I received an invitation to share my personal reminiscences of Leonid Vitalievich Kantorovich. This invitation does not make me hilarious, although I had many opportunities to write about L.V. and participate in publishing his selected works on mathematics and economics in Russia and abroad. The genre of memories drives the one who remembers to the person remembered, thus annihilating the difference between the two at least slightly. Of course, we all are humans, but there is a gap between being a historical personage and encountering with such a person sometimes. However, the future generations and historians of science will appreciate even minor facts and details that could help to better envisage the past from the ever renewable modernity.
It so happens that L.V.'s and ours families stem from Leningrad, and we had lived very close to one another on the Petrograd side. My father and L.V. were practically the same age. The main groups of the first inhabitants of Novosibirsk Akademgorodok were former residents of Moscow and Leningrad. The morals in Moscow and Petersburg have always been somewhat different, which drives the fellow countrymen together in Novosibirsk as well. L.V. was elected to a vacancy of Corresponding Member in economics for the Siberian Division of the Academy of Sciences of the USSR. But he was alien in spirit to the company of G.A. Prudenskii, the first director of the Institute of Economics and Industrial Engineering of the SDAS of the USSR. L.V. had been friendly with S.L. Sobolev and A.D. Alexandrov from the years in Leningrad State University and their unity and mutual support became stronger as time was elapsing.
Leningraders played a key role in creating the syllabuses of the Department of Mechanics and Mathematics and the Department of Economics of Novosibirsk State University. The main reason for that was the fact that the first years of the Siberian Division were also the times of propagation of the ideas of computational mathematics and cybernetics in the Soviet Union. Note that L.V. founded and held the first Chair of Computational Mathematics in LSU, as well as S.L. Sobolev did in Lomonosov Moscow State University. Together with L.A. Lusternik, the three of them delivered a program talk “Functional Analysis and Computational Mathematics” at the Third All-Union Mathematical Congress in 1956.
L.V. founded and took the Chair of Computational Mathematics in NSU. His staff provided the theoretical course in functional analysis as well as the applied course of computational mathematics. From 1963 on the course of mathematical analysis in NSU was delivered by Yu.G. Reshetnyak, a graduate of LSU and a student of A.D Alexandrov; and the course “Analysis III,” the euphemism for functional analysis in that time, by G.P. Akilov, a graduate of LSU and a student of L.V. who coauthored L.V. in writing the textbook Functional Analysis in Normed Spaces which remained revolutionary for many years. The course in the methods of optimization was included into the syllabus of the Department of Mechanics and Mathematics by the proposition of L.V. This course was delivered by G.Sh. Rubinstein, a graduate of LSU and a student of L.V. It was G.Sh. Rubinstein who invited me to take major in computational mathematics and became my supervisor. After graduation, I was assigned to the Institute of Mathematics as a postgraduate student of the Mathematical Economics Division headed by L.V. in 1968.
In 1970 L.V. arranged a Symposium on Modeling National Economy in Akademgorodok. One of the participants was T. Koopmans whom L.V. shared a Nobel prize with in 1975. They spent much time together. I was the third at their meetings as an interpreter and recall utmost frankness, mutual respect, and sympathy.
L.V. moved to Moscow in the end of 1970 but visited Novosibirsk rather often. In fact I was able to see how L.V. managed his staff only for two years. I recall his extraordinary ability to look sleepy at a seminar but to remember all that happened. Once I had to review an article on convex analysis by my friends A.D. Ioffe and V.M. Tikhomirov in Russian Mathematical Surveys. The review took a few meetings and at one of them I told some relevant results by A.Ya. Dubovutskii and A.A. Milyutin to make the picture complete. In a few months later there was a talk of another colleague who referred to the A.Ya. Dubovutskii and A.A. Milyutin method. L.V. intervened immediately: “Why Dubovutskii and Milyutin? This is due to Ioffe and Tikhomirov.” Definitely, I was inaccurate in my review.
L.V. has a strong principle about the freedom of research. Every novice can tackle whatever abstract mathematical theme of his own choice. But if L.V. saw no serious result in his estimation in two years, the freedom was over and L.V. appointed the looser to a down-to-earth applied problem of an economic nature. About a decade later when we were closer related, L.V. said that if a theoretician in the Steklov Mathematical Institute had not written a (second) doctor thesis by the age of 35, then there are no reason to keep the person and he or she should be transferred to some applied unit.
L.V.'s attitude to me was miraculously benevolent. He appointed me to the editorial board of the periodical series Optimization he had founded in Novosibirsk. After moving to Moscow, he wrote a memo to S.L. Sobolev, Editor-in-Chief of Siberian Mathematical Journal, in which he proposed that I should be authorized to take decisions on the articles submitted to L.V. as a member of the editorial board.
In 1973 my (second) doctor thesis was obstructed. It was overtly attacked only by N.N. Yanenko who was indignant to notice that the proof of one of my theorems was shorter that its statement. L.V. took the floor to rebuff the attack, but N.N. Yanenko immediately left the meeting hall. L.V. could kill time, he elongated words, growled, and kept silence for about five minutes. When N.N. Yanenko had decided that L.V.'s speech was over, he returned to the hall. Instantly L.V. revived and shouted something like that: “As far as those are concerned who say that do not understand the claim of a theorem but judge it by the length of the proof, I must confess that I prefer the views of those who do understand the content of the theorem.” My thesis was flunked but in a few years (after the reorganization of the Higher Attestation Committee) L.V. agreed to be an official reviewer (“opponent” in Russian) of my another (second) doctor thesis in LSU. Of course, no one planned that he should attend the public maintenance, but it happened so that one of my official reviewers, A.A. Milyutin, had caught a bleeding in the eyes and so he was unable to participate in the maintenance in person. L.V. asked his Moscow colleagues to prepare a review. He arrived in Leningrad and stayed at I.L. and I.V. Romanovkii's apartment (his daughter and son-in-law). L.V. perused the review and was annoyed with it. I.V. Romanoskii typed a new version on some portable typewriter that could produce only two carbon copies which were requested by the Higher Attestation Committee. So I had never seen this review but heard it at the maintenance. The maintenance in LSU was surely a memorable event. The Mathematics and Mechanics Faculty of LSU on the Vasilyevsky Island had once again seen their coryphaeuses—L.V. and S.L. Sobolev who had also arrived in a token of solidarity. There was no traditional banquet after the maintenance (the times were mouseless), which was a slight disappointment to L.V. Of course, we celebrated the event at my home during the next visit of L.V. to Akademgorodok.
Those days I had often been to Moscow—we live in two cities, and my daughters had studied in Moscow. L.V. had lived in a Stalin skyscraper a few years and then in a chick apartment in Cherëmushki near our humble khruschëvka we lived those days. L.V. invited me at home on my every visit to Moscow.
I recall the conversation in the skyscraper when I asked L.V. about his new staff. The conversation was in presence of L.V.'s wife Natalia Vladimirovna. L.V. pronounced two names. N.V. was surprised: “Lënya, there are quite a few other employees of yours!” L.V. smiled in response: “But I do understand that Sëma does not count the others as persons.” I still do not gather whether this was a compliment or rebuke.
In L.V.'s Cherëmushki apartment I met a few Moscow mathematicians. In particular, I got acquaintance with S.P. Novikov. L.V. liked to hear our verbose and harsh declarations. He poured vodka to our glasses and definitely enjoyed the proceedings. L.V. was an exquisitely educated person and never spoke of himself. This was the revelation of Petersburg's intelligence. The same features were characteristic of his LSU's cronies—S.L. Sobolev and A.D. Alexandrov.
Once I had asked L.V. to submit to Doklady an article by my friend who wrote a thesis under my supervision. L.V. had agreed. When the article had been published, I saw that it was submitted by P.S. Aleksandroff. I asked L.V. how this had happened. L.V. replied that he had asked P.S. Aleksandroff himself because the title of the article mentioned Kantorovich spaces which would made the submission by L.V. an indecent self-promotion.
In the fall of 1983, L.V. had last visited Akademgorodok to participate in the celebration of S.L. Sobolev on the occasion of his 75th birthday. The weather was damp and cold, and L.V. came to my home to lunch. My father and he started quite energetically to warm themselves up with Siberian vodka. Embolden, I straightforwardly asked L.V. what he considers the main achievement of his life. Without a doubt, he answered: “Most useful? Well! Linear programming.” Since the technical essence of this branch of science does not seem to have a proper scale of difficulty for a mathematician of his strength, I continued my interrogation: “And for the soul?” L.V. (a refined person who easily understood interlocutors) smiled and said what was expected: “For the soul? Indeed, K-spaces.”
It is hard to remember the last months of L.V.'s life when he died slowly from an inoperable cancer. I had visited him in hospital rather often and saw how the walking in the snowy front garden turned into stretching legs in a corridor and vanished eventually. But L.V. worked intensively: He gave interviews, wrote a message to the General Assembly of the Academy of Sciences of the USSR, dictated his farewell mathematical paper, and planned the future without himself. He was leaving this world as nobly and worthily as he lived.
L.V.'s life consists of Leningrad, Siberian and Moscow periods.
The main achievements in mathematics belong to the Leningrad period of L.V.'s life. In the 1930s he published more papers in pure mathematics whereas his 1940s were devoted to computational mathematics in which he was soon appreciated as a leader in this country.
N.N. Luzin, one of the first-rate mathematicians of those times and the founder of the famous “Lusitania,” wrote to L.V.: “You must know my attitude to you. I haven't complete knowledge of you but I guess a warm and admirable personality. One thing I know for certain, however: the range of your mental powers which, so far as I accustomed myself to guess people, open up limitless possibilities in science. I will not utter the appropriate word—what for? Talent—this would belittle you. You are entitled to get more.”
In 1935 L.V. made his major mathematical discovery—he defined K-spaces, i. e., vector lattices whose every nonempty order bounded subset had an infimum and supremum. The K-spaces have provided the natural framework for developing the theory of linear inequalities which was a practically uncharted area of research those days. The concept of inequality is obviously relevant to approximate calculations where we are always interested in various estimates of the accuracy of results. Another challenging source of interest in linear inequalities was the stock of problems of economics. The language of partial comparison is rather natural in dealing with what is reasonable and optimal in human behavior when means and opportunities are scarce. Finally, the concept of linear inequality is inseparable from the key idea of a convex set. Functional analysis implies the existence of nontrivial continuous linear functional over the space under consideration, while the presence of a functional of this type amounts to the existence of nonempty proper open convex subset of the ambient space. Moreover, each convex set is generically the solution set of an appropriate system of simultaneous linear inequalities.
At the end of the 1940s L.V. formulated and explicated the thesis of interdependence between functional analysis and applied mathematics. He distinguished the three techniques: the Cauchy method of majorants also called domination, the method of finite-dimensional approximations, and the Lagrange multiplier method for the new optimization problems motivated by economics. L.V. based his study of the Banach space versions of the Newton method on domination in general ordered vector spaces. Approximation of infinite-dimensional spaces and operators by their finite-dimensional analogs must be considered alongside the marvelous universal understanding of computational mathematics as the science of finite approximations to general (not necessarily metrizable) compacta.The novelty of the extremal problems arising in social sciences is connected with the presence of multidimensional contradictory utility functions. This raises the major problem of agreeing conflicting aims. The corresponding techniques may be viewed as an instance of scalarization of vector-valued targets.
From the end of the 1930s the research of L.V. acquired new traits in his audacious breakthrough to economics. L.V.'s booklet Mathematical Methods in the Organization and Planning of Production which appeared in 1939 is a material evidence of the birth of linear programming. Linear programming is a technique of maximizing a linear functional over the positive solutions of a system of linear inequalities. It is no wonder that the discovery of linear programming was immediate after the foundation of the theory of K-spaces.
The economic works of L.V. were hardly visible at the surface of the scientific information flow in the 1940s. But the problems of economics prevailed in his creative studies. During the Second World War he completed the first version of his book The Best Use of Economic Resources which led to the Nobel Prize awarded to him and T. Koopmans in 1975. The pioneering ideas of L.V. were legalized and introduced in the national economy.
The Council of Ministers of the USSR issued a top secret Directive No. 1990–774ss/op in 1948 which ordered “to organize in the span of two weeks a group for computations with the staff up to 15 employees in the Leningrad Division of the Mathematical Institute of the Academy of Sciences of the USSR and to appoint Professor Kantorovich the head of the group.” That was how L.V. was enlisted in the squad of participants of the project of producing an H-bomb.
The impressive diversity of the areas of research rests upon not only the personal traits of L.V. but also his methodological views. He always emphasized the innate integrity of his scientific research as well as mutual penetration and synthesis of the methods and techniques he used in solving the most diverse theoretic and applied problems of mathematics and economics. The characteristic feature of L.V.'s contribution is his orientation to the most topical and difficult problems of mathematics and economics of his epoch.
The classical research of the Newton method led to acknowledging L.V. as a top figure in computations. L.V.'s research was based on the general scheme of domination in K-spaces. These days the development of domination proceeds within Boolean valued analysis. The modern technique of mathematical modeling opened an opportunity to demonstrate that the principal properties of lattice normed spaces represent the Boolean valued interpretations of the relevant properties of classical normed spaces. The most important interrelations here are as follows: Each Banach space inside a Boolean valued model becomes a universally complete Banach–Kantorovich space in result of the external deciphering of constituents. Moreover, each lattice normed space may be realized as a dense subspace of some Banach space in an appropriate Boolean valued model.
Boolean valued analysis enables us to expand the range of applicability of K-spaces and more general modules for studying extensional equations. Many promising possibilities are open by the new method of hyperapproximation which rests on the ideas of infinitesimal analysis. The classical discretization approximates an infinite-dimensional space with the aid of finite-dimensional subspaces. Arguing within nonstandard set theory we may approximate an infinite-dimensional vector space with external finite-dimensional spaces. Undoubtedly, the dimensions of these hyperapproximations are given as actually infinite numbers. Infinitesimal methods also provide new schemes for hyperapproximation of general compact spaces. As an approximation to a compact space we may take an arbitrary internal subset containing all standard elements of the space under approximation. Hyperapproximation of the present day stems from L.V.'s ideas of discretization.
It was in the 1930s that L.V. became engrossed in the practical problems of decision making. Inspired by the ideas of functional analysis and order, L.V. attacked these problems in the spirit of searching for an optimal solution. L.V. was among the first scientists who formulated optimality conditions for rather general extremal problems. Classical remains his approach to the theory of optimal transport whose center is occupied by the Monge–Kantorovich problem. Another particularity of the extremal problems stemming from praxis consists in the presence of numerous conflicting ends and interests which are to be harmonized. In fact, we encounter the instances of multicriteria optimization whose characteristic feature is a vector-valued target. Seeking for an optimal solution in these circumstances, we must take into account various contradictory preferences which combine into a sole compound aim. Furthermore, it is impossible as a rule to distinguish some particular scalar target and ignore the rest of the targets without distorting the original statement of the problem under study.
The specific difficulties of practical problems and the necessity of reducing them to numerical calculations led L.V. to pondering over the nature of the reals. He viewed the members of his K-spaces as generalized numbers, developing the ideas that are now collected around the concept of scalarization. In the most general sense, scalarization is reduction to numbers. Since a number is a measure of quantity; therefore, the idea of scalarization is of importance to mathematics in general. L.V.'s studies on scalarization were primarily connected with the problems of economics he was interested in from the very first days of his creative path in science.
L.V.'s Siberian and Moscow periods were tied primarily with economics. Mathematics studies the forms of reasoning. The subject of economics is the circumstances of human behavior. Mathematics is abstract and substantive, and the professional decisions of mathematicians do not interfere with the life routine of individuals. Economics is concrete and declarative, and the practical exercises of economists change the life of individuals substantially. The aim of mathematics consists in impeccable truths and methods for acquiring them. The aim of economics is the well-being of an individual and the way of achieving it. Mathematics never intervenes into the private life of an individual. Economics touches his purse and bag. Immense is the list of striking differences between mathematics and economics.
Mathematical economics is an innovation of the twentieth century. It is then when the understanding appeared that the problems of economics need a completely new mathematical technique.
G. Cantor, the creator of set theory, remarked as far back as in 1883 that “the essence of mathematics lies entirely in its freedom.” The freedom of mathematics does not reduce to the absence of exogenous restriction on the objects and methods of research. The freedom of mathematics reveals itself mostly in the new intellectual tools for conquering the ambient universe which are provided by mathematics for liberation of humans by widening the frontiers of their independence. Mathematization of economics is the unavoidable stage of the journey of the mankind into the realm of freedom.
The nineteenth century is marked with the first attempts at applying mathematical methods to economics in the research by A. Cournot, K. Marx, W. Jevons, L. Walras, and his successor in Lausanne University V. Pareto. J. von Neumann and L.V., mathematicians of the first caliber, addressed the economic problems in the twentieth century. The former developed game theory, making it an apparatus for the study of economic behavior. The latter invented linear programming for decision making in the problems of best use of limited resources. These contributions of von Neumann and Kantorovich occupy an exceptional place in science. They demonstrated that modern mathematics opens up broad opportunities for economic analysis of practical problems. Economics has been drifted closer to mathematics. Still remaining a humanitarian science, it mathematizes rapidly, demonstrating high self-criticism and an extraordinary ability of objective thinking.
The principal discovery of L.V. at the junction of mathematics and economics is linear programming which is now studied by hundreds of thousands of people throughout the world. The term signifies the colossal area of science which is allotted to linear optimization models. In other words, linear programming is the science of the theoretical and numerical analysis of the problems in which we seek for an optimal (i. e., maximum or minimum) value of some system of indices of a process whose behavior is described by simultaneous linear inequalities. It is worth observing that to an optimal plan of every linear program there corresponds some optimal prices. The interdependence of optimal solutions and optimal prices is the crux of the economic discovery of L.V.
The present-day research has corroborated that the ideas of linear programming are immanent in the theory of K-spaces. It was demonstrated that the validity of one of the various statements of the duality principle of linear programming in an abstract mathematical structure implies with necessity that the structure under consideration is in fact a K-space. L.V.'s heuristic principle declares that the elements of K-spaces are instances of reals. It was discovered at the end od the twentieth century that this principle is in full agreement with the modern technologies of nonstandard set theory in Boolean valued analysis. The progress of the latter has demonstrated the fundamental importance of the so-called universally complete K-spaces. Each of these spaces turns out to present one of the possible noble models of the real axis and so such a space plays a similar key role in mathematics. The K-spaces provide new models of the reals, thus earning their eternal immortality. L.V.'s heuristics has received brilliant corroboration, which proves the integrity of science and inevitability of interpenetration of mathematics and economics.
L.V. deserves much more than the reminiscences of my meetings with him. 1982 was the year of the 70th anniversary of L.V., and I was involved in preparation of some biographical materials about L.V. This was the first date that I had seen and read practically all publications of L.V. Executing his will, I edited the two-volume set of his selected mathematical papers, a volume of his papers on mathematical economics, as well as the two-volumes of remembrances about him. His contributions and place in the world science became much clearer to me.
L.V.'s life itinerary is not a series of parades and decorations but a path of the perennial war against inertia, stagnation, ignorance, hatred, and misunderstanding. The epoch of the USSR in Russia was the time of collective triumphs, personal tragedies, bright victories, and grim cannibalism. The main moral loss of the Soviet society was the refusal from universal humanism. The excesses of the collectivist eschatology had never bypassed science. L.V. encountered many abominations in mathematics and economics. Careerism prevailed and flourished, and the main symptoms of careerism was antisemitism aggravated by intolerance to any form of dissidence. Antisemitism had never vanished in Tsarist Russia, since Russia was never a secular state. Some attempts at secularizing the society were made after the October revolution but soon them annihilated completely. The same fate was doomed to many other Utopian if not ephemeral dreams of Russian intelligentsia. The freedom of conscience and scientific outlook could not opposed Stalinism. The All-Union Communist Party (Bolsheviks) had attained the generic traits of a totalitarian sect which did not disappear in the USSR Communist Party after the dethronement of the personality cult. The domestic antisemitism was covertly appraised and even inspired by the party bonzes. Soon it became a rather effective mechanism of building a career in the years of the Jewish exodus from the country.
The negative processes did not bypass L.V. The theses of his coworkers and students were hampered or flunked, some obstacles awaited the publication of his books, his articles were procrastinated and his proposals were dillydallied. The utmost disaster was his short-term hospitalization in a lunatic asylum after his brave but vain attack against the pseudo-scientific attempt of the notorious “machine deciphering of Maya script.” Last year G.G. Ershova included the materials about L.V.'s Don Quixote attack against the haters of Yu.V. Knorozov in her book The Last Genius of the Twentieth Century.
In the times of “victorious and developed Socialism” the abomination often wore the cassocks of the “orthodox pops of Marxism” who tried to disavow L.V.'s economic ideas as well as their author. Mathematization of economics by L.V. deprived from the vanish of would-be professionalism all his opponents who could not cope with the challenges of the new realities. The unacceptability of L.V.'s conception for the top stratum of Soviet economists was due to the total lack of understanding of the role of optimal prices which was characteristic of the profaners of the Marxist theory of labor value. The novelty of L.V.'s ideas for “anti-Soviet” economists consisted in the fact that prices in L.V.'s theory are formed during the choice of an optimal plan of production rather than by a market. The market for L.V. is a mechanism of experimental determination of the optimal prices of production. L.V. was a greater scientist than any enlisted “Marxist.”
The contradistinction between the brilliant achievements and the instances of poor adaptation to the practical seamy side of life is listed among the dramatic enigmas by L.V. His life became a fabulous and puzzling humanitarian phenomenon. L.V.'s introversion, obvious in personal communications, was inexplicably accompanied by outright public extroversion. The absence of any orator's abilities neighbored his deep logic and special mastery in polemics. His innate freedom and self-sufficiency coexisted with the purposeful and indefatigable endurance that reached the power of a “wolf grip” in the case of necessity. L.V.'s freedom can hardly bewilder anyone as stemming from his essence, the gift of mathematics. His kindness and mildness were inborn. The tenacity and tremendous force of penetration were the acquired traits that he selected and cultivated conscientiously for the sake of rationality.
L.V.'s life is a turnpike of a scientist and citizen whose contributions are tied up with the fates of his next of kin and with service to his homeland irrespective of the prevalent ideological obstinacy. This lesson is of utmost import these days. Attempts at slandering and silencing L.V.'s life and legacy are doomed to vanish. Pygmies can never hide a giant. The genius of rationality in science, L.V. was ingeniously rational in choosing his world-line and path in science. He bequeathed us an exemplar of the best use of personal resources in the presence of restrictive internal and external constraints.
For me as well as for many others, the memory of Leonid Vitalievich Kantorovich has become one of the decorations and consolations of life.
March 13, 2021.
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