Some comments on information theory

  • Thread starter Thread starter Cleopatre VII
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I apologize for not responding sooner. I am thinking a lot on how I can do this, present my ideas in a simple more understandable way. I agree this is going no where and there is a huge communication gap. I need a couple of days to see if I can put together one final post that can better express my ideas. I am pretty sure you will be able to follow it or perhaps others will be able to and as a warning, it will be wordy, but I do not have your tools. I will give it one more shot and this can be put rest. Thank you for being patient.
 
Good. Just make sure that everything is clear and logically one thing follows from another. Do not assume that the reader is able to read your mind. Avoid logical jumps. Do not follow bad examples, like in the mathematical paper I have linked! This IS NOT A CLEARLY WRITTEN PAPER AT ALL! Take one small step at a time. Instead of writing one big paper, write small post, with one single step in a post. Then we can discuss this one step. After one step is cleared, you may take another step.
 
ark said:
Good. Just make sure that everything is clear and logically one thing follows from another. Do not assume that the reader is able to read your mind. Avoid logical jumps. Do not follow bad examples, like in the mathematical paper I have linked! This IS NOT A CLEARLY WRITTEN PAPER AT ALL! Take one small step at a time. Instead of writing one big paper, write small post, with one single step in a post. Then we can discuss this one step. After one step is cleared, you may take another step.
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Thanks.
 
Post #1

My paper is simply a 'thought experiment', a Gedankenexperiment. It starts with one simple math object and I also do bring in C's quotes in many places, but not yet.

What are illustrations 2.4.1 and 2.5.1? Erase the lines in them. They are the same thing. They are both Euler's infinite product from Theorem 7, in its form where each term of his infinite product is expressed as a prime power series. That is where the thought experiment starts. I took the right hand side of Euler's derivation, the harmonic series on the left, an equal sign in the middle, and the infinite product on the right. I am thinking only about the right hand side, the infinite product, in a thought experiment.

I was playing with Euler's product in my mind. Then I used the definition of the representation of any inverse n (inverse natural number) can be written uniquely as an infinite product where each term is a prime number and the exponent has choices (zero or a negative integer). There is one restriction though on the choice of exponents, and that is, you can only choose finitely many terms (prime numbers) with an exponent that is non-zero.

I began to create inverse natural numbers as infinite products, one at a time. First the infinite product for 1, all terms having the choice of exponents = 0. Instead of recording that choice in some list somewhere, I used illustration 2.4.1 (with no lines yet) to record that one number. The first line would be a straight line where the choice of exponents are all 0. Next is the number 1/2. What is its choices for exponents? Record it as one unique line in illustration 2.4.1. That is the process. One number, record it as one unique line.

After having completed that process a sufficient number times, especially if I create and record one number at a time in order, 1, 1/2, 1/3, 1/4, ..., I realize every inverse natural number does exist in Euler's infinite product. At this point I stop and make two observations.

1. I observe that every line I have drawn has a similar pattern. In the infinite progression to right for every line drawn, there is a point where every line drawn becomes the same. When creating one inverse n, because we have a constraint on what is possible to select for exponents, every line at some point becomes an infinite straight line to the right where every exponent = 0. I call that a signature that every inverse n possesses.

2. I observe that when looking at all of the unique lines/paths I have drawn, there are a bunch of lines/paths in there that exist in Euler's product, that will never have a line drawn through them. These lines/paths are there in Euler's infinite product. An example would be the straight line created by selecting the value of the exponent for each term as = -1, or all exponents as = -2, etc. Do they exist or not? They are there. There is an exact precise reason why they cannot exist even though I see them. The constraining definition for the representation of an inverse natural number as an infinite product, that only finitely many terms can be selected where the selection of the value of the exponent is non-zero. So I can see these infinite numbers in there, but they do no exist, even though they really are valid numbers. Well what to make of that?

That is my next post.
 
"What are illustrations 2.4.1 and 2.5.1? Erase the lines in them. They are the same thing. They are both Euler's infinite product from Theorem 7,"

I do not see it. Please explain.
 
ark said:
"What are illustrations 2.4.1 and 2.5.1? Erase the lines in them. They are the same thing. They are both Euler's infinite product from Theorem 7,"

I do not see it. Please explain.
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1) When I say, "What are illustrations 2.4.1 and 2.5.1? Erase the lines in them."

Means, as a starting point, just remove the lines as if none of the lines have been drawn yet.

2) What do I mean, "They are both Euler's infinite product from Theorem 7"

It is what I see when I look at the Euler Product.

2/1 x 3/2 x 5/4 x 7/6 x ...

The first term of Euler's product:
2/1 = 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...

The second term of Euler's product:
3/2 = 1 + 1/3 + 1/9 + 1/27 + 1/81 + ...

etc.

-------------

Most derivations of Euler's product include this in the derivation. One of many examples is Wolfram MathWorld's page on the Euler Product.

This link should take you to Wolfram MathWorld's page on the Euler Product, using a browser feature called "link to highlight. Clicking the link should take you to MathWorld's page on the Euler Product and display the exact line I have highlighted. If it does not work, just go to the equation labeled (3). Look at the third line in the equation labeled (3).

Wolfram MathWorld - Euler Product
 
ark said:
Thanks. That becomes my level. I am starting to understand what you may have in mind. Now, in Euler Product -- from Wolfram MathWorld
it is assumed that s>1. While it seems to me that you are taking s=1. Am I right?
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Yes, just s=1.

In many of these web pages (references) about the Euler product they jump steps and call it different things. I think in Euler's original paper, "'Variae observationes circa series infinitas'", there was no real valued variable in the exponent. But later that was a next step of his and many references actually call that next step with the real valued variable in the exponent, Euler's zeta function, the precursor to Riemann's zeta function.
 
How do you know that for s=1 the infinite product is converging? I plotted your product for first 1000 primes (every 10 primes) and it is not clear from the plot that the limit is finite, though the rate of growth seems to be slowly slowing down:

primes_products.jpg
So, how do you know that your expression has a meaning?
 
ark said:
In fact it looks like a plot of a Log function, which has no finite limit....
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I am not sure if I follow you. What expression of mine and where did I set any variable s to anything. I am only working with the original Euler Product from "Variae observationes circa series infinitas". In Euler's paper there is no variable. But having no variable is equal to setting the variable s = 1, in the reference I linked to in Wolfram. I don't see where I made any implication about something convergent or divergent. It is entirely possible that I did do that and don't realize it.
 
Didn't you say "Yes, just s=1."? Didn't you write "2/1 x 3/2 x 5/4 x 7/6 x ..."
I am simply saying your formula "2/1 x 3/2 x 5/4 x 7/6 x ..." seems to have no mathematical meaning. It is a nicely looking "expression", but does it represent something mathematically meaningful? It seems to me that it does not, and my graph seems to support this conclusion. How do you know that "2/1 x 3/2 x 5/4 x 7/6 x ..." is meaningful? Please, explain.
 
ark said:
In short: it seems to me that your 2/1 x 3/2 x 5/4 x 7/6 x ... is ∞. Do you think it represents a finite number?
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No. I think it represents the Euler Product.

And in response to your post #462:

I kind of get what your meaning is and I will try to explain myself. But it will require a couple of C's quotes that are your own interactions with the C's (two interactions) and then a real life example in mathematics that applies.

If I understand your meaning in regards to: "formula "2/1 x 3/2 x 5/4 x 7/6 x ..." seems to have no mathematical meaning".

I think you are saying "Just looking at the Euler Product as a stand-alone mathematical object/formula has no mathematical meaning". It only has mathematical meaning if you do something with it. Like Euler's Theorem 7 that derived the infinite product from the harmonic series or his next step, where including a real valued variable into the exponents can be written as a function and using values for s in that function, we can see that the result is either convergent or divergent and deduce that there is a specific range for the values of the variable s that defines convergence or divergence for the function. Those things have mathematical meaning, but the formula for Euler's infinite product as just some stand-alone mathematical object, on its own, has no inherent mathematical meaning. If my starting point was at least an equation, an equality, then it would have some mathematical meaning, because at least an equality expresses a mathematical relationship that gives it some mathematical meaning.

It seems to me that we are back to square one. The Euler infinite product as just an expression in itself is a ratio of two infinite numbers. If I would at least start with an equation, an equality, as a starting point, it would at the minimum show a mathematical relationship, it would have some mathematical meaning.

I think that is what your are getting at. I could be wrong.

But I think the stand-alone mathematical object (Euler's Product) is useful. It has all kinds of information in it. It has mathematical operations, it has mathematical relationships, it has the information within it of the 'Fundamental Theorem of Arithmetic' and the definition of an inverse natural number. It should have within it all of the elements of the harmonic series. It has all of that information within it and I think it is useful to analyze that information. Sometimes looking deeper, reveals something interesting and unexpected. That is where this thought experiment starts. The next step would be that I do see a piece of information in Euler's Product that is there, but it probably should not be there. There is an exact reason why it is there, but the only reason for ignoring it, is because of one single definition. If you remove that one constraint that is created by one definition it leads to all kinds of new interesting possibilities.

I think analyzing all information has value. I think you would agree to that, but maybe I am going entirely in the wrong direction and what I say makes no sense to you at all. I am not sure.

I was waiting to introduce two quotes from the C's sessions in a near future post. If we ever get to a point where you can understand my process and these conversations do continue, there will be many more C's sessions quotes, that I also think are pertinent and may reveal the meaning of some of their concepts. So many of the C's concepts lead to contradictions and confusion. I am not sure that you can make sense of any of my thought processes. I am pretty sure that one of the main reasons for you even having this conversation with me is external consideration expressed towards me, and I really do appreciate that. But I also know that with each interaction with me here you must evaluate if it is even worth your time.

You can ask questions about what I have written above for clarification. Am I way off in the point you are trying to make in regards to: "formula "2/1 x 3/2 x 5/4 x 7/6 x ..." seems to have no mathematical meaning"?

This post is getting a little too verbose and lengthy already, but I want to make one post tonight about those two C's sessions quotes that are interactions you have had with them and the pertinence of them to the conversation we are having and this process of mine that I am trying to demonstrate. In fact if I can find a way to be clear, the pertinence of those two C's sessions quotes becomes more and more important.

My tonight is about 6-8 hours from the timestamp of this post.
 
"No. I think it represents the Euler Product"

Do you agree with me that for s>1 the Euler product, as defined in Euler Product -- from Wolfram MathWorld, is convergent to a finite number, while for s=1, as in your case, it is just a formal expression that diverges to plus infinity?

The point is that we need to be clear about facts. Only after we are completely clear about our 3D reality, only then we can meaningfully attempt to cross the reality boundaries and wander into still unknown dream lands.

You see, 1+1+1+1 ..... = 2+2+2+.... = 3+3+3+... = ∞, so comparing such expressions is dangerous. You can try to deduce from that 1=3, which is evidently not true.

So, do you fully agree with my statement that I put in bold above?
 
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