[Sci] An Interview with V. Arnold

["VU HOANG LINH" <LINH@oplab.sztaki.hu>]

Hi all, 
after the long long discussion on a "poor" Vietnamese mathematics "professor", 
it is high time we relaxed our mind. The following article has been found in 
the Notices of the AMS. The brief version is suggested to your attention, 
(not only to mathematicians's). 
Professor Arnold's stories and opinions on the past and the present Mathematics, 
on Russian and Western Mathematics Educations are very interesting. I am looking 
forward to comments from you all.
My apology to VNSA admins and all subscribers for the large posting, 
V.H. Linh
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TITLE: An Interview with Vladimir Arnol'd
"http://www.ams.org/publications/notices/199704/arnold.html

Notices of the AMS
April 1997

An Interview with Vladimir Arnol'd

by S. H. Lui

Vladimir Arnol'd is currently professor of mathematics at both the
Steklov Mathematical Institute, Moscow, and Ceremade,
Universite de Paris-Dauphine. Professor Arnol'd obtained his
Ph.D. from the Moscow State University in 1961. He has made
fundamental contributions in dynamical systems, singularity theory,
stability theory, topology, algebraic geometry, magneto-hydrodynamics,
partial differential equations, and other areas. Professor Arnol'd has
won numerous honors and awards, including the Lenin Prize, the
Crafoord Prize, and the Harvey Prize.

This interview took place on November 11, 1995. 

Lui: Please tell us a little bit about your early
education. Were you already interested in mathematics as a child?

Arnol'd: The Russian mathematical tradition goes back to the old merchant
problems. Very young children start thinking about such problems even
before they have any knowledge of numbers. Children five to six years
old like them very much and are able to solve them, but they may be
too difficult for university graduates, who are spoiled by formal
mathematical training. A typical example is:

You take a spoon of wine from a barrel of wine, and you put it into
your cup of tea. Then you return a spoon of the (nonuniform!) mixture
of tea from your cup to the barrel. Now you have some foreign
substance (wine) in the cup and some foreign substance (tea) in the
barrel. Which is larger: the quantity of wine in the cup or the
quantity of tea in the barrel at the end of your manipulations?

Slightly older children, knowing the first few numbers, like the
following problem. Jane and John wish to buy a children's book.
However, Jane needs seven more cents to buy the book, while John needs
one more cent. They decide to buy only one book together but discover
that they do not have enough money. What is the price of the book?
(One should know that books in Russia are very cheap!)

Many Russian families have the tradition of giving hundreds of such
problems to their children, and mine was no exception. The first real
mathematical experience I had was when our schoolteacher I. V.
Morozkin gave us the following problem: Two old women started at
sunrise and each walked at a constant velocity. One went from A to B
and the other from B to A. They met at noon and, continuing with no
stop, arrived respectively at B at 4 p.m. and at A at 9 p.m. At what
time was the sunrise on this day?

I spent a whole day thinking on this oldie, and the solution (based on
what is now called scaling arguments, dimensional analysis, or toric
variety theory, depending on your taste) came as a revelation. The
feeling of discovery that I had then (1949) was exactly the same as in
all the subsequent much more serious problems--be it the discovery of
the relation between algebraic geometry of real plane curves and
four-dimensional topology (1970) or between singularities of caustics
and of wave fronts and simple Lie algebra and Coxeter groups (1972).
It is the greed to experience such a wonderful feeling more and more
times that was, and still is, my main motivation in mathematics.

Lui: What was it like studying at Moscow State
University? Can you tell us something about the professors
(Petrovskii, Kolmogorov, Pontriagin, Rokhlin, ... )?

Arnol'd: The atmosphere of the Mechmat (Moscow State University Mechanics and
Mathematics Faculty) in the fifties when I was a student is described
in detail in the book Golden Years of Moscow
Mathematics, edited by S. Zdravkovska and P. L. Duren and published jointly by
the AMS and LMS in 1993. It contains reminiscences of many people. In
particular, my article was on A. N. Kolmogorov, who was my supervisor.

The constellation of great mathematicians in the same department when
I was studying at the Mechmat was really exceptional, and I have never
seen anything like it at any other place. Kolmogorov, Gelfand,
Petrovskii, Pontriagin, P. Novikov, Markov, Gelfond, Lusternik,
Khinchin, and P. S. Alexandrov were teaching students like Manin,
Sinai, S. Novikov, V. M. Alexeev, Anosov, A. A. Kirillov, and me.
.....
.....
Lui: Can you tell us your philosophy of teaching
undergraduates and of supervising graduate students and how many you
have had in Russia and France?

Arnol'd: The number of Ph.D. theses defended under my supervision is something
like forty. I cannot give the exact number for several reasons. In the
``stagnation'' period, I was not allowed to supervise foreign graduate
students at Moscow University because I was not a party member. They
still were studying with me, but the official supervisor was some
friendly party member who also got paid for it. Some graduate students
had other supervisors but wrote their theses on topics discussed in my
seminars and were practically my students. Three examples are S. M.
Gusein-Zade, Yu. Iliashenko, and A. I. Neistadt. At present, I'm
working with two undergraduates and three graduates in Moscow and with
four graduates in Paris. Two or three more are supposed to start in
January.

I learn a lot from my students, especially undergraduates. I never
assign a thesis topic to my students. This is like assigning them a
spouse. I merely show them what is known and unknown.

My Moscow seminar, working even when I am abroad, consists of about
thirty mathematicians, mostly my former graduate students, but there
are always others. The seminar has existed for about thirty years, and
among the participants in different years were Ya. Sinai, V. Alexeev,
S. Novikov, M. Kontsevich, A. Goncharov, D. B. Fuchs, G. Tjurina, A.
Tjurin. ... 

Life in Moscow is so difficult that most students have to earn their
living independently of their scientific work. Some, for instance,
start their own businesses. The rate of crime is so high, however,
that in starting a business, one risks being killed. One of my
graduate students in Moscow, who has just finished his thesis but has
not defended it, disappeared a few weeks ago. We have doubts about
whether he is alive or not.

Lui: Do you have any mathematical heroes?

Arnol'd: I would mention Barrow, Newton (who was, however, a very unpleasant
person--see my book Huygens and Barrow, Newton and
Hooke published by Birkhauser, 1990), Riemann, Poincare;,
Minkowski, Weyl, Kolmogorov, Whitney, Thom, Smale, and Milnor.
One-half of the mathematics I know comes from the book of F. Klein Lectures
on the Development of Mathematics in the 19th
Century. I have also learned a lot from many mathematicians like Gelfand,
Rokhlin, S. Novikov, P. Deligne, Fuchs, and from my own students like
Khovanskii, Nekhoroshev, Varchenko, Zakaljukin, Vassiliev, Givental,
Goryunov, O. Scherbak, Chekanov, and Kazarian.

.....
......
Lui: Do you notice any differences in the way
people from different cultures do mathematics?

Arnol'd: I was unaware of these differences for many years, but they do exist.
A few years ago, I was participating in an International Science
Foundation (ISF) meeting in Washington, DC. This organization
distributes grants to Russian scientists. One American participant
suggested support for some Russian mathematician because ``he is
working in a good American style.'' I was puzzled and asked for an
explanation. ``Well,'' the American answered, ``it means that he is
traveling a lot to present all his latest results at all our
conferences and is personally known to all experts in the field.'' My
opinion is that ISF should better support those who are working in the
good Russian style, which is to sit at home working hard to prove
fundamental theorems which will remain the cornerstones of mathematics
forever!

Russian salaries are (and were) so small, that if someone is doing
mathematics, it means that for him it is the goal and not a means to earn
money. It is still possible to attain a
high reputation in the Western mathematical community by simply
rewriting (or modernizing) classical Russian achievements and ideas
unknown to the West.

The Russian attitude toward knowledge, science, and mathematics always
conforms to the old traditions of the Russian intelligentsiya. This word does 
not exist in other languages, since no other country
has a similar caste of scholars, medical doctors, artists, teachers,
etc., who find more reward from their contributions to society than
from personal or monetary gains.

My friend Vershik recently tried to obtain an American visa in Paris.
``What is your salary in St. Petersburg?'' asked the staff at the
American consulate. After hearing his honest reply, the staff asked,
``Do you wish to persuade us that you intend to return to St.
Petersburg at such a salary?'' Vershik answered, ``Of course. Money is
not all!'' The staff was so shocked that Vershik was given the visa
immediately.

I was applying for a visa a week earlier, and they put me on a waiting
list for three weeks. Their reasoning was that my papers must be
checked in Washington since I am a ``donkey''. I asked for an
explanation. ``Well,'' they replied, ``we have such names for every
crime: dog, cat, tiger, camel, and so on.'' They showed me the list,
and ``donkey'' is a pseudonym for a Russian scientist.

One other characteristic of the Russian mathematical tradition is the
tendency to regard all of mathematics as one living organism. In the
West it is quite possible to be an expert in mathematics modulo 5,
knowing nothing about mathematics modulo 7. One's breadth is regarded
as negative in the West to the same extent as one's narrowness is
regarded as unacceptable in Russia.

The French mathematical school was brilliant for several centuries, up
to the penetrating works of Leray, H. Cartan, Serre, Thom, and Cerf.
The Bourbakists claimed that all the great mathematicians were, using
the words of Dirichlet, replacing blind calculations by clear ideas.
The Bourbaki manifesto containing these words was translated into
Russian as ``all clear ideas were replaced by blind calculations.''
The editor of the translation was Kolmogorov. His French was
excellent. I was shocked to find such a mistake in the translation and
discussed it with Kolmogorov. His answer was: I had not realized that
something is wrong in the translation, since the translator described
the Bourbaki style much better than the Bourbakists did.
Unfortunately, Poincare left no school in France.

A typical example of the French narrow-mindedness is the recent
discussion at the Academy of Sciences. Gromov was a foreign associate
for many years, but he recently chose the French nationality and hence
could no longer remain a foreign associate. The problem was to
transfer him to be an ordinary fellow of the Academy. The French
mathematicians, however, were opposed to this, saying that ``those
places are for the really French people!'' In my opinion, all the
``really French'' candidates were incomparably below the level of
Gromov, who is one of the world's leading mathematicians. In the end,
Gromov is still not a fellow.

To teach in France is very difficult because of the formalized
Bourbaki training the students have. For example, at a written
examination in dynamical systems for fourth-year students at
Paris-Dauphine, one problem was to find the limit of the solution of a
system of Hamiltonian equations on the phase plane starting with some
given initial point when time goes to infinity. The idea was to choose
the initial point on a separatrix of a saddle, with the limit being
the saddle point.

Preparing the examination problem, I made an arithmetical error, and
the phase curve (the energy-level curve containing the initial point)
was a closed oval instead of the separatrix. The students discovered
this and concluded that there exists a finite time $T$ at which the solution 
returns to the initial point. Using the unicity
theorem, they were able to deduce that for any integer $n$ the value of the 
solution at time $nT$ is still the initial point. Then came the conclusion: 
since the limit
at infinite time coincides with the limit for any subsequence of times
going to infinity, the limit is equal to the initial point! This
solution was invented independently by several good students sitting at 
different places in the examination hall. In all
this reasoning, there are no logical mistakes. It is a correct deduction 
which one may also generate by a computer. It is apparent
that the authors understood nothing. It is awful to think what kind of 
pressure the Bourbakists put on
(evidently nonsilly) students to reduce them to formal machines! This
kind of formalized education is completely useless for any practical
problem and even dangerous, leading to Chernobyl-type events.
Unfortunately, this plague of formal deduction is propagating in many
countries, and the future of the mathematics infected by it is rather
bleak.

The United States has a different danger. No Russian professor is able
to solve correctly the problem they give in the Graduate Record
Examination, the official entrance examination for graduate studies:
find the closest pair to (angle, degree) among the pairs: (time,
hour), (area, square inch), and (milk, quart). Every American
immediately solves it correctly. The official explanation for the
correct response (area, square inch) is: one degree is the minimal
measure of angle, one square inch is the minimal measure of area,
while an hour contains minutes and a quart contains two pints. I
always wondered how it is possible for so many Americans to overcome
such difficulties and become great mathematicians. One physicist in
New York who solved the problem successfully told me that he had the
correct model of the degree of stupidity of the authors of such
problems.

H. Whitney told me that the problem (intended for fourteen-year-old
American school children) of whether 120% of the number 80 is a number
greater than, smaller than, or equal to 80 was correctly solved (in a
nationwide test) by 30% of the students. People making the test
thought that 30% of the school children understood percentages.
Whitney explained to me, however, that the number of those who really
understood was negligible with respect to the whole sample. Since
there were three possible answers, the statistical prediction for a
correct random choice was 33%, with a 5% uncertainty.

Recently, even the National Academy of Sciences decided that
scientific education in America should be enhanced. What they propose
is to eliminate from the curriculum unnecessary scientific facts too
difficult for American children and replace them by really fundamental,
basic knowledge, such as all objects have properties and all organisms have
nature! (See Nature 372:5606, December 8, 1994.) Undoubtedly, they will go 
far with this! Two years ago, I read in USA
Today that American parents have formed a list of really necessary
knowledge for children in each age category. At ten they have to know
that water has two phases, and at fifteen that the moon has phases and
rotates around the earth. In Russia we still teach children in primary
school that water has three phases, but the new Americanized culture
will undoubtedly win in the near future. There are, however, some
remarkable advantages in the free American system, where a high school
student may take, say, a course on the history of jazz instead of
algebra.

A few months before his death, Whitney, who was still very active at
the Institute for Advanced Study in Princeton, told me the story of
his mathematical studies. He was an undergraduate in violin at Yale,
and after the second year he was sent to one of the best centers in
Europe for music. Unfortunately, I have forgotten which city it was,
but in any case it was not far from the Alps, since he already was a
mountain climber. There, a student had to pass an exam in a subject
different from his own studies. Whitney asked his fellow students
which subject was the most fashionable then, and they told him quantum
mechanics. After his first class in quantum mechanics, Whitney
approached the famous lecturer (Pauli? Schroedinger? Sommerfeld?)
with the following words: ``Dear Professor, it seems to me that
something is wrong with your lectures. I'm the best student from Yale,
and still I am unable to understand a word in your lecture.'' The
lecturer, after being informed that Whitney was studying music,
answered quite politely, ``This is because you need some background,
such as calculus and linear algebra.'' ``Well,'' Whitney replied, ``I
hope these are not so brand new as your subject and someone has
already written textbooks on these subjects.'' The lecturer agreed and
mentioned the titles of some textbooks. (If someone knows about this
story, I would like to know the name of the city, lecturer, and
titles.) ``In three weeks,'' Whitney continued, ``I was understanding
his lectures, and at the end of the semester I switched from music to
mathematics.''

Kolmogorov also started as a nonmathematician--he was studying
history. His first paper, written when he was seventeen, was reported
at a seminar given by Bakhrushin at Moscow University. Kolmogorov came
to some conclusion based on an analysis of medieval tax records in
Novgorod. After his talk, Kolmogorov asked Bakhrushin whether he
agreed with the conclusions. ``Young man,'' the professor said, ``in
history, we need at least five proofs for any conclusion.'' Next day,
Kolmogorov switched to mathematics. The paper was rediscovered in his
archive after his death and is now published and approved by the
historians.

Lui: Any comments on the relation between pure
and applied mathematics?

Arnol'd: According to Louis Pasteur, there exist no applied sciences--what do
exist are the APPLICATIONS of sciences. The common opinion of both
pure mathematicians and theoretical physicists on the applied
mathematics community is that it consists of weak thinkers unable to
produce something scientifically important and of those who are more
interested in money than in mathematics. I do not think that this 
characteristic is fully deserved by the applied mathematics community. See my 
article ``Apology of applied mathematics'' in Russian Mathematical
Surveys, 1996. It summarizes my talk at the opening of the Hamburg
International Congress of Industrial and Applied Mathematics, July
1995. I think that the difference between pure and applied mathematics
is social rather than scientific. A pure mathematician is paid for
making mathematical discoveries. An applied mathematician is paid for
the solution of given problems.

When Columbus set sail, he was like an applied mathematician, paid for
the search of the solution of a concrete problem: find a way to India.
His discovery of the New World was similar to the work of a pure
mathematician. I do not think that the discoveries of Galileo (who was
immediately exploiting them in a businesslike American style) are less
important than, say, those of the pure philosopher Pascal. The real
danger is not the applied mafia itself, but the divorce between pure
mathematics and the sciences created by the (I would say criminal)
formalization of mathematics and of mathematical education. The
axiomatical-deductive Hilbert-Bourbaki style of exposition of
mathematics, dominant in the first half of this century, is now
fortunately giving place to the unifying trends of the Poincare
style geometrical mathematics, combining deep theoretical insight with
real world applications.

By the way, I read in a recent American book that geometry is the art
of making no mistakes in long calculations. I think that this is an
underestimation of geometry.

Our brain has two halves: one is responsible for the multiplication of
polynomials and languages, and the other half is responsible for
orientation of figures in space and all the things important in real
life. Mathematics is geometry when you have to use both halves. See,
for instance, ``The geometry of formulae'' by A. G. Khovanskii in the Soviet
Sci. Rev. Sect. C: Math. Phys. Rev. V4 (1984).