Wilson and Mathematical Statistics
Paul Samuelson took great pride in describing himself as the intellectual grandson of the great American physicist Josiah Willard Gibbs, on the grounds that he was a student of Edwin Bidwell Wilson, Gibbs's last protege.1 Gibbs's other protege had been the eminent mathematical economist Irving Fisher, whose PhD thesis had been submitted to Yale in 1891.
Wilson (1879—1964) was Professor of Vital Statistics in Harvard's Institute of Public Health and also a member of the Economics Department, teaching statistics and mathematical economics in alternate years. He had been trained as a mathematician at Yale and at the Ecole normale superieure in Paris before joining MIT's Mathematics Department in 1907. Wilson had been close to Gibbs in the last years of his life, from 1899 to 1902, and as a graduate student he was responsible for writing up Gibbs's lectures on vector calculus, in circulation since 1881, for publication as Vector Analysis (1901). This settled the notation that American physicists used for operations on vectors, as well as expounding the theory and some of its physical applications. However, in complete contrast to Gibbs, Wilson was loquacious and with a sharp wit and sometimes caustic tongue, and had no qualms about challenging authority. His mathematical interests were broad, covering geometry, algebra, and various applied fields.2 In 1902, while a student at the Ecole normale superieure, he criticized David Hilbert's attempt to set out new foundations for geometry in a paper provocatively titled “The So-called Foundations of Geometry” (1903), which questioned the way Hilbert used set theory and logic. As with his later criticisms of theorems in set theory, Wilson's ideas were complicated, out of tune with modern developments in mathematics, and did not always survive well. His early output included what was, for more than a decade after its publication, the only modern American advanced calculus textbook.3 In 1917, he was appointed Professor of Mathematical Physics, becoming head of the Physics Department. From 1920 to 1922, he was one of three academics who took over the running of MIT after the death of its president, Richard Maclaurin. He moved to Harvard in 1922. Working in many fields, publishing in mathematics, physics, aeronautical engineering, statistical theory, public health, and economics, and acquiring a knowledge of many other fields, he was managing editor of the Proceedings of the National Academy of Sciences from 1915 to 1964—almost half a century. He was described by Wesleyan University on the occasion of receiving an honorary degree as “the modern Renaissance man, taking all knowledge for his province.”4 As the only intelligent man known to Samuelson who enjoyed committee meetings, he was active in Harvard's administration, the National Academy of Sciences, and the Social Science Research Council, as well as many other bodies; he had contacts throughout the academic world and government.5 He was an important figure in American science.6The First World War was important for Wilson's career. Just before the war he had taught the theory of aerodynamics at MIT, and in 1916 his analysis of an aircraft's behavior when it encountered gusts of wind was included in a National Advisory Committee for Aeronautics Report to Congress. This work led to the publication of Aeronautics (1920). Samuelson described him as “a pioneer in writing down the stability conditions for the new-fangled aeroplane.”7 During the war, Wilson also developed an interest in statistics and public health, and after his move to Harvard he turned to mathematical statistics. In 1927, he arrived at an idea of confidence intervals that was very similar to that developed by Jerzy Neymann and Karl Pearson in the early 1930s, and which became one of the foundations of statistical inference. However, because he chose to work in the no-man's-land between sciences rather than develop any one discipline, his profile among statisticians was lower than it might have been.
This may have been one reason, along with Henry Schultz's connection with Columbia, why Samuelson's teachers in Chicago had claimed that he would learn mathematical statistics only by going to Columbia to study with Harold Hotelling.Samuelson's first formal contact with Wilson was in the course Topics in Statistical Theory in the spring term of his first year, shortly before he was scheduled to take his “generals.” As its title suggests, this was a theoretical course and it appears to have covered some mathematical economics. But, whatever the balance between mathematical economics and statistics in the classroom, Samuelson learned from Wilson outside the classroom as well. He wrote that, as one of the good students, he was able to talk to Wilson for an hour after each lecture, and that their conversations covered “any and every subject.”8 It is most likely in these after-class conversations that Wilson taught Samuelson about thermodynamics. This way, Samuelson would have picked up more of Wilson's attitudes toward economics and different types of research than could have been conveyed in the more formal setting of the classroom.
Though it was a course in statistics, Wilson appears to have covered consumer theory in a way that was pertinent to the paper Samuelson was to write later that year, and which was to become his first academic article, as well as to papers he was to write over the next few years. On July 14, 1936, Wilson wrote to his colleague John Black that he had “bored the class dreadfully going so slowly over the first 12 pages of the Introduction to Mathematical Economics."9,a The reason he proceeded so slowly through what was a textbook in mathematical economics was that it had become clear to him that “some of our high-powered mathematical economists did not know their fundamental definitions and would read right over a pair of statements which were contradictory and assume that both were right." It can be no coincidence that the previous year, Wilson had published an article in the Quarterly Journal of Economics criticizing Bowley, the book's author, for making a mathematical error:
such phraseology as: “There are certain simplifications if we suppose that the marginal utility of money be unaffected by the sale or purchase of a good, in other words, that the individuals have so much money that the particular deal does not sensibly affect its marginal utility"...
seems to be of doubtful validity because the change of the marginal utility of money nowhere enters into the proof of the theorems but only its rate of change—a change may be infinitesimal while a rate of change is finite.10What Wilson had done in this short article was to take a theorem in which Pareto had shown there would be an inverse relationship between a good and its price if the utility of each good was independent of all other goods being consumed and show that the same result could be proved under more general
a. The book he referred to here is Mathematical Groundwork ofEconomics (1924) by the British statistician and economist Arthur Bowley. assumptions: it did not matter whether or not the utilities of goods other than the one whose price was changed were dependent upon each other or not. Moreover, and this was the point where Wilson believed Bowley to have made a mistake, the result did not rely on constancy of the marginal utility of money.
Given that he discussed Bowley in such a detail and that his paper had just been published in the department's journal, it is hard to believe that he did not expose Samuelson and his fellow students to the problem of what was necessary to derive results in consumer theory, and that he would have made clear to them the value of using advanced mathematics in providing rigorous proofs.b
Wilson's course was attended intermittently by Schumpeter, whose interest was apparently to learn techniques he could use to analyze the mass of data he had assembled for his Business Cycles, on which he was then working. Schumpeter proved an irregular attender at the course, and eventually had to drop out. In April he wrote apologetically to Wilson,
I wish to write a line in explanation of the fact, which I greatly regret, that I have dropped sitting in on your course to which you generously admitted me and which I greatly enjoyed. As a matter of fact, I need your instruction very badly to fill the most shocking lacunae in my statistical armor, but work on my manuscript, which it is of the greatest importance to me to finish before the end of June or at all events in the summer, has been so slow as to send me into something like a panic.11
It is no surprise that Wilson replied Schumpeter with no need to apologize, going on to explain that his lectures were, in his opinion, “not of very great advantage to a person who works with actual statistical material.”12 In justifying this view, Wilson explained that the “modern mathematical methods” with which he was dealing had so far not proved themselves to be of real practical importance.
In saying this, he was expressing the view of someone with great experience in practical statistical analysis.If the techniques he was teaching had not proved their worth, what was the case for teaching them? Wilson's answer was the negative one, that it
b. Samuelson took Wilson's mathematical economics course in 1937. However, by this time, his first paper on consumer theory, tackling some of the problems Wilson had discussed in his paper, was published, and the idea on which his second paper rested had already occurred to him.
protected young mathematicians from being over-impressed with mathematical statistics.
I regard the course as valuable to the young mathematician and economist chiefly because of the protection it affords him in dealing with the contributions of mathematical statisticians.... People [mathematicians] who have an over-elaborate technique... make a very great impression on those who don't have it and may in fact publish remarkable theorems which persons acquainted with the facts in detail believe are of no particular importance if valid.13
He went on to explain that, given what was happening in economics, the course was particularly important for graduate students.
Now there is such a trend toward mathematical economics and mathematical statistics that unless our students working for the doctorate get further into mathematical statistics than may ever be necessary for them in their personal statistical work they won't have any adequate protection from the increasing number of persons who are using what is probably an excess of mathematics and they won't even be able in an all-around way to discuss their contributions. The educational value of the course for young people ought to be considerable.
Its value was, therefore, largely negative. This was, however, a situation that might change if the techniques he covered were in future to prove their worth.
The depth of Wilson's commitment to this view is indicated by the fact that it was more or less a corollary of critical remarks he had published a decade earlier, when writing in Science on the topic of statistical inference.
Many statisticians, he wrote, did not take responsibility for making sure that the formulas they were using were appropriate to the problems they were tackling and would give the right results. “They seem,” he wrote, “for some reason to believe that a mathematical formula is eternally true. Their attitude is Shamanistic. They go through with magic propitiatory rites, idolatrous of mathematics, ignorant of what it can and can not do for them.”14 It was not simply young scholars and those with limited training who were guilty: “I am not quite sure,” Wilson continued, “that the high priests of this pure and undefiled science do not somewhat aid and abet the idolatory.”Why was this? It was because statistics had not yet developed to the point where its foundations were as secure as those other branches of mathematics, because the premises were not yet understood. As a result it was impossible to prove when particular methods would and would not work: it was necessary to work this out case by case. This was consistent with Wilson's view, expressed to the Harvard agricultural economist John Black, that the way to train students to engage in practical work was to employ them in a project in which they would “work in detail on a lengthy piece of statistical analysis.” As an example he cited a project on Industrial and Related Agricultural Fluctuations, on which the department was about to embark.15 While some prior statistical training would be required, their main training would come through doing the research.
Wilson's course was one he had designed himself, drawing on his own ideas and materials scattered throughout the literature. If he was covering his own work, then it would have included inference and confidence intervals. Wilson covered characteristic functions, a way of representing probability distributions that makes it easier to undertake certain types of analysis.16 Though the course may have changed by then, correspondence with Lloyd Metzler indicates that two years later Wilson spent a lot of time on the problem of smoothing data.17 The course was a rigorous introduction to important topics in mathematical statistics, but it was not a standard course. As Wilson explained to Schumpeter later in the summer, “The material is concentrated all over the literature and what I do is to get it together and discuss it with the class. It is almost impossible to give any decent assigned reading. I find it hard enough to dig the stuff out of the memoirs [journals] myself.”18
This lack of a close fit with what they were studying elsewhere caused a problem for Harvard students, whose attendance was often irregular owing to the way the examination system was structured, for students were concerned more with passing their generals than with tackling courses like Wilson's. Samuelson, though he got an A-, was one of those students who failed to benefit from the course as much as Wilson thought he should have done. “The difficulty with him [Samuelson],” Wilson wrote, “was that he was so concerned about his ‘generals' that he could not concentrate on the course as well as a man like Levine.”19 In addition, because other students were concentrating in mathematics, Samuelson did not get the top grade in the course.20 He had, however, impressed Wilson enormously. Samuelson was “the most original and inquisitive of all the students”;21 he did not perform any better than the other students, but in Wilson's view, “he had the potentialities to be as good as anybody in the class.”22