JUST A PLAIN TEXAS FARM BOY
In the late 1920s, when Samuel S. Wilks left the family farm in Texas to study at the University of Iowa, mathematical research was scaling the heights of beautiful abstraction. Purely abstract fields like symbolic logic, set theory, point set topology, and the theory of transfinite numbers were sweeping through the universities. The level of abstraction was so great that any inspiration from real-life problems that might have given rise to the initial ideas in these fields had long since been lost. Mathematicians plunged into the axioms that the ancient Greek Euclid had declared were the foundations of mathematics; and they found unstated assumptions behind those axioms. They scrubbed mathematics clean of such assumptions, explored the fundamental building blocks of logical thought, and emerged with remarkable, seemingly self-contradictory ideas, like space-filling curves and three-dimensional shapes that touched everywhere and nowhere at the same time. They investigated the different orders of infinity and “spaces” with fractional dimensions. Mathematics was on the upsurge of an all-encompassing wave of sheer abstract thought, completely divorced from any sense of reality.
Nowhere was this drive into abstraction beyond practicality as great as in the mathematics departments of American universities. The publications of the American Mathematical Society were being recognized among the top echelon of mathematical journals in the world, and American mathematicians were pushing at the frontiers of abstractions beyond abstractions. As Sam Wilks was to state ruefully some years later, these departments, with their siren calls offering opportunities for pure thought, were siphoning off the best brains among American graduate students.
Sam Wilks’s first graduate mathematics course at Iowa was taught by R. I. Moore, the most renowned member of the university’s mathematics faculty. Moore’s course in point set topology introduced Wilks to this wonderful world of impractical abstraction. Moore made it clear that he disdained useful work, and insisted that applied mathematics was on a level with washing dishes and sweeping streets. This is an attitude that has plagued mathematics since the time of the ancient Greeks. There is a story told of Euclid tutoring the son of a nobleman and going over a particularly beautiful proof of a theorem. In spite of Euclid’s enthusiasm, the student seemed unimpressed and asked what use this might have. At which Euclid called over his slave and said, “Give the lad a copper. It seems he must gain from his knowledge.”
Sam Wilks’s bent for practical applications was answered by his thesis adviser, Everett F. Linquist, when he began looking for a Ph.D. thesis topic at Iowa. Linquist, who had worked in the mathematics of insurance, was interested in the newly developing field of mathematical statistics and proposed a problem from that area to Wilks. At that time, there was something slightly disreputable about mathematical statistics, at least among the mathematics departments of American and European universities. R. A. Fisher’s great pioneering work had been published in “out-of-the-way” journals
like the Philosophical Transactions of the Royal Society of Edinburgh. The Journal of the Royal Statistical Society and Biometrika were looked down upon as publications in which statistical collections of numbers were tabulated. Henry Carver at the University of Michigan had started a new. journal entitled the Annals of Mathematical Statistics, but its standards were much too low for most mathematicians to take notice of it. Linquist suggested an interesting problem in abstract mathematics that emerged from a method of measurement used in educational psychology. Wilks solved this problem and used it for a doctoral thesis, and the results were published in the Journal of Educational Psychology.
To the world of pure mathematics, this was not much of an achievement. The field of educational psychology was well below their horizon of interest. But a Ph.D. thesis is supposed to be a first tentative step into the world of research, and few students are expected to make important contributions with their theses. Wilks went to Columbia University for a year of postgraduate study (in the course of which he was expected to increase his ability to handle the cold, rarefied abstractions of important mathematics). In the fall of 1933, he arrived at Princeton University, having taken a position as an instructor in mathematics.
The Princeton mathematics department was immersed in the cold and beautiful abstractions as much as any other department in the United States. In 1939, The Institute of Advanced Studies was to be established nearby, and among its first members was Joseph H. M. Wedderburn, who had developed the complete generalization of all finite mathematical groups. Also at the institute were Hermann Weyl, who was famous for his work in nondimensional vector spaces, and Kurt Godel, who had developed an algebra of metamathematics. These men influenced the Princeton faculty,
which had its share of world-renowned mathematicians, prominent among them Solomon Lefshetz, who had opened the doors to the new abstract field of algebraic topology.26
In spite of the general bent toward abstraction among the Princeton faculty, Sam Wilks was fortunate to have Luther Eisenhart as chairman of the mathematics department. Eisenhart was interested in all kinds of mathematical endeavor and liked to encourage junior faculty members to follow their own inclinations. Eisenhart hired Wilks because he thought this new field of mathematical statistics held remarkable promise. Sam Wilks arrived in Princeton with his wife, pursuing a vision of applied mathematics that set him apart from the rest of the department. Wilks was a gentle fighter. He would disarm everyone with his “just folks” Texas farm boy attitude. He was interested in people as individuals and could persuade others to follow his vision, and he was extremely good at organizing work activities to accomplish difficult goals.
Wilks could often get to the heart of a problem and locate a way to solve it while others were still trying to understand the question. He was an extremely hard worker and persuaded others to work as hard as he did. Soon after arriving at Princeton, he became editor of the Annals of Mathematical Statistics, the journal started by Henry Carver. He raised the standards of publication and brought his graduate students into the editing of the journal. He persuaded John Tukey, a new faculty member with an initial interest in more abstract mathematics, to join him in statistical research. He took on a sequence of graduate students who went out to found or staff the new departments of statistics in many universities after World War II.
Wilks’s initial thesis on a problem in educational psychology led him to work with the Educational Testing Service, where he helped formulate the sampling procedures and the scoring techniques
used for college entrance and other professional school exams. His theoretical work established the degree to which weighted scoring schemes could differ and still produce similar results. He was in contact with Walter Shewhart27 at Bell Telephone Laboratories, who was beginning to apply Fisher’s experimental design theories to industrial quality control.
As the 1940s approached, perhaps Wilks’s most important work came from his consulting with the Office of Naval Research (ONR) in Washington. Wilks was convinced that experimental design methods could improve the weapons and the fire doctrines of the armed forces, and he found a receptive ear at the ONR. By the time the United States entered World War II, the army and the navy were both ready to apply statistical methods in the American version of operations research. Wilks set up the Statistical Research Group-Princeton (SRG-P) under the National Defense Research Council. The SRG-P recruited some of the brightest young mathematicians and statisticians, many of whom would make major contributions to the science in the years following the war. These included John Tukey (who swung over entirely to applications), Frederick Mosteller (who would go on to found the several statistics departments at Harvard), Theodore W. Anderson (whose textbook on multivariate statistics was to become the bible of that field), Alexander Mood (who was to go on to make major advances in the theory of stochastic processes), and Charles Winsor (who was to give his name to an entire class of estimation methods), among others.
Richard Anderson, working at that time as a graduate student with the SRG-P, describes attempts that were being made to find a method of destroying land mines. As the invasion of Japan neared, the American army learned that the Japanese had developed a nonmetallic land mine that could not be detected by any known means. They were planting these mines in random patterns across the beaches of Japan and along any possible invasion routes. Estimates of deaths from those land mines alone ran into the hundreds of thousands. It was urgent to find a way to destroy them. Attempts to use bombs dropped from airplanes against land mines in Europe had failed. Anderson and others from the SRG-P were set to designing experiments on the use of lines of explosive cord to destroy the mines. According to Anderson, one of the reasons why the United States dropped the atomic bomb on Japan was that all their experiments and calculations showed that it was impossible to destroy these mines by such means.
The group worked on the effectiveness of proximity fuses in antiaircraft projectiles. A proximity fuse sends out radar signals and explodes when it is close to a target. The group helped develop the first of the smart bombs that could be steered toward their target. They worked on range finders and different types of explosives. Members of the SRG-P found themselves designing experiments and analyzing data at ordnance laboratories and army and navy facilities all over the country. Wilks helped organize a second group, called Statistical Research Group-Princeton, Junior (SRG-Pjr), at Columbia University. Out of SRG-Pjr came “sequential analysis.” This was a means of modifying the design of an experiment while the experiment is still running. The modifications allowed by sequential analysis involve the very treatments being tested. Even in the most carefully designed experiment, it sometimes happens that the emerging results suggest that the original design should be changed to produce more complete results. The mathematics of sequential analysis allow the scientist to know which modifications can be made and which cannot, without affecting the validity of the conclusions.
The initial studies in sequential analysis were immediately declared top secret. None of the statisticians working on it were allowed to publish until several years after the war was over. Once the first papers on sequential analysis and its cousin, “sequential estimation,” began to appear in the 1950s, the method captured the imagination of others, and the field rapidly developed. Today, sequential methods of statistical analysis are widely used in industrial quality control, in medical research, and in sociology.
Sequential analysis was just one of the many innovations that came out of Wilks’s statistical research groups during World War II. After the war, Wilks continued to work with the armed forces, helping them improve the quality control of their equipment, using statistical methods to improve planning for future needs, and bringing statistical methods into all aspects of military doctrine. One of Wilks’s objections to the mathematicians who continued to inhabit their world of pure abstractions was that they were not being patriotic. He felt the country needed the brainpower they were siphoning away into these purposely useless abstractions. This brainpower needed to be applied, first to the war effort and then to the Cold War afterward.
There is no record of anyone getting angry at Samuel S. Wilks. He approached everyone he dealt with, whether a new graduate student or a four-star general of the army, with the same informal air. He was nothing but an old Texas farm boy, he would imply, and he knew he had a lot to learn, but he wondered if … . What followed this would be a carefully reasoned analysis of the problem at hand.
Sam Wilks struggled to make mathematical statistics both a respectable part of mathematics and a useful tool for applications. He tried to move his fellow mathematicians away from the cold world of abstraction for the sake of abstraction. There is indeed a fundamental beauty in mathematical abstractions. They so
attracted the Greek philosopher Plato that he declared that all those things that we can see and touch are, in fact, mere shadows of the true reality and that the real things of this universe can be found only through the use of pure reason. Plato’s knowledge of mathematics was relatively naive, and many of the cherished purities of Greek mathematics have been shown to be flawed. However, the beauty of what can be discovered with pure reason continues to entice.
In the years since Wilks was editor of the Annals of Mathematical Statistics, the articles that appear in the Annals28 and in Biometrika have become more and more abstract. This has also been true of articles in the Journal of the American Statistical Association (whose early issues were devoted to descriptions of government statistical programs) and the Journal of the Royal Statistical Society (whose early issues contained articles listing detailed agricultural and economic statistics from throughout the British Empire).
The theories of mathematical statistics, once thought by mathematicians to be too mired in messy practical problems, have become clarified and honed into mathematical beauty. Abraham Wald unified the work on estimation theory by creating a highly abstract generalization known as “decision theory,” wherein different theoretical properties produce different criteria for estimates. R. A. Fisher’s work on the design of experiments made use of theorems from finite group theory and opened up fascinating ways of looking at comparisons of different treatments. From this came a branch of mathematics called “design of experiments,” but the papers published in this field often deal with experiments so complicated that no practicing scientist would ever use them.
Finally, as others continued to examine the early work of Andrei Kolmogorov, the concepts of probability spaces and stochastic processes became more and more unified but more and more
abstract. By the 1960s, papers published in statistical journals dealt with infinite sets on which were imposed infinite unions and intersections forming “sigma fields” of sets—with sigma fields nested within sigma fields. The resulting infinite sequences converge at the point of infinity, and stochastic processes hurl through time into small bounded sets of states through which they are doomed to cycle to the end of time. The eschatology of mathematical statistics is as complicated as the eschatology of any religion, or more so. Furthermore, the conclusions of mathematical statistics are not only true but, unlike the truths of religion, they can be proved true.
In the 1980s, the mathematical statisticians awoke to a realization that their field had become too far divorced from real problems. To meet the urgent need for applications, universities began setting up departments of biostatistics, departments of epidemiology, and departments of applied statistics. Attempts were made to rectify this breaking apart of the once unified subject. Meetings of the Institute of Mathematical Statistics were devoted to “practical” problems. The Journal of the American Statistical Association set aside a section of every issue to deal with applications. One of the three journals of the Royal Statistical Society was named Applied Statistics.29 Still, the siren calls of abstraction continued. The Biometric Society, set up in the 1950s, had introduced a journal named Biometrics, which would publish the applied papers that were no longer welcome in Biometrika. By the 1980s, Biometrics had become so abstract in its contents that other journals, like Statistics in Medicine, were created to meet the need for applied papers.
The mathematics departments of American and European universities missed the boat when mathematical statistics arrived on the scene. With Wilks leading the way, many universities developed separate statistics departments. The mathematics departments missed the boat again when the digital computer arrived, disdaining it as a mere machine for doing engineering calculations. Separate computer science departments arose, some of them spun off from engineering departments, others spun off from statistics departments. The next big revolution that involved new mathematical ideas was the development of molecular biology in the 1980s. As we shall see in chapter 28, both the mathematics and the statistics departments missed that particular boat.
Samuel S. Wilks died at age fifty-eight in 1964. His many students have played major roles in the development of statistics during the last fifty years. His memory is honored by the American Statistical Association with the annual presentation of the S. S. Wilks Medal to someone who meets Wilks’s standards of mathematical creativity and engagement in the “real world.” The old Texas farm boy had made his mark.