Mathematics and Economics—Two Intellectual Stances
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 Russian mathematics 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 synthesizing
the exact and verbal methods of reasoning, I. M. Gelfand,
the last of the mathematical giants of the 20th century,
wrote in [
1]:
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...
Kantorovich's Itinerary
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 an interest group circle of G. M. Fikhtengolts (which the Russian name for
a students' research group)
and in a seminar on descriptive set theory.
It is natural that the early academic years formed his first environment that included
D. K. Faddeev, I. P. Natanson, S. L. Sobolev, and S. G. Mikhlin
whom Kantorovich was friendly during all his life.
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 assistant 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 Nikolai N. Luzin, written on April 29, 1934,
was found in the personal archive of Kantorovich not long ago [
2]
which demonstrates the attitude of one of the most eminent
and influential Russian mathematicians of that time, to the brilliance of
the young prodigy. Luzin wrote:
... 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
[
3]
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 [
4] 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
book on linear programming [
5,pp. 22-23]:
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 addressed
economics, with the exception of
the celebrated course of
functional analysis[
6]-"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
ceasing 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's Contributions
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 [
7].
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 [
8]:
...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.
Methodological Integrity
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,
1
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.
Invention, Meaning, and Place of K-spaces
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 [
9].
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 I
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.,
we can 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.
To be precise,
assume that X is a real vector space,
Y is a
Kantorovich space also known as a Dedekind complete
vector lattice or a complete Riesz
space.
Denote by L (X,Y) the space of linear operators from X to Y.
In case X is furnished with some Y-seminorm on X,
by L
(m)(X,Y) we mean
the
space of dominated operators from X to Y.
Given T:X→ Y and y ∈ Y, put {T ≤ y}:={x ∈ X | Tx ≤ y}
and ker(T):={T=0}:=T
−1(0).
Consider another real vector space W.
It is well known that
(i)
(∃ 𝔵) 𝔵A=B ↔ ker(A) ⊂ ker(B);
(ii)
Kantorovich's Theorem
Let W be ordered by some positive cone W+ and
A(X)−W+=W+ − A(X)=W, i.e.,
A(X) is coinitial in W. Then
(∃𝔵 ≥ 0) 𝔵A=B ↔{A ≤ 0} ⊂ {B ≤ 0}.
Kantorovich's theorem was the first product of the most important
methodological position that is now referred to as
Kantorovich's heuristic principle.
It is worth noting that his definition of a semiordered
linear space contains the axiom of Dedekind completeness which
was denoted by I
6. Kantorovich demonstrated the role of
K-spaces by widening the scope of the Hahn-Banach Theorem.
The heuristic principle turned out applicable to this fundamental
Dominated Extension Theorem; i.e., we 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 the space.
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.
Attempts at formalizing Kantorovich's heuristic
principle started in the middle of the twentieth century
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.
The diversity of Kantorovich's contributions
implements the methodological integrity of his contributions.
It is no wonder so that he tried to apply semiordered spaces to
numerical methods in his earliest papers.
In the note of 1936 [
10]
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 heuristic 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 a 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 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. Vopenka.
The Boolean valued status of the notion of Kantorovich space was first
demonstrated
by Gordon's Theorem [
11] communicated by Kantorovich and
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, each theorem of ZFC, i.e., Zermelo–Fraenfkel Set Theory with Choice,
about reals
has a full analog for the corresponding
K-space.
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 various spaces and classes of linear and nonlinear operators,
operator algebras, etc.
Miraculously, it is the “bizarre” decomposability axiom of Kantorovich
which guarantees these opportunities.
Impact on Economics and Linear Programming
It was in the 1930s that Kantorovich became engrossed in the practical
problems of decision making. Inspired by the ideas of functional
analysis and order, Kantorovich. attacked these problems in the spirit of
searching for an optimal solution. Kantorovich 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 Kantorovich 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. Kantorovich'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.
Kantorovich'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 Kantorovich, 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 Kantorovich 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
Kantorovich's economic discovery.
Returning to the background ideas of K-space theory
in his last mathematical paper [
12],
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 the elements of such a space, finite- or
infinite-dimensional, as values of a norm in constructing
Banach-type 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.
Consolidation of Mind
Ideas rule the world.
John Maynard Keynes completed this banal statement with a touch
of bitter irony.
He finished his most acclaimed treatise [
13]
in a rather aphoristic manner:
Practical men, who
believe themselves to be quite exempt from any intellectual influences, are usually the slaves of some
defunct economist.
Political ideas aim at power, whereas economic ideas aim at
freedom from any power.
Political economy is inseparable from not only the economic practice but also
the practical policy. The political content of economic teachings implies their
special location within the world science.
Changes in epochs, including their technological achievements and
political utilities, lead to the universal proliferation of spread of the
emotional attitude to economic theories, which drives economics in the position
unbelievable for the other sciences.
Alongside noble reasons for that, there is one rather cynical:
although the achievements of exact sciences drastically change the life of the mankind,
they never touch the common mentality of humans as vividly and sharply as any statement about
their purses and limitations of freedom.
Science is "supersensible," implying that its
content cannot be wholly revealed without humans.
Located in the very center of culture,
science reminds of the Tower of Babel, the naive but heroic and grandiose
project of the peoples of the Earth.
Drive to freedom, innate in humans, lives in
the unsatisfiable striving for knowledge.
"We must know, we will know"-this centenarian motto of David Hilbert
resides comfortably in the treasure-trove of common sense.
Georg 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 exogenic 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 Antoine Augustin Cournot, Karl Marx,
William Stanley Jevons, Léon Walras, and his successor in Lausanne University
Vilfredo Pareto.
John von Neumann and Leonid Kantorovich, 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
the 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 turn in the mentality of the mankind that was effected by
von Neumann and Kantorovich is not always comprehended to full extent.
There are principal distinctions between the exact and humanitarian
styles of thinking. Humans are prone to reasoning by analogy
and using incomplete induction, which invokes the illusion of
the universal value of the tricks we are accustomed to.
The differences in scientific technologies are not distinguished
overtly, which in turn contributes to self-isolation and deterioration of
the vast sections of science.
The methodological precipice between
economists and mathematics was well described by
Alfred Marshall). The founder of the Cambridge school of neoclassicals,
“Marshallians,”
wrote in his
magnum opus [
14]:
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...
It is obvious that there is no room in economics
for long trains of deductive reasoning.
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” [
15]
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.
Kantorovich's Memes
Kantorovich 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. Kantorovich. 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. 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 Kantorovich. 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.
In the times of “victorious and developed Socialism” the
abomination often wore the cassocks of the «orthodox pops of
Marxism” who tried to disavow Kantorovich's economic ideas as well as their
author. Mathematization of economics 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 Kantorovich'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 Kantorovich's ideas for “anti-Soviet” economists consisted
in the fact that prices in his theory are formed during the choice
of an optimal plan of production rather than by a market. The market
in his theory is a mechanism of experimental determination of the optimal
prices of production. Kantorovich was a greater scientist than any enlisted
“Marxist.”
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.
References
- [1]
Gelfand I. M.
“Leonid Kantorovich and the Synthesis of Two Cultures.”
In: Kantorovich L. V. Selected Works. Part I. Amsterdam, Gordon and Breach, 1996, pp. 7–9.
- [2]
Reshetnyak Yu. G. and Kutateladze S. S.“A Letter of N. N. Luzin to L. V. Kantorovich,”
Vestnik Ross. Acad. Nauk, 72:8 (2002), 740–742.
- [3]
Kantorovich L. V.
Mathematical Methods of Organizing and Planning Production.
Leningrad State University (1939) [Russian];
Management Sci., 6:4 (1960), 366–422 [English].
- [4]
Kantorovich L. V.
The Best Use of Economic Resources. Oxford etc.: Pergamon Press
(1965).
- [5]
Dantzig G. B. Linear Programming and Extensions.
Princeton University Press (1963).
- [6]
Kantorovich L. V. and Akilov G. P.
Functional Analysis. Oxford etc., Pergamon Press
(1982).
- [7]
Kantorovich L. V. and Krylov V. I.
Approximate Methods of Higher Analysis.
New York and Gröningen, Interscience/P. Noordhoff (1958).
- [8]
Kantorovich L. V. “Functional Analysis and Applied Mathematics,”
in Kantorovich L. V. Selected Works.
Part II. Amsterdam, Gordon and Breach, 1996, pp. 171–280.
- [9]
Kantorovich L. V.
“Semi-ordered Linear Spaces and Their Application to the Theory of
Linear operators,”
in Kantorovich L. V. Selected Works. Part I.
Amsterdam, Gordon and Breach, 1996, pp. 213–216.
- [10]
Kantorovich L. V. “On One Class of Functional Equations,”
Dokl. Akad. Nauk SSSR, 4:5 (1936), 211–216.
- [11]
Gordon E. I. “Real Numbers in Boolean-Valued Models
of Set Theory, and K-Spaces,”
Soviet Math. Doklady, 18 (1977), 1481–1484.
- [12]
Kantorovich L. V.
“Functional Analysis (Basic Ideas),” Siberian Math. J., 28:1 (1987), 1–8.
- [13]
Keynes J. M. The General Theory of Employment, Interest and Money. Cham, Palgrave Macmillan
(2016).
- [14]
Marshall A. Principles of Economics.
8th edition, Macmillan and Co., Ltd., 1920.
Appendix C: The Scope and Method of Economics. §1 and § 3.
- [15]
Davidson P.
“Is ‘Mathematical Science’ an Oxymoron When Used to Describe Economics?”
J. Post Keynesian Economics, 25: 4 (2003), 527–546.
Footnotes:
1Kantorovich wrote about “my spaces” in his personal notes.