MATHEMATICS AND ECONOMICS OF
KANTOROVICH

On the Occassion of the Centenary of
Leonid Kantorovich (1912–1986)

Leonid Kantorovich

January 19, 2012 is the centenary of the birth of Leonid Vital'evich Kantorovich, a renowned mathematician and economist. He was a prodigy who graduated from university in 18, became a professor in 20, was elected a full member of the Department of Mathematcis of the Academy of Sciences of the USSR, and was awarded with a Nobel Prize in economics. These extraordinary events of life deserve some attention in their own right. But they hardly lead to any useful conclusions for the general audience in view of their extremely low probability. This is not so with the creative legacy of a human, for what is done for the others remains unless it is forgotten, ruined, or libeled. The date of felicitation is a good reason for inventory of memories. Recollecting the contributions of persons to culture, we preserve their spiritual worlds for the future....

Linear Programming. 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 was in 1939 that Kantorovich formulated the basic ideas of the new area of science.

The term “linear programming” was minted in 1951 by Tjalling C. Koopmans, an American economist with whom Kantorovich shared in 1975 the Nobel Prize for research of optimal use of resources. The most commendable contribution of Koopmans was the ardent promotion of the methods of linear programming and the strong defence of Kantorovich’s priority in the invention of these methods.

In the USA the independent research into linear optimization models was started only in 1947 by George B. Dantzig who also noted the priority of Kantorovich.

It is worth observing that to an optimal plan of every linear program there corresponds some optimal prices or “objectively determined estimators.” Kantorovich invented this bulky term by tactical reasons in order to enhance the “criticism endurability” of the concept. The conception of optimal prices as well as the interdependence of optimal solutions and optimal prices is the crux of the economic discovery of Kantorovich.

Matematics and 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 decision 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.

Homo sapiens has always been and will stay forever homo economicus. Practical economics for everyone as well as their ancestors is the arena of common sense. Common sense is a specific ability of a human to instantaneous moral judgement. Understanding is higher than common sense and reveals itself as the adaptability of behavior. Understanding is not inherited and so it does nor belong to the inborn traits of a person. The unique particularity of humans is the ability of sharing their understanding, transforming evaluations into material and ideal artefacts.

Culture is the treasure-trove of understanding. The inventory of culture is the essence of outlook. Common sense is subjective and affine to the divine revelation of faith that is the force surpassing the power of external proofs by fact and formal logic. The verification of statements with facts and by logic is a critical process liberating a human from the errors of subjectivity. Science is an unpaved road to objective understanding. The religious and scientific versions of outlook differ actually in the methods of codifying the artefacts of understanding.

The rise of science as an instrument of understanding is a long and complicated process. The birth of ordinal counting is fixed with the palaeolithic findings hat separated by hundreds of centuries from the appearance of a knowing and economic human. Economic practice precedes the prehistory of mathematics that became the science of provable calculations in Ancient Greece about 2500 years ago.

It was rather recently that the purposeful behavior of humans under the conditions of limited resources became the object of science. The generally accepted date of the birth of economics as a science is March 9, 1776—the day when there was published the famous book by Adam Smith An Inquiry into the Nature and Causes of the Wealth of Nations.

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 The General Theory of Employment, Interest, and Money 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.

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, Carl Marx, William Stanley Jevons, Léon Walras, and his successor in Lausanne University Vilfredo Pareto.

John von Neumann and Leonid Kantorovich, mathematicians of the first calibre, 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.

Universal Heuristics. The integrity of the outlook of Kantorovich was revealed in all instances of his versatile research. The ideas of linear programming were tightly interwoven with his methodological standpoints in the realm of mathematics. Kantorovich viewed as his main achievement in this area the distinguishing of Dedekind complete vector lattes called K-spaces} or Kantorovich spaces in the literature of the Russian provenance, since Kantorovich wrote about “my spaces” in his personal memos.

Kantorovich observed in his first short paper of 1935 in Doklady on the newly-born area of ordered vector spaces: “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.”

This was the first formulation of the most important methodological position that is now referred to as Kantorovich’s heuristic principle. The abstract theory of $K$-spaces, linear programming, and approximate methods of analysis were particular outputs of Kantorovich’s universal heuristics.

More recent research has corroborated that the ideas of linear programming are immanent in the theory of K-spaces. It was demonstrated that the validity of one of the various statements of the duality principle of linear programming in an abstract mathematical structure implies with necessity that the structure under consideration is in fact a K-space.

The Kantorovich heuristic principle is connected with one of the most brilliant pages of the mathematics of the twentieth century—the famous problem of the continuum. Recall that some set A has the cardinality of the continuum whenever A in equipollent with a segment of the real axis. The continuum hypothesis is that each subset of the segment is either countable of has the cardinality of the continuum. The continuum problem asks whether the continuum hypothesis is true or false.

The continuum hypothesis was first conjectured by Cantor in 1878. He was convinced that the hypothesis was a theorem and vainly attempted at proving it during his whole life. In 1900 the Second Congress of Mathematicians took place in Paris. In the opening session Hilbert delivered his epoch-making talk “Mathematical Problems.” He raised 23 problems whose solution was the task of the nineteenth century bequeathed to the twentieth century. The first on the Hilbert list was open the continuum problem. Remaining unsolved for decades, it gave rise to deep foundational studies. The efforts of more than a half-century yielded the solution: we know now that the continuum hypothesis can neither be proved nor refuted.

The two stages led to the understanding that the continuum hypothesis is an independent axiom. Göodel showed in 1939 that the continuum hypothesis is consistent with the axioms of set theory, and Cohen demonstrated in 1963 that the negation of the continuum hypothesis does not contradict the axioms of set theory either. Both results were established by exhibiting appropriate models; i. e., constructing a universe and interpreting set theory in the universe.

Cohen’s method of forcing was simplified in 1965 on using the tools of Boolean algebra and the new technique of mathematical modeling which is based on the nonstandard models of set theory. The progress of the so-invoked Boolean valued analysis has demonstrated the fundamental importance of the so-called universally complete K-spaces. Each of these spaces turns out to present one of the possible noble models of the real axis and so such a space plays a similar key role in mathematics. The spaces of Kantorovich implement new models of the reals, this earning their eternal immortality.

Kantorovich heuristics has received brilliant corroboration, this proving the integrity of science and inevitability of interpenetration of mathematics and economics.

Memes for the Future. The contradistinction between the brilliant achievements and the childish unfitness for the practical seamy side of life is listed among the dramatic enigmas by Kantorovich. His life became a fabulous and puzzling humanitarian phenomenon. Kantorovich’s introvertness, obvious in personal communications, was inexplicably accompanied by outright public extravertness. The absence of any orator’s abilities neighbored his deep logic and special mastery in polemics. His innate freedom and self-sufficiency coexisted with the purposeful and indefatigable endurance in the case of necessity. He bequeathed us a magnificent example of the best use of personal resources in the presence of restrictive internal and external constraints.

The memes of Kantorovich have been received as witnessed by the curricula and syllabi of every economics or mathematics department in any major university throughout the world. The gadgets of mathematics and the idea of optimality belong to the tool-kit of any practicing economist. The new methods erected an unsurmountable firewall against the traditionalists that view economics as a testing polygon for the technologies like Machiavellianism, flattery, common sense, or foresight.

Economics as an eternal 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 impeccably rigorous methods.

Calculation will supersede prophesy.

S. Kutateladze

January 1, 2011


A fuller version is available at http://arxiv.org/abs/1101.0984.


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© Kutateladze S. S. 2011