Re: [HM] Mathematical proofs


Subject: Re: [HM] Mathematical proofs
From: Alexander Zenkin (alexzen@com2com.ru)
Date: Mon Jan 10 2000 - 17:26:53 EST


[Mary Catherine Leng]
>
> I am doing some philosophical work on the process of mathematical
> proof, and as a result I am looking for historical examples of
> mathematicians talking about how they went about discovering or
> proving particular mathematical results.
> ...

[John Conway]
:
: Let me immodestly report on one of my own. I had just, with
: Peter Kleidman and Robert Wilson, found a construction that seemed
: to give a very large number of projective planes. The problem was
: to determine just how many of them were really distinct.
: ...
:
: It was an afternoon in late winter, when it gets dark by about
: 4pm in England. I went home at about that time, and deliberately
: decided to just sit and think in the dark, and in the event I did
: so for 3 or 4 hours, thinking very hard indeed (so hard that I
: sweated considerably!).
: ...

[Walter Felscher]
|
| While I am very skeptical about the possibilities to discover
| insights on what is characteristic, or typical, for the ways in which
| mathematicians find their proofs, I will add a most curious story
| about how I actually dreamt a proof once.
| ...

[Antreas P. Hatzipolakis]
$ ...
$ Thanks to Prof. Walter Felscher and John Conway who shared with us their
$ recollections about their discoveries.
$
$ It would be very interesting to read recollections of other distinguished
$ mathematicians who inhabit this list.

                             ----------
Dear Antreas,

    It is really fine and very interesting not only for mathematical
historians but also for philosophers, educators, and mathematicians
themselves to know how such distinguished mathematicians found some of
their famous discoveries. First of all because in mathematics, during a
very long time, there exists a quite strange situation, which is best
described by Einstein's words: "The tendency to 'objectivity'
<rationality> put a taboo upon all 'personal' <intuition>". That tendency
led the mathematics to the position when scientific work results are
religiously and carefully saved (the notorious 'objectivity') for future
generations, but the individual creativity "technologies" (the notorious
'personal'), generating these scientific achievements and discoveries,
are threw away, rejected, and forgotten without regret and foresight. So,
every new generation is forced to waste time and force in order to invent
afresh this "technologies" beginning, practically, with an empty place.

    Imagine a society where goods are religiously and carefully saved,
but the technologies themselves for their manufacture are destroyed and
threw away. It is obvious that such the society has not a future. Alas,
almost all our science is just a such "society".

    I believe that today one of reasons for such the dangerous situation
is a false paradigm that the only "technology" for mathematical
knowledge production is a formal deduction within the framework of
formal systems of modern meta-mathematics, or, shortly, a
"bourbakization" (Prof. V.Arnold's term) of mathematics against which
already G.Berkeley, H.Poincare, Von Neumann, etc. warned.

    But look here, as far back as the 16th -17th centuries, the
Portuguese mathematician Pedro Nunes and John Wallis wrote (see Joao
Filipe Queiro's [HM]-message on 08 Dec 1999): "O how well had it been if
those Authors, who have written in Mathematics, had delivered to us
their Inventions, in the same way, and with the same discourse, as they
were found out!" - Unfortunately, they are not heard ad interim.

    In this connection, the problem of new knowledge generation is
especially interesting for me, mathematician, since I really have a
quite non-standard, but very modern TECHNOLOGY which allows TO DREAM
mathematical discoveries and their PROOFS :-) - and to deliver that
"esoteric" process itself to any "advanced" child :-)

    So, let me report some words on this technology. I hope that it will
be interesting in connections with the [HM]-question (30 Nov 1999
17:59:22) of the philosopher Mary Catherine Leng as to "the processes by
which specific mathematicians went about proving specific theorems ",
and with a lot of stories about mathematical "discoveries in dreams"
told by John Conway (1 Dec 1999), Walter Felscher (1 Dec 1999), and
others.

    So, according to the known Poincare's opinion, the most liked line
of work for all mathematicians is to generalize a non-trivial theorem of
some of their predecessors. All mathematicians estimate most of all such
results where some problem is generalized from an only particular case,
say, m = 0 to all values m = 1, 2, 3, ...

    In the very end of 70s, I FORMULATED a quite non-trivial generalization
of the Classical Waring's Problem (m=0) to any m = 1, 2, 3, ... But to
FORMULATE something in Mathematics does not mean to PROVE that. And
therefore, I began to look for a proof. But during 2-3 months without
visible success.

    As it is well known, the Classical Waring's Problem was formulated
by English mathematician E.Waring as far back as 1770. Many outstanding
mathematicians was trying to prove the Waring's Hypothesis (Euler,
Lagrange, Gauss, etc.). But it was only David Hilbert who gave, at last,
a complete solution of the Classical Waring's Problem in 1909. Many
number-theorists believe, that D.Hilbert's result is one of the most
important achievement of the Number Theory-XX, since all modern analytical
Number Theory started from the D.Hilbert's ideas underling his solution.

    Of course, I had known quite well all these achievements, but it was
perceived by me as a quite far "history". However one day, after a lot
of hours of my fruitless attempts to find out a proof of the Generalized
Waring's Problem, I suddenly distinctly realized that such the proof
would be a generalization of that famous D.Hilbert's proof for m=0 to
any case m>=1.

    I never was an immodest person, therefore that thought became a
fatal for me: I understood well that to engage in intellectual
competition with the Great Hilbert himself and, moreover, to generalize
his most famous number-theoretical achievement is simply senseless and
that I will not be able to do my proof ever, "by definition". So, I
simply stopped to engage in that problem. - No thought on the problem
during 2-3 months! I can swear on Holy Bible! :-)

    But one fine morning I awaked with a happy FEELING that I HAVE the
PROOF. I set at the table and began TO WRITE the proof. I was writing
during about 7 hours almost not thinking and without any coffee-breaks!
Sometimes, it seems to me that I write from dictation of someone
standing behind my back. Next day I typed the writing without even one
correction and inscribed all quite complex (three-storied) mathematical
formulas (about 3/4 of the text). The typewritten manuscript was about
18 pages. You see, it was not possible to have all this in a normal head
in advance and even not to guess about it.

    Next week, I was reporting that PROOFS of that GENERALIZED (for any
m=1,2,3, ...) Waring's Problem in the Number Theory Seminar (the head:
academician I.M.Vinogradov) of Steklov's Mathematical Institute (the
famous MIAN of the Academy of Science of the fUSSR). The time (2 hours)
of the one meeting turned out not enough, and my report was continuing
all next meeting. As a whole, my report on that PROOF was lasting about
FOUR hours! Without even one correction, objection and long discussions.
As every mathematician knows well, the I.M.Vinogradov's School and his
Seminar were most respected in the Soviet analytical Number Theory
community.

    So, the "Voice", dictated me that PROOF that fine day (not an idea,
a way, or a scheme of the proof! But just the COMPLETE 18 pages PROOF
with all formulas!), certainly, does not know mistakes. :-)

    Today you can read that PROOF in one of the most respected Soviet
Mathematics journal:
 A.A.Zenkin, Generalization of Hilbert-Waring's Theorem. - Vestnik of
Moscow University, Ser.1, Math., Mech., 1983, No. 2, 11-19.

    In the following years, the "VOICE of NIGHT" did not leave me. I
became quite quickly familiar with that new "technology" of scientific
discoveries: going to bed, I imagined a problem most distinctly, and, in
the next morning, wrote (from dictation!) its complete solution.

    Being a scientist, I realized some of that "VOICE of NIGHT"
technologies in my Cognitive Computer Graphics (CCG) System for NEW
scientific knowledge discovery, and the corresponding color-musical
CCG-images (so-called PYTHOGRAMS) of high abstract mathematical objects
and conceptions made much more effective the initial stage of my
contacts with the "VOICE of NIGHT" (the clear imagination of a problem
before going to bed).

    By means of such the way, a lot of real unique scientific
discoveries was made, proved, and then approved by mathematical
community.

    For example, a NEW kind of FINITE mathematical objects, so-called
invariant sets of the Generalized Waring's Problem, Z(m,r), m=1,2,3, ... ,
r = 2, 3, 4, ..., constructed from a common (!), finite (!) Natural Numbers,
was discovered. I think, philosophers and psychologists will appreciate
rightly the EPISTEMOLOGICAL sense of such the discovery in the very
beginning of the well-known series 1, 2, 3, ... which is investigated by
the Humankind during about 5000 years, and which is not only the foundation
from which, according to Poincare, Bourbaki, etc., all modern Mathematics
is derived, but is also the important element of the Humankind
Intellectual Culture as a whole.

    Further, these new mathematical objects themselves (the Theorem on
their finiteness) CCG-SHOWED a NEW UNIVERSAL property of that common
finite Natural Numbers, a NEW kind of logical and mathematical
statements (so-called EA-Theorems), a NEW METHOD (so-called
Super-Induction), basing on such the EA-Theorems, for rigorous proving
common mathematical statements by means of the Pythograms, and so on.
I am emphasize here once more (see my message "About some epochal
FOM-jokes" to the [FOM]-list (Wed, 12 Aug 1998 00:29:16)) that the
Super-Induction method is a natural generalization of the Classical
Mathematical Induction B.Pascal's method which already during about 300
years is a basis for any rigorous mathematical, logical, and even
philosophical inductive reasonings.

    Some of these mathematical discoveries and the CCG-PROCESS ITSELF of
their production and proof are quite in detail described in my
monograph "Cognitive Computer Graphics. Some Applications in Theory of
Natural Numbers "
(Synopsis for epistemological philosophers and cognitive psychologists
can be seen at

http://www.com2com.ru/alexzen/papers/synopsis&contnt-e.html )

    Many years ago, by means of the CCG-visualization, a new kind of
GEOMETRICAL objects was discovered. It occurred that the common INFINITE
series of squares of Natural Numbers { 1, 4, 9, 16, ... } generates in
CCG-space (in the 2D-CCG-representation of the common 1D-series of the
common Natural Numbers) FINITE VIRTUAL PARABOLAS which are moving and
leaping along the common series of the common Natural Numbers. Some of
them, being described mathematically, represent an isolated, solitary
wave with an increasing, but finite amplitude which is running along the
series of Natural Numbers, i.e., a VIRTUAL PARABOLIC SOLITON. Such the
objects exist VIRTUALLY in the series {1,2,3, ...} and are not visible and
not perceived. Therefore nobody could see them ever. And only under the
CCG-visualization, they become ACTUAL, visible and accessible to
mathematical investigations.
    This old CCG-discovery can be seen (in direct sense) at the
WEB-address:

http://www.com2com.ru/alexzen (the CCG-movie in the very beginning of
the page)

and at:

http://www.com2com.ru/alexzen/vgeom/vgeom.htm

    Last year, known Russian mathematician of Steklov's Mathematical
Institute, academician Boris Stechkin, inspired by the virtual parabolic
solitons (he had been attending some my CCG-lectures), made the
following unique, elegant, and simply beautiful discovery in a
collaboration with other academician of Steklov's Mathematical
Institute, Prof. Yu. Matiyasevich.

     As is known, as far back as 1841, well-known German mathematician
Moebius (recall the famous one-sided Moebius band!) was using a graphic
representation of a HORIZONTAL parabola (even by the modulus 16 as to
which especial sense see WEB-addresses shown below) as a nomogram for
multiplication of Natural Numbers. That his work layed the foundation of
modern nomography. But nobody from that time guessed to stand that
graphic parabola VERTICALLY. It was first made by Boris Stechkin and Yu.
Matiyasevich. And it turned out that if to draw ALL chords connecting
all integer points of the parabola, lying on its opposite branches, then
such chords will intersect the vertical axis of the parabola in the
integer points which are PRODUCTs of ALL integers. In such the case, ALL
integer points of the vertical axis of the parabola, which are not
intersected by the chords, are the PRIME NUMBER points. So, we have a
very graceful and elegant COGNITIVE model of the famous ERATOSTHENES
SIEVE.

    Some meta-mathematical experts can say "It is too trivial in
comparison with, say, a theory of unattainable transfinite ordinals!" -
It is difficult to raise an objection. But to talk profusely about, say,
TRANSFINITE numbers not knowing elementary (not to confuse with
"trivial") properties of common FINITE Natural Numbers 1,2,3, ... , is
quite strange position for a true scientist. As well as to talk about a
transfinite induction knowing nothing about the super-induction which is
really a very natural generalization of the common mathematical
induction by B.Pascal (see a lot of discussions as to the
super-induction at [FOM]- and [HM]-lists).

    However that may be, I am sure that such the simple, understandable,
and beauty geometrical model of the prime numbers genesis, being shown
to modern CHILDREN, will generate a lot of "other distinguished
mathematicians" (see above) in the XXI Century :-)

    Moreover, this model permits to re-formulate some unsolved Number
Theory problems in a more simple and graphic form. It allows to hope to
find out new ways to their solutions.

    BTW, I believe that an explicit formulating of the connection
between the coordinates (-m, m^2), (n, n^2) of the endpoints of a
Stechkin-Matiyasevich'es chord, the coordinates (0, mn) of the
cross-point of that Stechkin-Matiyasevich'es chord and the parabola
axis, and the coordinates ((n-m)/2, -mn) of the cross-point of the
tangents drawn through these endpoints (see Clark Kimberling's
[HM]-message "Intersecting tangents to a parabola" on the Date: 3 Dec 99
12:40:41) will give a non-trivial relation between the product and the
sum of the corresponding integer points of the parabola. Maybe, some
school boy/girl will make here a new CCG-discovery as to a Nature of
Prime Numbers? :-)

    As is well known, one of the most ancient (from the Pythagoras time)
and important problem of Philosophy is a problem about the connection
between the Finity and the Infinity. New unique CCG-discovery presented
in the form of VISUAL CCG-image at

http://www.com2com.ru/alexzen/ccg/techniq.htm#TechFig02
http://www.com2com.ru/alexzen/ccg/techniq.htm#TechFig03

    allows to comprehend a new aspect of that connection: viz how ONE
but INFINITY entity (the common parabola) is structurally transformed
into INFINITY family of FINITE entities (finite parabolas).

    As is well known, one of the most ancient (from the Pythagoras time)
and important problem of Mathematics is a problem about the strict law
generating the series of the Prime Numbers:

2, 3, 5, 7, 11, 13, ...

    Modern Mathematics does know how to solve that Problem. However,
already now you have a possibility to look at the CCG-image of this law
at the address

http://www.com2com.ru/alexzen/ccg/techniq.htm#TechFig05

    Of course, taking into account that "to be looking at THAT" and "to
be having seen THAT" is quite different things.
    But I am sure that if such the law exists really then some school
boy/girl will ever see and formulate that law by means of the
CCG-Technology :-)

    Unfortunately, today by virtue of the well-known situation in the
Russia as to the basic science, I have not any possibility to continue
this investigations normally, and my CCG-System is in desperate
S.O.S.-straits. Alas, my mind today is occupied by unavoidable problems
that have a little attitude to scientific "proofs in dreams".
Practically, I have no time and possibility to contact with my dear
"VOICE of NIGHT". :-(

    Therefore, in order not to lose my CCG-Technology for scientific
discoveries totally, I handed over it to my good friends - to the
recognized Russian PAINTER, Alexander F.Pankin and to the recognized
Russian CHEMIST, Valery A.Nikanorov.

    Recently, using the CCG-technology, they have made an other unique
MATHEMATICAL discovery, which might seem impossible in the very end of
the XX Century. VIZ, they have discovered a new kind of VIRTUAL
GEOMETRICAL objects like the virtual parabolas which also were not ever
known in Mathematics.

    It occurred that the common Classical series of the famous
Fibonacci's Numbers (liked very much by artists and philosophers from
Leonardo's time) generates a kind of VIRTUAL FIBONACCI's TRIANGLES,
which are moving and leaping along the common series of the common
Natural Numbers as well as the virtual parabolic solitons. Only the
CCG-visualization of these objects became them actual, visible and
accessible to mathematical investigations.

    Today this CCG-discovery can be also seen directly at the
WEB-address:

http://www.com2com.ru/alexzen/vgeom/vgeom.htm

    I think that last CCG-discoveries lays the foundation of a NEW
direction in Mathematics-XXI - a VIRTUAL GEOMETRY OF NATURAL NUMBERS. Of
course, there exists nothing in the sublunar(y) World what would not
already have been existing in Ancient Greece. I mean the well-known
triangular, square, rectangular, etc., numbers of ancient Greek
Mathematics which might be called predecessors of these new virtual
geometrical objects.

    Such is, shortly, our CCG-Technology for mathematical
CCG-discoveries "in dreams". More precisely, - in modern human-computer,
multi-media, color-musical "dreams".

 Best regards,

 AZ

P.S. Any suggestions and collaborations as to that S.O.S.-straits of
the CCG-System are welcome. We are ready to hand over the
CCG-Technology to any Mathematical, Philosophical, Psychological,
Aesthetical, etc. education communities which wish to teach a new
generation of distinguished mathematicians and philosophers to make
their own scientific discoveries.

    My "Voice of Night" prompts me that all this can be realized already
during the modern generation life time. Or actually much sooner. :-)

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Prof. Alexander A. Zenkin,
Doctor of Physical and Mathematical Sciences,
Leading Research Scientist of the Computing Center
of the Russian Academy of Sciences,
Member of the AI-Association and the Philosophical Society of the Russia,
Full-Member of the Creative Union of the Russia Painters.
The last solely owing to the "Artistic "PI"-Number Gallery":
http://www.com2com.ru/alexzen/gallery/Gallery.html
e-mail: alexzen@com2com.ru
WEB-Site: http://www.com2com.ru/alexzen
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

"Infinitum Actu Non Datur" - Aristotle.
"Drawing is a very useful tool against the uncertainty of words" - Leibniz.
"Beauty is the first test: there is no permanent place in the world for
ugly mathematics." - G.H.Hardy



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