For a long period, a very special role in Moscow mathematics was played by Andrei Kolmogorov. Being an ideologue of set theory, axiomatization of science, and the foundations of mathematics.
pag9
In the early 1960s, the anti-mathematical aggressiveness of the new class of professional computational mathematicians increased. They began a campaign against pure mathematics, claiming that the true development of mathematics is only computational mathematics. Among the older generation of mathematicians, this was definitely the opinion of A. N. Tikhonov and A. S. Kronrod. People working in computational mathematics would say that the community of pure mathematicians consists of practically insane people, speaking their own weird language, totally unintelligible to physicists and applied mathematicians, and they, the ”pure ones”, would soon become attractions in zoological gardens.
pag11
I decided to spend some years studying theoretical physics. I began with quantum field theory, but soon understood that one should begin with the basics, not at the end. My decision was motivated by the great respect I had for physics. The lectures of Einstein, Feynman, Landau, and several other prominent physicists deeply impressed me. The clarity and simplicity of their exposition of mathematical methods were in sharp contrast with what one could read in texts written by mathematicians, with very few exceptions.
I had first observed the natural way in which mathematical notions arise when in my youth I was studying topology at the peak of its development in the work of the most outstanding topologists; the complicated and deep algebraic apparatus seemed to appear easily and naturally from qualitative geometry and analysis, yielding a double intuition about the same objects.
pag12
I was attracted by the beauty and power of physics. In 1965-1970 I
systematically studied the entire course of physics textbooks. Except for two or
three books (on statistical physics and quantum electrodynamics) I
studied the books of the Landau-Lifshits series.
Even earlier I had seen that the circle of physicists was not only richer than the circle of mathematicians in terms of science, but also more honest. This was the situation in the USSR, not in the West.
pag14
I would say that in the first half of the 1960s the attacks on pure
mathematics by applied mathematicians had
not developed very far. One of the main reasons for this was the remarkable
discovery of new particles by means of Lie group theory and related concepts.
pag15
The community of pure mathematics in the West developed the following viewpoint: to earn money I teach mathematics at the university; this is my duty to society. The rest of the time I study my pure mathematics. They lived for a number of decades with that viewpoint. In our country it was not like that, this approach did not work: no one wanted to teach. Except for a small number of leading universities, conditions for those who were teaching were bad. The teaching load was too big, teachers could not even think of trips abroad, and there was no time for research. Be that as it may, the western community of mathematicians broke away from society further and deeper than ours did. Even in the brilliant centers of applied mathematics, say, at the Courant Institute in New York, with time the community increasingly came to understand applied mathematics as a set of rigorous proofs and questions of logical justification.
pag16
In the process of study, I realized that theoretical physics, if learned
systematically, from the fundamentals to contemporary quantum theory, is
integral, extensive and profound mathematical knowledge, and is remarkably
adapted to describing the laws of nature, working with them and effectively
obtaining results. One cannot help but agree with Landau: to understand this, it
is necessary to study his socalled ”theoretical minimum”’ entirely.
This was the backbone determining one’s level of scientific culture. A person
who has failed to learn it has a deficient idea of theoretical physics. Such
people could turn out to be harmful for science; they should not be allowed into
theoretical physics. Their influence would contribute to the downfall of
education.
Unfortunately, the mathematicians’ community of the time did not acquire even
the basic elements of this knowledge, and this was true even of those who called
themselves applied mathematicians. For example, I quickly discovered that
virtually none of the experts in partial differential equations knew precisely
what the energy-momentum tensor was and could never clearly define the notion in
mathematical terms.
pag17
Second half of the 20th century: excessive formalization of mathematics.
When I read the works of the 1920s and 1930s in set theory, I noticed that,
despite the abstract subject, these works were written clearly and lucidly. The
authors tried to explain their thoughts and do it as simply as possible.
Studying topology in the 1950s, I saw that the best books and articles by famous
topologists (Seifert-Trelfall, Lefschetz, Morse, Whitney, Pontryagin, Serre,
Thom, Borel, Milnor, Adams, Atiyah, Hirzebruch, Smale, and others) were written
very clearly. The subject itself was not simple, but no one
wanted to confuse you further. The subject was expounded as simply as possible
to help the reader understand it. But then a different kind of source began to
appear-for example, in my early youth, I saw that in the monograph of my teacher
M. M. Postnikov, where his best works were presented, the content became
burdened by unnecessary formalization, which made it difficult to understand.
With time the quantity of such texts increased. This process went ahead
especially rapidly if there was a great deal of algebra and category theory in
them.
pag20
Formalization of a science’s language in the Bourbaki style is not useful: it is not like Hilbert’s formalization, which makes understanding easier. It is a parasitic formalization, making understanding more complicated, hindering mathematical unity and its unity with the applications. I believe that over-formalized literature appeared, in particular, because it was possible to predict its success among broad sections of algebra-oriented pure mathematicians. It is necessary to go against the current in order to retain a transparent general scientific style, which could secure the unity of mathematics and unite mathematics with physics and applications. This can only be done by a few mathematicians today. Today’s community does not understand.
pag22
I can understand that Fermat’s last theorem and the four color problem, solved in the same period, are worth a long proof and have been checked. But it is simply preposterous to live in a world of exceedingly long proofs which no one reads. This is a road to nowhere, a bizarre end of the Hilbert program.
...
The existence of a crisis in the mathematicians’ community, with its educational system and approach to science, should be distinguished from the question: Is there a crisis of mathematics as a science? Perhaps there is no crisis; simply the best works in a number of fields are now done by other people, coming from physics?
pag24
Frankly speaking, I should say this about my program: I have spent many years studying theoretical physics in the search for new situations where topological ideas could be useful in applications and natural sciences. The new topology that physicists created is a wonderful thing, but I have studied theoretical physics enough to know that this field is not a branch of physics; let those who studied nothing believe it is. Physics is the science of the phenomena of nature, phenomena that can be observed in reality. The Platonic physics is a set of ideal concepts separate from reality. A large group of talented theoretical physicists was fascinated by Platonic physics and imperceptibly moved far away from reality. In the last quarter of the 20th century, their belief that real physics, following the experience of the previous 75 years, would move on and confirm the most beautiful theories, ceased to be justified. Confirmation of supersymmetry in physics of elementary particles bogged down for 25-30 years. It still does not exist, although the supersymmetry hypothesis strongly improves the mathematical theory. Quantum gravitation and all its manifestations-strings and so on-are extremely far from any possible real world confirmation. At the same time, these theories proved to be so beautiful that they gave rise to many results and ideas in pure mathematics. The departure from real physics of such a talented theoretical community exposes physics and deprives it of the stratum capable of combining the realism of physics and state of the art contemporary mathematics.
pag26
Here we approach the key question, the main cause of the crisis in physics and mathematics, namely the disintegration of education. Will the still existing generations of competent mathematicians and theoretical physicists be able to teach equally competent young heirs for the 21st century? The key to everything lies in education, and the difficulties of the problem, symptoms of disintegration, begin in elementary and secondary school and continue at the university.
...
mathematical ideologues in a number of countries (in the USSR it was Kolmogorov) began to carelessly destroy established schemes of phased teaching of mathematics and introduced ideas of set theory ”for everyone”.
pag29
the situation of disintegration that emerged in the late USSR. It would be possible to overcome the difficulties related to low wages: one could work in the West and return when the conditions are tolerable. What happened was much worse: from the very beginning it became clear that there was nowhere to return: nobody was waiting for you in Russia, all the positions having been taken by false scientists. Such was the process of disintegration in the USSR/Russia.
In the West, however, there was also an abrupt fall in the level of college and school education in physics and mathematics in the last 20-25 years, and in the USA the decline in school education was apparently particularly steep. I can clearly see that contemporary education cannot produce a theoretical physicist capable of passing Landau’s theoretical minimum.
...
Democratic progress in education led to the same results in physics and mathematics as the Brezhnev regime did. The conclusion is very simple: we are undergoing a profound crisis.
pag30
The fall in the level of mathematics and physics education in various computer based sciences is also obvious to everyone. These sciences are being reoriented to provide services for business and commerce.
pag31
(Editor’s Note. The expression ”Landau theoretical minimum”, well known among
Russian physicists, stands for the basic facts of physics anyone aspiring to do
research under Landau was supposed to know quite well.
These facts, roughly speaking, consist of the complete contents of the numerous
volumes of the famous L. D. Landau-E. M. Lifshitz physics course. For more about
this, see the interview with S. P. Novikov in this volume.)