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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#181

Post by hzhou321 »

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…

I am not a mathematician (physicist). I think the concept of infinity is a con that mathematicians have pulled on us (as there isn't an easy reality to map on to). I can understand arbitrarily big set; however, I never managed to make the jump from arbitrarily finite to infinity. Mathematicians made that jump and glossed over, then continue to show the difference between countable infinity and infinity beyond. Since I never could make that jump, it all sounds nonsense to me.

I have a similar (and I believe to be equivalent) problems with infinitesimal as well. How does arbitrarily small but non-zero become infinitesimal? Since I have problem with infinitesimal, I find the differentiation of real numbers and rational numbers equally non-sensical.

So mathematician, take a pause, could you explain what is countable infinity? Since you never can finish counting (all the natural numbers), how does it make the set countable? What do we mean exactly by countable here?

Wikipedia refers to the idea of one-to-one correspondence. But since you can never exhaust the correspondence, what do we mean by one-to-one? Give me any unique real, I'll give you a unique natural number, and we can go on forever, so how does that not count as one-to-one correspondence?

Unlike mathematicians, physicists are fine with unresolved :)

PS: I guess countable can be defined as there is a definite way of ordering the set, which is true for natural numbers but questionable for real numbers. I still don't see how the ordering connects to the size of infinity and one to one correspondence. Even for the set of real numbers, I can have an algorithm continuously generate random numbers (discarding re-occurring ones so it will be a unique sequence) and prove there is an order of the set (non-exhaustively defined, same as the set of natural number). The ordering may not be describable though. But non-describable ordering is still an ordering, right? Just as a real number that cannot be exhaustively described is still a number. I don't have to describe it, I can hand-wave it just as the way mathematicians hand-waved the infinity.

Re: How real are real numbers? (2004)

#183

Post by klodolph »

maverick_iceman wrote: post #170
klodolph wrote: post #35

Earlier quoted context omitted.

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.

Let's avoid equivocating here: "the continuum" is sometimes used to refer to the real numbers, but "continuity" in this context is a property of functions between metric spaces (or possibly topological spaces). "The continuum hypothesis" and "continuous functions" are actually from completely different branches of mathematics.

This happens fairly often in mathematics, where similar-sounding terms are used to describe completely different concepts, or the same term sometimes means different things in context, or sometimes an Adjective Noun is neither described by Adjective nor by Noun.

Re: How real are real numbers? (2004)

#184

Post by D_Alex »

Infinity is a weird thing, isn't it?

Now: one of the proofs in the paper relied on an assumption that all possible computer programs are countable, which I think implies that they are finite in length. But it is fairly trivial to generate computer programs that are infinitely long, say by assigning characters or expressions in some language to the digits of transcendental numbers such as pi. It is also possible to generate infinitely many such programs, simply by using pi/2, pi/3... etc.

Now, the proof as presented fails, since these programs cannot be ordered by size.

Can the proof be modified to take account of this? I don't know... comments invited.

Re: How real are real numbers? (2004)

#185

Post by bo1024 »

D_Alex wrote: post #184

Infinity is a weird thing, isn't it? Now: one of the proofs in the paper relied on an assumption that all possible computer programs are countable, which I think implies that they are finite in length. But it is fairly trivial to generate computer programs that are infinitely long, say by assigning characters or expressions in some language to the digits of transcendental numbers such as pi. It is also possible to ge…

I'm not sure it's so easy to define or generate an infinitely-long program. Such a thing doesn't sound to me like it would be either possible in practice or equivalent to a Turing Machine in theory.

For example, you suggest an assignment of expressions to digits of pi. Now how would you run such a program? Presumably by generating the digits of pi, interpreting them as expressions, and evaluating the expressions, etc.

But the program you used to do that was finite. So are you running the infinite program? I think it's more fair to say you are running the finite one.

Re: How real are real numbers? (2004)

#186

Post by cvoss »

hzhou321 wrote: post #181
mbid wrote: post #45

Earlier quoted context omitted.

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…

I am not a mathematician (physicist). I think the concept of infinity is a con that mathematicians have pulled on us (as there isn't an easy reality to map on to). I can understand arbitrarily big set; however, I never managed to make the jump from arbitrarily finite to infinity. Mathematicians made that jump and glossed over, then continue to show the difference between countable infinity and infinity beyond. Since…

I find it helpful to think of countability in terms of the following game: You have a set of items in mind. You propose a (non-terminating) scheme for listing all the items in the set. An adversary attempts to name any item X, hoping your scheme misses it. However, you then show your scheme does, in fact, get to X after a _finite_ amount of time. The set is said to be countable if you prove that your adversary cannot win this game.

Yes, the counting process is non-terminating. But every item gets counted after only finite time.

Re: How real are real numbers? (2004)

#187

Post by jonreem »

hzhou321 wrote: post #181
mbid wrote: post #45

Earlier quoted context omitted.

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…

I am not a mathematician (physicist). I think the concept of infinity is a con that mathematicians have pulled on us (as there isn't an easy reality to map on to). I can understand arbitrarily big set; however, I never managed to make the jump from arbitrarily finite to infinity. Mathematicians made that jump and glossed over, then continue to show the difference between countable infinity and infinity beyond. Since…

A set being "countably infinite" only means that you can write a function that maps each distinct entry in the set to exactly one natural number (0, 1, 2, etc.) without duplicates. That's it.

So for example, the set of natural numbers is countably infinite and we know this because we can write a function that maps each natural number to exactly one natural number: the id function.

We can extend this and say that the set of even natural numbers is countably infinite because it has a mapping function of x => x / 2.

The same is true for all integers (natural numbers + negative numbers): x => if (x You can even write a function that maps all rational numbers to the natural numbers, since each rational number can be written as a fraction of two integers. (Figuring out the function is a fun exercise but it is also easy to google)

However, you can't write a function that maps any real number to a natural number. The easiest to understand proof of this is Cantor's Diagonal Argument[0], which is a proof by contradiction that shows that any attempted function must exclude some real numbers. Therefore, the real numbers are not countably infinite, and we call them uncountably infinite.

EDIT: In response to your edit, Cantor's Diagonal Argument basically shows that for any given function (and you have to define the function completely ahead of time - that's key) I can give you a real number that is not included in the domain of your function.

[0]: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

Re: How real are real numbers? (2004)

#188

Post by lngnmn »

Numbers? Real?

Neither molecular biology nor the sane part of physics has any of em.

Btw, it is heuristic - if there are numbers involved then it is human made. Reality as it is has no such notion. Biology does not count.

Numbers require an observer, which is a by-product of the processes in vastly complex brain structures of the cortex, and cannot be the basis of anything in the underlying universe.

Any good (which means Eastern) philosophy arrived at these simple conclusions millennia ago.

Re: How real are real numbers? (2004)

#189

Post by D_Alex »

bo1024 wrote: post #185
D_Alex wrote: post #184

Infinity is a weird thing, isn't it? Now: one of the proofs in the paper relied on an assumption that all possible computer programs are countable, which I think implies that they are finite in length. But it is fairly trivial to generate computer programs that are infinitely long, say by assigning characters or expressions in some language to the digits of transcendental numbers such as pi. It is also possible to ge…

I'm not sure it's so easy to define or generate an infinitely-long program. Such a thing doesn't sound to me like it would be either possible in practice or equivalent to a Turing Machine in theory. For example, you suggest an assignment of expressions to digits of pi. Now how would you run such a program? Presumably by generating the digits of pi, interpreting them as expressions, and evaluating the expressions, etc…

Yes - I was just thinking the same thing. I have not come to a conclusion one way or another on whether it is possible to de-couple the generating program from the generated programs for the purpose of analysing the proof. It seems that there should be a way to do it... unfortunately I cannot spend more time on this now.

Re: How real are real numbers? (2004)

#190

Post by psyc »

Are they real? Well, when a human says something about a thing, they can never be completely sure whether they're saying something about an actual thing with an independent existence, or whether they're only saying something about what they say.
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