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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#191

Post by psyc »

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

In all seriousness, I don't know that we know for sure that the former doesn't in fact exactly mean the latter.

Re: How real are real numbers? (2004)

#192

Post by psyc »

tejon wrote: post #172
CuriouslyC wrote: post #154

Earlier quoted context omitted.

Infinite values (e.g. the density of a black hole, the size of the universe) and infinitesimals (e.g. continuous space-time) are assumed to be real, rather than inaccurate but useful approximations.

Not a physicist but I thought the Planck length specifically denies infinitesimality? And one of the proposed solutions to the black hole information problem is that due to local relativistic effects, they never actually reach singularity in finite time.

The Planck length isn't a minimum length or size, and doesn't have a whole lot of real significance other than as it results from a particular choice of units.

Re: How real are real numbers? (2004)

#193

Post by hzhou321 »

jonreem wrote: post #187
hzhou321 wrote: post #181

Earlier quoted context omitted.

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…

Why a pre-definable function matters here? We have allowed a real number (that cannot be exhaustively described) in, so why a real function (that you cannot exhaustively define) has to be excluded? Isn't it unfair that I am only allowed to use a finitely definable function while you can choose a non-pre-finitely-describable real number? Isn't that a loop proof -- that the set of real numbers is uncountable simply because it is defined to be uncountable, and it is larger infinity than that of natural numbers simply it is defined to be bigger (as what bigger means in the context of infinity is otherwise undefined).

Re: How real are real numbers? (2004)

#194

Post by hzhou321 »

cvoss wrote: post #186
hzhou321 wrote: post #181

Earlier quoted context omitted.

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…

Since it is non-terminating, you didn't really prove every item gets counted after finite time, did you?

Refer to my other reply, you asserted a requirement of predefined (describable) counting scheme here. Why that requirement has any relevance here (in the context of infinity)?

Re: How real are real numbers? (2004)

#195

Post by drdeca »

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…

> Since you never can finish counting

Not what countable means. The meaning meant is the one about having a map from the set to the integers, where no two things of the set map to the same integer.

The correspondence is meant as either a full thing, or at least a well defined specification which could be applied to any of the things, if you want to get philosophical about ontology or something.

Also, the specification can't depend on things like, what reals have been given as input "previously".

Say that in this game, you have two obligations:

1: for any n,m I give, you must tell me what the mth binary digit of the real number associated with n is, in the mapping you are considering (or, if there is no real number associated with that natural number.).

2: For any finite collection of the binary digits of the real number I am choosing, you must tell me the first natural number such that the corresponding real number matches all the digits I specified.

With these rules, I can choose my real number (specifying more and more digits) such that for any natural number, I will be able to show that my real number doesn't correspond to that natural number, or any lower one.

Therefore, there is no natural number that corresponds to my number in the matching system you are providing.

(To win, I repeat this procedure:

Ask what the first natural not ruled out already by my specification of my number is. (Call that n)

Ask what the nth digits of the real associated with n is.

Inform you that the nth digit of my number turns out to have the other value for its nth digit, so no natural less than or equal to n corresponds to my number.

Repeat.

This will only ever tell you more about my number, and will not result in me changing my mind about any of the digits of my number, yet there is no natural number which won't ever be ruled out as potentially corresponding to my number.

Therefore, no natural corresponds to my number in your system.

So I win.)

Re: How real are real numbers? (2004)

#196

Post by hn_throwaway_99 »

prmph wrote: post #164
hn_throwaway_99 wrote: post #122

Earlier quoted context omitted.

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?

I don't think it's simpler that way at all, mainly because the ways in which some "infinities" are larger than others, which explains why it's more probable to pick an irrational number. The rational numbers are countable, while the real numbers are not.

Re: How real are real numbers? (2004)

#197

Post by hzhou321 »

drdeca wrote: post #195
hzhou321 wrote: post #181

Earlier quoted context omitted.

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…

> Since you never can finish counting Not what countable means. The meaning meant is the one about having a map from the set to the integers, where no two things of the set map to the same integer. The correspondence is meant as either a full thing, or at least a well defined specification which could be applied to any of the things, if you want to get philosophical about ontology or something. Also, the specificatio…

Why should you get to choose your real number -- specifying more and more digits -- implies an un-exhaustive process, while I am not allowed to do the same? An unfair game will have unfair winners, how would it mean anything? If we both are allowed an un-exhaustive process of specifying what we have, this goes back to the counting game. As we never can finish, how does it make countable (or not)?

Re: How real are real numbers? (2004)

#198

Post by [deleted] »

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…

[deleted]

Re: How real are real numbers? (2004)

#199

Post by jawarner »

D_Alex wrote: post #189
bo1024 wrote: post #185

Earlier quoted context omitted.

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.

Does there have to be a generating program? The set of all finite programs is countable, so it could not possibly describe the uncountable set of real numbers - this would be a surjection from a countable set to an uncountable set. In particular only the subset of computable numbers [1] can be described by the countable set of finite computer programs.

Also, any infinite program either passes through a finite number of instructions or never terminates. So any program which does not have a finite representation will never terminate.

[1] https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf

Re: How real are real numbers? (2004)

#200

Post by sirclueless »

CuriouslyC wrote: post #171
CogitoCogito wrote: post #160

Earlier quoted context omitted.

Assumed to be real by whom? Not all physicists believe the same thing. Also why is this unfortunate? Why does it matter if they do or do not "believe" it? Are they able to make useful predictions with their models? Are they able to better understand physics? If so, why worry about their personal beliefs?

If we want to put physics on the faith table, I'm cool with whatever anyone wants to believe, and more power to them. But you can't be objectively right on the faith table - that's the price of admission. If we want to be "true" and fully rational then we need to try to accurately represent our degree of knowledge about the world. Thus we shouldn't be making strong statements about things with an absence of evidence.

I wouldn't say assuming infinite density of black holes involves an absence of evidence. It is a hypothesis made in advance of evidence, and it leads to specific predictions that should be falsifiable, and in that sense it's considered scientifically sound.

As for implications that can't be falsified, for example ones that (to butcher Douglas Adams) "rather involve being on the other side of the event horizon," scientists can and do feel free to disregard those. Contradictory assumptions by scientists do not imply that one of them is "wrong" unless their theories imply contradictory observable phenomena. In which case there is probably a fruitful experiment to be done.

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