The great Polish-American mathematician Mark Kac (1914–1984) possessed one of the most dazzling minds of the 20th century. Kac made fundamental contributions to many branches of mathematics and physics, including statistical mechanics, potential theory, and the theory of Brownian motions. Kac’s name is immortalized in the Feynman-Kac equation used in quantum field theory, and (surprisingly) also in mathematical finance.
In pioneering probability theory, Kac paved the way for a radical new way of thinking and ushered in the first generation of scientists trained to think probabilistically — making room for uncertainty, be it scientific or otherwise. This probabilistic mode of judgment is all the more necessary today as the complexity of our world grows, leaving ordinary folks (and even some scientists) struggling to cope with the vicissitudes of life.
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 and his entire family, along with millions of other Jews, was killed by the Nazis. Despite the ordeal, Kac went on to lead a long and creatively fertile life — one he considered 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 (1976).
Marc Kac on Creativity and the Inner Life
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.
On Geniuses and Magicians
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.
Kac points to the nuclear physicist Hans Bethe as an example of an “ordinary genius” and to Richard Feynman as a “magician.” One might also add the likes of the pioneer computer scientist and code-breaker, Alan Turing and the information theorists, Claude Shannon, to the list of the “magicians”. Of there, there is always everybody’s favorite magician – the inimitable Albert Einstein.
On the Enchantment of Mathematics
Mathematics is an ancient discipline. 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 …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.”
Related: “The influence of Mark Kac on probability theory” by Harry Kesten, Cornell University.