General Information Framework

[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).