Useless Math

For the discussion of math. Duh.

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Re: Useless Math

Postby Yakk » Thu Dec 16, 2010 5:58 pm UTC

Yes? Well, there are real numbers for whom knowing the distribution is sufficient to make it computably enumerable (like 0.1111111...)

See:
https://secure.wikimedia.org/wikipedia/ ... erable_set

Ie, for a set to be enumerable, you have to be able to recognize an element of the set reliably. (you don't need to be able to recognize elements of the not-set reliably)

For a sequence to be enumerable, you need to be able to recognize (index, element) pairs reliably.

Computationally recursive sets (or sequences) are sets (sequences) for whom you can reliably recognize an element, and recognize a non-element, of the set.

I suspect that many computationally enumerable decimal expansions are recursive, as you can just test (x,0) through (x,9) in parallel and generate the guaranteed halt even if your question is "is this not the digit". This leads me to believe that my definition of "enumerable sequence" to be off, or it to not have much meaning... (even with unbounded 'digits', you can just go forever until you find the actual element. This halts if the set is enumerable, if after an indeterminate period of time...)
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Re: Useless Math

Postby doogly » Thu Dec 16, 2010 6:08 pm UTC

Yakk wrote:Ie, for a set to be enumerable, you have to be able to recognize an element of the set reliably. (you don't need to be able to recognize elements of the not-set reliably)

So let's say that is the rule, each digit is 0-9 with some distribution. You give me a candidate number. I analyze the moments of the distribution. I can now tell it if it fits that rule, so it seems like for any distribution, these actually are inumerable.

It seems more like non-enumerable numbers are just ones with a vague description. You can't enumerate the number I'm thinking of now, but just because I'm not telling you anything about it. Has nothing to do with the property of the number. If I tell you how I chose it, you can choose it too. (it was 7, by the way).

If it is really inherently non-enumerable, then how are either of us talking about it? Maybe the problem is with a dialectical model of math thinking that I do.
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Re: Useless Math

Postby WarDaft » Thu Dec 16, 2010 6:11 pm UTC

Computable enumerable reals are not all computable reals.

When enumerating a real, you give up knowing how accurate you are in exchange for the knowledge that, after sufficient yet undecidable amounts of time, you will eventually know each digit.

Enumerating Turing machines do not halt.

I could give examples of Turing machines that do and do not enumerate numbers (not the actual state tables, that would be large and hard to read), and indeed, people have even thought up non-enumerable reals, but I can't give a description that does them justice.
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Re: Useless Math

Postby Yakk » Thu Dec 16, 2010 6:40 pm UTC

doogly wrote:
Yakk wrote:Ie, for a set to be enumerable, you have to be able to recognize an element of the set reliably. (you don't need to be able to recognize elements of the not-set reliably)

So let's say that is the rule, each digit is 0-9 with some distribution. You give me a candidate number. I analyze the moments of the distribution. I can now tell it if it fits that rule, so it seems like for any distribution, these actually are inumerable.
The number being talked about was actually a specific number, not any element of the distribution.

That can tell you if it belongs to a set described by a distribution. It cannot pick a specific number.
It seems more like non-enumerable numbers are just ones with a vague description. You can't enumerate the number I'm thinking of now, but just because I'm not telling you anything about it. Has nothing to do with the property of the number. If I tell you how I chose it, you can choose it too. (it was 7, by the way).

If it is really inherently non-enumerable, then how are either of us talking about it? Maybe the problem is with a dialectical model of math thinking that I do.

If by "vague description" you mean "you couldn't recognize it if you had the digits written out", then yes, non-enumerable numbers have a "vague description".

Because computational enumerability is about recognizing elements. If it is easy to recognize elements reliably (easy as in possible with some fixed algorithm), then it is computationally enumerable.

The point of this is, of course, that the reals have "uncountable" elements. Meanwhile, the cardinality of the computationally enumerable reals is no more than the cardinality of Turing Machines, which is countable. So "almost all" real numbers cannot be recognized, or their decimal digits generated, by any algorithm.

There is at least one interesting member of the set of non-enumerable numbers:
https://secure.wikimedia.org/wikipedia/ ... s_constant
which happens to have a short program you can create whose output "converges" to it, yet is still non-enumerable.

Sometimes I think the universe is just messing with us.
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Re: Useless Math

Postby WarDaft » Thu Dec 16, 2010 6:55 pm UTC

Yakk wrote:which happens to have a short program you can create whose output "converges" to it, yet is still non-enumerable.

Sometimes I think the universe is just messing with us.

Perhaps we're using a difference in terminology, because I've usually seen 'computably enumerable real' used to describe a real whose digits are enumerated by a not-necessarily halting Turing machine, rather than the set of reals for which there exist halting Turing machines that compute representations. Under this definition, Chaitin's Constant is a computably enumerable real, but not a computable real.

It was this larger definition I intended to mean in my post, as if we did not exclude Chaitin's constant, we would have more useful numbers in our set than otherwise necessary.
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Re: Useless Math

Postby Yakk » Thu Dec 16, 2010 7:08 pm UTC

The problem with Chaitin's constant's enumeration is that you never, ever, every can know if a given digit has stabilized yet. (well, you can for the first few digits, but you can prove that you cannot beyond that).

So it isn't really "enumerating" the value as much as it is "converging" in a non-computational sense to the value (as if you let it run "to infinity", it would reach Chaitin's constant). We can just never know if it is finished doing a given digit (because knowing if it is finished with a digit is equivalent to knowing if a certain class of Turing machines halt -- this can be known for some classes, but eventually it cannot be known.)

In fact there is this result "For each specific consistent effectively represented axiomatic system for the natural numbers, such as Peano arithmetic, there exists a constant N such that no bit of Ω after the Nth can be proven to be one or zero within that system." -- ie, your very axiom system runs out of juice at some point, and you can no longer prove if the Turing Machine that is generating the digits of Chaitin's constant has finished editing a digit.

When you talk about a non-halting Turing machine outputting an enumeration, you generally mean that it writes it down and then is guaranteed to not back up and erase it later.

On the other hand, you could take this as "the axiom system is overly vague". ;)
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Re: Useless Math

Postby WarDaft » Thu Dec 16, 2010 7:14 pm UTC

Well, yes, that's sort of the point really. It's an exact formulation of a way to, essentially, be extremely vague about a number. This admits a lot more numbers to be described in your system, and so a lot more otherwise useful numbers not in the Real compliment of that set, thus making a much less useful set of real numbers. That's exactly why I mentioned it.
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Re: Useless Math

Postby pizzazz » Fri Dec 17, 2010 12:25 am UTC

Ok, I think I got these enumerable v. non-enumerable numbers down. Just to check, is this correct: Some reals, such as e and pi, can be approximated with various formulae (such as the Gregory-Leibniz series), and for these formulae, given K steps or applications, we can calculate the error, and so know that the first N digits are correct, and for any N, there exists a finite K that provides an estimate accurate to N. These reals are enumerable.

For some reals, there exists no such method. Any approximation will have unknown error, and so we cannot tell how many digits are correct. These reals are non-enumerable.

Is that right?

[I'll post thread-relevant comments once this is confirmed].

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Re: Useless Math

Postby WarDaft » Fri Dec 17, 2010 1:09 am UTC

Perhaps by some definitions, but that was not the set I was referring to when I brought it up.

The set of numbers I was referring to, was the set that specifically does not have a known method for calculating the error. All you know is, eventually, you will get every digit right. You don't know how long that will take for any given digit, in fact, it can be more difficult to know a digit is correct than it is to solve the halting problem. Furthermore, the enumeration methods do not actually halt. If you want an approximation, you actually interrupt the calculation before it finishes, and you cannot be sure how close it is to the limit value.

We take the set of real numbers for which there does not exist one of these seemingly useless functions that will tend towards the value after an undecidable amount of time.
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Re: Useless Math

Postby firechicago » Fri Dec 17, 2010 1:16 am UTC

pizzazz wrote:Ok, I think I got these enumerable v. non-enumerable numbers down. Just to check, is this correct: Some reals, such as e and pi, can be approximated with various formulae (such as the Gregory-Leibniz series), and for these formulae, given K steps or applications, we can calculate the error, and so know that the first N digits are correct, and for any N, there exists a finite K that provides an estimate accurate to N. These reals are enumerable.

For some reals, there exists no such method. Any approximation will have unknown error, and so we cannot tell how many digits are correct. These reals are non-enumerable.

Is that right?

[I'll post thread-relevant comments once this is confirmed].


There's another important difference, which is that for reals like e and pi not only can we estimate the error in our approximation, but given an arbitrarily large amount of computing power we can make that error arbitrarily small. For example, if I tell you that my nonenumerable number begins with .123456789, it's trivial to construct a program which will "calculate" that number to within 10^-9, but you can't make the error any smaller than that. Similarly, with Chaitin's constant, after a certain point you can never know if the nth digit of the binary is correct, and so you can never approximate it to within 2^-n.

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Re: Useless Math

Postby ImTestingSleeping » Fri Dec 17, 2010 2:44 am UTC

I would think that mathematics in higher dimensions would be rather useless. Would I be wrong with that statement?

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Re: Useless Math

Postby the tree » Fri Dec 17, 2010 2:48 am UTC

ImTestingSleeping wrote:I would think that mathematics in higher dimensions would be rather useless. Would I be wrong with that statement?
Pretty wrong, yeah. For instance, I'm doing coding theory at the moment, which is largely in an arbitrary amount of dimensions.

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Re: Useless Math

Postby skullturf » Fri Dec 17, 2010 2:53 am UTC

I remember when I was learning more about mathematics, being almost disappointed at how mundane (in a sense) higher dimensions really are. To most mathematicians, having more than three dimensions just means you have more than three coordinates. That's pretty much it. (We can't necessarily visualize it, but we just go by analogy.)

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Re: Useless Math

Postby pizzazz » Fri Dec 17, 2010 4:52 am UTC

firechicago wrote:
pizzazz wrote:Ok, I think I got these enumerable v. non-enumerable numbers down. Just to check, is this correct: Some reals, such as e and pi, can be approximated with various formulae (such as the Gregory-Leibniz series), and for these formulae, given K steps or applications, we can calculate the error, and so know that the first N digits are correct, and for any N, there exists a finite K that provides an estimate accurate to N. These reals are enumerable.

For some reals, there exists no such method. Any approximation will have unknown error, and so we cannot tell how many digits are correct. These reals are non-enumerable.

Is that right?

[I'll post thread-relevant comments once this is confirmed].


There's another important difference, which is that for reals like e and pi not only can we estimate the error in our approximation, but given an arbitrarily large amount of computing power we can make that error arbitrarily small. For example, if I tell you that my nonenumerable number begins with .123456789, it's trivial to construct a program which will "calculate" that number to within 10^-9, but you can't make the error any smaller than that. Similarly, with Chaitin's constant, after a certain point you can never know if the nth digit of the binary is correct, and so you can never approximate it to within 2^-n.


Ah, ok, thanks.

I'm just guessing, but since these numbers were defined in this thread with Turing machines, is there some way they could be connected to the limits of computers?

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Re: Useless Math

Postby z4lis » Fri Dec 17, 2010 4:59 am UTC

ImTestingSleeping wrote:I would think that mathematics in higher dimensions would be rather useless. Would I be wrong with that statement?


Tons of math, abstract and applied, use large numbers of dimensions. For instance, a way of solving a linear ODE of degree n is to convert it to a system of ODEs with n variables. And matricies would be rather useless if we couldn't make them any larger than 3 by 3. :P
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Re: Useless Math

Postby antonfire » Fri Dec 17, 2010 3:13 pm UTC

WarDaft wrote:
antonfire wrote:Chaitin's constant is an example, for the same sorts of reasons that the halting function is uncomputable.
Quite to the contrary. Chaitin's constant is ennumerable, but not computable.
My mistake.

Still, it's not terribly hard to come up with an explicit example of a real number which is not computably enumerable.

For instance, instead of talking about the halting probability, say, talk about the probability that a random Turing machine gives you a computably enumerable number. (That is, essentially, whether each position of the tape is eventually constant.) I'm pretty sure this works.

And even if it doesn't, Cantor's diagonal argument is constructive. Given a list of real numbers, it gives you an explicit number which is not on that list.
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Re: Useless Math

Postby charonme » Tue Dec 21, 2010 9:17 am UTC

Atmosck wrote:what are some examples of pieces of mathematics that have no known application to the empirical world

I believe your trouble lies in your lack of a precise definition of "an application to the empirical world". It couldn't be too vague because once you give an example, you prove about it that it has an application - namely to write your paper.

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Re: Useless Math

Postby EstLladon » Tue Dec 21, 2010 10:53 am UTC

Math doesn´t study the ¨real world¨. It actually studies how our minds work. It studies consistent systems. Any conceivable consistent systems. It studies how to organize our thoughts. Not all of them, but some part that we call ¨rational thoughts¨. Even physics stopped caring about real world some time ago. It stopped caring about how world actually works. Now it cares about totally different question - ¨How can we describe real world effects with our limited minds?¨. I mean what does wave-particle dualism even mean? It means that in some cases the equations that we came up with to describe our made up ¨ideal particles¨ work better to describe real world effects, and in some cases it is equations that describe our made up ¨ideal waves¨ that work better. It does not say anything about actual nature of light. I would even say that this nature is totally ununderstandable to our minds. Math also studies limitations of our minds - we know that there are and always will be things we have no way to calculate or even understand. And these limitations are interesting by themselves. And coming up with new mathematical structures with interesting properties is good - since all we can use to try describe real world is these structures we never know beforehand what structures we will need. Also they help to look at world from different perspective, which helps a lot.

In short I think that the Idea that we can fully understand something real is patently ridiculous. World is big and we are small. But it is useful to try to describe it. Because the more we describe the more we can do.

And we can be sure that we never run out of things to describe and invent.
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Re: Useless Math

Postby alphachap » Tue Dec 21, 2010 9:47 pm UTC

I don't know if I agree or not with your thesis.
But I do have an example: the discovery of non-Euclidean geometries.
At the time, many were blaming mathematicians for inventing useless things having nothing to do with reality.
Non-Euclidean geometries were a prime example of such a useless and fanciful construct, useless since the real world was of course euclidean and could not even in principle be otherwise (many philosophers has said so).
But it was actually the other way around, to the utter surprise of many.
Soon after it was realised that the actual geometry of space is not euclidean, that mass distort space, that light bends around stars, that space and time can not be separated (there is no universal time) but form a 4-dimensional non-euclidean continuum, that spinning stars and black-holes twist space (frame dragging).

(Around 1830, the Hungarian mathematician János Bolyai and the Russian mathematician Nikolai Ivanovich Lobachevsky separately published treatises on hyperbolic geometry.)

http://en.wikipedia.org/wiki/Non-Euclidean_geometry
http://en.wikipedia.org/wiki/Hyperbolic_geometry
http://en.wikipedia.org/wiki/Frame_dragging
http://en.wikipedia.org/wiki/Gravitational_lens

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Re: Useless Math

Postby Superisis » Fri Jan 21, 2011 6:31 pm UTC

The problem is that if you're building you proof on "maths is not inherently connected to the real world and any connection we find is accidental because look this, this, and this discovery is completely useless" you will have all the examples (mentioned previously by others) of apparently useless discoveries that are now useful (hence a claim to being connected to the real world) however it is impossible to prove the opposite. In other words, we know that maths that is useless (i the real world) today could be useful tomorrow. We cannot prove that maths that is useless today cannot be useful tomorrow. Arguably one could say that given enough time one could find a real world application for any section of mathematics. And since maths is all one system, built on simpler assumptions which have real world applications, it's not impossible to consider machines using the same logic system using those avant-garde maths in aid of their simpler maths which has the real world application. If that was understandable at all.

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Re: Useless Math

Postby ConMan » Mon Jan 24, 2011 11:36 pm UTC

pizzazz wrote:I'm just guessing, but since these numbers were defined in this thread with Turing machines, is there some way they could be connected to the limits of computers?

Yes and no. Turing machines are theoretical constructs, but any modern computer (and using one of an incredibly large number of programming languages), if you could somehow give it access to an infinite amount of storage space (and remember that even the amount of data on the internet is nothing compared to infinity) and an infinite amount of time to run its programs (while of course any halting Turing machine only needs a finite amount of time to run, you need the infinite amount of time to be able to run all of the Turing machines that run for an arbitrary amount of time before halting), could emulate any Turing machine. The infinite storage is a pretty strong requirement, but once you take that restriction away then certainly anything you can't do on a computer you can't do on a Turing machine. Turing machines are sort of the bridge between real computers and pure theoretical arithmetic, or at least they are if you accept the Church-Turing thesis, without which it's kind of hard to do much interesting.
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