# Herbert Ellis Robbins

### Born: 12 January 1915 in New Castle, Pennsylvania, USA

Died: 12 February 2001 in Princeton, New Jersey, USA

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**Herbert Robbins**'s parents were Mark Louis Robbins and Celia Klemansky, and he had a younger sister Francie. Herbert's father died when he was thirteen years old leaving the family in financial difficulties. Herbert was not particularly interested in mathematics at high school; it was literature which really fired his interest at that time [4] or [5]:-

He entered Harvard University in 1931, when only sixteen tears old, and took courses on literature but he also took a calculus course [4] or [5]:-I used to go down to the public library after school, and come home with an armful of books. I'd read them all before the next day? I must have read every book in the library.

Robbins graduated with an A.B. from Harvard in 1935 and remained there undertaking graduate studies. He undertook research in topology, although he ended up studying this topic almost by accident [4] or [5]:-Having just entered Harvard with practically no high school mathematics, I knew calculus would be useful if I ever wanted to study any of the sciences. At the end of my freshman year, much to my surprise I was asked by the mathematics department to join the Harvard math team. Marston Morse was our coach. We met with him on several occasions to prepare for the competition ...

His doctoral thesis, written with Whitney as his advisor, was entitledHassler Whitney had come back from a topology conference in Moscow around1936, and in a talk at Harvard on some of the topics discussed at the conference, he mentioned an unsolved problem that seemed to be important. Since I was then a graduate student looking for a special field to work in - not particularly topology, since I hadn't even taken a course in the subject - I asked Whitney to let me work on it. That's how I got started.

*On the Classification of the Maps of a 2-Complex into a Space*. He was awarded a Ph.D. in 1938 and published the main results of the thesis in the

*Transactions*of the American Mathematical Society in 1941. Witold Hurewicz describes the paper as follows:-

After the award of his doctorate he informed Marston Morse, who was then at the Institute for Advanced Study at Princeton, of his success. He immediately invited Robbins to be his assistant for the academic year 1938-39 which Robbins happily accepted. However, he badly needed a permanent position, particularly since he was financially supporting his mother and sister. During that year Richard Courant contacted Morse asking if he could recommend someone for the position of Instructor in Mathematics at New York University (NYU) - Morse recommended Robbins and he began work at NYU in 1939 [4] or [5]:-The difficult problem of characterizing homotopy classes of mappings by combinatorial properties is solved in this paper for the case of the mappings of a2-complex into an arbitrary space T. The necessary and sufficient condition for the homotopy of two mappings involves the use of chains of K with coefficients from the fundamental group of T and the2-dimensional homotopy group of T.

The collaboration between Courant and Robbins onMy salary as an instructor at NYU remained fixed at $2,500a year during1939-1942. That was my sole support; there were no NSF grants then. Some time during the beginning of my first year at NYU, Courant said to me: "I've been given a little money to work up some old course material into a book on mathematics for the general public. Would you like to help me with it? I can pay you $700-$800for your assistance." I was in no position to turn down extra money for a legitimate enterprise, and the idea of communicating my ideas about mathematics to the educated layman appealed to me.

*What is Mathematics?*was a close one, and at one stage Robbins moved near to Courant's home in New Rochelle so that they could work whenever Courant had some free time. However, Robbins had a nasty shock when the final page proofs came from the printers as his name did not appear as an author. Despite having done a large part of the writing, he had quite a battle to have his name included as a joint author and, even after he succeeded, Courant held the copyright for the book and forwarded some money to Robbins each year as his share of the royalties although he never knew how many copies had sold and whether what he received was fair.

*What is Mathematics?*, published in 1941, was a best-seller. Donald Spencer writes:-

Given Robbins' initial interest in literature, this comment from a description of a new edition brought out in 1996 (and still in print) is interesting:-The book is an elementary approach to modern mathematics in which processes of thought, rather than mechanical routines, are emphasized, and is a model of lucid exposition. From a level approximately that of a sound high-school training, the development proceeds by direct paths to some of the best content of mathematics; and fundamental ideas are made strikingly clear by well-chosen, simple examples. The text is illustrated with nearly300diagrams, and much of the clarity arises from a generous use of geometrical interpretation. ... Although it is inevitable in a work of this scope that there should be a few historical statements with which specialists may disagree, it is nevertheless a work of extraordinary perfection.

It was during his time at NYU that Robbins first became involved in statistics. Courant invited William Feller to give a course on probability and statistics at NYU and, at the last minute, after the course was advertised, Feller was unable to come. Courant asked Robbins to give the course [4] or [5]:-The best mathematics is like literature - it brings a story to life before your eyes and involves you in it, intellectually and emotionally. 'What is Mathematics?' is like a fine piece of literature - it opens a window onto the world of mathematics for anyone interested to view.

In December 1941, following the Japanese attack on the U.S. fleet at Pearl Harbour, the United States entered World War II. Robbins joined the United States Naval Reserve in 1942, not as a mathematician but "as a reasonably able-bodied person." During this period of war service, he overheard [3]:-It must have been a pretty terrible course because I knew nothing about either subject.

Robbins married Mary Dimock in 1943; they had two daughters Mary Susannah and Marcia. When he left the United States Naval Reserve in 1946 he had no academic job and decided to leave mathematics and look for an alternative career. He bought a farm in Vermont with back pay he received from the Navy. However, he received a phone call from Harold Hotelling who had just been appointed Professor of Mathematical Statistics at the University of North Carolina, Chapel Hill and was building up a new Department of Mathematical Statistics [4] or [5]:-... a conversation between two senior naval officers concerning the effect of random scatter on bomb impacts. Because of his lack of appropriate security clearance, he was prevented from pursuing this problem ... Nevertheless, his work on the naval officers' problem led to the fundamental papers['On the measure of a random set'(1944)and 'On the measure of a random set II'(1944)]in the field of geometric probability.

Robbins joined the new department, which also had appointed William Cochran to develop the graduate programme in statistics [4] or [5]:-Hotelling offered me an associate professorship in this newly created department. I thought he'd telephoned the wrong Robbins ... Hotelling insisted that there was no mistake, even though I told him that I knew nothing about statistics. He didn't need me as a statistician; he wanted me to teach measure theory, probability, analytic methods, etc. to the department's graduate students. Having read my paper in the Annals of Mathematical Statistics, Hotelling felt that I was just the sort of person he was looking for.

Robbins was promoted to professor at the University of North Carolina, Chapel Hill in 1950. During six years at Chapel Hill [3]:-At Chapel Hill, I attended seminars and got to know several very eminent statisticians. Soon I began to get some idea about what was going on in that subject and finally, at age thirty-two, I became really interested in statistics.

He spent the year 1952-53 as a Guggenheim Fellow at the Institute for Advanced Study at Princeton, then took up an appointment as Professor of Mathematical Statistics at Columbia University. In 1955 Robbins and his wife Mary were divorced. He married Carol Hallett in 1966; they had two sons, Mark Hallett and David Herbert, and a daughter Emily Carol. Robbins spent two years 1966-68 as Professor of Mathematics at the University of Michigan, Ann Arbor, before returning to Columbia. He was again a Guggenheim Fellow in 1975-76 and he spent the year at Imperial College in London, England. He remained as Professor of Mathematical Statistics at Columbia until he retired in 1985 [3]:-... Robbins not only studied and developed an increasingly deep interest in statistics, but he also made a number of profound contributions to his new field: complete convergence, compound decision theory, stochastic approximation, and the sequential design of experiments, to name a few.

Some further details are given in [8]:-During this period[at Columbia]he published over100papers on a variety of topics in probability and statistics. His most notable contributions include the creation of the empirical Bayes methodology, the theory of power-one tests, and the development of sequential methods for estimation, hypothesis testing, and comparative clinical trials.

His work on empirical Bayes methods is discussed in detail in [2] where Bradley Efron writes:-In his research on empirical Bayes methods he showed that certain problems, which usually were treated separately, could profitably be combined so that data concerning one problem provided information useful in solving the others. ... Robbins' paper with his student Sutton Monro on Stochastic Approximation provided an analogue of an iterative method due to Isaac Newton for finding the root of a function, even when the function's equation is unknown and the evaluation of the function involves experimental error. The process they introduced has become a prototype for many iterative algorithms for on-line control of engineering systems.

During the 1960s and for part of the 1970s, Robbins main area of research was on sequential analysis. David Siegmund explains in [7] that his contributions here were of a somewhat different nature to his earlier work:-Herbert Robbins ranks high on anyone's list of influential postwar statisticians. Among his many fruitful ideas, empirical Bayes, which he named as well as developed, has had the biggest effect on statistical thinking. ... current scientific trends favor a greatly increased role for empirical Bayes methods in practical data analysis.

We mentioned above that Robbins retired from Columbia University in 1985. He had reached the age of seventy and a collection of 48 of his 133 papers (up to 1984) were reprinted in the book... unlike stochastic approximation and empirical Bayes, where his first articles were arguably unprecedented, sequential analysis was a rapidly developing subject when Robbins made his first contributions. Consequently, his recurrent contributions can be seen as a dialogue with his contemporaries that has influenced succeeding generations. ... Herbert Robbins' research in sequential analysis was a major intellectual achievement that has changed forever the way we think about a large class of challenging problems.

*Herbert Robbins: Selected Papers*. A reviewer wrote:-

This, however, was certainly not the end of his academic career for he had another twelve years as Professor of Mathematical Statistics at Rutgers University before finally retiring for the last time in 1997. He died of esophageal cancer in 2001 at the Princeton Medical Center in Princeton, New Jersey.Specialists all over the world esteem Herbert Robbins as a mathematician and statistician of outstanding versatility, creativity and of daring originality.

Finally, let us quote from [6] Robbins advice to future statisticians:-

... for the future I recommend that we work on interesting problems, avoid dogmatism, contribute to general mathematical theory or concrete practical applications according to our abilities and interests and, most important, formulate for ourselves a canon of humanistic values that will inspire and justify our work on a higher level than that of the well-trained and useful technician.

**Article by:** *J J O'Connor* and *E F Robertson*

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