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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#51

Post by mcherm »

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…

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 choice of computer and programming language) -- which is so ornery a number that we cannot (even in THEORY) figure out a single digit of it (other than its being between 0 and 1).

Consider that great big set. It's still countable. The thing that makes "real numbers" bigger than integers, the "extra" that real numbers have must be very, very peculiar.

Re: How real are real numbers? (2004)

#52

Post by klodolph »

AnimalMuppet wrote: post #39
maverick_iceman wrote: post #16

Earlier quoted context omitted.

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.)

The Standard Model has discrete energy levels but continuous spacetime.

Re: How real are real numbers? (2004)

#53

Post by mcherm »

upquark wrote: post #31
FabHK wrote: post #25

Earlier quoted context omitted.

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.

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

Um... roughly the same. Is Robert Mugabe a king? Since we didn't give a clear and precise definition of "king" you can't really say.

Re: How real are real numbers? (2004)

#54

Post by ska »

mrcactu5 wrote: post #37

i think it's no coincidence Godel's proof of uncountable reals, comes around the same time as Lebesgue integration. As mathematicians started exploring what the serious use of Fourier series

I think you are mixing up Godel (incompleteness of axiomatic systems) and Cantor (uncountable reals).

Timing wise Cantors proof was done the year before Lebesque was born; Cantor was a generation before Lebesgue, and Fourier a generation before that iirc. Godel's is a generation younger than Lebesgue. Lesbesgue and Borel were working at the same time - Godel was very young when he published his incompleteness theorem in the 1930s.

Re: How real are real numbers? (2004)

#56

Post by enugu »

upquark wrote: post #21
FabHK wrote: post #14

Earlier quoted context omitted.

True, but the argument that one set is larger still works.

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.

Re: How real are real numbers? (2004)

#57

Post by hammock »

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.…

This makes sense. It feels analogous to economics (and economic models) - something fake we make up to be able to create functions and theories and so on

Re: How real are real numbers? (2004)

#58

Post by btilly »

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…

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.

Re: How real are real numbers? (2004)

#59

Post by 7373737373 »

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.

Really? Interesting. I would have expected it to be something like epsilon.

Re: How real are real numbers? (2004)

#60

Post by enugu »

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.

We have the holographic principle from physics which says that the amount of information in a finite volume of space is discrete.

If true, then we can think of the continuity just as a convenient interface. All experimental questions can be answered by a simulation with a discrete state space.

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