Question from a Texan Textbook

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Re: Question from a Texan Textbook

#61  Postby Evolving » Sep 10, 2015 3:23 pm

That doesn't cover the negative rationals, but that's the idea anyhow!
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Re: Question from a Texan Textbook

#62  Postby The Plc » Sep 10, 2015 3:33 pm

scott1328 wrote:The real numbers are defined in terms of Dedekind cuts.


Worth a mention to add that they are also the rigorous completion of the rational numbers. That is, a space X containing a subspace Y that isometric 'distance-preserving' to the rational numbers, such that each Cauchy sequence (a sequence in which the distance between the points can be made arbitrarily small) in X converges to point within X.

Nicko, you have defined a one-to-one correspondence between the positive integers and a subset of the rationals, the rationals of the form {1/n}. But the question is: is there no one-to-one correspondence between the positive integers and the entire set of rationals? The question is of course, yes.

Although, note, it is a fact that the countable union of countable sets is countable. It is possible to prove from this that the rational numbers are countable by observing that the rationals are countable union of the sets of the kind you just described.
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Re: Question from a Texan Textbook

#63  Postby The Plc » Sep 10, 2015 3:39 pm

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


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Suppose that we have two sets X and Y and a function f defining a correspondence between them. f is surjective if for each and every point y of the set Y, there exists a point x of the set X such that f corresponds x with y.

So for simple example, in considering only functions sending the reals to the reals , the function f(x)=x^2 is not surjective, since no negative real is the square of a real number.
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Re: Question from a Texan Textbook

#64  Postby scott1328 » Sep 10, 2015 4:55 pm

I challenge the assertion that non-computable reals contain any information whatsoever, they cannot even be approximated. So exactly how much information is in a string of digits that cannot be written down? Do they exist. Yes. As abstractions. Numbers only exist as abstractions.
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Re: Question from a Texan Textbook

#65  Postby crank » Sep 10, 2015 6:07 pm

scott1328 wrote:I challenge the assertion that non-computable reals contain any information whatsoever, they cannot even be approximated. So exactly how much information is in a string of digits that cannot be written down? Do they exist. Yes. As abstractions. Numbers only exist as abstractions.

Any string of digits is information. An infinite string contains infinite information. It's known exactly how much information is in a string of digits. Convert to binary, #digits is #bits of information. This is why I keep saying 'incompressible'. The value of pi isn't, you can specify pi with a very few bits, like the ascii for 'pi'. If there is redundancy in the string, there will be fewer bits. So you understand how '2' isn't real? That you can't count discreet objects with reals?
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Re: Question from a Texan Textbook

#66  Postby scott1328 » Sep 10, 2015 7:54 pm

crank wrote:
scott1328 wrote:I challenge the assertion that non-computable reals contain any information whatsoever, they cannot even be approximated. So exactly how much information is in a string of digits that cannot be written down? Do they exist. Yes. As abstractions. Numbers only exist as abstractions.

Any string of digits is information. An infinite string contains infinite information. It's known exactly how much information is in a string of digits. Convert to binary, #digits is #bits of information. This is why I keep saying 'incompressible'. The value of pi isn't, you can specify pi with a very few bits, like the ascii for 'pi'. If there is redundancy in the string, there will be fewer bits. So you understand how '2' isn't real? That you can't count discreet objects with reals?

Pi is compressible. It can be represented by the program that calculates it. which is far from infinite. A non-computable number has no information because you cannot express its digits at all.
And yes for the fucking 17th time 2 is a real number and yes you can count discrete objects with real numbers.
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Re: Question from a Texan Textbook

#67  Postby crank » Sep 10, 2015 9:30 pm

scott1328 wrote:
crank wrote:
scott1328 wrote:I challenge the assertion that non-computable reals contain any information whatsoever, they cannot even be approximated. So exactly how much information is in a string of digits that cannot be written down? Do they exist. Yes. As abstractions. Numbers only exist as abstractions.

Any string of digits is information. An infinite string contains infinite information. It's known exactly how much information is in a string of digits. Convert to binary, #digits is #bits of information. This is why I keep saying 'incompressible'. The value of pi isn't, you can specify pi with a very few bits, like the ascii for 'pi'. If there is redundancy in the string, there will be fewer bits. So you understand how '2' isn't real? That you can't count discreet objects with reals?

Pi is compressible. It can be represented by the program that calculates it. which is far from infinite. A non-computable number has no information because you cannot express its digits at all.
And yes for the fucking 17th time 2 is a real number and yes you can count discrete objects with real numbers.


I know pi is compressible, that's why I said it wasn't incompressible, see the red text above.

If you can count discreet objects with reals, what is 4.594837 of a discreet object? It's meaningless, reals are continuous. In other words, using discreet objects is cheating, you get to evade what is at issue here, that you can't ever tell if A is equal to B by some continuous measure, it's impossible without infinite precision, which you refuse to address, and which can't exist in the real world, which is the argument you are making. Try to show me a 2 that is real, and and I'll say prove to me it's equal to 2.000.... That's why discreet objects are a cheat.

And uncomputable numbers can be expressed to however many digits you want/can, e.g., random number generation schemes, Chaitin's own Omega number, there are many ways of getting uncomputables. You can't express one in it's entirety, obviously. You're the one arguing they exist in the real world, not me.

You said "In any case, real numbers exist as a class in exactly the same way as 0 and 1: abstractions defined into existence." I don't really know what that is supposed to mean, one thing it doesn't mean is that reals exist in the real world. Saying that 1 and 2 exist in the real world, therefore reals exist, is wrong even if you let '1' and '2' be reals, this is seriously wrong. For example, if A is defined as B and C and D and E, showing me B doesn't let you say A exists. Or, showing me B and C and D, still doesn't mean A exists. Showing me an integer doesn't prove reals exist, that is patently absurd, it proves that that integer exists, or to be generous, that some reals exist. You haven't proven that the class exists, all the integers make up zero percent of the reals. All the integers and all the rationals make up zero percent of the reals.
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Re: Question from a Texan Textbook

#68  Postby Calilasseia » Sep 10, 2015 9:55 pm

What part of "integers are a proper subset of the reals" do you not understand? They are simply numbers for whom, in my expression above (or, more properly, expansions thereof to include non-negative powers of 10 in the sum), all the coefficients ai in all terms of the form ai/10-i are zero.
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Re: Question from a Texan Textbook

#69  Postby scott1328 » Sep 10, 2015 10:04 pm

All numbers real or otherwise are abstractions. They are defined into existence as abstract objects; the definition for the class of reals is as rigorous as that of the integers. You cannot point to a physical manifestation of an integer any more than you can to a rational number, computable number, or uncomputable number. I do not claim that any number exists in the physical world. I also do not believe that the foregoing is suitable basis to conclude a digital, discrete universe that Chaitin is claiming.
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Re: Question from a Texan Textbook

#70  Postby crank » Sep 10, 2015 10:19 pm

Calilasseia wrote:What part of "integers are a proper subset of the reals" do you not understand? They are simply numbers for whom, in my expression above (or, more properly, expansions thereof to include non-negative powers of 10 in the sum), all the coefficients ai in all terms of the form ai/10-i are zero.


Who is that to? If it's to me, I never said otherwise. 2 can be an integer, 2.000000... isn't the same thing, how can you prove 2.0000... with zeros out to some finite number of places, how can you prove that is equal to 2 the integer?. .999999 = 1, but .99999 isn't an integer. Like saying the number of syllables in a word is 2.000..., if you are counting discreet objects, integers apply, but reals don't, you would be using a continuous variable for a discreet object. I don't know how to express this technically, but they're must be some way to say it. In the context of the discussion, where the infinite precision of reals is at issue, to say the number of some discreet objects is 2, therefor reals exist, makes no sense. It's not trout we're discussing. A single trout can prove trouts exist. Showing an integer exists doesn't prove reals exist. Put another way, prove something with integers, doesn't necessarily prove it for reals.
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Re: Question from a Texan Textbook

#71  Postby crank » Sep 10, 2015 10:28 pm

scott1328 wrote:All numbers real or otherwise are abstractions. They are defined into existence as abstract objects; the definition for the class of reals is as rigorous as that of the integers. You cannot point to a physical manifestation of an integer any more than you can to a rational number, computable number, or uncomputable number. I do not claim that any number exists in the physical world. I also do not believe that the foregoing is suitable basis to conclude a digital, discrete universe that Chaitin is claiming.

That's not the argument we were having. Now you say no numbers exist in the real world. OK, then you agree with the assertion reals don't exist in the real world, which is the original claim you said was nonsense. I don't agree about all numbers not existing, but then, that's an argument that was around before Plato and hasn't been solved yet. If you would have said that at the beginning without all the arguing, we wouldn't have gone through all of this.
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Re: Question from a Texan Textbook

#72  Postby scott1328 » Sep 10, 2015 10:36 pm

crank wrote:
Calilasseia wrote:What part of "integers are a proper subset of the reals" do you not understand? They are simply numbers for whom, in my expression above (or, more properly, expansions thereof to include non-negative powers of 10 in the sum), all the coefficients ai in all terms of the form ai/10-i are zero.


Who is that to? If it's to me, I never said otherwise. 2 can be an integer, 2.000000... isn't the same thing, how can you prove 2.0000... with zeros out to some finite number of places, how can you prove that is equal to 2 the integer?. .999999 = 1, but .99999 isn't an integer. Like saying the number of syllables in a word is 2.000..., if you are counting discreet objects, integers apply, but reals don't, you would be using a continuous variable for a discreet object. I don't know how to express this technically, but they're must be some way to say it. In the context of the discussion, where the infinite precision of reals is at issue, to say the number of some discreet objects is 2, therefor reals exist, makes no sense. It's not trout we're discussing. A single trout can prove trouts exist. Showing an integer exists doesn't prove reals exist. Put another way, prove something with integers, doesn't necessarily prove it for reals.

2 and 2.00000000.... are exactly equal and they are both real numbers (and integers)
1 = 1.000000..... = .99999999.... are all exactly equal and they are all real numbers (and integers)

Real number doesn't imply "infinite" precision although the VAST majority of real numbers cannot be expressed with finite precision. I don't know where you are getting these ideas from.

Could it be that you are hung up on the irrational numbers? The Pythagoreans didn't like them much either.
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Re: Question from a Texan Textbook

#73  Postby scott1328 » Sep 10, 2015 10:40 pm

crank wrote:
scott1328 wrote:All numbers real or otherwise are abstractions. They are defined into existence as abstract objects; the definition for the class of reals is as rigorous as that of the integers. You cannot point to a physical manifestation of an integer any more than you can to a rational number, computable number, or uncomputable number. I do not claim that any number exists in the physical world. I also do not believe that the foregoing is suitable basis to conclude a digital, discrete universe that Chaitin is claiming.

That's not the argument we were having. Now you say no numbers exist in the real world. OK, then you agree with the assertion reals don't exist in the real world, which is the original claim you said was nonsense. I don't agree about all numbers not existing, but then, that's an argument that was around before Plato and hasn't been solved yet. If you would have said that at the beginning without all the arguing, we wouldn't have gone through all of this.

I am arguing with your benighted terminology. you keep saying integers are not real numbers, but they are, you keep saying other very odd things. Like 2.0000... is not the same as 2, that discrete objects cannot be counted with real numbers, they can. For some reason you seem to think that the "non-existance" of reals implies a discrete universe, it doesn't.

Or at least that is what you appear to be saying to me.
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Re: Question from a Texan Textbook

#74  Postby crank » Sep 10, 2015 11:16 pm

scott1328 wrote:
2 and 2.00000000.... are exactly equal and they are both real numbers (and integers)
1 = 1.000000..... = .99999999.... are all exactly equal and they are all real numbers (and integers)

Real number doesn't imply "infinite" precision although the VAST majority of real numbers cannot be expressed with finite precision. I don't know where you are getting these ideas from.

Could it be that you are hung up on the irrational numbers? The Pythagoreans didn't like them much either.

All real numbers have infinite precision, that's why you need to specify 2.00... Just because all digits past some particular place is zero, doesn't mean it isn't there.
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Re: Question from a Texan Textbook

#75  Postby The_Metatron » Sep 10, 2015 11:21 pm

Evolving wrote:
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.

Thanks. I never heard that one before.
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Re: Question from a Texan Textbook

#76  Postby The_Metatron » Sep 10, 2015 11:26 pm

Calilasseia wrote:
[Reveal] Spoiler:
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.
[Reveal] Spoiler:
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
.

Primus and I were discussing this idea once when he was asking me about π. It ended with him understanding that there is no precise way to write that number except for "π". Anything else is an approximation, although we can get pretty goddamned close to it, we never get quite there.

But, if you write "π", that is balls-on-accurate.

Is this the same concept?
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Re: Question from a Texan Textbook

#77  Postby crank » Sep 10, 2015 11:34 pm

scott1328 wrote:
I am arguing with your benighted terminology. you keep saying integers are not real numbers, but they are, you keep saying other very odd things. Like 2.0000... is not the same as 2, that discrete objects cannot be counted with real numbers, they can. For some reason you seem to think that the "non-existance" of reals implies a discrete universe, it doesn't.

Or at least that is what you appear to be saying to me.

If you're doing some kind of mathematics that is restricted to integers, then to me, the 2 used there is not real, it's integer. Yes, integers are a subset of reals, but when you're doing something where reals don't apply, then you can't use 2.000..., you use 2. There's some fundamental difference when using an integer verses using a real, even if they're the same number. It's the difference between continuous and discrete, how an integer can only change by whole multiples of 1, and reals can change by infinitesimals. I apologize that I don't know the technical language, but I can't believe these distinctions aren't made in some way.

I think, in my far from qualified way, that infinite precision can't be real, from the informational aspects, which you don't agree exist, but there's a lot of mathematicians and physicists who do. Also, because of how there's a lot of physics that imply a quantized universe, I don't even know if that could technically be equivalent to 'discrete universe', though it has some real basic similarities if it isn't equivalent.
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Re: Question from a Texan Textbook

#78  Postby crank » Sep 10, 2015 11:40 pm

The_Metatron wrote:
Calilasseia wrote:
[Reveal] Spoiler:
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.
[Reveal] Spoiler:
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
.

Primus and I were discussing this idea once when he was asking me about π. It ended with him understanding that there is no precise way to write that number except for "π". Anything else is an approximation, although we can get pretty goddamned close to it, we never get quite there.

But, if you write "π", that is balls-on-accurate.

Is this the same concept?


What about the various infinite series expansions, would you consider the standard way to write them, with the sigma and all, a precise way to write it?
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Re: Question from a Texan Textbook

#79  Postby Calilasseia » Sep 11, 2015 1:57 am

Frequently, they're the only way to write them.

Though in the case of π, Bailey, Borwein and Plouffe alighted upon a neat proof, that it's possible to generate any desired digit of π without having to precompute earlier digits. Though their proof, I remind everyone, is set in base 16 (hexadecimal), and geared to computer generation of the digits. Although they express their formula as an infinite sum, each term yields a specific digit, once the series is manipulated to produce what is termed a "spigot algorithm". More details about this [url=https://en.wikipedia.org/wiki/Bailey–Borwein–Plouffe_formula]here[/url]

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Re: Question from a Texan Textbook

#80  Postby Kaleid » Sep 11, 2015 2:38 am

Seems it's a Unicode problem with those damn dashes, Cali:

Code: Select all
[url=https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula]here[/url]


here
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