## Language Negation and Randomness

What did you say? And what did you mean by it?

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wtf
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### Re: Language Negation and Randomness

Hobbes' Choice wrote: "Show me a set of numbers about which nothing can be said please!"
Yes but I already answered that question. Twice now. You made a comment about zero and I responded to that. What on earth are you talking about here?

Hobbes' Choice
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### Re: Language Negation and Randomness

wtf wrote:
Hobbes' Choice wrote: "Show me a set of numbers about which nothing can be said please!"
Yes but I already answered that question. Twice now. You made a comment about zero and I responded to that. What on earth are you talking about here?
Well Daaah.
Your response said something meaningful, obviously.

wtf
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### Re: Language Negation and Randomness

Hobbes' Choice wrote: Well Daaah.
Your response said something meaningful, obviously.
Awesome football game. Back to what's really important.

I see your point. I'll do my best to respond. As I understand it, we are having this dialog:

Me: If x is a noncomputable number, you can not say anything meaningful about x.

You: You just said something meaningful about x! You just defeated your own point!

I'll respond the best I can.

The OP wrote:
Justintruth wrote: "Clearly nothing can be said about (some set)...."

...implying it is random.
My response was to point out that there's a mathematical definition of randomness such that it's sensible to say that nothing sensible can be said. And that this point of view is indeed part of modern abstract probability theory.

If you have the set of even numbers, you can think of a unary predicate or property E such that E(n) is true just in case n is an even number, and false otherwise. Likewise there's a predicate Prime(x), and Square(x), and all the other describable subsets of the natural numbers.

Now, what is a predicate? In formal logic it's a finite string of symbols chosen from some alphabet. The alphabet is finite in the case of natural language, and is generally taken to be at most countably infinite in the abstract case.

It's not hard to prove that there are only countably many predicates. But there are uncountably many subsets of the natural numbers. That's Cantor's theorem (related to but much more general than Cantor's diagonal argument). It says that the cardinality of the collection of subsets of a set, is strictly greater than the cardinality of the set.

The conclusion is that there are far more sets of natural numbers than properties that characterize them. A set that can not be characterized by a property or predicate is essentially random. Nothing meaningful can be said about it, for the reason that there simply are not enough meaningful statements to go around! If a "meaningful statement" is a finite string of symbols, there are far fewer meaningful statements than there are sets of natural numbers.

There are several related forms of the same argument. There are only countably many Turing machines but uncountably many real numbers, so most real numbers are uncomputable. Kolmogorov complexity is another approach.

Clearly what I've just talked about is very meaningful. But it's "meta" meaningful. It's a meaningful statement ABOUT meaningful statements. It does not live within the universe of discourse, which is statements about the natural numbers such as: "n is even." "n is prime." "n is the exponent in a counterexample to Fermat's Last Theorem." That last predicate was proven by Wiles to describe the empty set. Each of those is a statement that is true or false about each natural number, and therefore defines a particular subset of the natural numbers.

So that would be my point. The statement, "There aren't enough predicates on the natural numbers," is what I said. The statement "Oh but that itself is meaningful" is a predicate ABOUT predicates of natural numbers. I am making a meaningful statement about meaningful statements of natural numbers.

That does NOT contradict the statement I made that there are some sets of natural numbers that can not be described by any meaningful statements about numbers. My outer statement is a meaningful statement about meaningful statements of numbers; not a meaningful statement about numbers.

I guess I'm repeating myself now.

Perhaps from the standpoint of the philosophy of language I'm wrong, and the point you are making is regarded as important. If this is where you're coming from, I'd be happy to be educated on the subject. I just told you what I know about it.

I hope this has been helpful.

Londoner
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### Re: Language Negation and Randomness

wtf wrote:
My response was to point out that there's a mathematical definition of randomness such that it's sensible to say that nothing sensible can be said.
Just looking at the word, I would ask what the opposite of 'random' would be? I would think it would involve 'selection' or 'purpose'.

Suppose I was asked to 'give a selection of numbers' and I picked: 2,4,6,8. I would say that as a group they are not random in that all those symbols were examples of numbers; I have purposely not included any letters or punctuation marks.

Now it is also true that we might look at those numbers and see a pattern of relationships between them, but since that pattern has no meaning in the context of 'give a selection of numbers' then I don't think it makes them non-random.

To put it another way, if somebody saw significance in the fact that each number was 2 more than the last, they would be misunderstanding the meaning of the sample.

Justintruth
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### Re: Language Negation and Randomness

Ok, just two things to focus....
... You can't compute without an input of energy. That energy needs to go somewhere. Heat is the usual output. I don't believe that statement. Nor do I know what it means to compute without erasing. Erasing is one of the basic operations of a Turing machine.
https://en.m.wikipedia.org/wiki/Landauer's_principle

Second here is a paper on the derivation of the Boltzman distribution.

In the last half of the third paragraph it describes how the Boltzman distribution is derived from certain assumptions about "distinguishability". As far as I can see it says two different situations -microstates- can be distinguished from each other even if they are identical - sometimes. Other times not. So my question is very broad. Under what circumstances can two identical situations be distinguished from each other?

How this issue is resolved has consequences in what is predicted in terms of macroscopic physical properties.

You might want to look at the Bose-Einstein distribution and the Fermi-Dirac distribution for contrast but I will focus only on the Boltzman distribution.

If you want me to focus somewhere else no problem.

I think this is not a case of me knowing something no one else does. More probably it is me just missing something that I need to understand settled science / math / philosophy whatever.

Justintruth
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### Re: Language Negation and Randomness

[quote]...Kolmogorov complexity is another approach...

Great post. Got out my mental notepad. You seem to really know this material. How much do you usually get an hour to explain it...?

Anyway... it seems to me that the Kolmogorov approach disconnects the notion of randomness from language because if you program a Turing machine you can just count the number of bits in the program and compare it to the number in the sequence output. It's a property of the initial and final length of the tape...at least the written on part. What has that got to do with language?

I have a million questions....suppressing them

Justintruth
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### Re: Language Negation and Randomness

Imagine a torus made of a continuous homogenous material. Establish a 3 d Cartesian coordinate system with the xy plane bisecting the torus like you slice a bagel. Imagine that the torus is stationary in this system.

Now Imagine a second coordinate system rotating relative to the first about the z axis. Now that second coordinate system should be moving relative to the first and the torus.

I think the surface of the torus must be frictionless also.

Now in one frame the torus is rotating and in another it is not but the torus is identical in both frames. No way to tell which is which.

Look, we don't need the torus. Imagine the coordinate labels are missing or slip so when you rotate one plane the coordinates stay still.

If the points in the plane are distinguishable then I can say that that point rotated and is now here. But if I cannot distinguish the points then there is no way. If I capture what is there with a camera. Turn around and you rotate the plane then I turn back there is nothing to distinguish the new situation from the old. Don't I need a difference to establish the motion? Locomotion is a species of motion. Motion is change. There is no change here.

Somehow the notion that the points that make up a plane are distinguishable is required to define a rotation. If the points are indistinguishable then we can't define a rotation.

One to one correspondence requires that I be able to distinguish one point from another. But if I have a set of n indistinguishable points how do I know what the number n is. Even countable infinities require that I be able to distinguish the elements to establish the correspondence but if I claim the are identical then...?

Maybe there is only one identical point? Identity? One? They call one the identity element in multiplication because it maps something into itself instead of something else but 6=1 uunless I can distinguish the ones in the six. There is something about distinguishing that creates number.

Now usually we distinguish the points by their position. That establishes some kind of distinguishability. But that's like socks not shoes. The points are like socks not shoes.

I think that the points in a plane must be distinguishable. I can set up a 1 to 1 correspondence between the Cartesian cross product and the points of the plane but given two Cartesian cross products and a rotation relation between them how do I know which coordinates to assign a stationary relationship to those points. If the locations are discreet it doesn't work but with a continuum any rotation maps the points into themselves and there is no way to know whether the point mapped into is the same one as the point mapped from.

It's not just that motion is relative. It's that the elements of the set that is the plane are or are not distinguishable.

I guess my question is can I have a set of n indistinguishable elements and what is the difference between that set and a set with only one element?

If the points in a plane are placed into a bag and all relations of one being separated from the other in a space are removed how many points do I have? One?

Can I have more than one point at the same place? If there is nothing I can say to distinguish them except that they are two...or 10..or an infinite countable or otherwise...then aren't they the same point?

I think perhaps I am confused between the notion of a material particle and a point?

Anyway, I won't be offended if you don't respond to this feel free to focus on any one or two things. No pun intended.

wtf
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### Re: Language Negation and Randomness

Justintruth wrote:Imagine a torus made of a continuous homogenous material.
You're making me hungry again!
Justintruth wrote: Now in one frame the torus is rotating and in another it is not but the torus is identical in both frames. No way to tell which is which.
There is no preferred frame of reference in the universe. Isn't all of this relativity 101? The bagel is rotating with respect to one coordinate system but not the other. I don't know much about physics but my understanding is that all this was worked out by Einstein. There is no preferred frame of reference, and the speed of light is the same in all reference frames. From that, everything follows.

I actually sort of agree with you that in the simple example of a rotating plane, if you look at the plane before the rotation and then after, you can't actually see any difference. If you say that "the points moved," the question is, How do you know? They all look the same. If you just showed up and saw the plane, how would you know if it's the original plane or if it had recently been rotated? There is no way to tell.

So I agree with your philosophical point, but I don't see how this changes math or physics. If I'm facing east and rotate through an angle of pi/2 radians, I'm now facing north. That seems indisputable.

I don't actually know what the philosophical answer is. I know that Poincaré was probably all over this because he pre-discovered relativity before Einstein and was also a philosopher. Maybe he had some thoughts about it.

More later.

Arising_uk
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### Re: Language Negation and Randomness

Justintruth wrote:...

Now in one frame the torus is rotating and in another it is not but the torus is identical in both frames. No way to tell which is which. ...
How do you know one's rotating then?

wtf
Posts: 622
Joined: Tue Sep 08, 2015 11:36 pm

### Re: Language Negation and Randomness

Londoner wrote: Just looking at the word, I would ask what the opposite of 'random' would be? I would think it would involve 'selection' or 'purpose'.

Suppose I was asked to 'give a selection of numbers' and I picked: 2,4,6,8. I would say that as a group they are not random in that all those symbols were examples of numbers; I have purposely not included any letters or punctuation marks.

Technically the opposite of random would be "compressible." Let me explain.

Before explaining I should put this in context. The mathematical ideas I'm talking about are not necessarily related to the various philosophical or linguistic meanings of the word random. We might ask, are the things that happen to us determined? Or are they random? That's a perfectly good use of the word; even though it has nothing at all to do with the mathematical meaning.

Even within mathematics there are various different concepts of randomness. Statistical randomness, etc. Everything I'm saying is in the context of one particular meaning of randomness, namely incompressibility.

Ok, so instead of the set {2, 4, 6, 8}, which is too small to be interesting to us, suppose we had the set {2, 4, 6, 8, 10, 12 ..., 100,000,000}. In other words, All the even numbers between 2 and 100 million, inclusive.

Now I have just described a set of 50 million numbers with the 57 characters in italics above. That's very efficient compression. I could send those 57 characters to you over the Internet, and you would receive those 57 characters and be able to perfectly regenerate a copy of the original set of 50,000,000 even numbers.

So we are not just talking about abstractions. This is about data compression on the Internet. Figuring out how to send 57 characters instead of 50,000,000 numbers.

I hope this is clear. Sets that can described using far fewer characters than the size of the set, are compressible. Sets that are not compressible are random. And there are degrees in between. The fewer characters it takes to describe a set, the less random it is.
Londoner wrote: Now it is also true that we might look at those numbers and see a pattern of relationships between them, but since that pattern has no meaning in the context of 'give a selection of numbers' then I don't think it makes them non-random.
If thre's any pattern at all, and if you can describe the pattern using fewer characters than there are numbers in the set, then you can achieve communications efficiency by transmitting the description of the pattern, rather than the set, across the Internet.

And what is another word for a description of a pattern? How about algorithm? Sets that are generated by short algorithms are not random. I don't need to send you all the infinitely many digits of pi. I can just send you one of the many known algorithms and you can generated the digits for yourself. The digits are highly compressible. They are not random.

[However do note that the digits of pi are highly statistically random. That's a completely different mathematical meaning of the word random].
Londoner wrote: To put it another way, if somebody saw significance in the fact that each number was 2 more than the last, they would be misunderstanding the meaning of the sample.
If I understand you correctly than I disagree. "Start with 2. Keep adding 2. Stop at 100,000,000" is only 48 characters long. You compressed my data better than I did. You understand this perfectly.

The meaning isn't the algorithm. The meaning is the set they generate. We have two different algorithms that uniquely describe the same set of numbers. How the set is generated is not where the meaning lies.

It's interesting that you see the meaning in the method. "Start with 2 and keep adding 2" is not the same as "Even numbers starting with 2." You see these as having different meanings.

And perhaps you are right in some circumstances. For example, I disagree that the ends justify the means. In other words even if two methods achieve the same goal, one method might be moral and the other immoral. So in the real world, the meaning is often in the method as much as in the result.

Justintruth
Posts: 119
Joined: Sun Aug 21, 2016 4:10 pm

### Re: Language Negation and Randomness

wtf wrote:
Londoner wrote: Just looking at the word, I would ask what the opposite of 'random' would be? I would think it would involve 'selection' or 'purpose'.

Suppose I was asked to 'give a selection of numbers' and I picked: 2,4,6,8. I would say that as a group they are not random in that all those symbols were examples of numbers; I have purposely not included any letters or punctuation marks.

Technically the opposite of random would be "compressible." Let me explain.

Before explaining I should put this in context. The mathematical ideas I'm talking about are not necessarily related to the various philosophical or linguistic meanings of the word random. We might ask, are the things that happen to us determined? Or are they random? That's a perfectly good use of the word; even though it has nothing at all to do with the mathematical meaning.

Even within mathematics there are various different concepts of randomness. Statistical randomness, etc. Everything I'm saying is in the context of one particular meaning of randomness, namely incompressibility.

Ok, so instead of the set {2, 4, 6, 8}, which is too small to be interesting to us, suppose we had the set {2, 4, 6, 8, 10, 12 ..., 100,000,000}. In other words, All the even numbers between 2 and 100 million, inclusive.

Now I have just described a set of 50 million numbers with the 57 characters in italics above. That's very efficient compression. I could send those 57 characters to you over the Internet, and you would receive those 57 characters and be able to perfectly regenerate a copy of the original set of 50,000,000 even numbers.

So we are not just talking about abstractions. This is about data compression on the Internet. Figuring out how to send 57 characters instead of 50,000,000 numbers.

I hope this is clear. Sets that can described using far fewer characters than the size of the set, are compressible. Sets that are not compressible are random. And there are degrees in between. The fewer characters it takes to describe a set, the less random it is.
Londoner wrote: Now it is also true that we might look at those numbers and see a pattern of relationships between them, but since that pattern has no meaning in the context of 'give a selection of numbers' then I don't think it makes them non-random.
If thre's any pattern at all, and if you can describe the pattern using fewer characters than there are numbers in the set, then you can achieve communications efficiency by transmitting the description of the pattern, rather than the set, across the Internet.

And what is another word for a description of a pattern? How about algorithm? Sets that are generated by short algorithms are not random. I don't need to send you all the infinitely many digits of pi. I can just send you one of the many known algorithms and you can generated the digits for yourself. The digits are highly compressible. They are not random.

[However do note that the digits of pi are highly statistically random. That's a completely different mathematical meaning of the word random].
Londoner wrote: To put it another way, if somebody saw significance in the fact that each number was 2 more than the last, they would be misunderstanding the meaning of the sample.
If I understand you correctly than I disagree. "Start with 2. Keep adding 2. Stop at 100,000,000" is only 48 characters long. You compressed my data better than I did. You understand this perfectly.

The meaning isn't the algorithm. The meaning is the set they generate. We have two different algorithms that uniquely describe the same set of numbers. How the set is generated is not where the meaning lies.

It's interesting that you see the meaning in the method. "Start with 2 and keep adding 2" is not the same as "Even numbers starting with 2." You see these as having different meanings.

And perhaps you are right in some circumstances. For example, I disagree that the ends justify the means. In other words even if two methods achieve the same goal, one method might be moral and the other immoral. So in the real world, the meaning is often in the method as much as in the result.

http://www.mdpi.org/lin/similarity/similarity.htm

Apparently there is a chemist that believes that symmetry is the key.

Is there a formal mathematical definition of symmetry and what is its relation to Kolmogorov entropy?

Is symmetry required to say anything about something other than you can't say anything about it?

Without the ability to distinguish? One cannot set up the difference between 1 and 0 so there can be no information unless one can distinguish.

Probability theory does not establish what is likely nor the spectrum of possibilities. It starts to calculate only once these are established. Random means there are no symmetries on which to base any description of possibilities. I can say there is a collection of particles in random motion in a gas because of the temporal symmetry of each particle in its rest frame. So there is a random motion but the particle number is conserved.

In a continuum there is no particle number because the particles are uncountable. This leads to resolution of the torus motion problem because the natural identity of the torus based on the symmetries of certain transformations does not alow one to distinguish different positions. However the different positions in the torus can be associated with its being ontologically and so a purely metaphysical identity is created that allows us to distinguish - or rather is the distinction - between "this" point or that. They are not the same points but are distinct. Once that distinction/identity is allowed the torus can move based on it. This means that the material of a continuous substance is not a natural property.

But I can say still certain ontological statements like being is independent of nature. Being may be related to a kind of temporal symmetry of itself that establishes time. Without the fact of existence being symmetric inherently then being itself looses identity and becomes ontologically random in the sense of unsayability. Perceptions of this may underly the Oneness or identity of being.

Wittgenstein's idea that about everything one can say nothing may be equivalent to a statement that about nothing one can say nothing. The symmetry of being allows us to say that if everything ceases to exist then there is nothing. Without it one cannot even say that.

So there are two negation. The negation of things (seinde or "beings" following Heidegger ) and the negation of being itself which eliminates the possibility of saying anything.

This radical nihilation may be related to a notion of randomness and the denial of the ability for anything to be may "be" the most extreme form of randomness. An infinite regress occurs and may be related to the mathematical definition of the limit which is required to define the continuum. Nothingness may be as infinite as being. It would be interesting to know how the empty set and the topology of a continuum are related. Perhaps there is always a subset of a continuum and if the set of subsets becomes empty there is no continuum.

Lucky for us as it appears we are inherently non random just by the symmetry of our temporarily.

It may mean that nothing can be said about life after death however. Not even that it isn't. It may be that we can die only for others.

wtf
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Joined: Tue Sep 08, 2015 11:36 pm

### Re: Language Negation and Randomness

This all seems a long way from ...
Justintruth wrote:I keep running into phrases like this in the study of probability:

"Clearly nothing can be said about (some set)...."

...implying it is random.

I wonder if randomness can be defined somehow as a set expressing an arrangement about which nothing can be said.
I tossed out a mathematical model of what that phrase might mean; and it's my belief that this is what they meant. If they're talking about probability theory they likely intend a technical interpretation of the words. I hope some of what I've written is helpful.

Regarding the philosophy I'm afraid I can't help.

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