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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#101

Post by E6300 »

deepsun wrote: post #67
mcherm wrote: post #51

Earlier quoted context omitted.

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…

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 m…

> 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.

Re: How real are real numbers? (2004)

#102

Post by IngoBlechschmid »

upquark wrote: post #92
IngoBlechschmid wrote: post #88

Earlier quoted context omitted.

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

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.)

Re: How real are real numbers? (2004)

#103

Post by JacksonGariety »

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…

Why do you suppose there is a difference between things and the names we give them? What sort of basis is there for that?

EDIT: How do you know that a large number is "there" if you can't count to it? What would it mean for a large number to "be there."

Being-there is something special. ;-)

Re: How real are real numbers? (2004)

#104

Post by mbid »

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.

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.

If you don't trust me, maybe you'll trust wikipedia: https://en.wikipedia.org/wiki/Constructivism_(mathematics)#C....

Re: How real are real numbers? (2004)

#105

Post by eli_gottlieb »

mannykannot wrote: post #99
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 propo…

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.

Re: How real are real numbers? (2004)

#106

Post by gjm11 »

Smaug123 wrote: post #18
enugu wrote: post #17

Earlier quoted context omitted.

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 infi…

Nitpick: the infinite countable union of countable well-ordered sets is countable. This statement is immune to the failure of Choice.

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.)

Re: How real are real numbers? (2004)

#108

Post by xyzzyz »

j2kun wrote: post #94
xyzzyz wrote: post #4

Earlier quoted context omitted.

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 dan…

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.

[1] - http://perso.ens-lyon.fr/yassine.hamoudi/wp-content/uploads/...

Re: How real are real numbers? (2004)

#109

Post by mannykannot »

eli_gottlieb wrote: post #105
mannykannot wrote: post #99

Earlier quoted context omitted.

"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 propo…

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.

[1] http://mathworld.wolfram.com/Banach-TarskiParadox.html

Re: How real are real numbers? (2004)

#110

Post by nightcracker »

xyzzyz wrote: post #108
j2kun wrote: post #94

Earlier quoted context omitted.

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 dan…

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 eve…

The Church-Turing hypothesis only applies to functions that apply on discrete data. If the data itself is also continuous it does not apply.
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