Question from a Texan Textbook

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Re: Question from a Texan Textbook

#41  Postby crank » Sep 10, 2015 2:10 pm

Calilasseia wrote:
crank wrote:
Calilasseia wrote:Actually, I can write down an expression that constitutes the definition of a real number. Viz:

Real Number Definition.jpg


where for all possible values of i, ai is an element of the set {0,1,2,3,4,5,6,7,8,9}.

If ai=0 for all i>N, where N is some suitable finite integer, then you have a rational number. But not all rational numbers fall into this category. 1/7 being a classic example. The decimal expansion of 1/7 continues indefinitely:

(1/7) = 0.124857124857124857124857124857124857124857124857124857124857...

But in this and other cases, the coefficients ai are cyclically periodic for all i>N, where N is some suitable finite integer.

Oh, and see also, Dedekind Cuts.


That's the definition of a rational, not a real, isn't it?


No. What part of the words "infinite sum" did you fail to derive from the expression I provided?

crank wrote:Most reals require an infinite string of integers, incompressible, you have to stipulate each one to define it.


Which is precisely what that expression says if you read it.

Plus, since the expression converges to a finite limit for all choices of coefficients as defined, and indeed converges to a finite limit in the interval [0,10] (courtesy of 100 being a part of the expression), it satisfies all the conditions required to be a definition of a real number. Indeed, setting all the ai to 9, which results in the largest of the numbers thus defined, results in the expression converging to the value:

9[1/(1-[1/10])] = 9[10/9] = 10.

Plus, as I already stated, there are rational numbers whose decimal expansions are infinite. I provided the example of 1/7: indeed, this is true for all rationals of the form 1/p, where p is a prime number greater than 2.

EDIT: 5 is an exception, having the decimal expansion 0.2. All other primes apart from 2 and 5 have infinite decimal expansions.

OK, I was trying to read too much into the definition. Is it anything more than saying the reals have a decimal representation? I thought there must be something more that I'm not getting, because this is trivial. How is this relevant to Chaitin's claim? I must be missing something because this doesn't seem to have anything to do with anything I've said, maybe it pertains to something in Chaitin's work, I don't know, I never claimed to have serious understanding of any of this, in fact just the opposite.
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Re: Question from a Texan Textbook

#42  Postby The_Metatron » Sep 10, 2015 2:12 pm

What the blue fuck is "onto correspondence"?


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Re: Question from a Texan Textbook

#43  Postby Evolving » Sep 10, 2015 2:18 pm

The_Metatron wrote:What the blue fuck is "onto correspondence"?


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Also known as a surjection: the function maps the domain (the set of integers, say) on to the whole of the other set (the set of rational numbers): it uses up the whole of that set and doesn't leave any gaps.
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Re: Question from a Texan Textbook

#44  Postby crank » Sep 10, 2015 2:24 pm

Newmark wrote:
crank wrote:
Calilasseia wrote:Actually, I can write down an expression that constitutes the definition of a real number. Viz:

Real Number Definition.jpg


where for all possible values of i, ai is an element of the set {0,1,2,3,4,5,6,7,8,9}.

If ai=0 for all i>N, where N is some suitable finite integer, then you have a rational number. But not all rational numbers fall into this category. 1/7 being a classic example. The decimal expansion of 1/7 continues indefinitely:

(1/7) = 0.124857124857124857124857124857124857124857124857124857124857...

But in this and other cases, the coefficients ai are cyclically periodic for all i>N, where N is some suitable finite integer.

Oh, and see also, Dedekind Cuts.

That's the definition of a rational, not a real, isn't it? Most reals require an infinite string of integers, incompressible, you have to stipulate each one to define it.


No, Cali gave a correct definition of a (quite generic) real number above, which indeed can be viewed an infinite string of integers. It is not the definition of a rational, which severely makes me doubt your ability to identify "bad math". However, this definition does not cover all reals r, only those for which 0 <= r <= 10, but there should be sufficiently many non-computable ones in that range for Cali's point to be valid...

[Edit: Ninja'd by Cali...]


As I said in my response to cali, I was trying to read more into the definition, I didn't realize it was so basic. As for understanding bad math, any interval will have the same number, the same infinities, as any other, unless there's weird constructions that makes that not true. Any interval, like 0-1, or -infinity to infinity, 6-43.5, have a 1 to 1 mapping. Discussing that some range should have sufficiently many non-computables doesn't really make sense.
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Re: Question from a Texan Textbook

#45  Postby Evolving » Sep 10, 2015 2:28 pm

crank, does this way of looking at it help?

In the real world (and I am speaking as a physicist here, not a mathematician), where a quantity is defined as a continuum rather than discrete variables (the height of my children for instance, as opposed to how many of them I have), it's impossible to identify one real number (whether it be an integer, a rational or an irrational number) which exactly gives the value of that quantity. All we can do is say that the value is within a certain interval (or volume).

That is not just because of the limitations on our measuring equipment, but it is true at a fundamental level because of Heisenberg's uncertainty principle in its various expressions, and it has to do with the wavelike nature of very small physical entities. In a similar way (sort of), you can't say exactly, to the molecule, where a water wave is.

For this reason we can't even make the interval arbitrarily small within which we know that the value is to be found: there's a limit set by Planck's constant, and it means that the value we are seeking not only can't be measured with total accuracy, it doesn't even have a single value: it's spread out, smeared, over the interval (or volume).

In that sense I suppose it makes sense to say that a particular number doesn't exist in the real world: there is no physical quantity which has exactly that number.

Is that what you meant?
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Re: Question from a Texan Textbook

#46  Postby Calilasseia » Sep 10, 2015 2:30 pm

scott1328 wrote:Chaitin is arguing for a discrete, digital universe: a model of the universe. In a digital universe a subset of the real numbers exist: the so called computable numbers. Virtually every real number that You have or ever will encounter is a computable number: square root of two, pi, the base of the natural logarithm e, phi the golden ratio and a host of others. Because these numbers are computable, they do not entail an infinity of information; because they are computable their digital expansion can be expressed in a finite amount of time to any finite precision. Our physical universe can be neatly modeled by computable real numbers. But computable real numbers are countably finite: there is a one-to-one mapping of computable reals to the natural numbers. But, even so there uncountably more non computable numbers than computable.

In any case, real numbers exist as a class in exactly the same way as 0 and 1: abstractions defined into existence.


At this point, I have to step in and add some rigour.

In Turing's original 1936 paper, where he defined the concept of computable number, he defined such a number as any number whose computation by a Turing machine would halt in finite time. Specifically, if such a machine is fed with the integer N as its input, it generates the first N decimal places of that number. This definition was later superseded because it was subject to rounding issues, and instead, a move was made to use Turing machines to define computable functions, for which a Turing machine halts and yields an integer answer if fed with input for which the function is defined, and which does not halt if fed with input for which the function is not defined. Usually, such a function is defined in terms of an input vector consisting of integer components:

x = [x1, x2, x3, ... xn], where n is a finite integer

And a computable function is a function f(x) such that given any tuple x, f(x) yields an integer result after a finite number of operations if f(x) is defined for that tuple, and does not do so if f(x) is not defined for that function. Restricting the functions to working with tuples of integers avoids the rounding problem, which centres upon the fact that for many real-valued functions, it is impossible to know in advance how far to extend the computation beyond N digits before being able to round the result to N digits consistently in accordance with appropriate rounding algorithms. This itself is a consequence of Turing's proof of the falsity of Hilbert's conjecture with respect to the Entscheidungsproblem, which was the motivation behind the development of Turing machines in the first place.

Whilst this restriction to tuples of integers might seem overly restrictive, it transpires that there is a proof in existence, that functions of this type are sufficient to express all computations that are possible, including computations involving the reals, though this proof, being a part of the theory of μ-recursive functions, is beyond the scope of this post, though its postulates can be demonstrated to be equivalent to the postulates in Turing's paper.

So, having established the above results, the definition of a computable number was then modified, and recast in terms of computable functions, and an equivalent concept, in the form of a computable Dedekind cut, was also established.

But, the salient point I wish to drive home here, is that computability takes many forms. In the case of real numbers other than the rationals (including the integers), there are algebraic irrationals, which can be represented as the roots of a finite polynomial equation, and for which finite computations obviously exist to generate the computed approximations. Then there are the transcendental irrationals, which cannot be thus represented, and which include numbers that are the products of such functions as trigonometric or logarithmic functions, themselves represented for computational purposes as a finite number of terms of the defining infinite series. In the case of the transcendental irrationals, many of these are still computable in this sense.

The fun part, after all this, is that the computable numbers have been demonstrated to form a countable set. Since the reals as a whole are uncountable (see Cantor's diagonal proof), then the uncomputable numbers are vastly more abundant, in the Cantor sense.

It's important to separate out computability from the other features of real numbers, such as whether they are algebraic or transcendental irrationals, which is where some confusion can arise if diligence is not exerted with respect to this. :)
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Re: Question from a Texan Textbook

#47  Postby crank » Sep 10, 2015 2:31 pm

Newmark wrote:
crank wrote:
Veida wrote:You're making much too much out of the simple fact that most reals are irrational, crank.

All I said was this guy Chaitin had an interesting idea. Then I started getting really bad maths hurled at me and no attempt to answer the simple points I raised, which I can't leave alone, that goes against my nature. And it's not irrationals but uncomputable numbers. Any of which contains an infinite amount of information, which is one reason to say they can't really exist.


Why can't it contain an infinite amount of information, exactly? That we cannot measure anything with infinite precision I have no problem with buying, but that isn't really the same thing...

For something to contain infinite information makes things like the 2nd law and entropy and quantum mechanics go crazy. Like I said, I'm no expert, this stuff fascinates me, doesn't mean I understand much of it.
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Re: Question from a Texan Textbook

#48  Postby crank » Sep 10, 2015 2:37 pm

Evolving wrote:crank, does this way of looking at it help?

In the real world (and I am speaking as a physicist here, not a mathematician), where a quantity is defined as a continuum rather than discrete variables (the height of my children for instance, as opposed to how many of them I have), it's impossible to identify one real number (whether it be an integer, a rational or an irrational number) which exactly gives the value of that quantity. All we can do is say that the value is within a certain interval (or volume).

That is not just because of the limitations on our measuring equipment, but it is true at a fundamental level because of Heisenberg's uncertainty principle in its various expressions, and it has to do with the wavelike nature of very small physical entities. In a similar way (sort of), you can't say exactly, to the molecule, where a water wave is.

For this reason we can't even make the interval arbitrarily small within which we know that the value is to be found: there's a limit set by Planck's constant, and it means that the value we are seeking not only can't be measured with total accuracy, it doesn't even have a single value: it's spread out, smeared, over the interval (or volume).

In that sense I suppose it makes sense to say that a particular number doesn't exist in the real world: there is no physical quantity which has exactly that number.

Is that what you meant?

That's basically it, but see my previous response also, it's not just that infinite precision is needed, it needs to be combined with the fact that uncomputables cannot be compressed, meaning the only way represent one is with an infinite string of integers that are completely random, that's where the infinite information content comes in.
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Re: Question from a Texan Textbook

#49  Postby Calilasseia » Sep 10, 2015 2:41 pm

Evolving wrote:crank, does this way of looking at it help?

In the real world (and I am speaking as a physicist here, not a mathematician), where a quantity is defined as a continuum rather than discrete variables (the height of my children for instance, as opposed to how many of them I have), it's impossible to identify one real number (whether it be an integer, a rational or an irrational number) which exactly gives the value of that quantity. All we can do is say that the value is within a certain interval (or volume).

That is not just because of the limitations on our measuring equipment, but it is true at a fundamental level because of Heisenberg's uncertainty principle in its various expressions, and it has to do with the wavelike nature of very small physical entities. In a similar way (sort of), you can't say exactly, to the molecule, where a water wave is.

For this reason we can't even make the interval arbitrarily small within which we know that the value is to be found: there's a limit set by Planck's constant, and it means that the value we are seeking not only can't be measured with total accuracy, it doesn't even have a single value: it's spread out, smeared, over the interval (or volume).

In that sense I suppose it makes sense to say that a particular number doesn't exist in the real world: there is no physical quantity which has exactly that number.

Is that what you meant?


In short, one has to draw a rigorous distinction between abstract existence, courtesy of the existence of formal definitions, and concrete existence, namely whether there exists an observable instantiation of the entity in question.

It seems that Chaitin is suggesting that whilst the non-computable numbers certainly exist in the abstract, courtesy of those formal definitions, he's suggesting that they do not have concrete instantiations. Which is rather interesting, given that Chaitin was, along with Kolmogorov, instrumental in defining a measure of complexity, which was later demonstrated to be uncomputable.
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Re: Question from a Texan Textbook

#50  Postby Evolving » Sep 10, 2015 2:43 pm

crank wrote:...it's not just that infinite precision is needed, it needs to be combined with the fact that uncomputables cannot be compressed, meaning the only way represent one is with an infinite string of integers that are completely random, that's where the infinite information content comes in.


I'm not sure that's a very convincing argument, though, crank (and I admit this may simply be a physicist's mistrust of mathematicians trying to establish something about the real world by being clever with mathematical concepts): if there were no uncertainty principle and a physical quantity really could take on an exact value (never mind whether we could measure it in practice with that degree of precision), why does it matter whether we can express the number in our mathematical language?
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Re: Question from a Texan Textbook

#51  Postby crank » Sep 10, 2015 2:47 pm

I said "uncomputables cannot be compressed", I'm not sure that is correct, I'm pretty sure it's mostly correct, in that you can't compress them such that a finite amount of info is enough to specify one. And I'm not positive non-computable are totally random, but incompressible noncomputables must be, [I'm almost totally sure on that one]. Oh, look, is that a robin?
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Re: Question from a Texan Textbook

#52  Postby Nicko » Sep 10, 2015 2:49 pm

scott1328 wrote:The set of natural numbers is a proper subset of the integers which is in turn a proper subset of the rationals. The rationals are considered countable because there is a one-to-one mapping of the natural numbers to the rationals and vice versa. And not, merely because the natural numbers never "run out"


I guess I'm just a glutton for embarrassment, but here's what I meant by that.

Say that we map the integer 1 onto the rational number 1/n.

Then we map the integer 2 onto the rational number 1/n-1.

3 onto 1/n-2.

And so on.

It's perfectly clear that we will need an infinite amount of integers before we even get to the rational number 1.

Happily, we just happen to have an infinite amount of integers on hand. So, no problem.

Am I still looking at this the wrong way?
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Re: Question from a Texan Textbook

#53  Postby Evolving » Sep 10, 2015 2:53 pm

Nicko: that's what I meant with my post no. 7, following up on my post no. 5, after posting which I noticed exactly the point that you have just made!
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Re: Question from a Texan Textbook

#54  Postby Nicko » Sep 10, 2015 2:54 pm

Evolving wrote:Nicko: that's what I meant with my post no. 7, following up on my post no. 5, after posting which I noticed exactly the point that you have just made!


So, I do get what's going on now?
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Re: Question from a Texan Textbook

#55  Postby crank » Sep 10, 2015 2:57 pm

Evolving wrote:
crank wrote:...it's not just that infinite precision is needed, it needs to be combined with the fact that uncomputables cannot be compressed, meaning the only way represent one is with an infinite string of integers that are completely random, that's where the infinite information content comes in.


I'm not sure that's a very convincing argument, though, crank (and I admit this may simply be a physicist's mistrust of mathematicians trying to establish something about the real world by being clever with mathematical concepts): if there were no uncertainty principle and a physical quantity really could take on an exact value (never mind whether we could measure it in practice with that degree of precision), why does it matter whether we can express the number in our mathematical language?

We're getting way beyond my pay grade, we passed that point a while back. But why let that stop me? If there existed such things, having infinite detail, you could represent all knowledge with a stick of the right length. All possible knowledge. Now I really don;t know what I am talking about, but entropy, the second law, and quantum mechanics get uppity when you don't deal with information properly. I don't think they like infinite information. That stick would give you the complete history of every particle in the universe from the beginning to the end, it would contain all the information in the universe. It's a dangerous stick, dangerous to our concepts of information, entropy, the 2nd law and quantum mechanics.
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Re: Question from a Texan Textbook

#56  Postby Evolving » Sep 10, 2015 2:58 pm

Nicko wrote:
Evolving wrote:Nicko: that's what I meant with my post no. 7, following up on my post no. 5, after posting which I noticed exactly the point that you have just made!


So, I do get what's going on now?


:lol:

I think so, yes. The nub of both of our epiphanies is that, for any element in the domain, you have to be able, in principle, to name the element in the co-domain that it maps to. The counting can take arbitrarily long to get there, but it can't take infinitely long.
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Re: Question from a Texan Textbook

#57  Postby crank » Sep 10, 2015 3:00 pm

Nicko wrote:
scott1328 wrote:The set of natural numbers is a proper subset of the integers which is in turn a proper subset of the rationals. The rationals are considered countable because there is a one-to-one mapping of the natural numbers to the rationals and vice versa. And not, merely because the natural numbers never "run out"


I guess I'm just a glutton for embarrassment, but here's what I meant by that.

Say that we map the integer 1 onto the rational number 1/n.

Then we map the integer 2 onto the rational number 1/n-1.

3 onto 1/n-2.

And so on.

It's perfectly clear that we will need an infinite amount of integers before we even get to the rational number 1.

Happily, we just happen to have an infinite amount of integers on hand. So, no problem.

Am I still looking at this the wrong way?

Never mind, I have no idea what that's all about, don't want to
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Re: Question from a Texan Textbook

#58  Postby Evolving » Sep 10, 2015 3:02 pm

crank wrote:
Evolving wrote:
crank wrote:...it's not just that infinite precision is needed, it needs to be combined with the fact that uncomputables cannot be compressed, meaning the only way represent one is with an infinite string of integers that are completely random, that's where the infinite information content comes in.


I'm not sure that's a very convincing argument, though, crank (and I admit this may simply be a physicist's mistrust of mathematicians trying to establish something about the real world by being clever with mathematical concepts): if there were no uncertainty principle and a physical quantity really could take on an exact value (never mind whether we could measure it in practice with that degree of precision), why does it matter whether we can express the number in our mathematical language?

We're getting way beyond my pay grade, we passed that point a while back. But why let that stop me? If there existed such things, having infinite detail, you could represent all knowledge with a stick of the right length. All possible knowledge. Now I really don;t know what I am talking about, but entropy, the second law, and quantum mechanics get uppity when you don't deal with information properly. I don't think they like infinite information. That stick would give you the complete history of every particle in the universe from the beginning to the end, it would contain all the information in the universe. It's a dangerous stick, dangerous to our concepts of information, entropy, the 2nd law and quantum mechanics.


Well, this may be a bit of a cop-out, but given that, in fact, there is an uncertainty principle, it doesn't really matter!
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Re: Question from a Texan Textbook

#59  Postby crank » Sep 10, 2015 3:09 pm

Evolving wrote:
Well, this may be a bit of a cop-out, but given that, in fact, there is an uncertainty principle, it doesn't really matter!

That's a cop-out, but then I have no idea what I'm talking about. I defer to the mathematicians and physicists. You could, possibly, take it as a proof that the universe must be quantized, which I think is Chaitin's claim, the reals can't exist, therefore the universe has to be quantized, if it wasn't, then these information problems would plague. One way to see that is how you get all kinds of infinities when calculating certain values when using continuous maths.
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Re: Question from a Texan Textbook

#60  Postby scott1328 » Sep 10, 2015 3:10 pm

Nicko wrote:
scott1328 wrote:The set of natural numbers is a proper subset of the integers which is in turn a proper subset of the rationals. The rationals are considered countable because there is a one-to-one mapping of the natural numbers to the rationals and vice versa. And not, merely because the natural numbers never "run out"


I guess I'm just a glutton for embarrassment, but here's what I meant by that.

Say that we map the integer 1 onto the rational number 1/n.

Then we map the integer 2 onto the rational number 1/n-1.

3 onto 1/n-2.

And so on.

It's perfectly clear that we will need an infinite amount of integers before we even get to the rational number 1.

Happily, we just happen to have an infinite amount of integers on hand. So, no problem.

Am I still looking at this the wrong way?


Here is a diagram of how the mapping works Nicko.
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