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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#41

Post by petercerno »

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…

Think about a real number with digits: 0.a_1 a_2 a_3 ... in which a_i encodes whether Turing Machine with index i (or program i) halts or not. Since Halting problem is not computable, we will never be able to compute the digits of this real number. Never in our universe.

Re: How real are real numbers? (2004)

#42

Post by AnimalMuppet »

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…

1-3: You can define a (mathematical) real number that cannot be interpreted as a position in the universe that can be physically realized, taking the Planck Length into account.

4: At the precision of the Planck Length, you have two integers that are the closest physically-meaningful values whose ratio approximates pi.

In both cases, you're confusing mathematical abstractions with what is physically realizable in a discrete system. You can't (meaningfully) do that.

Re: How real are real numbers? (2004)

#43

Post by upquark »

petercerno wrote: post #41
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…

Think about a real number with digits: 0.a_1 a_2 a_3 ... in which a_i encodes whether Turing Machine with index i (or program i) halts or not. Since Halting problem is not computable, we will never be able to compute the digits of this real number. Never in our universe.

Yes but you defined the number just fine, right? "Compute" is a separate issue from "define", you can't compute a lot of things. Is this going in the intuitionistic logic direction?

Re: How real are real numbers? (2004)

#44

Post by scythe »

>In addition to this mathematical soul-searching regarding real numbers, some physicists are beginning to suspect that the physical universe is actually discrete [Smolin, 2000] and perhaps even a giant computer [Fredkin, 2004, Wolfram, 2002]. It will be interesting to see how far this so-called “digital philosophy,” “digital physics” viewpoint can be taken.

Here is how far: Everything written in words about the physical universe is, by necessity, discrete. Thus all information that can be encoded in human languages is discrete. Any non-discrete behavior of the physical universe which causes a change in the discrete information available to us, must, by assumption, have a component which is orthogonal to all of the prior discrete information (otherwise it is fully discrete). Since this component is independent of all previously available information, it looks like randomness.

In other words: from the viewpoint of a discrete (linguistic) observer, the behavior of a continuous universe looks identical to that of a discrete universe that contains random fluctuations.

What is interesting, then, is that observationally, our discrete observable universe is full of random fluctuations. Speculation as to their true continuous underpinnings is, however, unfalsifiable, unless the randomness itself can be made to disappear. I usually turn the question around: is it inconceivable that there would be a continuous universe with inhabitants that used a discrete language?

So:

>According to these ideas the amount of information in any physical system is bounded

"the amount of observable information in any physical system" -- any unobservable continuous information shows up as unpredictable changes in the observable information.

Re: How real are real numbers? (2004)

#45

Post by mbid »

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…

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 in M.

Re: How real are real numbers? (2004)

#46

Post by xigency »

enugu wrote: post #17
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 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?

Re: How real are real numbers? (2004)

#47

Post by eatitraw »

upquark wrote: post #43
petercerno wrote: post #41

Earlier quoted context omitted.

Think about a real number with digits: 0.a_1 a_2 a_3 ... in which a_i encodes whether Turing Machine with index i (or program i) halts or not. Since Halting problem is not computable, we will never be able to compute the digits of this real number. Never in our universe.

Yes but you defined the number just fine, right? "Compute" is a separate issue from "define", you can't compute a lot of things. Is this going in the intuitionistic logic direction?

Even though you defined the number, you can't really do usual stuff with it: e.g. you can't compare it to other numbers.

IMO, these kind of numbers are no more "real" than infinitesimals: https://en.wikipedia.org/wiki/Hyperreal_number

Re: How real are real numbers? (2004)

#48

Post by enugu »

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…

Yes, when you restrict to computable numbers, then there are countably many reals. Countable in classical sense. You wont have an computable enumeration, again of because of diagonalization.

Re: How real are real numbers? (2004)

#49

Post by ska »

klodolph wrote: post #35
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 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…

All models are wrong. Some models are useful.

Some of them extremely useful :)

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