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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#31

Post by upquark »

FabHK wrote: post #25
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…

There are more people on earth than, say, kings. That's true, even though I can't enumerate all non-kings, and even if the set of non-kings is somehow ill-defined.

Still not sure I follow. Set of non-kings is not ill-defined, not in the same way as set of numbers non-describable by any formal system is.

Re: How real are real numbers? (2004)

#32

Post by FabHK »

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.

Re: How real are real numbers? (2004)

#33

Post by surement »

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!)

Also the author of Meta Math! The Quest for Omega [0], where this topic is discussed at length.

[0] https://arxiv.org/abs/math/0404335

Re: How real are real numbers? (2004)

#34

Post by pacman128 »

maverick_iceman wrote: post #16
Smaug123 wrote: post #10

Earlier quoted context omitted.

But in our universe, if it's discrete on a macroscopic level, then it's either inherently a natural number (it can be counted), or it's intelligently created.

Our best theories, General Relativity and the Standard Model, say that the world is a continuum.

Do they really say this? They assume this just as Newtonian mechanics does. And like Newtonian mechanics give reasonable answers in the domains they describe. How would they change if there was a cutoff at some infinitesimal scale?

Re: How real are real numbers? (2004)

#35

Post by klodolph »

maverick_iceman wrote: post #16
Smaug123 wrote: post #10

Earlier quoted context omitted.

But in our universe, if it's discrete on a macroscopic level, then it's either inherently a natural number (it can be counted), or it's intelligently created.

Our best theories, General Relativity and the Standard Model, say that the world is a continuum.

I'm a little uncomfortable with the language that the theories "say that the world is" X. General Relativity and the Standard Model both model the world using real numbers, but they're both known to be wrong, and the fact that they are continuous is not a great reason to claim that the universe is continuous.

On the other hand, observations about Lorentz symmetry holding at distances on the order of the Planck scale put a wrench in a bunch of discrete spacetime theories. I don't really understand the math, though.

All this is somewhat tangential to the issue of real numbers. Real numbers are not necessary for continuity.

Re: How real are real numbers? (2004)

#36

Post by zodiac »

Smaug123 wrote: post #13
upquark wrote: post #12

Earlier quoted context omitted.

"Real numbers that cannot be defined with language" is not a well-defined set though. Just like "integers that cannot be described in less than 100 words" is not well-defined (I could reach a contradiction by pointing to the "smallest integer that cannot be described in less than 100 words").

A sentence which fails to specify a real uniquely… fails to specify a real uniquely. Borel's talking about numbers which can be defined uniquely: that is, picked out, identified. Your Berry-paradox description doesn't identify an integer.

But if we assume that we can say a sentence either specifies a real number uniquely or does not, the Berry paradox number is uniquely specified.

Re: How real are real numbers? (2004)

#39

Post by AnimalMuppet »

maverick_iceman wrote: post #16
Smaug123 wrote: post #10

Earlier quoted context omitted.

But in our universe, if it's discrete on a macroscopic level, then it's either inherently a natural number (it can be counted), or it's intelligently created.

Our best theories, General Relativity and the Standard Model, say that the world is a continuum.

I'm not sure that I'd agree with you on the Standard Model. To me, it's view of the universe is pretty discrete.

(That is, presuming that you're speaking of the particle physics Standard Model, not something in cosmology.)

Re: How real are real numbers? (2004)

#40

Post by CogitoCogito »

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 whereas the real numbers come about from an abstract "completion" of (say) the rational numbers in a very specific mathematical sense.

My point is that deriding "imaginary numbers" (as many do) is total nonsense if you accept real numbers.

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