Content uploaded by Semen Samsonovich Kutateladze
Author content
All content in this area was uploaded by Semen Samsonovich Kutateladze on Sep 18, 2012
Content may be subject to copyright.
arXiv:math/0702632v5 [math.HO] 24 Apr 2009
THE TRAGEDY OF MATHEMATICS IN RUSSIA
S. S. KUTATELADZE
Abstract. This is a brief overview of the so-called “case of Academician
Luzin.”
Tsunami swept over the Russian mathematical community in 1999 after publi-
cation of the book The Case of Academician Nikola˘ı Nikolaevich Luzin [1]. For the
first time it revealed the complete sho rthand notes of the meetings of a notorious
sp e c ial Commission of the Academy of Sciences of the USSR.
N. N. Luzin (1883–1950) was one of the founding fathers o f the Moscow math-
ematical school. The list of his students contains Full Members of the Academy
P. S. Aleksandr off (1886–1982), A. N. Kolmogorov (1903–1987), M. A. Lavrentiev
(1900– 1980), P. S. Novikov (1901– 1975); Corresponding Members L. A. Lyusternik
(1899– 1981), A. A. Lyapunov (1911 –1973), D. E. Menshov (1892–1988), A. Ya.
Khinchin (1894–19 59), L. G. Shnirelman (1905–1938); and many other mathemati-
cians.
The Commission was convened after the article “Enemies under the Mask of
a Soviet Citizen” in the Pravda newspaper on July 3, 1936. Luzin was accused
of all theo retically possible instances of misconduct in science and depicted as an
enemy that combined “moral unscrupulousness and scientific dishonesty with deeply
concealed enmity and hatred to every bit of the Soviet life.” It was alleged that he
publishes “would-be scientific papers,” “feels no shame in declaring the discoveries
of his students to be his own achievements,” stands close to the ideology of the
“black hundred”, orthodoxy, and monarchy “fascist-type modernized but slightly.”
All Russian scientists of the elder generation knew about the Pravda editorial and
the savage dissolution of “luzinism.” The newly-published archive files open to
the public that some students of Luzin were the active participants of the political
assault on their teacher. The key role was played by P. S. Aleksandroff who headed
the Moscow topological school. Also active at the meetings of the Commission were
A. N. Kolmogorov, L. A. Lyusternik, A. Ya. Khinchin, and L. G. Shnirelman. The
political a ttacks on Luzin were vigorously supported by members of the Commission
S. L. Sobolev (1907–1989) and O. Yu. Schmidt (1891–1956). A. N. Krylov (1863–
1945) and S. N. Bernstein revealed valor in the vig orous defence of Luzin. The final
clause of the official Resolution of the Commission read as follows: “Everything
of the above, s ummarizing the overwhelming materia l evidence in possession of
Date: April 24, 2009.
I am very grateful to W. A. J. Luxemburg for attracting my attention to the inadvertent
omission in the preprint of this article of a reference to the revealing article [2] by G. G. Lorentz
(1910–2006). Also, I acknowledge that this essay summarizes my position at the informal lobby
discussions at the Mini-Simposium on Convex Analysis which was held at Lomonosov Moscow
State University, February 2-4 (2007). I am especially grateful to Professor V.M. Tikhomirov,
Chairman of the Mini-Simposium, for endurance, friendliness, and hospitality.
1
2 S. S. KUTATELADZE
the Academy of Sciences, completely ascertains the characteristics of Luzin in the
Pravda newspaper.”
All participants of the events of 1936 we discuss had left this world. They
seemingly failed to know that the files of the Commission are all safe and intact.
Today we are aware in precise detail of what happened at the meetings of the
Commission a nd around the whole case . The mathematica l community painfully
reconsiders the events and rethinks the role of the s tudents of Luzin in his political
execution.
P. S. Novikov and M. A. Lavrentiev were not listed as participants of the public
persecution of Luzin (despite the fa ct that both were mentioned at the meetings of
the C ommission among the persons robb ed by Luzin). It transpires now why M. A.
Lavrentiev was the sole author of a memorial article in Russian Mathematical Sur-
veys on the occas ion of the 90th anniversa ry of the birth of Luzin. He also included
this article in the collection of his papers on the general issues of science and life
[3, 4]. M. A. Lavrentiev was the chairman of the editorial board of the selected
works of Luzin which were published by the decision of the Academy of Sciences
of the USSR after the death of Luzin on the occasion of the 70th anniversary of
the birth of Luzin. P. S. Aleksandroff and A. N. Kolmo gorov were absent from the
editorial board.
Practically the same are the comments on their relationship with L uzin which
were left by P. S. Aleksandroff and A. N. Kolmogorov. Their statements are still
shared to some extent by their numerous students. It is customary to e mphasize
that Luzin was not so great a mathematician as his students that had p e rsecuted
him. Some moral fault is persistently incriminated to Luzin in the untimely death
of M. Ya . Suslin (189 4–1919) from typhus fever. Luzin is often blamed fo r all his
disasters at least partly. He is said to deserve all punishments and if not all then it
is not his s tudents’ fault but stalinism and the curse of the epoch. These arguments
reside in the minds of not only the elders but also the youngsters. The best of them
view the Luzin case a s the mutual tragedy of all participants.
However, we should distinguish the personal trag e dy of Luzin from the tragedy
of the Moscow school and the tragedy of the national mathematical community.
The students of Luzin who participated in the persecution of the teacher never
considered their own fates tragical.
P. S. Aleksandroff wrote in his reminiscences [6]:
“Knowing L uzin in his green creative years, I got acquaintance with a truly
inspired teacher and scholar who lived only by science and in the name of science.
I met a person who resided in the sphere of the sublime human treasures which is
forbidden for any rotten ghost or spirit. When a human being leaves this sphere
(and Luzin had le ft it once), he is doomed to surrender to the forces that were
described by Goethe as follows:
Ihr f¨uhrt in’s Leben uns hinein,
Ihr lasst den Armen Schuldig werden
Dann ¨uberlasst Ihr ihn der Pein,
Denn jede Schuld r¨acht sich auf Erden.
Into our life you lead us in,
The wretch’s guilt you bring to birth,
Then bring affliction down on sin,
For all guilt takes revenge on Earth.
THE TRAGEDY OF MATHEMATICS IN RUSSIA 3
In his terminal years Luzin saw the bottom of the sour bowl of the revenge that
was described by Goethe.”
1
It is worth observing that Khinchin, hostile to Luzin, commented on the accusa-
tions that Luzin drove Suslin to death [5]: “Suslin is called the student perished by
N. N. Luzin. Why, when a man dies from typhus fever this is a rather exaggerated
expression. In fact Suslin could possibly get typhus fever in Ivanovo. Furthermore ,
in the common opinion it was N. N. who tried and expelled Suslin from Ivanovo.
However, the transfer from Mo scow to Ivanovo I view as a favor to Suslin who was
not hostile to Luzin in those days.”
Narrating his reminiscences of P. S. Aleksandroff, A. N. Kolmogorov told in 1982
[7]: “My e ntire life as a whole was full of happiness.” Neither he nor Aleksandroff
nor other participants of the persecution of Luzin had ever treated the “case of
Luzin” as a common trage dy with Luzin. They were correct in this judgement but
on the grounds completely different from those they declared.
If Luzin were guilty then his fault would belong to the sphere of the personal
mathematical relations between a tea cher and a student. No convincing evidence
of Luzin’s plagiarism was ever submitted. The alleged accusations that he asc ribe d
to H. Lebesgue (1 875–1 941) or kept a grip of Suslin’s results are poorly disguised
and baseless. To prove the scientific misconduct o f Luzin it was a lleged that Luzin
played the underdog and flattered Lebesgue by attributing to Lebesgue his sieve
method. On the other hand, Lebesgue wrote in his preface to the Luzin boo k on
analytic sets as follows: “Anyone will be astonished to find out from Luzin’s book
that I had incidentally invented the sieve method and was the first to construct an
analytic set. However, nobody could be more ama z ed than me. Mr. Luzin feels
himself happy only when he has managed to ascribe his own discoveries to someone
else” [8]. The students were “more pious than the Pope.”
It is ea sy to assume the genuine or imaginary injustice and prejudice of Luzin in
citing his students as well as the genuine or imaginary feebleness of Luzin in over-
coming mathematical obstacles. We may agree to see hypoc risy in Luzin’s decision
to vote against P. S. Aleksandroff in the elections to a vacancy of an academician
despite his personal letter of support of Aleks androff to A. N. Kolmogorov. Well,
there is nothing untypical of the academic manners or extraordinary in Luzin’s
conduct, is there? It is the true background of the “case of Luzin,” isn’t it?
Available is the following testimony of W. Sierpinski (1882–1969), a famo us
Polish ma thematician who was declared to be a “blatant black hundredist” a t the
meetings of the Commission of the Academy of Sciences of the USSR on the “Case
of Luzin”: “ When I was in Moscow in September, 193 5, Mr. Aleksandroff as sured
me that the apprehensions of Luzin are purely ima ginary and that he re spects
Luzin, his former teacher. In my presence Aleksandroff shook hands with Luzin
and declared that he would always be a friend of Luzin” [9].
The pretentious re c onciliation of P. S. Aleksandro ff with Luzin which was de-
scribed by Sierpinski and which was later publicly refuted by P. S. Aleks androff
is in no way similar to the refusa l of Luzin to suppor t the election of P. S. Alek-
sandroff as an academician, isn’t it? It is in general belief that this refusal was
the reason for A. N. Kolmogorov to slap the face of Luzin publicly in 1 946. Luzin
1
Aleksandroff cited the poem Harfenspieler dated as of 1795 by Johann Wolfgang von Goethe
(1749–1832) and gave a rough translation into Russian. The lines in English here belong to Vernon
Watkins (1906–1967).
4 S. S. KUTATELADZE
was twenty years older than A. N. Kolmog orov. L uzin was a teacher of A. N. Kol-
mogorov and carried the heavy burden of political accusations that were impos e d
on Luzin with participation of P. S. Aleksandroff and A. N. Kolmogorov. Luzin was
granted “mercy” and accepted at the country house of A. N. Kolmogorov and P. S.
Aleksandroff in Komarovka before the elections.
2
Everyone at the meeting remem-
bered the most important matter that Luzin was victimized and must surrender to
the noble victors, didn’t he? It transpires now, doesn’t it? We can compare the
internal academic ma tters, say Luzin’s misconduct a nd even plagiaris m, with the
accusations of subversive activities a gainst the Soviet life, can’t we?
These grave and vexed questions...
I must emphasize explicitly that in my opinion all mora l accusations against
Luzin are absolutely inconvincible. That which was submitted as pro ofs was inad-
equate even in the times of the Commission neither for P. L. Kapitsa (1894–1984 ),
nor V. I. Vernadsky (1863–1945), nor A. Denjoy (1884–1974), nor Lebesgue, nor
many other elder persons.
The objection of Ka pitsa was expressed on July 6 in his letter to V. M. Molotov
who was the Chairmen of the Council of the People’s Commissars of the USSR.
Vernadsky wrote in his diary on the next day “Letters to Luzin, Chaplyg in, and
Fersma n about him. Majority treats as demonstrated the slander and insinuations.
2
V. M. Tikhomir ov wrote about the meeting in Komarovka: “The correspondence of L. S.
Pontryagin and his student and friend I. I. Gordon reveals that Luzin was accepted and served
a meal in Komarovka” [7, p. 83]. The relevant excerpt of a letter of Pontryagin of December 24,
1946 [13, Letter No. 49] reads as follows: “You are interested in a joint work of Kolmogorov and
Luzin. This should be narrated rather than written since the voice is needed to express everything
fully. Kolmogorov told me in summer that his only inconvenience as regards the election of Alek-
sandroff is the fact that Aleksandroff had become an indisputable candidate four months before
the voting. Pusics [= the collective nickname of Kolmogorov and Aleksandroff (S. K.)] made an
enormous preliminary work in the sense of entering into vari ous agreements with academicians.
For instance, there was a promise to Vinogradov to support Lavrentiev in reward for Vinogradov’s
support of Aleksandroff. It seemed indeed that everyone will vote for Aleksandroff. For exam-
ple, Bernstein hi mself nominated Aleksandroff at a meeting of the institute; well, in actuality, he
nominated Chebotarev too. Kolmogorov had reached an agreement with the bosses that he would
be nominated to the expert commission. The first glimpse of disappointment was the fact that
he was not nominated to the commission. However, he hoped that this was not very important.
After the session of the expert commission there were a few closed meetings of academicians in
which they discussed all candidates. It was at this stage that Kolmogorov became aware that
none of the members of the expert commi ssion supported Aleksandroff. Moreover, Bernstein ve-
hemently objected and said that Aleksandroff had a harmful area of research. The behavior of
Bernstein seems far from comprehensible to me by now; maybe, he simply had a quarrel with Pu-
sics. All the rest is rather clear. Lavrentiev turned out somehow to be an indisputable candidate
and needed no supp ort from Kolmogorov who was out off the commission at that. Therefore,
Vinogradov needed neither Kolmogorov nor Pusic. As regards Sobolev and Khristianovich, the
former hates Pusic [= the nickname of Aleksandroff (S. K.)] for Sobolev’s dismissal from the
directorship; while the latter is Sobolev’s friend and crony. In these circumstances, there was no
hope of success. The only possibility remained that some mathematicians among academicians
would support Aleksandroff; physicists wanted to support him, but they surely could never try
to confront all mathematicians. Luzin became the hope of Pusics. He was invited to Komarovka
and promised his support. However, he spoke against Aleksandroff at the final closed meeting.
Departing from this meeting, Kol mogorov was absolutely upset and stung. He came to Luzin and
said that he would have nothing in common with Luzin ever since. Luzin pretended that he did
not understand anything and began to talk as follows: ‘Dear me, calm down. Forget it. You are
ill. Relax.’ T his is what must be narrated with expression. Kolmogorov then answered him: ‘So
what shall I do to you: spit at your physiognomy or slap your mug?’ A fter a short thought, he
dared the latter.”
THE TRAGEDY OF MATHEMATICS IN RUSSIA 5
M[ay] b[e], he [is needed] abroad but not at home. I am afr aid that this disgusting
article will affect him much. Many conversations and many impressions.” On the
same day he sent a letter to Academicia n A. E. Fersman (1883 –1945), a member of
the Commission. Vernadsky wrote: “ I think that such an episode would eventually
be perilous to the Academy were it led to the expulsion of N. N. [Luzin] from the
Academy or any similar actions. We would slide down the slippery slope” [10].
Leb e sgue’s letter of August 5, 1936 is in order now. I remind that Lebesgue was
elected in 19 29 to the Academy of Sciences of the USSR for his outstanding con-
tribution to mathematics. The great Lebesgue, the author of that very “Lebesgue
integral” which is indispensable in modern mathematics, was in the state of utmost
indignation and anger. He wrote: “You will see that it was not yesterday when the
attacks on Luzin began with the aim of firing him and emptying place for Alek-
sandroff. You will see there that I was already mixed in this by contrasting ‘my’
science, which is bourgeoise and useless, to analysis situs [topolo gy], a pr oletarian
and useful science. Since the former was the science of Luzin; whereas the latter,
the science of Aleksandroff. What is curious is tha t he begins as Ury sohn whose
papers he inherited at the same starting point that was mine. With the only dif-
ference that Urysohn c ited me whereas Aleksandroff has never cited me anymore
since he must now speak badly of me in his struggle against Luzin!” [9].
Another evidence of Sierpinski: “I share the opinion and the same opinion is
shared by my Polish colleagues tha t the presence of Aleksandroff, Khinchin, K ol-
mogorov, and Shnirelman who confronted their former teacher in the mos t dishon-
est ma nner and slanderous ly accused him is intolerable a t any meeting of decent
persons” [9].
The method of political insinuations and slander was used against the old Mus-
covite pro fessorship many years before the ar ticle in Pravda. The declaration of
November 21, 1930 of the “initiative group” of the Moscow Mathematical Society
which consis ted of L. A. Lyusternik, L. G. Shnirelman, A. O. Gelfond (1906–1968),
and L. S. Pontryagin (1908–1988) claimed that “there appeared active counter-
revolutionaries among mathematicians” [5]. Some of these were pointed out, namely,
D. F. Egorov (1869–1931), a teacher of Luzin. Shortly before Egorov had been ar-
rested, and Luzin decided it wise to leave the university (he was later accused of
this removal by his students). In his life’s-description, dated as of the late 1970s,
Academician Pontryagin wrote [11]: “The two public actions, in 1936 as regards
Luzin a nd in 1939 as regards elections, were the important stages o f my uprising
as a public perso n. In my opinion both were the struggle for rightful ends.”
This is inconsistent with the position of Luzin who wrote in his letter of 1934 to
L. V. Kantorovich (1912– 1986) after the ugly declaration signed by Gelfond that
his choice in Moscow for the forthcoming election of corre sponding members of the
Academy “will be Gelfond who has recently made a discovery worthy of a genius”
[12].
A broad campaign agains t Luzin and “luzinism” waged over this country in
1936. Fortunately, Luzin was not repressed nor expelled from the Academy. Some
historians opine that there was a relevant oral direction of Joseph Stalin.
3
3
It was disclosed recently that the above-mentioned letter of Kapitsa to Molotov was mul-
tiplied in 16 copies for the members of the Political Bureau of the All-Union Communist Party
(Bolsheviks) and discussed over with other letters in support of Luzin.
6 S. S. KUTATELADZE
However, the badge of an enemy under the mask of a Soviet citizen was pin-
pointed to Luzin during 14 years up to his death. The monstrosity over Luzin is
absolutely incomparable with the alleged acc us ations of moral misconduct.
The human passions and follies behind the 1930s tragedy of mathematics in
Russia are obvious. But was there a mathematical background? Some roots are
visible.
We are granted the blissful world that has the indisputable property of unicity.
The solitude of reality was perceived by our ancestors as the ultimate proof of
unicity. This argument resided behind the incessant attempts at proving the fifth
postulate of Euclid. The same gives grounds for the common search of the unique
best solution of any human problem.
Mathematics has never liberated itself from the tethers of experimentation. The
reason is not the simple fact that we still complete proofs by declaring “obvious.”
Alive and rather popular are the views o f mathematics as a toolkit for natural
sciences. These stances may be expressed by the slogan “mathematics is experi-
mental theo retical physics.” Not less popular is the dual claim “theoretical physics
is experimental mathematics.” This s hort digress ion is intended to point to the
interconnections of the trains of thought in mathematics and natural sciences.
It is worth observing that the dogmata of faith and the principles of theology are
also well reflected in the history of mathematical theories. Varia tional calculus was
invented in search of better understanding of the principles of mechanics, r e sting
on the religious views of the universal beauty and harmony of the act of creation.
The twentieth century marked an important twist in the content of mathematics.
Mathematical ideas imbued the humanitarian sphere and, primarily, politics, so c iol-
ogy, and economics. Social events are principally volatile and possess a high degree
of uncertainty. Economic processes utilize a wide rang e of the admissible ways of
production, orga nization, and management. The nature of nonunicity in economics
transpires: The genuine interests of human beings cannot fail to be contradictory.
The unique solution is an oxymoron in any nontrivial problem of economics which
refers to the distribution of goods between a few agents. It is not by chance that
the social sciences a nd instances of humanitarian mentality invoke the numerous
hypotheses of the best organization of production and consumption, the most just
and e quitable social structure, the codices of rational behavior and moral conduct,
etc.
The twentieth century became the age of freedom. P lurality and unicity were
confronted as collectivism a nd individualism. Many particular phenomena of life
and culture reflect their distinction. The dissolution of monarchism and tyranny
were accompanied by the rise of parliamentarism and democracy. Quantum me-
chanics and Heisenberg’s uncertainty incorpo rated plurality in physics. The waves
of mo dernism in poetry and artistry should be also listed. Mankind had changed
all valleys of residence and dream.
In mathematics the quest for plurality led to the abandonment o f the overwhelm-
ing pressure of unicity and categoricity. The latter ideas were practically absent,
at least minor, in Ancient Greece and sprang to life in the epoch of absolutism and
Christianity. Cantor was a harbinger of mighty changes, claiming that “das Wesen
der Mathematik liegt gerade in ihre r Freiheit.” Paradoxically, the resurrection of
freedom expelled mathematicia ns from Cantor’s paradise.
THE TRAGEDY OF MATHEMATICS IN RUSSIA 7
Nowadays we are accustomed to the unso lvability and undecidability of many
problems. We s ee only minor difficulties in accepting no ns tandard models and
modal logics. We do not worry that the problem of the continuum is undecidable
within Zermelo–Fra e nkel set theory. However simple nowadays, these s tances o f
thought seemed opportunistic and controversial at the times of Luzin. The success-
ful breakthroughs of the great students of Luzin were based on the rejection of his
mathematical ideas. This is a psychological partly Freudian background of the case
of Luzin. His gifted students smelled the nece ssity of liberation from description
and the pertinent blissful dre ams of Luzin which were proved to be undecidable
in favor of freedom for mathematics. His students were misled and consciously or
unconsciously tr ansformed the noble desire for freedom into the primitive hatred
and monstrosity. This transformation is a popular fixation and hobby horse of the
human be ings through the ages.
Terrible and unbearable is the lightheaded universal fun of putting blame entirely
on Luzin for the crimes in mathematics in which he was hardly guilty with the barely
concealed intention to revenge his genuine and would-be private and personal sins.
We should try and understand that the ideas of desc ription, finitism, intuitionism,
and similar heroic attempts at the turn of the twentieth century in search of the sole
genuine and ultimate foundation were unavoidable by way of liberating mathematics
from the illusionary dreams of categoricity. The collapse of the eternal unicity and
absolutism was a triumph and tragedy of the mathematical ideas of the first two
decades of the last century. The blo ssom of the creative ideas of Luzin’s students
stemmed partly from his mathematical illusions in description.
The struggle against Luzin had some mathematical roots that were impos sible to
extract and explicate those days. We see clearly now that the epoch of probability,
functional analysis, dis tributions, and topology began when the idea of the ultimate
unique foundation was ruined for ever. G¨odel had explained some trains of thought
behind the phenomenon, but the mathematicians par excellence felt them with
inborn intuition and challenge of mind.
It is the tragedy of mathematics in Russia that the noble endeavor for freedom
had launched the political monstrosity of the scientific giants disguised into the
cassocks of Torquemada.
History and decedents are out of the courts of justice. Scientists and or dinary
persons must see and collect facts. Never accuse the passed away, but calmly and
openly point out that which was in reality. E xplain the difference between moral
accusations and political insinuations to the youth. Demonstrate the difficulty and
necessity of the repairing of mistakes and repentance. Show how easy it is to forgive
oneself and accuse the others.
We must work out and trans fer to the next generations the objective views of the
past. Of its successes and tragedies. With love and doubts, with the understanding
of our unfortunate fate and the honor of objectivity. It is the personal faults and
failures that we are to accuse and repair fir st of all. They knew even in Ancient
Rome that we should tell nothing o r good about the dead. Fa c ts did never pass
away. Luzin was accused by the Moscow Mathematical Society and the Academy
of Sciences. Thes e scientific institutions are alive.
Any attempt at discerning morality in the past immorality is dangerous since it
feeds this immorality by creating the comfortable environment of immorality in the
present and future. The stamina of a scientist by belief is a discontinuous function.
8 S. S. KUTATELADZE
Evil and genius coexist fro m time to time. Mathematics does not inoculate mor ality.
Manuscripts do not burn. . . .
References
[1] Demidov S. S. and Levshin B. V. (Eds.) The Case of Academician Nikola˘ı Nikolaevich Luzin.
St. Petersburg: Russian Christian Humanitarian Institute, 1999.
[2] Lorentz G. G. “Mathematics and Politics in the Soviet Union f rom 1928 to 1953.” Journal
of Approximation Theory, 116 (2002), 169–223.
[3] Lavrentiev M. A. “Nikola˘ı Nikolaevich Luzin (on the 90th Anniversary of His Birth).”
Uspekhi Mat. Nauk, 29:5 (1974), 177–182.
[4] Lavrentiev M. A. Science. Progress in Technology. Cadres. Novosibi rsk: Nauka Publishers,
1980.
[5] Yushkevich A. P. “The ‘Case’ of Academician N. N. Luzin.” In: Science under Repressions.
Leningrad: Nauka Publishers, 1991, 377–394.
[6] Aleksandroff P. S. “Pages of the Autobiography.” Uspekhi Mat. Nauk, 34:6 (1979), 219–249.
[7] Tikhomirov V. M. Andre˘ı Nikolaevich Kolmogorov. Moscow: Nauka Publishers, 2006.
[8] Lebesgue H. “Introduction to the Book by N. N. Luzin Lectures on Analytic Sets and Appli-
cations.” Uspekhi Mat. Nauk, 40:3 (1985), 9–14.
[9] Dugac P. “The ‘Case’ of Luzin and French Mathematicians.” Istoriko-mat. issledovaniya,
5(40) (2000), 119–142.
[10] Vernadsky V. I. Dairies. 1935–1941. Book 1. 1935–1938. Moscow: Nauka Publishers, 2006.
[11] Pontryagin L. S. The Life Description of Lev Sem¨enovich Pontryagin as Compiled by H im-
self. Born 1908, Moscow. Moscow: IPE “Prima V,” 1998.
[12] Reshetnyak Yu. G. and Kutateladze S. S. “A Letter of N. N. Luzin to L. V. Kantorovich.”
Vestnik Ross. Akad. Nauk, 72:8 (2002), 740–742.
[13] Gordon Eu. I. “Letters from L. S. Pontryagin to I. I. Gordon.” Istoriko-Mat. Issledovaniya,
9(44) (2005), 27–208.
Sobolev Institute of Mathematics
4 Koptyug Avenue
Novosibirsk, 630090 RUSSIA
E-mail address: sskut@member.ams.org
