FOM: Visual Mirror-Like Proofs.

[Alexander Zenkin]

 Dear Colleagues,

   Some months ago (Feb-March'99) some of you were taking an active part
in the very interesting, useful, and topical [FOM]-discussion concerning
the use of VISUAL PROOFS in logic, mathematics and mathematical
education. Today, it is really a very important problem in connexion
with the developement of such new directions as Visual Mathematics,
Experimental Mathematics, Visual Inference in Logic, Cognitive Computer
Visualization of Mathematical Abstractions in Number Theory, New
Multi-Media Technologies in Mathematical Education, etc.
    As is known, the Continuum Problem is one of the basic problems of
Mathematics. Using the Cognitive Computer Visualization approach, I
visualized the Continuum Problem (the Problem - not one of a lot of
Continuum Hypothesis formulations) and obtained a lot of quite
unexpected and non-trivial results. In particular, the result concerning
the existence of Cantor's minimal transfinite number omega. As is well
known, the original Cantor's introduction of the omega in Mathematics
was an act of trust. Modern axiomatic set theories did not go much
forward. Using a so-called VISUAL MIRROR-LIKE PROOFs, we obtained the
following absolutely rigorous ONTOLOGICAL CONDITIONAL statement.

THEOREM. IF there EXISTS the common geometrical point (IN ANY SENSE
which we wish!)
THEN the Cantor's omega EXISTS IN THE SAME SENSE.

COROLLARY. IF the Cantor's omega DOES NOT EXIST as an individual thing
(IN ANY SENSE which we wish!) THEN the common geometrical point DOES NOT
EXIST as an individual thing too IN THE SAME SENSE.

   These statements make clear the ontological status and the
ontological correlation of such the important concepts of the F.O.M.
(Foundations of Mathematics) as the geometrical point and the Cantor's
minimal transfinite integer omega.

   More details are discribed in the paper "COGNITIVE (SEMANTIC)
VISUALIZATION OF THE CONTINUUM PROBLEM AND MIRROR-SYMMETRIC PROOFS IN
THE TRANSFINITE NUMBERS THEORY" that is accessible at the WEB-Sites of
the "VISUAL MATHEMATICS" e-journal:
http://members.tripod.com/vismath1/zen/index.html

  All comments as to the authenticity of VISUAL PROOFS presented in the
paper are welcome.

Alexander Zenkin.

> ############################################
> Prof. Alexander A. Zenkin,
> Doctor of Physical and Mathematical Sciences,
> Leading Research Scientist of the Computer Center
> of the Russian Academy of Sciences,
> Member of the Philosophical Society of the Russia,
> Full-Member of the Creative Unity of the Russia Painters.
> e-mail: alexzen at com2com.ru
> WEB-Site   http://www.com2com.ru/alexzen
> "Artistic "PI"-Number Gallery":
>http://www.com2com.ru/alexzen/galery/Galery.html
> ############################################
> "Infinitum Actu Non Datur" - Aristotle.
> "Drawing is a very useful tool against the uncertainty of words" -
Leibniz.