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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#131

Post by TezlaKoil »

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'm a trained mathematician as well.

Mayhap.

>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.

Re: How real are real numbers? (2004)

#132

Post by TheOtherHobbes »

xigency wrote: post #46
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…

What about a continuum of symbolic logic systems?

How do you find the edges?

Re: How real are real numbers? (2004)

#133

Post by smsm42 »

btilly wrote: post #30
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…

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…

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?

Re: How real are real numbers? (2004)

#134

Post by chadcmulligan »

j2kun wrote: post #98
ThrustVectoring wrote: post #84

Earlier quoted context omitted.

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.

I thought you could have calculus using 'computable numbers' that is reals that have a rule. It's been on my list for a while to look into it - there's some notes here for the curious http://math.stackexchange.com/questions/963061/can-the-set-o...

Re: How real are real numbers? (2004)

#135

Post by gerdesj »

"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.

Re: How real are real numbers? (2004)

#136

Post by kazagistar »

upquark wrote: post #119
enugu wrote: post #111

Earlier quoted context omitted.

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…

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.

Re: How real are real numbers? (2004)

#137

Post by throwaway5752 »

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?

Re: How real are real numbers? (2004)

#138

Post by TheOtherHobbes »

stcredzero wrote: post #127
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…

I think Borel has confused names with things. Pedantic positivist?

Or simply a pragmatist.

Reals are a useful sensory abstraction. Countability is another sensory abstraction, although for most people it's a rather less useful one.

There's no particular obligation for sensory abstractions to be logically watertight - if only because logic is an abstraction itself.

Re: How real are real numbers? (2004)

#139

Post by gerdesj »

bo1024 wrote: post #126
gerdesj wrote: post #121

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, sp…

> 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. Nope, because he used the diagonalization technique to make sure the new text was not in the original set, which supposedly contained all texts.

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 ,

So, how do we proceed to define reals in French?

Re: How real are real numbers? (2004)

#140

Post by gerdesj »

surement wrote: post #124
gerdesj wrote: post #121

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, sp…

> I think that the set of possible French texts is exactly as large as the set of reals. How? The set of French texts is countable and the set of reals is uncountable.

0,1,2 etc are French words. All reals are French words 8)
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