You are here

Mark Kac’s First Publication: A Translation of "O nowym sposobie rozwiązywania równań stopnia trzeciego" – Biographical Introduction

David Derbes (University of Chicago Laboratory School, retired)

Mark Kac (pronounced “cots”, 1914–1985) was a pioneer in the development of the modern theory of probability.1 He was a star among the constellation of great Polish mathematicians who emerged in the early part of the twentieth century.2 Kac’s first language was Russian, since he was born in an area then under the control of the Russian Empire. As a child he learned to speak Hebrew, the language of instruction at his primary school (his father was the principal), and French. He was eleven before he learned Polish, which was required by his new school, the Krzemieniec Lyceum of his home town. Intending to study engineering at Jan Kazimierz University (Lwów), he was diverted into mathematics by the events related below, and he earned a doctorate (1937) there under Hugo Steinhaus (a student of D. Hilbert’s, 1911). Thanks to a postgraduate scholarship at Johns Hopkins awarded in December 1938, Kac acquired his fifth language, English, and escaped the Nazis’ murder of his immediate family among the other Jewish citizens of Krzemieniec. His career was spent almost entirely in the United States: Hopkins, Cornell, war work at the Radiation Lab of MIT, Rockefeller University, and finally the University of Southern California. He pioneered the application of probability measures to topics in unexpected areas: number theory with P. Erdös [Erdös and Kac 1940]; the physics of ideal gases with G. Uhlenbeck and P. Hemmer [Kac, Uhlenbeck, and Hemmer 1963], and, closely related to work by R. Feynman, the “Feynman-Kac formula” [Kac 1949] within quantum theory (in particular, quantum field theory).3

Photo of Mark Kac taken by Paul Halmos
Figure 1. Photograph of Mark Kac taken by Paul R. Halmos in 1962.
Who's That Mathematician? Images from the Paul R. Halmos Photograph Collection, page 26.

Kac’s willingness to look at things in a new way, and the simplicity and elegance of his expression, serve as models for all who work in the mathematical sciences.4 On two occasions (1950 and 1968) Kac won the Chauvenet Prize, the MAA’s highest award for expository writing; the second for the justly famous essay “Can one hear the shape of a drum?” [Kac 1966]. A master of the well-chosen example, Kac preferred the specific and concrete to the general and abstract. It would be contrary to the spirit of this good man not to provide such examples. Here is his oft-quoted distinction between types of genius:

There are two kinds of geniuses: the “ordinary”, and the “magicians”. An ordinary genius is a fellow 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. . . . Richard Feynman is a magician of the highest caliber [Kac 1985, xxv.].

Beyond his mathematical achievements, Kac was admired and loved for his quick wit and his genial, kind personality; he didn’t take himself too seriously. Concerning his swift tongue, a friend related this anecdote from 1952:

Kac went to Pasadena to lecture at the California Institute of Technology. Richard Feynman was in the audience. After the lecture, Feynman got up and announced: “If all mathematics disappeared, it would set physics back precisely one week.” Without a pause Kac responded: “Precisely the week in which God created the world” [Cohen 1986, 1147].

Illustrative of his kindness and lack of pretension, Kac told the story of examining a “not terribly good” doctoral candidate. Kac asked him to describe the behavior of the function \(\frac{1}{z}\) in the complex plane.

The student replied, “The function is analytic, sir, in the whole plane except at \(z = 0\), where it has a singularity,” he answered, and it was perfectly correct. “What is this singularity called?” I continued. The student stopped in his tracks. “Look at me,” I said. “What am I?” His face lit up. “A simple Pole, sir,” which is in fact the correct answer [Kac 1985, 126].

To this can be added an anecdote of Ulam’s:

I remembered him in Poland as very slim and slight, but here he became rather rotund. I asked him, a couple of years after his arrival [in America], how it had happened. With his characteristic good humor, he replied: “Prosperity!” [Ulam 1991, 269].


[1] Most of the biographical details which follow come from Kac’s autobiography [Kac 1985]. Also helpful are [Cohen 1986], [McKean 1990], [Niss 2018], and [Stroock 2015].

[2] Consider only these few names (in order of birth): J. Łukasiewicz (1878–1956), W. Sierpiński (1882–1969), S. Banach (1882–1945), J. Schauder (1899–1943), A. Zygmund (1899–1992), A. Tarski (1901–1993), S. Ulam (1909–1984), and S. Eilenberg (1913–1998).

[3] Kac and Feynman overlapped for a few years at Cornell. Kac was working with the ideas of N. Wiener on Brownian motion and other stochastic processes when he attended a talk by Feynman on his doctoral work. He realized that their methods, involving a generalized form of integration, had much in common; for example, the diffusion equation describing Brownian motion and the Schrödinger equation of quantum mechanics turn into each other when the time variable \(t\) is mapped to \(\pm it\).

[4] A lecture series given to undergraduates at Haverford and Bryn Mawr in 1958 provides a charming and undergraduate-accessible introduction to Kac’s work; it was written up as an MAA Carus monograph [Kac 1959a]. An overview for professionals in the physical sciences is given in [Kac 1959b]. For a survey of mathematics aimed at a more general audience, see [Kac and Ulam 1968], written with his friend and fellow Polish-American, Stanislaw Ulam. This was originally commissioned to be an appendix in the Encyclopaedia Britannica [Ulam 1991, 268].

David Derbes (University of Chicago Laboratory School, retired), "Mark Kac’s First Publication: A Translation of "O nowym sposobie rozwiązywania równań stopnia trzeciego" – Biographical Introduction," Convergence (April 2021)