Deary, deary me...
Moderators: Calilasseia, ADParker
scott1328 wrote:The real numbers are defined in terms of Dedekind cuts.
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.
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?
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.
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.
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.
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.
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.
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.
Evolving wrote:
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.
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
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.
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?
[url=https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula]here[/url]
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