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.
^{4}
on linear programming:
_{6}. 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.

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}

*
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:

*
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.

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 analysis^{5}—“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}

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:

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 note^{9}
of 1936 he described the background for his approach as follows:

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 Theorem^{10}
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:

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:

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...

On 18 Jul 2003, 01:18.

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© Kutateladze S. S. 2004