MATH ROOTS OF THE LUZIN CASE

The human passions and follies behind the 1930s tragedy of mathematics in Russia are obvious: love and hatred, jealousy and admiration, vanity and modesty, generosity and careerism, etc. But was there a mathematical background? Some roots are visible.

We are granted the blissful world that has the indisputable property of unicity. The solitude of reality was perceived by our ancestors as the ultimate proof of unicity. This argument resided behind the incessant attempts at proving the fifth postulate of Euclid. The same gives grounds for the common search of the unique best solution of any human problem.

Mathematics has never liberated itself from the tethers of experimentation. The reason is not the simple fact that we still complete proofs by declaring “obvious.” Alive and rather popular are the views of mathematics as a toolkit for natural sciences. These stances may be expressed by the slogan “mathematics is experimental theoretical physics.” Not less popular is the dual claim “theoretical physics is experimental mathematics.” This short digression is intended to point to the interconnections of the trains of thought in mathematics and natural sciences.

It is worth observing that the dogmata of faith and the principles of theology are also well reflected in the history of mathematical theories. Variational calculus was invented in search of better understanding of the principles of mechanics, resting on the religious views of the universal beauty and harmony of the act of creation.

The twentieth century marked an important twist in the content of mathematics. Mathematical ideas imbued the humanitarian sphere and, primarily, politics, sociology, and economics. Social events are principally volatile and possess a high degree of uncertainty. Economic processes utilize a wide range of the admissible ways of production, organization, and management. The nature of nonunicity in economics transpires: The genuine interests of human beings cannot fail to be contradictory. The unique solution is an oxymoron in any nontrivial problem of economics which refers to the distribution of goods between a few agents. It is not by chance that the social sciences and instances of humanitarian mentality invoke the numerous hypotheses of the best organization of production and consumption, the most just and equitable social structure, the codices of rational behavior and moral conduct, etc.

The twentieth century became the age of freedom. Plurality and unicity were confronted as collectivism and individualism. Many particular phenomena of life and culture reflected their distinction. The dissolution of monarchism and tyranny was accompanied by the rise of parliamentarism and democracy. Quantum mechanics and Heisenberg’s uncertainty incorporated plurality in physics. The waves of modernism in poetry and artistry should be also listed. Mankind had changed all valleys of residence and dream.

In mathematics the quest for plurality led to the abandonment of the overwhelming pressure of unicity and categoricity. The latter ideas were practically absent, at least minor, in Ancient Greece and sprang to life in the epoch of absolutism and Christianity. Cantor was a harbinger of mighty changes, claiming that “das Wesen der Mathematik liegt gerade in ihrer Freiheit.” Paradoxically, the resurrection of freedom expelled mathematicians from the Cantor paradise.

Nowadays we are accustomed to the unsolvability and undecidability of many problems. We see only minor difficulties in accepting nonstandard models and modal logics. We do not worry that the problem of the continuum is undecidable within Zermelo–Fraenkel set theory. However simple nowadays, these stances of thought seemed opportunistic and controversial at the times of Luzin. The successful breakthroughs of the great students of Luzin were based on the rejection of his mathematical ideas. This is a psychological partly Freudian background of the Luzin case. His gifted students smelled the necessity of liberation from description and the pertinent blissful dreams of Luzin which were proved to be undecidable in favor of freedom for mathematics. His students were misled and consciously or unconsciously transformed the noble desire for freedom into the primitive hatred and cruelty. This transformation is a popular fixation and hobby horse of the human beings through the ages.

Terrible and unbearable is the lightheaded universal fun of putting the blame entirely on Luzin for the mathematical crimes that he was hardly guilty of with the barely concealed intention to revenge his genuine and would-be private and personal sins. We should try and understand that the ideas of description, finitism, intuitionism, and similar heroic attempts at the turn of the 20th century in search of the sole genuine and ultimate foundation were unavoidable by way of liberating mathematics from the illusionary dreams of categoricity. The collapse of the eternal unicity and absolutism was a triumph and tragedy of the mathematical ideas of the first two decades of the last century. The blossom of the creative ideas of Luzin’s students stemmed partly from his mathematical illusions in description.

The struggle against Luzin had mathematical roots which were impossible to extract and explicate those days. We see clearly now that the epoch of probability, functional analysis, distributions, and topology began when the idea of the ultimate unique foundation was ruined for ever. Gödel had explained some trains of thought behind the phenomenon, but the mathematicians par excellence felt them with inborn intuition and challenge of mind.

It is the tragedy of mathematics in Russia that the noble endeavor for freedom had launched the political monstrosity of the scientific giants disguised into the cassocks of Torquemada.

S. Kutateladze

28.02.2007


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