THE PATH AND SPACE OF
KANTOROVICH ^*

Mathematical economics belongs to or at least borders the realm of applied mathematics. The epithets “pure” and “applied” for mathematics have many deficiencies provoking endless discussion and controversy. Nevertheless the corresponding brand names persist and proliferate in scientific usage, signifying some definite cultural phenomena. Scientometricians and ordinary mathematicians, pondering over this matter, usually state that the hallmark of the mathematics of this country as opposed to, say, the American mathematics, is a prevalent trend to unity with intention when possible to emphasize common features and develop a single infrastructure. Any glimpse of a gap or contradistinction between pure and applied mathematics usually brings about the smell of collision, emotion, or at least discomfort to every specialist of a Russian provenance. At the same time the separate existence of the American Mathematical Society and the Society for Industrial and Applied Mathematics is perfectly natural for the States.

It is rarely taken into account that these special features of social life are linked with the stances and activities of particular individuals.

Leonid Vital'evich Kantorovich (1912–1986) will always rank among those Russian scholars who maintained the trend to unity of pure and applied mathematics by an outstanding personal contribution. He is an exception even in this noble company because of his extraordinary traits stemming from a quite rare combination of the generous gifts of a polymath and practical economist. Describing the exceptional role of Kantorovich in synthesis of the exact and verbal methods of reasoning, I. M. Gelfand, the last of the mathematical giants of the 20th century, wrote:²

   Only a very few people of the 20th century turned out to be capable of the required synthesis of the two cultures—mathematics and the human sciences. Among them I can name Andrei Kolmogorov who always understood the world as a unified whole. This was also understood, perhaps on a more naive level, with a strong technocratic influence, by John von Neumann. In the field of social sciences, closer to the humanities, this synthesis was effected by Leonid Kantorovich.

   When I say synthesis I don't mean that the two parts of Kantorovich’s heritage are two sides of his personality, that he is sometimes a mathematician, sometimes a specialist in the human sciences. Such combinations occur often: they do not concern us here. What I mean is the all-prevailing light of the spirit that appears in all his creative work...

I will now recall some facts of the life of Kantorovich so as to present at least a rough draft of the list of events and achievements.

Kantorovich was born in the family of a venereologist at St. Petersburg on January 19, 1912 (January 6, according to the old Russian style). The boy’s talent revealed itself very early. In 1926, just at the age of 14, he entered St. Petersburg (then Leningrad) State University (SPSU). Soon he started participating in a circle of G. M. Fikhtengolts for students and in a seminar on descriptive function theory. It is natural that the early academic years formed his first environment: D. K. Faddeev, I. P. Natanson, S. L. Sobolev, S. G. Mikhlin, and a few others with whom Kantorovich was friendly during all his life also participated in Fikhtengolts’s circle.

After graduation from SPSU in 1930, Kantorovich started teaching, combining it with intensive scientific research. Already in 1932 he became a full professor at the Leningrad Institute of Civil Engineering and an associate professor at SPSU.

From 1934 Kantorovich was a full professor at his alma mater. His close connection with SPSU and the Leningrad Division of the Steklov Mathematical Institute of the Academy of Sciences lasted until his transition to Novosibirsk on the staff of the Institute of Mathematics of the Siberian Division of the Academy of Sciences of the USSR (now, the Sobolev Institute) at the end of the 1950s.

The letter of Academician N. N. Luzin, written on April 29, 1934, was found in the personal archive of Kantorovich not long ago. ³ This letter demonstrates the attitude of Luzin, one of the most eminent and influential mathematicians of that time, to the brilliance of the young prodigy. Luzin remarked:

   ... you must know my attitude to you. I haven't complete knowledge of you but I guess a warm and admirable personality.

   However, one thing I know for certain: 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...

Kantorovich had written practically all of his major mathematical works in his “Leningrad” period. Moreover, in the 1930s he mostly published articles on pure mathematics whereas the 1940s became his season of computational mathematics in which he was soon acknowledged as an established and acclaimed leader.

At the end of the 1930s Kantorovich revealed his outstanding gift of an economist. His booklet Mathematical Methods in the Organization and Planning of Production is a material evidence of the birth of linear programming. The economic works of Kantorovich were hardly visible at the surface of the scientific information flow in the 1940s. However, 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 Tjalling C. Koopmans in 1975.

The priority of the Kantorovich invention was never questioned. George B. Dantzig wrote in his classical book⁴ on linear programming:

   The Russian mathematician L. V. Kantorovich has for a number of years been interested in the application of mathematics to programming problems. He published an extensive monograph in 1939 entitled Mathematical Methods in the Organization and Planning of Production...
    Kantorovich should be credited with being the first to recognize that certain important broad classes of production problems had well-defined mathematical structures which, he believed, were amenable to practical numerical evaluation and could be numerically solved.
    In the first part of his work Kantorovich is concerned with what we now call the weighted two-index distribution problems. These were generalized first to include a single linear side condition, then a class of problems with processes having several simultaneous outputs (mathematically the latter is equivalent to a general linear program). He outlined a solution approach based on having on hand an initial feasible solution to the dual. (For the particular problems studied, the latter did not present any difficulty.) Although the dual variables were not called “prices,” the general idea is that the assigned values of these “resolving multipliers” for resources in short supply can be increased to a point where it pays to shift to resources that are in surplus. Kantorovich showed on simple examples how to make the shifts to surplus resources. In general, however, how to shift turns out to be a linear program in itself for which no computational method was given. The report contains an outstanding collection of potential applications...
    If Kantorovich’s earlier efforts had been appreciated at the time they were first presented, it is possible that linear programming would be more advanced today. However, his early work in this field remained unknown both in the Soviet Union and elsewhere for nearly two decades while linear programming became a highly developed art.

In 1957 Kantorovich was invited to join the newly founded Siberian Division of the Academy of Sciences of the USSR. He agreed and soon became a corresponding member of the Division of Economics in the first elections to the Siberian Division. Since then his major publications were devoted to economics, with the exception of the celebrated course of functional analysis⁵—“Kantorovich and Akilov” in the students’ jargon.

The 1960s became the decade of his recognition. In 1964 he was elected a full member of the Department of Mathematics of the Academy of Sciences of the USSR, and in 1965 he was awarded the Lenin Prize. In these years he vigorously propounded and maintained his views of interplay between mathematics and economics and exerted great efforts to instill the ideas and methods of modern science into the top economic management of the Soviet Union, which was almost in vain.

At the beginning of the 1970s Kantorovich left Novosibirsk for Moscow where he was still engaged in economic analysis, not ceasizing his efforts to influence the everyday economic practice and decision making in the national economy. These years witnessed a considerable mathematical Renaissance of Kantorovich. Although he never resumed proving theorems, his impact on the mathematical life of this country increased sharply due to the sweeping changes in the Moscow academic life on the eve of Gorbi’s “perestroika.” Cancer terminated his path in science on April 7, 1986. He was buried at the Novodevichy Cemetery in Moscow.

Kantorovich started his scientific research in rather abstract and sophisticated sections of mathematics such as descriptive set theory, approximation theory, and functional analysis. It should be stressed that at the beginning of the 1930s these areas were most topical, prestigious, and difficult. Kantorovich’s fundamental contribution to theoretic mathematics, now indisputable and universally acknowledged, consists in his pioneering works in the above-mentioned areas. Note also that in the “mathematical” years of his career he was primarily famous for his research into the approximate methods of analysis, the ancient euphemism for the computational mathematics of today.

The first works of Kantorovich on computational mathematics were published in 1933. He suggested some approaches to approximate solution of the problem of finding a conformal mapping between domains. These methods used the idea of embedding the original domains into some one-parameter family of domains. Expanding in a parameter, Kantorovich found out new explicit formulas for approximate calculation of conformal mappings between multiply-connected domains.

In 1933 one of Kantorovich’s teachers, V. I. Smirnov included these methods in his multivolume treatise A Course of Higher Mathematics which belongs now to the world-class deskbooks.

Kantorovich paid much attention to direct variational methods. He suggested an original method for approximate solution of second order elliptic equations which was based on reduction of the initial problem to minimization of a functional over some functions of one variable. This technique is now called reduction to ordinary differential equations.

The variational method was developed in his subsequent works under the influence of other questions. For instance, his collocation method was suggested in an article about calculations for a beam on an elastic surface.

A few promising ideas were proposed by Kantorovich in the theory of mechanical quadratures which formed a basis for some numerical methods of solution of a general integral equation with a  singularity.

This period of his research into applied mathematics was crowned with a joint book with V. I. Krylov Methods for Approximate Solution of Partial Differential Equations whose further expanded editions appeared under the title Approximate Methods of Higher Analysis.

Functional analysis occupies a specific place in the scientific legacy of Kantorovich. He has been listed among the classics of the theoretic sections of this area of research as one of the founders of ordered vector spaces. Also, he contributed much to making functional analysis a natural language of computational mathematics. His article “Functional analysis and applied mathematics” in Russian Mathematical Surveys (1948) made a record in the personal file of Kantorovich as well as in the history of mathematics in this country. Kantorovich wrote in the introduction to this article:⁶

    ...there is now a tradition of viewing functional analysis as a purely theoretic discipline far removed from direct applications, a discipline which cannot deal with practical questions. This article is an attempt to break with this tradition, at least to a certain extend, and to reveal the relationship between functional analysis and questions of applied mathematics, an attempt to show that functional analysis can be useful to mathematicians dealing with practical applications.
   Namely, we would try to show that the ideas and methods of functional analysis can be readily used to construct and analyze effective practical algorithms for solving mathematical problems with the same success as they were used for the theoretic studies of the problems.

The mathematical ideas of this article remain classical by now: The method of finite-dimensional approximations, estimation of the inverse operator, and, last but not least, the Newton–Kantorovich method are well known to the majority of the persons recently educated in mathematics.

The general theory by Kantorovich for analysis and solution of functional equations bases on variation of “data” (operators and spaces) and provides not only estimates for the rate of convergence but also proofs of the very fact of convergence.

As an instance of incarnation of the idea of unity of functional analysis and computational mathematics Kantorovich suggested at the end of the 1940s that the Mechanics and Mathematics Department of SPSU began to prepare specialists in the area of computational mathematics for the first time in this country. He prolonged this line in Novosibirsk State University where he founded the chair of computational mathematics which delivered graduate courses in functional analysis in the years when Kantorovich hold the chair. I had a privilege of specialization in functional analysis which was offered by the chair of computational mathematics in that unforgettable span of time.

It should be emphasized that Kantorovich tied the progress of linear programming as an area of applied mathematics with the general demand for improving the functional-analytical techniques pertinent to optimization: the theory of topological vector spaces, convex analysis, the theory of extremal problems, etc. Several major sections of functional analysis (in particular, nonlinear functional analysis) underwent drastic changes under the impetus of new applications.

The scientific legacy of Kantorovich is immense. His research in the areas of functional analysis, computational mathematics, optimization, and descriptive set theory has had a dramatic impact on the foundation and progress of these disciplines. Kantorovich deserved his status of one of the father founders of the modern economic-mathematical methods. Linear programming, his most popular and celebrated discovery, has changed the image of economics.

Kantorovich authored more than 300 articles. When we discussed with him the first edition of an annotated bibliography of his publications in the early 1980s, he suggested to combine them in the nine sections:

   (1) descriptive function theory and set theory;

   (2) constructive function theory;

   (3) approximate methods of analysis;

   (4) functional analysis;

   (5) functional analysis and applied mathematics;

   (6) linear programming;

   (7) hardware and software;

   (8) optimal planning and optimal prices;

   (9) the economic problems of a planned economy.

The impressive diversity of these areas of research rests upon not only the traits of Kantorovich 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. I leave a thorough analysis of the methodology of Kantorovich’s contribution a challenge to professional scientometricians. It deserves mentioning right away only that the abstract ideas of Kantorovich in the theory of Dedekind complete vector lattices, now called Kantorovich spaces or K- spaces, ⁷ turn out to be closely connected with the art of linear programming and the approximate methods of analysis.

Kantorovich told me in the fall of 1983 that his main mathematical achievement is the development of K-space theory within functional analysis while remarking that his most useful deed is linear programming. K-space, a beautiful pearl of his scientific legacy, deserves a special discussion.

Let us look back at the origin of K-space. The first work of Kantorovich in the area of ordered vector spaces appeared in 1935 as a short note in Doklady.⁸ Therein he treated the members of a K-space as generalized numbers and propounded the heuristic transfer principle. He wrote:

   In this note, I define a new type of space that I call a semiordered linear space. The introduction of such a space allows us to study linear operations of one abstract class (those with values in these spaces) in the same way as linear functionals.

It is worth noting that his definition of a semiordered linear space contains the axiom of Dedekind completeness which was denoted by I₆. Therefore, Kantorovich selected the class of K-spaces, now named after him, in his first article on ordered vector spaces. He applied K-spaces to widening the scope of the fundamental Hahn–Banach Theorem and stated Theorem 3 which is now known as the Hahn–Banach–Kantorovich Theorem. This theorem claims in fact that the heuristic transfer principle is applicable to the classical Dominated Extension Theorem; i.e., one may abstract the Hahn–Banach Theorem on substituting the elements of an arbitrary K-space for reals and replacing linear functionals with operators acting into this space.

The diversity of Kantorovich’s contributions combines with methodological integrity. It is no wonder so that Kantorovich tried to apply semiordered spaces to numerical methods in his earliest papers. In a note⁹ of 1936 he described the background for his approach as follows:

   The method of successive approximations is often applied to proving existence of solutions to various classes of functional equations; moreover, the proof of convergence of these approximations leans on the fact that the equation under study may be majorized by another equation of a simple kind. Similar proofs may be encountered in the theory of infinitely many simultaneous linear equations and in the theory of integral and differential equations. Consideration of semiordered spaces and operations between them enables us to easily develop a complete theory of such functional equations in abstract form.

There is no denying that the classical method of majorants which stems from the works of A. Cauchy acquires its natural and final form within K-space theory.

Inspired by some applied problems, Kantorovich propounded the idea of a lattice-normed space or B_K-space and introduced a special decomposability axiom for the lattice norm of a B_K-space. This axiom looked bizarre and was often omitted in the subsequent publications of other authors as definitely immaterial. The principal importance of this axiom was revealed only within Boolean valued analysis in the 1980s. As typical of Kantorovich, the motivation of B_K-space, now called Banach–Kantorovich space, was deeply rooted in abstractions as well as in applications. The general domination method of Kantorovich was substantially developed by himself and his students and followers and occupies a noble place in the theoretic toolkit of computational mathematics.

The above-mentioned informal principle was corroborated many times in the works of Kantorovich and his students and followers. Attempts at formalizing the heuristic ideas by Kantorovich started at the initial stages of K-space theory and yielded the so-called identity preservation theorems. They assert that if some algebraic proposition with finitely many function variables is satisfied by the assignment of all real values then it remains valid after replacement of reals with members of an  arbitrary K-space.

Unfortunately, no satisfactory explanation was suggested for the internal mechanism behind the phenomenon of identity preservation. Rather obscure remained the limits on the heuristic transfer principle. The same applies to the general reasons for similarity and parallelism between the reals and their analogs in K-space which reveal themselves every now and then. The omnipotence and omnipresence of Kantorovich’s transfer principle found its full explanation only within Boolean valued analysis in the 1970s.

Boolean valued analysis (the term was minted by G. Takeuti) is a branch of functional analysis which uses special set-theoretic models with truth-values in an arbitrary Boolean algebra. Since recently this term has been treated in a broader sense of nonstandard analysis, implying the tricks and tools that stem from comparison between the implementations of a mathematical concept or construct in two distinct Boolean valued models.

Note that the invention of Boolean valued analysis was not connected with the theory of vector lattices. The appropriate language and technique had already been available within mathematical logic by 1960. Nevertheless, the main idea was still absent for rapid progress in model theory and its applications. This idea emerged when P. J. Cohen demonstrated in 1960 that the classical continuum hypothesis is undecidable in a rigorous mathematical sense. It was the Cohen method of forcing whose comprehension led to the invention of Boolean valued models of set theory which is attributed to the efforts by D. Scott, R. Solovay, and P. Vopěnka.

The Boolean valued status of the notion of Kantorovich space was first demonstrated by Gordon’s Theorem¹⁰ dated from the mid 1970s. This fact can be reformulated as follows: A universally complete K-space serves as interpretation of the reals in a suitable Boolean valued model. (Parenthetically speaking, every Archimedean vector lattice admits universal completion.) Furthermore, every theorem of ZFC about real numbers has a full analog for the corresponding K-space. ¹¹ Translation of one theorem into the other is fulfilled by some precisely-defined Escher-type procedures: ascent, descent, canonical embedding, etc., i.e., by algorithm, as a matter of fact. Thus, Kantorovich’s motto: “The elements of a K-space are generalized numbers” acquires a rigorous mathematical formulation within Boolean valued analysis. On the other hand, the heuristic transfer principle finished its auxiliary role of a guiding nature in many studies of the pre-Boolean-valued K-space theory and becomes a powerful and precise method of research within Boolean valued analysis.

Further progress of Boolean valued analysis revealed that this translation (transfer or interpretation) making new theorems from available facts is possible not only for K-spaces but also for practically all objects related to them such as B_K-spaces, various classes of linear and nonlinear operators, operator algebras, etc. A. G. Kusraev proved that the heuristic transfer principle for B_K-spaces (to within elementary stipulations) reads formally as follows:¹² Each Banach–Kantorovich space embeds in a Boolean valued model, becoming a Banach space. In other words, a B_K-space is a Boolean valued interpretation of a Banach space. Moreover, it is the “bizarre” decomposability axiom of Kantorovich that guarantees the possibility of this embedding.

Returning to the background ideas of K-space theory in his last mathematical paper,¹³ Kantorovich wrote just before his death:

   One aspect of reality was temporarily omitted in the development of the theory of function spaces. Of great importance is the relation of comparison between practical objects, alongside algebraic and other relations between them. Simple comparison applicable to every pair of objects is of a depleted character; for instance, we may order all items by weight which is of little avail. That type of ordering is more natural which is defined or distinguished when this is reasonable and which is left indefinite otherwise (partial ordering or semiorder). For instance, two sets of goods must undoubtedly be considered as comparable and one greater than the other if each item of the former set is quantitatively greater than its counterpart in the latter. If some part of the goods of one set is greater and another part is less than the corresponding part of the other then we can avoid prescribing any order between these sets. It is with this in mind that the theory of ordered vector spaces was propounded and, in particular, the theory of the above-defined K-spaces. It found various applications not only in the theoretic problems of analysis but also in construction of some applied methods, for instance the theory of majorants in connection with the study of successive approximations. At the same time the opportunities it offers are not revealed fully yet. The importance for economics is underestimated of this branch of functional analysis. However, the comparison and correspondence relations play an extraordinary role in economics and it was definitely clear even at the cradle of K-spaces that they will find their place in economic analysis and yield luscious fruits.
   The theory of K-spaces has another important feature: their elements can be treated as numbers. In particular, we may use elements of such a space, finite- or infinite-dimensional, as a norm in construction of analogs of Banach spaces. This choice of norms for objects is much more accurate. Say, some function is normed by a dozen of its maxima on parts of an interval rather than the maximum of this function on the whole interval.

Observe that this excerpt of the Kantorovich article draws attention to the close connection of K-spaces with the theory of inequalities and economic topics. It is also worth noting that the ideas of linear programming are immanent to K-space theory in the following rigorous sense: The validity of each of the universally accepted formulations of the duality principle with prices in some algebraic structure necessitates that this structure is a K-space.

Magically prophetic happens to be the claim of Kantorovich that the elements of a K-space are generalized numbers. The heuristic transfer principle of Kantorovich found a brilliant justification in the framework of modern mathematical logic. Guaranteeing a profusion of unbelievable models of the real axis, the spaces of Kantorovich will stay for ever in the treasure-trove of the world science.

Alfred Marshall (1842–1924), the founder of the Cambridge school of neoclassicals, “Marshallians,” wrote in his magnum opus:

   The function then of analysis and deduction in economics is not to forge a few long chains of reasoning, but to forge rightly many short chains and single connecting links... ¹⁴
   It is obvious that there is no room in economics for long trains of deductive reasoning. ¹⁵

At the same time, there is no gainsay in ascribing the beauty and power of mathematics to the axiomatic method which consists ideally in deriving new bits and bobs of knowledge from however lengthy chains of formal implications. The conspicuous discrepancy between economists and mathematicians in mentality has hindered their mutual understanding and cooperation. Many partitions, invisible but ubiquitous, were erected in ratiocination, isolating the economic community from its mathematical counterpart and vice versa.

This status quo with deep roots in history was always a challenge to Kantorovich, contradicting his views of interaction between mathematics and economics. His path in science is well marked with the signposts conveying the slogan: “Mathematicians and Economists of the World, Unite!” His message has been received as witnessed by the curricula and syllabi of every economics department in a major university throughout the world.

Despite the antediluvian opinion that “the mathematical scientist emperor of mainstream economics is without any clothes,”¹⁶ the gadgets of mathematics and the idea of optimality will come in handy for a practical economist. Calculation will supersede prophecy. Economics as a boon companion of mathematics will avoid merging into any esoteric part of the humanities, or politics, or belles-lettres. The new generations of mathematicians will treat the puzzling problems of economics as an inexhaustible source of inspiration and an attractive arena for applying and refining their formal methods.

The years of Kantorovich’s life dim in the past. Yet Kantorovich’s path in science, lit with his phenomenal personality and seminal ideas, becomes clearer and brighter helping us to chart new roads between mathematics and economics. Most of them will lead to the turnpike of linear programming Kantorovich was the first to traverse...

^* A talk at the closing session of the “Kantorovich Memorial” in the St. Peterburg Department of the Steklov Institute on January 12, 2004.

© Kutateladze S. S. 2004