Harvard University (1909–1913)
“ I was nearly fifteen years old, and I had decided to make my try for the doctor’s degree in biology”
After graduating from college, Wiener went on to enter graduate school at Harvard University (where his father worked ) to study zoology. This despite the objections of Leo, who “was rather unwilling to concur in it. He had thought it might be possible for me to go to medical school” (Wiener, 1953). However, the emphasis on laboratory work combined with Wiener’s poor eyesight made zoology a particularly difficult specialization for him. His rebellion was not long lasting, and after a time, Wiener decided to follow his father’s advice and instead take up philosophy.
As usual the decision was made by my father. He decided that such success as I had made as an undergraduate at Tufts in philosophy indicated the true bent of my career. I was to become a philosopher.
Wiener was offered a scholarship to the Sage School of Philosophy at Cornell University, and transferred there in 1910. However, after a “black year” (Wiener, 1953) of feeling insecure and out-of-place, he transferred back to Harvard Graduate School in 1911. Originally intending to work with philosopher Josiah Royce (1855–1916) for his Ph.D. in mathematical logic, due to the latter’s onset illness, Wiener had to recruit his former professor at Tufts College — Karl Schmidt — to take his place. Schmidt, who Wiener himself later stated was “then a young man, vigorously interested in mathematical logic” was the person who inspired him to investigate a comparison between the algebra of relatives of Ernst Schroeder (1841–1902) and that of Whitehead and Russell’s Principia Mathematica (Wiener, 1953):
There was a lot of formal work to be done on this topic which I found easy; though later, when I came to study under Bertrand Russell in England, I learned that I had missed almost every issue of true philosophical significance. However, my material made an acceptable thesis, and it ultimately led me to the doctor’s degree.
His dissertation in philosophy, highly mathematical, was in formal logic. The essential results of his dissertation were published the following year in the 1914 paper “A simplification in the logic of relations” in the Proceedings of the Cambridge Philosophical Society. The coming fall, Wiener traveled to Europe to do postdoctoral work in the hopes that he might eventually land a permanent position on the faculty of one America’s most prominent universities.
Postdoctoral work (1913–1915)
Following his doctoral dissertation, defense and graduation from Harvard, Wiener — then 18 years old — was awarded one of the school’s prestigious one-year graduate fellowships to study abroad. His chosen destination was Cambridge, England.
Cambridge University (1913–1914)
“Leo Wiener hand-delivered his son to Bertrand Russell”
Norbert Wiener first arrived at Trinity College, Cambridge in September of 1913. Traveling with him was his entire family, spearheaded by his father Leo who had seized on the opportunity to take a year of sabbatical from Harvard and join his son in Europe. As Conway & Siegelman (2005) describe, “Young Wiener strode through the great gate of Trinity College, Cambridge, the Mecca of modern philosophy and the new mathematical logic, with his father in lock-step behind him”.
Wiener went to Cambridge to continue his study of philosophy with one of the authors of the Principia Mathematica which had been the focus of his dissertation at Harvard. Lord Bertrand Russell (1872–1970) — at that point in his early forties — was by 1913 considered the foremost philosopher of the Anglo-American world following the praise of his’ and Alfred North Whitehead’s monumental three-volume work, published in 1910, 1912 and 1913. The Principia or “PM” as it is often known, was at that point the most complete and coherent piece of mathematical philosophy to date. Renowned still for its rigor, the work among other efforts, infamously grounded the theory of addition to logic by proving, in no less than thirty pages, the validity of the proposition that 1+1 = 2.
Despite having been brought up at the heft of a polyglot “Harvard Don”, Wiener’s first impression of Russell’s fierce personality left something to be desired, as he would soon communicate to his father in letter-form:
Russell’s attitude seems to be one of utter indifference mingled with contempt. I think I shall be quite content with what I shall see of him at lectures
Russell’s impression of Wiener, or at least what he let him on to believe, appeared mutual. “Apparently, young Wiener did not “sense data” or do philosophy the way the titan of trinity prescribed it” (Conway & Siegelman, 2005):
Excerpt, letter from Norbert to Leo Wiener (1913)
My course-work under Mr. Russell is all right, but I am completely discouraged about the work I am doing under him privately. I guess I am a failure as a philosopher [...] I made a botch of my argument. Russell seems very dissatisfied [...] with my philosophical ability, and with me personally. He spoke of my views as "horrible fog", said that my exposition of them was even worse than the views themselves, and [...] accused me of too much self-confidence and cock-sureness [...] His language, though he excused himself, it is true, was most violent.
As with his father Leo, sadly, Russell’s opinion of Norbert, then 18 years old, was not as harsh as he himself had believed. In his private papers, Russell indeed noted approvingly of the boy and after reading Norbert’s dissertation commented that it was “a very good technical piece of work”, giving the young student a copy of the third volume of the Principia as a gift (Conway & Siegelman, 2005).
The single most important take-away of Wiener from his work with Russell, however, was neither physical nor related to philosophy. Rather, it was the Lord’s suggestion that the young Wiener look up four papers from 1905 by physicist Albert Einstein, which he would later make use of. Wiener himself at time time singled out G.H. Hardy (1877–1947) as having the most profound influence on him (Wiener, 1953):
Hardy’s course […] was a revelation to me […] [in his] attention to rigor […] In all my years of listening to lectures in mathematics, I have never heard the equal of Hardy for clarity, for interest, or for intellectual power. If I am to claim any man as my master in my mathematical thinking, it must be G.H. Hardy.
In particular, Wiener credited Hardy for introducing him to the Lebesgue integral which “lead directly to the main achievement of my early career”.
Göttingen University (1914)
One experience richer, Wiener in 1914 continued to Göttingen University. He arrived in the spring after briefly stopping by to visit his family in Munich. Although only staying for a single term, his time there would be crucial to his further development as a mathematician. He assumed the study of differential equations under David Hilbert (1862–1943), perhaps the foremost mathematician of his era whom Wiener would later laud as “the one really universal genius of mathematics”.
Wiener remained in Göttingen until the outbreak of World War I in June of 1914, when he decided to return to Cambridge and continue his studies of philosophy with Russell.
Prior to being hired at MIT — an institution he would remain with for the rest of his life — Wiener worked a number of somewhat odd jobs, in various industries and cities in America. He officially returned to the United States in 1915, briefly living in New York City while continuing studies in philosophy at Columbia University with philosopher John Dewey (1859–1952). After that, he went on to teach philosophy courses at Harvard and next accepted a job as an engineer apprentice at General Electric. After that, he joined Encyclopedia Americana in Albany, New York after his father had secured him a job as a staff writer there, “convinced that with my clumsiness I could never really make good at engineering“ (Wiener, 1953). He also worked briefly for the Boston Herald.
With America’s entry into World War I, Wiener was eager to contribute to the war effort, and attended a training camp for officers in 1916, but ultimately failed to earn a commission. In 1917 he tried again to join the military, but was rejected due to his poor eyesight. The next year, Wiener was invited by mathematician Oswald Veblen (1880–1960) to contribute to the war effort by working on ballistics in Maryland:
I received an urgent telegram from Professor Oswald Veblen at the new Proving Ground in Aberdeen, Maryland. This was my chance to do real war work. I took the next train to New York, where I changed for Aberdeen
His experiences at the Proving Ground transformed Wiener, according to Dyson (2005). Before arriving there, he was a 24-year-old mathematical prodigy who had been discouraged away from mathematics due to the failings of his first teaching job at Harvard. Afterwards, he was re-invigorated by the applications of his teachings on real world problems:
We lived in a queer sort of environment, where office rank, army rank, and academic rank all played a role, and a lieutenant might address a private under him as ‘Doctor’, or take orders from a sergeant. When we were not working on the noisy hand-computing machines which we knew as ‘crashers’, we were playing bridge together after hours using the same computing machines to record our scores. Whatever we did, we always talked mathematics.
In his extensive bibliography of published writings, Wiener’s first two publications in mathematics appeared in the 17th issue of the Proceedings of the Cambridge Philosophical Society in 1914, the latter of which is now lost:
- Wiener, N. (1914). “A Simplification of the Logic of Relations”. Proceedings of the Cambridge Philosophical Society 17, pp. 387–390.
- Wiener, N. (1914). “A Contribution to the Theory of Relative Position”. Proceedings of the Cambridge Philosophical Society 17, pp. 441–449.
The first work, which regarded mathematical logic, was according to Wiener “presented on 23 February 1914 by G. H. Hardy” despite “exciting no particular approval on the part of Russell”. In the note, Wiener introduces the “dissymmetry between the two elements of an ordered pair by using the null set”. The work, which was the main result of his Ph.D. thesis at Harvard, proved how the mathematical notion of relations can be defined by set theory, thereby showing that the theory of relations does not require any distinct axioms or primitive notions.
Wiener’s most well known mathematical contributions were however mostly made between the ages of 25 to 50, in the years 1921–1946. As a mathematician, Chatterji (1994) singles out Wiener’s skillful utilization of the Lebesgue type integration theory (which Hardy had introduced him to in Cambridge) as a unique hallmark of his art. The Lebesgue integral extends the traditional integral to a larger class of functions and domains.
Following the end of World War I, Wiener tried to secure a position at Harvard, but was rejected, likely to the university’s anti-semitism at the time, often attributed to the influence of Department Head G. D. Birkhoff (1884–1944). Instead, Wiener assumed the position of lecturer at MIT in 1919. From that point on, his research output increased significantly.
In the first five years of his career at MIT, he published 29 (!!) single-authored journal papers, notes and communications in various subfields of mathematics, including:
- Wiener, N. (1920). “A Set of Postulates for Fields”. Transactions of the American Mathematical Society 21, pp. 237–246.
- Wiener, N. (1921). “A New Theory of Measurement: A Study in the Logic of Mathematics”. Proceedings of the London Mathematical Society, pp. 181–205.
- Wiener, N. (1922). “The Group of the Linear Continuum”. Proceedings of the London Mathematical Society, pp. 181–205.
- Wiener, N. (1921). “The Isomorphisms of Complex Algebra”. Bulletin of the American Mathematical Society 27, pp. 443–445.
- Wiener, N. (1923). “Discontinuous Boundary Conditions and the Dirichlet Problem”. Transactions of the American Mathematical Society, pp. 307–314.
The Wiener process (1920-23)
Wiener first became interested in Brownian motion when was in Cambridge studying under Russell, who directed him to the “miracle year” work of Albert Einstein. In his 1905 paper Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhended Flüssigkeiten suspendierten Teilchen (“On the Motion of Small Particles Suspended in a Stationary Liquid, as Required by the Molecular Kinetic Theory of Heat”), Einstein modeled the irregular motion of a pollen particle as being moved by particular individual water molecules. This “irregular motion” had first been observed by botanist Robert Brown in 1827, but had not yet been investigated formally in mathematics.
Wiener approached the phenomenon from the perspective that “it would be mathematically interesting to develop a probability measure for sets of trajectories” (Heims, 1980):
A prototype kind of problem Wiener considered is that of the drunkard's walk: a drunk man is at first leaning against a lamp post; he then takes a step in some direction-it may be a short step or a long step; then he either stands still maintaining his balance or takes another step in some direction; and so on. The path he takes will in general be a complicated path with many changes in direction. [...]Assuming the man has no a priori preference for any particular direction or particular step size and may move fast or slowly according to his whim, is there some way to assign a probability measure to any particular set of trajectories?- Excerpt, John von Neumann and Norbert Wiener by Steve Heims (1980)
Wiener extended Einstein’s formulation of Brownian motion to describe such trajectories, and so established a link between the Lebesgue measure (a systematic way of assigning numbers to subsets) and statistical mechanics. That is, Wiener provided the mathematical formulation for describing the one-dimensional curves left behind by Brownian processes. His work, now often referred to as the Wiener process in his honor, was published in a series of papers developed in the period 1920–23:
- Wiener, N. (1920). “The Mean of a Functional of Arbitrary Elements”. Annals of Mathematics 22 (2), pp. 66–72.
- Wiener, N. (1921). “The Average of an Analytic Functional”. Proceedings of the National Academy of Sciences 7 (9), pp. 253–260.
- Wiener, N. (1921). “The Average of an Analytic Functional and the Brownian Movement”. Proceedings of the National Academy of Sciences 7 (10), pp. 294–298.
- Wiener, N. (1923). “Differential Space”. Journal of Mathematics and Physics 2, pp. 131–174.
- Wiener, N. (1924). “The Average Value of a Functional”. Proceedings of the London Mathematical Society 22, pp. 454–467.
As Wiener himself testified, although neither of these papers solved physical problems, they did however provide a robust mathematical framework which was later used by von Neumann, Bernhard Koopman (1900–1981) and Birkhoff to address problems in statistical mechanics originally posed by Willard Gibbs (1839–1903).