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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#91

Post by c-smile »

option wrote: post #55

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.

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?

Re: How real are real numbers? (2004)

#92

Post by upquark »

IngoBlechschmid wrote: post #88
upquark wrote: post #66

Earlier quoted context omitted.

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…

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…

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

Re: How real are real numbers? (2004)

#93

Post by Smaug123 »

btilly wrote: post #58
mbid wrote: post #45

Earlier quoted context omitted.

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…

Yeah, I'm a trained mathematician as well. 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.

Are you happy with the stronger "there is no surjection from any set to its power set"?

Re: How real are real numbers? (2004)

#94

Post by j2kun »

xyzzyz wrote: post #4
whatshisface wrote: post #3

"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…

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-T…

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?

Re: How real are real numbers? (2004)

#95

Post by [deleted] »

mbid wrote: post #45
btilly wrote: post #30

Earlier quoted context omitted.

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…

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…

[deleted]

Re: How real are real numbers? (2004)

#96

Post by upquark »

enugu wrote: post #56
upquark wrote: post #21

Earlier quoted context omitted.

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 la…

> In my view, every real number is well-defined... How so? The set of definable numbers in any formal langauage might not be clear concept. But you are making a stronger statement. For any given language, like for instance ZFC, we can say that definable numbers are a countable subset. Hence measure zero.

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.

For instance, Chaitin's constant is a perfectly well-defined number, albeit uncomputable by construction: https://en.wikipedia.org/wiki/Chaitin%27s_constant

Re: How real are real numbers? (2004)

#97

Post by j2kun »

CogitoCogito wrote: post #40

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

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.

Re: How real are real numbers? (2004)

#98

Post by j2kun »

ThrustVectoring wrote: post #84
CogitoCogito wrote: post #40

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.…

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…

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.

Re: How real are real numbers? (2004)

#99

Post by mannykannot »

whatshisface wrote: post #3

"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…

"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.

Re: How real are real numbers? (2004)

#100

Post by j2kun »

Sharlin wrote: post #28

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!)

Despite how much I enjoyed the article, I was a bit dismayed that he gave no mention of Kolmogorov.
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