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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#161

Post by MrMoenty »

j2kun wrote: post #94
xyzzyz wrote: post #4

Earlier quoted context omitted.

only that in a single symbolic system we can't have expressions for all of them at once. I don't think that's true. There are only countably many different symbolic systems[1], and as we can only express countably many numbers in each, we don't leave the realm of countable. [1] - A "symbolic system" must at least come with a procedure to tell whether a sequence is a part of it, and, unless you disbelieve the Church-T…

Doesn't this (and by extension, Curch-Turing) rely on the assumption that the universe is discrete? Provided it were not, and one were able to harness infinite precision, one could presumably make a new symbolic system based on it. Not saying I believe it, just teasing out assumptions. If one is arguing whether the universe is continuous and using the Church-Turing thesis as justification for something, there's a dan…

You are right, this reasoning depends on only allowing formal systems with finitely many symbols. Turing himself actually gave justification for this in a beautiful footnote in his 1936 paper [1]:

page 249:

> I shall also suppose that the number of symbols which may be printed is finite. If we were to allow an infinity of symbols, then there would be symbols differing to an arbitrarily small extent.

And then in the footnote:

> If we regard a symbol as literally printed on a square we may suppose that the square is 0 [1] https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf

Re: How real are real numbers? (2004)

#162

Post by prmph »

CogitoCogito wrote: post #159
prmph wrote: post #151

Earlier quoted context omitted.

An interesting idea that follows from this: what other kinds of "numbers" might we come up with if we relax our logical blinders? I have this concept of "materialization" and wonder if there is a formal mathematical term for it. Complex numbers are actual, in the sense that they can be used in calculations that finally would give us a number we can make sense of (materialization), even if we cannot actually imagine a…

> An interesting idea that follows from this: what other kinds of "numbers" might we come up with if we relax our logical blinders? It depends on exactly what you mean by this, but I would argue that we do this absolutely everywhere. For example, matrices have similar properties to numbers. You can add/subtract them, you can multiply them, you can (sometimes) divide them. Depending upon how you restrict your set of m…

Started reading about dual numbers after reading your reply; interesting concept

Re: How real are real numbers? (2004)

#163

Post by marcosdumay »

j2kun wrote: post #94
xyzzyz wrote: post #4

Earlier quoted context omitted.

only that in a single symbolic system we can't have expressions for all of them at once. I don't think that's true. There are only countably many different symbolic systems[1], and as we can only express countably many numbers in each, we don't leave the realm of countable. [1] - A "symbolic system" must at least come with a procedure to tell whether a sequence is a part of it, and, unless you disbelieve the Church-T…

Doesn't this (and by extension, Curch-Turing) rely on the assumption that the universe is discrete? Provided it were not, and one were able to harness infinite precision, one could presumably make a new symbolic system based on it. Not saying I believe it, just teasing out assumptions. If one is arguing whether the universe is continuous and using the Church-Turing thesis as justification for something, there's a dan…

Physics would have something to say about infinite precision.

But yes, with those assumptions, it does not hold. It is not as much that the argument is circular, but that the only tech we know how to use (symbols) limits our expressiveness.

Re: How real are real numbers? (2004)

#164

Post by prmph »

hn_throwaway_99 wrote: post #122
textfile wrote: post #65

Earlier quoted context omitted.

This is a mind-blowing. So rational numbers are an invention of humans. That is to say, natural numbers simply don't exist unless and until they are explicitly defined. Am I understanding correctly?

Someone else responded to me in a recent Hacker News discussion that really clarified this in my head: A real number essentially has an infinite number of digits after the decimal place - the difference between a rational and an irrational number is that the digits end up in a repeating pattern in a rational number (you can think of rationals that terminate as really having an infinite number of zeros, e.g. 1.5 as 1.…

Wouldn't it be simpler to explain it this way: there are infinitely many reals between any two different reals, and thus the theoretical probability of picking any number between them at random is 1/infinity, which we think if as 0.

But in that case, why is is more probable to pick an irrational number?

Re: How real are real numbers? (2004)

#165

Post by sjolsen »

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…

>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 write expressions for

It depends, as they say, on what the definition of is is. What does it mean for a number to exist if it cannot be described? What does it mean for a construction to exist if it cannot be constructed?

>if you confuse extant with useful you might end up believing that some random large integers aren't "there!"

The question isn't whether it's useful to claim the existence of uncomputable reals; the question is whether it's meaningful. A construction needn't be useful to be meaningful, but surely it must be meaningful to be useful.

Re: How real are real numbers? (2004)

#166

Post by alexbecker »

7373737373 wrote: post #59
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.

Epsilon is a variable, not a number, so that wouldn't make sense. The probability is less than epsilon for all epsilon > 0. The only non-negative real (and we know the probability must be some non-negative real) satisfying that is 0.

Re: How real are real numbers? (2004)

#167

Post by alexbecker »

CuriouslyC wrote: post #120
gertef wrote: post #80

Earlier quoted context omitted.

The measure of epsilon is 0. Proof: what else could it be? If it's not 0, there's a smaller number, contradicting your (intuitive) definition of epsilon.

Wouldn't it be more accurate to describe epsilon as an infinitesimal?

There are no infinitesimals in the reals. You can define a mathematical structure containing infinitesimals [0], but it's not the reals.

[0] Specifically, the dual numbers: https://en.wikipedia.org/wiki/Dual_number

Re: How real are real numbers? (2004)

#168

Post by nine_k »

sjolsen wrote: post #165
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…

>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 write expressions for It depends, as they say, on what the definition of is is. What does it mean for a number to exist if it cannot be described? What does it mean for a construction to exist if it cannot be constructed? >if you confuse extant with useful you might end up believing…

The standard delta-epsilon type of reasoning crucially depends on existence of arbitrary reals. Many geometry proofs / lines of reasoning crucially depend on the ability to position a point on a line at an arbitrary distance from another point.

All numbers ever written are rationals and thus countable. But those endless irrational numbers make a lot of ways of reasoning simpler, or possible at all.

Re: How real are real numbers? (2004)

#170

Post by maverick_iceman »

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…

>Real numbers are not necessary for continuity.

You know of any continuum that doesn't include the real numbers? That will contradict the continuum hypothesis.

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