General Information Framework

[moshek]

Reiman hytpotesis : Hilbert 8 th problem

Dear Organic,

Thank you for sharing with us more of your new creation.
I will study that file abouy Complementary Logic
and respond to you soon.

Until than here below what is consider today
the highst mountain in mathematics:


The Rieman hypothesis


8. Problems of prime numbers
Essential progress in the theory of the distribution of prime numbers has lately been made by Hadamard, de la Vallée-Poussin, Von Mangoldt and others. For the complete solution, however, of the problems set us by Riemann's paper "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse," it still remains to prove the correctness of an exceedingly important statement of Riemann, viz., that the zero points of the function (s) defined by the series

all have the real part 1/2, except the well-known negative integral real zeros. As soon as this proof has been successfully established, the next problem would consist in testing more exactly Riemann's infinite series for the number of primes below a given number and, especially, to decide whether the difference between the number of primes below a number x and the integral logarithm of x does in fact become infinite of an order not greater than 1/2 in x. Further, we should determine whether the occasional condensation of prime numbers which has been noticed in counting primes is really due to those terms of Riemann's formula which depend upon the first complex zeros of the function (s).
After an exhaustive discussion of Riemann's prime number formula, perhaps we may sometime be in a position to attempt the rigorous solution of Goldbach's problem, viz., whether every integer is expressible as the sum of two positive prime numbers; and further to attack the well-known question, whether there are an infinite number of pairs of prime numbers with the difference 2, or even the more general problem, whether the linear diophantine equation

ax + by + c = 0

(with given integral coefficients each prime to the others) is always solvable in prime numbers x and y.
But the following problem seems to me of no less interest and perhaps of still wider range: To apply the results obtained for the distribution of rational prime numbers to the theory of the distribution of ideal primes in a given number-field k—a problem which looks toward the study of the function k(s) belonging to the field and defined by the series

where the sum extends over all ideals j of the given realm k, and n(j) denotes the norm of the ideal j.
I may mention three more special problems in number theory: one on the laws of reciprocity, one on diophantine equations, and a third from the realm of quadratic forms.

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

David Hilbert ( 1900 Paris)

 

[Organic]

More about Complementary Logic (CL):

If you look at: http://www.geocities.com/complementarytheory/CompLogic.pdf

You will see that i use the word 'Balance'.

According GIF point of view, set is only a framework, which means a set with no name (or no property).

Let us notate a set with no name as: }{

By CL there are at least two opposite concepts: {} and {__}.

In this case the minimal GIF structure is:

{}          {__}
|            | 
|            |
|____________|
      |
     } {

where:

{} = Empty set

{__} = Full set

}{ = Balanced framework

From this point of view, the set's concept always transcendent its contents, therefore the minimal CL structure that we see above can be a building block for any Organic-Complex system.

And by using the word 'organic' i mean that any CL system is based on its inherent balance.

The useful information of CL is in the open interval of ({},{__}).


Organic

 

[phoenixthoth]

aren't "balanced network" and }{ names?

 

[Organic]

Anything within some model cannot be the concept itself.

For example: Any model of simplicity cannot be the simplicity itself.

Any model is only a dummy variable of the concept itself.

This is the unclosed gap between theory and reality.


Maybe this is the deep meaning of Godel's Incompleteness Theorems.


I believe that Evolution can exist only because of the gap between potential existence (some model) and actual existence (some reality).


So, in a model we have no choice but to give to any concept some name, otherwise we cannot deal with this concept in our model (which is definitely not the concept itself).



Organic

 

[moshek]

Some way to look on Godel theorems.

Organic Hi,

I read the two papers about your Complementary logic
and t 4 break points model. and it look very interesting.

Thank you.

Well we know that there are fuzzy logic,
or 3 stage logic e.t.c

In your case it look a little different.

It seem that object have in themselves
some combination of all the possible true
so it is some kind of a duality that you are pointed to.

I agree with you that one of the main problem
in mathematics today beside Rieman Hypotesis H(8),
is how to take Godel theorem and find a new
but a positive interpretation.

So what do you say?

Moshek
[?]

 

[moshek]

Hilbert(3)

3. The equality of two volumes of two tetrahedra of equal bases and equal altitudes

In two letters to Gerling, Gauss expresses his regret that certain theorems of solid geometry depend upon the method of exhaustion, i. e., in modern phraseology, upon the axiom of continuity (or upon the axiom of Archimedes). Gauss mentions in particular the theorem of Euclid, that triangular pyramids of equal altitudes are to each other as their bases. Now the analogous problem in the plane has been solved.6 Gerling also succeeded in proving the equality of volume of symmetrical polyhedra by dividing them into congruent parts. Nevertheless, it seems to me probable that a general proof of this kind for the theorem of Euclid just mentioned is impossible, and it should be our task to give a rigorous proof of its impossibility. This would be obtained, as soon as we succeeded in specifying two tetrahedra of equal bases and equal altitudes which can in no way be split up into congruent tetrahedra, and which cannot be combined with congruent tetrahedra to form two polyhedra which themselves could be split up into congruent tetrahedra.


David Hilbert ( 1900 Paris)

 

[Organic]

Hi Moshek,

As much as i know, 2 or 3 valued logic and also fuzzy logic do not connect concepts like symmetry-degree, information's clarity-degree, redundancy, uncertainty and complementarity into one organic information-system, which you can find in Complementary Logic system.

To know the limits of some system is a very important insight.

Without this insight i think we can't know if what we are doing is running in circles, or moving in a progressive path.

So, from this point of view, Godel's Incompleteness Theorems are positive Theorems, because they discover the unclosed gap between any model to reality itself.




Organic

 

[moshek]

I agree with you Organic,

But I am afraid that some mathematicians
believe that Godel theorem ( 1931)
say that mathematics has a limit!

Well?


Moshek

P.S : A student of Hilbert, Dehn (1901)solved
the 3 th' problem in a negative way.

 

[Organic]

Dear Moshek,

In my opinion any model has limits, and these limits is what we call reality.

 

[moshek]

I agree with you on that Organic.

Like the discovery of the existence
of Irrational number by Hippasus
by using self similarity inside the pentagon
that was contradiction to
"Everything is a number" by the Pythagoreans.
( He was also a Pythgoreans)

And not by the root(2) as can be misunderstood
in the 10 book of the Element.
Or in the dialog Theatetus by Plato ( 399 B.C)

 

[Organic]

moshek,

Do you mean that Hippasus reseached fractals more then 2000 years ago?

 

[moshek]

taken from: http://www.tmth.edu.gr/en/aet/1/59.html


MATHEMATICIAN
HIPPASUS OF METAPONTUM (fl. 5th century BC)

Life
Mathematician from Metapontum in Magna Graecia (south Italy) and disciple of Pythagoras, Hippasus established the "mathematical section" of the Pythagorean school. He is cited by Diogenes Laertius, Iamblichus and Suidas. Fragments of his work survive.


Work
Hippasus discovered that the ratio of the side to the diameter of a regular pentagon is an incommensurable number (The pentagon was a sign of recognition among the Pythagoreans.) His teaching differed from that of the orthodox Pythagoreans, in that he believed that the origin of the world was material (fire), whereas the Pythagoreans held it to be immaterial (numbers).

"Mysteries": Treatise published (according to Diogenes Laertius) under the name of Pythagoras.

Hippasus constructed vessels with different quantities of water and metal discs of different thicknesses and performed experiments in acoustics. His experiment with copper discs confirmed the proportionals of acoustic resonance.

Iamblichus tells us that Hippasus established a circle of "acousmatists", a group that studied the science of acoustics and its applications.

 

[Organic]

Hi moshek,

What do you think about us (self awared complex systems) as associators between potential existence (some model) and actual existence (some reality).