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Probability Theory Pioneer Mark Kac on the Duality of the Creative Life, the Singular Enchantment of Mathematics, and the Two Types of Geniuses

“Creative people live in two worlds. One is the ordinary world which they share with others and in which they are not in any special way set apart from their fellow men. The other is private and it is in this world that the creative acts take place.”

Probability Theory Pioneer Mark Kac on the Duality of the Creative Life, the Singular Enchantment of Mathematics, and the Two Types of Geniuses

The great Polish-American mathematician Mark Kac (August 3, 1914–October 26, 1984) possessed one of the most dazzling minds of the twentieth century. In pioneering probability theory, he paved the way for a radical new conception of truth and ushered in the first generation of scientists trained to think probabilistically — a more accurate assessment of knowledge, making room for uncertainty, be it scientific or otherwise. This probabilistic mode of judgment is all the more necessary today as the growing complexity of the world is swirling us into exponentially increasing uncertainty, which we attempt to tame through artificial absolutism.

Mathematics literally saved Kac’s life. His student work earned him a post-doctoral fellowship to study abroad, so he left Poland for Johns Hopkins University in December of 1938. World War II broke out months later. His entire family, along with millions of other Jews, was killed by the Nazis.

Kac went on to lead a long and creatively fertile life — one he considered, despite this unfathomable share of misfortune, a tremendously fortunate one. “I must pay tribute to that powerful but capricious lady, Chance, who chose to bestow her beneficence on my personal life even though I spent much of my mathematical life trying to prove that she does not really exist,” he wrote with his characteristic mix of wit and wisdom in Enigmas of Chance: An Autobiography (public library) — a small, wonderful 1976 book I discovered via a passing mention in an interview with the trailblazing astronomer Vera Rubin. (Here is further proof of my longstanding conviction that literature is the original internet — such citations, allusions, and cross-references between books are the wondrous “hyperlinks” connecting human knowledge throughout our “common record.”)

Mark Kac
Mark Kac

Kac writes:

Creative people live in two worlds. One is the ordinary world which they share with others and in which they are not in any special way set apart from their fellow men. The other is private and it is in this world that the creative acts take place. It is a world with its own passions, elations and despairs, and it is here that, if one is as great as Einstein, one may even hear the voice of God. The two worlds are intimately and intricately connected. Jealousy, the desire for recognition and competitiveness, for example, are part of the ordinary world but they are among the forces which propel into the second. Similarly, dreams and triumphs in the second have a way of merging with less than lofty thoughts of rewards in the first.

With an eye to the particular challenge that autobiography presents to the creative person, he adds:

To create a coherent and truthful picture of life in the two disparate and yet interrelated worlds is a nearly impossible task.

In discussing his great heroes and influences, Kac delineates another dichotomy in creative culture — the bifurcation of brilliance by degree and by kind:

In science, as well as in other fields of human endeavor, there are two kinds of geniuses: the “ordinary” and the “magicians.” An ordinary genius is a [person] that you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what he has done, we feel certain that we, too, could have done it. It is different with the magicians. They are, to use mathematical jargon, in the orthogonal complement of where we are and the working of their minds is for all intents and purposes incomprehensible. Even after we understand what they have done, the process by which they have done it is completely dark. They seldom, if ever, have students because they cannot be emulated and it must be terribly frustrating for a brilliant young mind to cope with the mysterious ways in which the magician’s mind works.

He points to the nuclear physicist Hans Bethe as an example of an “ordinary genius” and to Richard Feynman as a “magician.” (Kac’s distinction appears in James Gleick’s superb biography of Feynman and there is a high probability that it inspired the title of BBC’s documentary about the legendary physicist, No Ordinary Genius.) I would add Alan Turing to the “magicians” category, and of course Albert Einstein.

Illustration by Vladimir Radunsky for On a Beam of Light: A Story of Albert Einstein by Jennifer Berne

In the postscript, Kac considers what lends mathematics its enduring enchantment — what renders people besotted with it:

Mathematics is an ancient discipline. For as long as we can reliably reach into the past, we can find its development intimately connected with the development of the whole of our civilization. For as long as we have a record of man’s curiosity and his quest for understanding, we find mathematics cultivated and cherished, practiced and taught. Throughout the ages it has stood as an ultimate in rational thought and as a monument to man’s desire to probe the workings of his own mind.


The urge to understand and to create mathematics has always been remarkable, considering that those who have devoted their lives to the service of this aloof and elusive mistress could expect neither great material rewards nor widespread fame.

A champion of nuance, Kac challenges Bertrand Russell’s famous assertion that “mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.” Kac finds it to be a “dull and rather miserable picture of mathematics” and yet “not wholly wrong,” “just hopelessly incomplete and one-sided.” Instead, he considers what grants mathematics its richness as a mode of illuminating reality:

I am reminded of something Balthazaar van der Pol, a great Dutch scientist and engineer who was also a fine musician, remarked to me about the music of Bach. “It is great,” he said, “because it is inevitable and yet surprising.” I have often thought about this lovely epigram in connection with mathematics… The inevitability is, in many cases, provided by logic alone, but the element of surprise must come from an insight outside the rigid confines of logic.

It warrants noting that the altogether marvelous Enigmas of Chance is part of a series of scientists’ autobiographies funded by the Alfred P. Sloan Foundation, which has probably done more for science and its social life than any other entity in the past half-century. Complement it with this beautiful love letter to mathematics by the pioneering 19th-century English mathematician James Joseph Sylvester, the illustrated life of the eccentric mathematical genius Paul Erdos, and the great English mathematician John Horton Conway on tinkering, thinkering, and the art of being a professional nonunderstander.

Published May 4, 2016




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