FOM: Computer pictures of mathematical abstractions

Alexander Zenkin alexzen at
Thu Feb 5 19:17:54 EST 1998

Name: Alexander A. Zenkin
Position: Leading Scientific Co-Worker (Emploee), Professor of Computer
Institution: Computer Center of the Russian Academy of Sciences, Moscow,
Research interest: Number Theory, Logic, Foundations of Mathematics,
Cognitive Computer Visualisation of Mathemarical Abstractions
More information:

I agree, but not entirely, with Harvey Friedman (Date:  Tue, 3 Feb 1998
08:14:45 +0100) when he agree with John Steel:

If one is doing pictures of set theory, then a picture is an
interpretation (in the informal sense of meaning-assignment) of the
language of set theory.

Such the interpretation can't be a kind of a 1-1-correspondens between a
set of sentences in the language of set theory and its picture, because
the picture that visualizes an abstract conceptions contains as a rule
much more information than even the author of the picture have had in
his mind. For example, Americans D.Lehmer and E.Lehmer are known as the
eminent specialists in the most abstract field of the modern mathematics
- in analytical Number Theory. As far back as 1980, they visualized
rather complex trigonometrical sums in the form of "very beautiful
computer pictures". By the authors words, this pictures prompted them
(to the high professionals in that field of Number Theory!) "some
interesting and unexpected hypotheses that were successfully proved
further". As to their emotions inspiring by the computer pictures, the
unusual for respected mathematical journal title of their paper
"Picturesque exponential sums" (J.Reine Angev. Math., v.318, 1-19
(1980)) speaks for itself.
Following the Lehmers, during 1979-1985  I have worked out the computer
system DSNT (the Dialogue System for investigations in additive Number
Theory) and have visualized the famous Classical Waring Problem (CWP)
concerning representation of natural numbers by sums of s r-th powers of
non-negative integers. The meaningful pictures, generated by DSNT
(so-called Pythograms of Number-Theoretical abstractions), allowed me
first to see and then to prove a great numbers of non-trivial,
conceptually new Number-Theoretical results. For example, the following
nice theorem (here the parameters n, m, r, s are natural numbers, and
N(m,r,s) and Z(m,r,) are sets of natural numbers).

   Theorem 1.  For any m>=0, r>=2 there exist 1) a finite number g(m,r),
and 2) a finite invariant set Z(m,s), such that for any s > g(m,r)
N(m,r,s)={s+z : z (- Z(m,r)}.

This theorem gives the complete solution of a so-called Generalized
Waring Problem (GWP) which was seen, formulated and proved by means of
the DSNT-system. GWP holds for all m = 1,2,3, ┘, and only by m=0 we have
well-known CWP which was formulated in 1770 and solved by D.Hilbert in
1909. Near 300 years, Euler, Lagrange, Gauss, Legandre, Hilbert, Hardy,
Vinogradov and many many other outstanding mathematicians investigated
the CWP (m=0) and even did not guessed on the very existence of the
1,2,3,┘ -levels of the Problem. Only the computer visualization of a
Number-Theoretical essence of the CWP, just the computer pictures, the
pythograms of mathematical abstractions allowed to see such the
possibility itself. (More info in mathematical and visual form - at my

As concerns knowledge-generating possibilities of such the pictures,
there is a very remakable regularity: the more abstract the conception
you picture (visualize) is, the more probable to see new and unexpected
aspects of the conception is, even if the conception seems to be well
known for you (as you think a priori). Maybe, it is a result of an
unknown non-rational interaction  between the picture and a professional
semantic memory of its creator (at a level of a powerful creative

Once more interesting related fact. I think, Gotfried Leibniz understood
well and appreciate highly the meaning of the visualization of
mathematical abstractions. It is his words:"I think that figures are
very useful means against the uncertainty of the words" (Coll., vol.2,
pp. 392, the last Russian edition of the 90s). Further, as is known, he
attend the lectures of the German mathematician and astronomer Erchard
Weigel (1625-1699) which "thought out wittily figures for a
representation of moral things (l.c., pp.393). Of course, "moral" is not
"mathematical" but G.Leibnitz is not a man which spends so many time for
nothing. "This figures are useful, - writes he, - in order to awake a
thought" (pp.393).

But as to the use of such the "figures" of the XVII Century as well as
the computer pictures of the XX Century for the proof of mathematical
theorem, the opinion of mathematical community is unanimous:

Gotfried Leibniz: "But this figures serve rather for a memory, for a
retention and an allocation in order ideas, then for an assertion and an
acquisition of a demostrative knowledge" (l.c., pp. 393).
Harvey Friedman: "There is no notion of true in a given picture. There
is only "formally derivable from a given picture. And this is always
Steven G.Krantz: "I've never used any kind of picture to aid my
thoughts. ┘But the pictures do not prove anything" ( "The Immortality of
Proof". - Notices of AMS, January 1994, Vol. 41, no. 1, pp. 10-13).
My results obtained by a so-called Cognitive Computer Graphics (CCG)
allow me to make some other statement: under certain conditions,
CCG-pictures can be strict arguments in the regorous mathematical
proofs. However, for this aim I was forced to discover a new method for
proving general mathematical statements of the form: "for any n P(n)".
Now, I can prove some of mathematical theorems by means of an ostensive
showing the corresponding CCG-images. For example, using such the method
(called as superinduction method), I have formulated and proved the
generalization (for m=1) of the Lagrange theorem (1770) for sums of
squares (m=0), the Wieferiches theorem (1909) for sums of cubes (m=0),
the famous theorem of Balasubramainan, Deshouillers, and Dress (1986)
for sums of biquadrates (m=0),  and many other number-theoretical
theorems. I see well that it is not easy to believe that it is possible
to discover an alternative method to the well-known method of the
complete mathematical induction in the very ending the XX Century.
Therefore, I would like to emphasize here that all these results have
been published and approved by international number-theoretical
community. (For more info see my web- site.)

In conclusion of the message, I would like to express my hard confidence
that the computer visualiztion of mathematical abstraction of the most
high level will allows us to understand much better many difficult
problemss of mathematics, and, in particular, of the Foundations of

And the Harvey Friedman idea to visualize the continuum seems me very
fine and therefore very perspective.

Alexander Zenkin

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