| actually be used in mathematics, it's hard to say the concept
| has much mathematical validity. [Which makes its mathematical
| _usefulness_ a philosophical pickle.]
|
| > Math is about defining a system and playing it out
|
| A lot of mathematicians would consider this a very limited view
| of the subject.
| contravariant wrote:
| >A lot of mathematicians would consider this a very limited
| view of the subject.
|
| I'm not sure it's as limited as you think it is.
|
| It's not very specific though I'll give you that.
| asdf_snar wrote:
| > A lot of mathematicians would consider this a very limited
| view of the subject.
|
| Could you provide an example of such a mathematician?
| tomtomtom777 wrote:
| > in general even in principle mathematicians can only deal
| with a countable subset of the reals. So (metamathematically)
| if almost every single real can't actually be used in
| mathematics, it's hard to say the concept has much
| mathematical validity.
|
| Let me give it try:
|
| > let R be the set of real numbers. Let x be an element of R.
|
| There it is, I've dealt with and used every single real
| number.
|
| The notion that reals are only real when they are
| individually described (Borel's credo in the article) is an
| odd one. Sure, almost every real number is a "Mathematical
| fantasy" as the author calls it, but that doesn't mean
| Mathematicians can't use them or deal with them.
|
| Mathematics _excels_ in dealing with imaginary things. To
| extent Borel 's credo to question the Mathematical validity
| of these Mathematical fantasies seems to defy the entire idea
| of Mathematics.
| fractionalhare wrote:
| _> A lot of mathematicians would consider this a very limited
| view of the subject._
|
| Maybe? A lot of mathematicians actually do describe it just
| like that.
|
| In fact that's one of the most common ways the difference
| between mechanical arithmetic and research mathematics is
| described on forums like r/math. And a lot of posts on Math
| Overflow have precisely the flavor of defining things and
| then playing around with what consequences emerge.
| montecarl wrote:
| If it is not possible to compute most reals, then is there a
| number system, with only computable numbers that extends the
| rational numbers to include computable reals such as Pi, e,
| sqrt(2)?
| andromaton wrote:
| Yes, computable numbers (0)
| 0:https://en.m.wikipedia.org/wiki/Computable_number
| [deleted]
| rurban wrote:
| So the answer is unexpected: real enough.
| dmch-1 wrote:
| Real numbers are a convenient abstraction which help us think
| about certain things. They don't have to be any more real than
| that. (Have not read the article though, sorry if this comment is
| irrelevant).
| Smaug123 wrote:
| I know it's completely unrelated to the article, but I would like
| to take a moment to plug an absolutely beautiful proof of the
| uncountability of [0,1], which I first saw in
| http://people.math.gatech.edu/~mbaker/pdf/realgame.pdf .
|
| Briefly: Alice and Bob play a game. Alice starts at 0, Bob starts
| at 1, and they alternate taking turns (starting with Alice), each
| picking a number between Alice and Bob's current numbers. (So
| start with A:0, B:1, then A:0.5, B:0.75, A:0.6, ... is an example
| of the start of a valid sequence of plays.) We fix a subset S of
| [0,1] in advance, and Alice will win if at the end of all time
| the sequence of numbers she has picked converges to a number in
| S; Bob wins otherwise. (Alice's sequence does converge: it's
| increasing and bounded above by 1.)
|
| It's obvious that if S = [0,1] then Alice wins no matter what
| strategy either of them uses: a convergent sequence drawn from
| [0,1] must converge to something in [0,1].
|
| Also, if S = (s1, s2, ...) is countable then Bob has a winning
| strategy: at move n, pick s_n if possible, and otherwise make any
| legal move. (Think for a couple of minutes to see why this is
| true: if Bob couldn't pick s_n at time n, then either Alice has
| already picked a number bigger, in which case she can't ever get
| back down near s_n again, or Bob has already picked a number b
| which is smaller, in which case she is blocked off from reaching
| s_n because she can't get past b.)
|
| So if [0,1] is countable then Alice must win no matter what
| either of them does, but Bob has a winning strategy;
| contradiction.
| saiojd wrote:
| I don't follow why Bob can play any legal move when s_n is not
| possible. The proposition is that, if S is countable, Bob has a
| winning strategy. Suppose S = (s1, s2, s3), which is finite and
| therefore countable. On turn 1 Alice picks s1. Since s1 is
| already taken, Bob picks any legal move, which means any number
| in [0, 1]. Suppose he picks s3. Then, Alice picks s2 and wins?
| lmkg wrote:
| They must continue playing. There is no termination condition
| for the game. They always continue playing an infinite number
| of steps.
|
| Alice picks s2. Then Bob picks some number less than s3, so
| the sequence does not converge to s3. Then Alice picks some
| number larger than s2, so the sequence does not converge to
| s2. Any number that is picked after a finite number of steps,
| cannot be the value that the sequence converges to.
| saiojd wrote:
| Ok, that makes more sense. The part I still don't get is
| how we can pick a number x such that s2 < x < s3 without
| directly assuming that [0, 1] is uncountable.
| adamnemecek wrote:
| Proofs by contradiction are not accepted in some flavors of
| mathematics, like constructive mathematics. In fact,
| constructive mathematicians believe in a definition of real
| numbers that is more in line with what this paper talks about.
| millimeterman wrote:
| I'm not familiar with this specific proof, but reading the
| description it sounds perfectly constructive.
|
| A true proof by contradiction would be "Assume [0, 1] is not
| uncountable. That leads to a contradiction, so [0, 1] is not
| not uncountable. By the law of the excluded middle, [0, 1] is
| therefore uncountable."
|
| This proof seems to be "Assume [0, 1] is countable. That
| leads to a contradiction, so [0, 1] is not countable." That's
| not a proof by contradiction - it's just how you prove a
| negative.
|
| This is a nice blog post about the distinction:
| https://existentialtype.wordpress.com/2017/03/04/a-proof-
| by-...
| fractionalhare wrote:
| Nice comment. A succinct explanation of the difference
| between negation and contradiction can also be found in
| this r/math comment: https://reddit.com/r/math/comments/cu9
| gsk/_/exse1ps/?context...
|
| The terminology gets abused a lot, even by professors. It's
| actually hard to find theorems which really require proof
| by contradiction.
| a1369209993 wrote:
| > [0, 1] is not not uncountable. By the law of the excluded
| middle, [0, 1] is therefore uncountable.
|
| Nitpick: that's not excluded middle, that's double negation
| elimination. They're both non-constructive and (IIRC)
| equivalent in most formalizations, but they're not the same
| thing.
| ogogmad wrote:
| Proofs that derive a contradiction are perfectly acceptable
| in constructive mathematics. Indeed, the real numbers are
| uncountable in constructive mathematics. What is not allowed
| is elimination of double negatives, which is not present in
| the above proof.
| adamnemecek wrote:
| I'm not sure that real numbers are uncountable in
| constructive mathematics as they are not computable.
| ogogmad wrote:
| See Andrej Bauer's proof that they are indeed
| uncountable:
| https://mathoverflow.net/questions/30643/are-real-
| numbers-co...
| adamnemecek wrote:
| I'm still unconvinced.
| fractionalhare wrote:
| Is there a specific issue you have with this proof? With
| respect, I think you may be conflating a constructive
| proof of the uncountability of the reals with a proof
| that the set of all computable reals is countable. These
| are two different claims.
|
| A statement about the reals which is provable under
| constructive mathematics does not reduce to a statement
| about the computable reals and vice versa. A set can be
| constructively definable without all of its elements
| being constructively enumerable.
| ogogmad wrote:
| Given any explicit sequence of real numbers, it's always
| possible to generate a number not in that sequence. In
| fact, this can be done algorithmically using essentially
| Cantor's method.
| millimeterman wrote:
| A simple diagonalization argument is perfectly
| constructive.
| fractionalhare wrote:
| Yes, and to clarify what I predict might be a point of
| contention about the necessity of "contradiction" in
| Cantor's argument, some of the comments in this thread
| might be helpful: https://reddit.com/r/math/comments/70xe
| on/why_is_cantors_dia...
| Davidzheng wrote:
| Ok I think what happens here is that we assume for the real
| numbers an increasing sequence which is bounded converges. I
| think that a number system with this property is probably
| already hard to construct in constructive math.
| Davidzheng wrote:
| So the problem that constructive math has for real numbers
| is that it's hard to show real numbers exist at all, not
| that it's uncountable if you have its properties. Also the
| deeper point is that you have all the properties of the
| real numbers you can actually derive the law of excluded
| middle. (Coming from the supremum axiom)
| adamnemecek wrote:
| This is really interesting. I'm not surprised by axiom of
| choice being based on real numbers. I've been thinking
| about that lately but I don't have a good intuition why.
| Do you have a link?
| ogogmad wrote:
| > So the problem that constructive math has for real
| numbers is that it's hard to show real numbers exist at
| all
|
| No. They exist in any topos with a Natural Number Object.
| In other words, they exist in all varieties of
| constructive mathematics that admit a set of natural
| numbers.
| ogogmad wrote:
| No. The real numbers are constructed as located Dedekind
| cuts, or modulated Cauchy sequences. There are no issues
| with this construction in constructive mathematics.
|
| You are indeed right that you can't show that an increasing
| and bounded sequence of real numbers has a limit. But you
| can prove constructively that if a sequence of real numbers
| is _Cauchy_ then it has a limit. And it is that last
| property which is the defining property of the real numbers
| (Cauchy completeness).
| achillesheels wrote:
| I wonder if this further demonstrates the reality of the
| complex domain and the anachronism of the real number line? In
| that, numbers are "circularly" or ultimately geometrically
| countable and not sequentially?
| ogogmad wrote:
| The complex numbers are also uncountable.
| fractionalhare wrote:
| It does not demonstrate that, no. Complex numbers are no more
| "real" than real numbers are. They are uncountable, as
| another commenter said. In general they share many properties
| with real numbers because the sets C and R^2 are isomorphic
| and isometric with respect to each other.
| avani wrote:
| In a similar vein, Cantor's original diagonalization proof of
| uncountability is one of the most satisfying results in real
| analysis. From
| https://www.cs.virginia.edu/luther/blog/posts/124.html:
|
| "Any real number can be represented as an integer followed by a
| decimal point and an infinite sequence of digits. Let's ignore
| the integer part for now and only consider real numbers between
| 0 and 1. Now we need to show that all pairings of infinite
| sequences of digits to integers of necessity leaves out some
| infinite sequences of digits.
|
| Let's say our candidate pairing maps positive integer i to real
| number r_i. Let's also denote the digit in position i of a real
| number x as x_i. Thus, if one of our pairings was (17,
| 0.651249324...) then r_17^4 would be 2. Now, consider the
| special number z, where z_i is the bottom digit of r_i^i + 1.
|
| The number z above is a real number between 0 and 1 and is not
| paired with any positive integer. Since we can construct such a
| z for any pairing, we know that every pairing has at least one
| number not in it. Thus, the lists aren't the same size, meaning
| that the list of real numbers must be bigger than the list of
| integers."
| kmill wrote:
| Before the diagonalization argument, Cantor had another proof
| of the uncountability of the reals. Looking it up again, it's
| actually pretty much the Alice-Bob game!
|
| See theorem 2: https://www.maa.org/sites/default/files/pdf/up
| load_library/2...
|
| (To see the correspondence, it's helpful to think of limits
| as being a kind of game. You claim the limit is a particular
| value, another party challenges you with an epsilon, and then
| you respond to the challenge by saying an index in the
| sequence after which all the entries are within epsilon of
| the claimed limit.)
| [deleted]
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