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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#141

Post by kazagistar »

gertef wrote: post #81
FabHK wrote: post #32

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.

The encoding of all books written so far (and will ever be written in finite time), is a rational number. Don't need Reals

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.

Re: How real are real numbers? (2004)

#142

Post by surement »

throwaway5752 wrote: post #137

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 1…

> "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.

Re: How real are real numbers? (2004)

#143

Post by kazagistar »

dbcurtis wrote: post #26

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…

/me goes to wikipedia

> Theoretical significance

> There is currently no proven physical significance of the Planck length.

Re: How real are real numbers? (2004)

#144

Post by throwaway5752 »

surement wrote: post #142
throwaway5752 wrote: post #137

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 1…

> "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.

Re: How real are real numbers? (2004)

#145

Post by snarfy »

Smaug123 wrote: post #114
snarfy wrote: post #112

I tend to think of real numbers as a composite made of whole numbers and an operator.

Which operator?

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).

Re: How real are real numbers? (2004)

#147

Post by hawkice »

throwaway5752 wrote: post #144
surement wrote: post #142

Earlier quoted context omitted.

> "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…

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.

Re: How real are real numbers? (2004)

#148

Post by marplebot »

surement wrote: post #142
throwaway5752 wrote: post #137

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 1…

> "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.

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).

Re: How real are real numbers? (2004)

#149

Post by gerdesj »

joe_the_user wrote: post #125
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…

It depends how you conceptualize things. 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 "Richar…

"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.

Re: How real are real numbers? (2004)

#150

Post by cgh »

protonfish wrote: post #74
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…

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.

They may not be part of our physical world but that does not mean they don't exist independently:

https://plato.stanford.edu/entries/platonism-mathematics/#Ex...

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