Princeton Alumni Weekly, May 9, 1958
by Virginia Chaplin
Editors Note: This is probably the longest article to appear in PAW's 58 years of publication and in the Editor's judgment, one of the very best. Like the undergraduates on our cover, most Princetonians have hastened by Fine Hall as if it were a graveyard, not pausing to look in on the chill, abstract world of higher mathematics. (The Mathematics Department's undergraduate teaching of the "lower" mathematics is mainly done in other buildings.) Several years ago one undergraduate journalist somehow stumbled in and departed in haste, remarking on the "brilliant but unintelligible lecturers with foreign accents . . . the European, or demi-God, theory of instruction." Over the years only a few liberal arts alumni have managed to develop much affection for the mathematics they were forced to take: from his own experience Woodrow Wilson thought it was possible only by "a painful process of drill" to "insert" mathematics into "the natural, carnal man," Adlai Stevenson '22 remembered that "a page of mathematical equations still makes me shudder," Scott Fitzgerald '17 reflected bitterly that "conic sections" had flunked him out. But the instruction of undergraduates is only one function of the true university, and this fine article is focussed on the relationship of Princeton mathematicians not to the world of teaching but to the world of learningto what Wilson called the "big" Princeton rather than the "little" Princeton. For the Mathematics Department has usually been called our best, world famous for its research discoveries and for training graduate mathematicians. The story of the recruitment of this enclave of scholars and of its splendid record in basic research is the product of two months' investigation by Miss Virginia Chaplin, an editor at the Princeton University Press. The name Chaplin is not unknown at Princeton: she is the daughter of Hugh Chaplin '09, sister of Hugh Chaplin Jr. '44 and the niece of Duncan Chaplin '17 and the late Maxwell Chaplin '13. The sketches, by Dick Snedeker '51, are of the equations in the leaded windows of Fine Hall (except the Moebius band opposite).
At the turn of the century America's standing in the field of mathematics was virtually nonexistent. During the next two decades mathematical scholarship in this country rose by leaps and bounds to a position of equality with that of European nations. The mathematics enterprise at Princeton, set in motion by Woodrow Wilson and Henry Burchard Fine in 1905, profoundly influenced this remarkable growth of American mathematics during the early 20th century and put the University on the map as a center for creative mathematical research. In 1936 Harald Bohr of Copenhagen, speaking before a distinguished international gathering of mathematicians, described Princeton as "the mathematical center of the world." Today the University, living up to a great tradition, is playing a leading role in the spectacular mathematical developments of the modern era.
Before looking at Princeton's outstanding mathematical tradition, we should take a very brief and peripheral journey into the inner world of mathematics.
Mathematics is the oldest and the most rapidly advancing science today. Whereas a theoretical physicist can be expected to have a working understanding of at least half of his subject, the present-day mathematician is acquainted with only about one-fourth of the entire spectrum of mathematical knowledge. Due to the astonishing recent developments in the field, modern mathematics has a curiously dual nature. On the one hand, it is intimately related to experience and the physical world; on the other, it is incredibly abstract and removed from everyday life.
The ideas of mathematics are ultimately rooted in human experience: simple arithmetic arose as a means of transacting business; geometry began as a set of rules devised for land surveying in Egypt; and calculus, originated as a method for measuring areas and volumes bounded by curves and curved surfaces (Kepler's first attempts at integration arose in connection with the measurement of wine barrels). Algebra, too, has strong empirical ties.
Over the centuries mathematics has gradually freed itself from earthly bonds and climbed to unbelievable heights of abstraction and generalization. In the process of working through certain physical problems, questions arose such as "what kind of a number is the square root of two?" or "what kind of a number is pi (π) ?" Mathematicians began to devote more and more of their time to these purely mathematical side issues not directly related to the problem at hand. It became their primary task to construct a theoretical system that would be logically air-tight, faultlessly consistent, and that would eventually embrace all kinds of mathematical problems.
Today research mathematicians explore a rich idea-world of their own invention. They no longer start out with Euclidian axioms, which are supposedly self-evident truths directly perceived by the sensesfor instance, by looking at geometric objects. Instead, they make up their own sets of rules and arbitrary assumptions and proceed to build from there by the time-honored methods of rigorous logical deduction and verification. Their goal is consistency rather than absolute truth. Their world is an infinite one, in which they move about freely, unfettered by the demands or limitations of the material universe.
Mathematicians have by no means fully succeeded in creating a perfect structure, free of all paradox. In fact, Professor Kurt Godel of the Institute for Advanced Study ingeniously proved that mathematics is forever incapable of proving its own consistency. Startling implausibilities and contradictions occasionally loomfor instance, the discovery by two Polish mathematicians, about twenty-five years ago, that a solid sphere could be divided into five pieces and reassembled into two new spheres, each as "big" as the original sphere. Such temporary stumbling blocks do not seriously discourage researchers or slow the onward march of modern mathematics. Today there are over 1,000 distinct, completely consistent algebras (each with its own set of rules) and many geometries.
One of the most fascinating and active branches of modern mathematical research is topologythe geometry of form without size or shape. Workers in this area tackle such ticklers as: how to generate varied "spaces" from relatively simple components. For example, imagine a large number of rods (each about a foot long) distributed uniformly around a circle of stiff wire (about two feet in diameter)the rods attached to the wire at their centers so that they can pivot into any desired directions at right angles to the wire. If the rods are set parallel to one another, they form a two-sided (cylindrical) band; but, if they are then turned progressively (see fig. below) from some starting point so that the total turning amounts to 180', they form a one-sided ("Moebius") band.* [* The German mathematician A. F. Moebius discovered that if one takes a long strip of paper, gives it a half-twist, and connects the ends to make a band, one creates a seeming impossibilityan object that has only one side. A child of four can paint an ordinary paper band blue on one side and red on the other, but not even Picasso could do that to a Moebius band.] According to topologists, these bands represent two quite different "fibre bundles" based on a circle, the rods representing the fibres and the wire the circle (base). Going further, topologists have imagined three-dimensional space, with a single point added "at infinity," as a bundle of circular fibres based on the surface of a sphere, any two of the fibres caught together like two consecutive links in a chain.
The development of mathematics has involved not only specialization, but also generalization, resulting in a tremendous amount of overlapping among the various specialties. Logicians cooperate on certain problems with algebraists, topologists with differential geometers, analysts with statisticians, and so on.
But where do these abstract specialties overlap human experience? Can pure mathematics be useful in the modern world, or is it like a kite that has been let loose, wandering aimlessly for a while, soon to disappear entirely from human sight?
It is quite obvious that mathematics pervades and dominates the empirical disciplines, so much so that it has been called the "language" of science. As each science advances, its methods become more and more mathematical. But many people do not realize that progress against difficult technological problems has time and again depended on the discovery and exploitation of "useless" mathematical ideas. In the seventeenth century the French mathematicians P. Fermat and B. Pascal worked out a theory of probability, which lay dormant for two hundred years before the biologist Mendel found in it the basis for his famous law of heredity. Until the early nineteenth century it was thought that all algebraic multiplication had to be commutative (that is, AB = BA). When Ireland's W. R. Hamilton introduced a noncommutative system, nobody became excited. But one hundred years later, in the 1920's, Hamilton's algebra turned out to have extraordinary applications in quantum physics. A further example is the absolute differential calculus developed by Ricci and Levi-Civita in the 1890's. This branch of mathematics supplied the perfect tool for Einstein's general relativity in 1916. In each instance, the useful ideas were discovered by men intent only upon the development of mathematics, without any specific applications in view.
Today mathematics is an indispensable research tool in studies of shock waves, heat flow in metals, advanced weather forecasting, control of automatic machinery, prediction of economic trends, and a wide range of business and technological fields.
What are mathematicians like as people? And what is the nature of their creative experience?
In the entire world there are only about l,500 research mathematicians. They are a close-knit group, on the whole: they read the same journals, attend the same meetings, and make frequent visits to each other's universities. Word of important new ideas or significant researches sometimes spreads with startling rapidity and in mysterious ways. Princeton's associate professor John Moore was surprised one day to receive a letter from a mathematician in China, inquiring about some points Moore had raised at a seminar at Fine Hall only a few weeks past.
One cannot safely generalize about the mathematical personality, except to say that mathematicians are often men of modest nature, fond of music, lovers of mountain climbing, and always logical thinkers. Some have been famous for prodigious memories and the ability to solve long and difficult problems in their heads. Some find it relaxing to invent more complicated variations of already complicated gamesthree-dimensional chess, for example.
Describing the mathematician's creative experience, Marston Morse of the Institute says: ". . . discovery in mathematics is not a matter of logic. It is rather the result of mysterious powers which no one understands, and in which unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty's sake and pulls it down to earth. The creative scientist lives in a `wildness of logic,' where reason is the handmaiden and not the master."
In the mid-eighteenth century a young tutor described the philosophy of English dissenting academies in these words: "Formerly, if a man was well versed in the learned and dead languages ... and master of the distinctions of the divinity school, he passed, for a considerable scholar . . . whereas now 'tis mathematical learning that carries the bell."
The young College of New Jersey in America carried on this progressive spirit of its English forerunners. In 1760 entering freshmen were required to have not only a knowledge of Latin and Greek, but also an understanding of the rules of "vulgar arithmetic." As underclassmen, students got their full share of algebra, trigonometry, geometry, and conic sections along with their classical studies. A member of the class of 1845, while an undergraduate, exclaimed: ". . . piety and mathematics rated extravagantly high in the course." By 1853 a feature article in a Boston newspaper commented that at Princeton the study of mathematics was carried on "to an extent not excelled by any other college in the country."
The typical American attitude towards mathematicsand one not uncommon todaywas voiced by Woodrow Wilson in 1903 in an address introducing the preceptorial system. Wilson maintained that the natural man inevitably rebels against mathematics. He called the subject a mild form of torture that could only be learned by painful processes of drill and that was as necessary for the attainment of adulthood as the "measles."
At this turning point in the history of American higher education, the total number of active mathematicians in the United States was negligible. Significant original contributions came from a very few scattered men of genius, such as George W. Hill and Willard Gibbs. Nothing else was produced here worthy of notice by the rest of the world. Physicists, chemists, and astronomers based their experimental work on the theories of great European thinkers like Newton, Gauss, Poincaré, and others. On the continent there were at least thirty research professors of mathematics, who enjoyed limited teaching duties, good salaries, and prestige. In America, on the other hand, every position in the field was essentially that of a teacher whose primary duties were to inflict a grim subject on reluctant students via cut-and-dried recitations. Various factors in the American culture of that generation worked against the development of mathematical scholarship: a lack of encouragement and guidance, ignorance of the contemporary state of the science, and strong external pressures in other directions, such as business or diplomatic service.
Despite Wilson's distaste for the world of cosines and tangents and his feeling that the preceptorial system was not suited to mathematical instruction, he nevertheless favored the idea of distinguished new appointments to the mathematics faculty to balance those planned for other departments. To his close friend and colleague, Henry Burchard Fine, fell the responsibility of selecting the new personnel for the Departments of Science and Mathematics.
The name of Harry Fine brings to mind an old saying that "the investigator of mathematical truth comes to all other questions with a decided advantage." Fine was an excellent mathematician. But over and above this, he was an educator with vision, an administrator, and a man of great personal vigor and charm. Princeton's rapid development from a small college into a great university and the sudden blossoming of creative mathematical research in America owe much to this one man. His greatest asset was a remarkable ability to recognize young scholars destined for leadership in their fields.
As an undergraduate at Princeton Harry Fine specialized in the classics, graduating with first honors in 1880. He was one of a small group that discussed philosophy with Dr. McCosh. Fine's close and life-long friendship with Woodrow Wilson began when both were members of the editorial board of the Princetonian. In later years, Fine declined several government positions offered to him by Wilson, including the post of Ambassador to Germany.
The period between his graduation in 1880 and his appointment as assistant professor of mathematics in 1885 was crucial in its effect on the scientific and mathematical history of the entire country. While a graduate student in experimental science at Princeton, Fine decided to specialize in mathematics. He then spent considerable time studying in Germany under two leading mathematicians of the day, Felix Klein and Leopold Kronecker, receiving his Ph.D. in Leipzig in 1885. This experience no doubt generated Fine's strong convictions concerning the goals of a university, the importance of creative research, and the stimulating effects of contact with scholars of inventive genius. After exposing himself to the main currents of modern mathematics, he returned to America to teach mathematics at Princeton.
At various times during a full and energetic career on this campus, Fine served as Chairman of the Mathematics Department (twenty-four years), Dean of the Faculty (six years), Dean of the Departments of Science (eighteen years), and Head of the Faculty Committee on Athletics (twenty-five years). Throughout the two-year interim between the Wilson and Hibben administrations, Fine managed the affairs of the University. At several points along the way, he turned down calls to the presidency of other institutions of higher learning, including M.I.T. In 1891 Fine helped to found the New York Mathematical Society, which became the American Mathematical Society in 1894, and he served as its president in 1911-1912.
The institution of the preceptorial system marked the beginning of Princeton's mathematical tradition. In his search for topflight mathematicians, Fine set out to comb the world for promising material. In 1905 he called to preceptorships Oswald Veblen, Gilbert Ames Bliss, and John Wesley Young. These men were in their prime and at a point in their lives when recognition meant the most to them. At the same time, Fine brought in the noted British scientist and astronomer, James Jeans of Cambridge, as Professor of Applied Mathematics. Eddington, too, was approached, but declined the call in order to accept a post at the Greenwich Observatory. Potential greatness at home was not overlooked. Fine promoted Luther Pfahler Eisenhart to a preceptorship and had his eye on a brilliant young undergraduate James Waddel Alexander, whom he made an instructor several years later.
In his wise choice of replacements, Fine managed to improve the quality of the group from year to year. When Bliss left for the University of Chicago and Young went to Dartmouth, they were replaced by two younger and equally brilliant mathematicians, George David Birkhoff and the Scotsman, J. H. M. Wedderburn. When Birkhoff accepted teaching duties at Harvard, Fine called in Pierre L. Boutroux, a graduate of the University of Paris. Thomas H. Gronwall and Einar Hille of Sweden were among others who brought distinction to the Mathematics Department under Fine.
During the first two decades of its existence, the Department contained a very large proportion of the best mathematicians in America at the time, many of them recruited from Europe. Those who stayed at Princeton only a few years were deeply impressed by Fine and made substantial contributions to the Princeton mathematical group as a growing organism. The Princeton organism, in turn, had a greater influence on the mathematics movement throughout America because of the rapid turnover of its members.
The young mathematicians who came to Princeton during these years were characterized by their burning interest in mathematics for its own sake. In spite of heavy undergraduate teaching loads and without benefit of high salaries, special opportunities for research, or prestige, they nevertheless pursued with enthusiasm and singleness of purpose the advancement of pure mathematics. Many of them left a permanent mark on their science. Eisenhart, Veblen, Bliss, Birkhoff, and Alexander traveled by way of Princeton to the National Academy of Sciences, Wedderburn and Jeans to the Royal Society of London.
Research programs sprouted in such specialized areas as differential geometry, algebraic geometry and topology, the theory of matrices and of hypercomplex numbers, differential equations, and the logical foundations of mathematics. In the applied realm a group coordinated its efforts in a mathematically oriented attack on some central problems of physics.
The system of postdoctoral research fellowships, initiated by the great foundations in the twenties, furthered the rapid development of mathematical research on the campus. A survey in 1926 showed that in the number of National Research Council fellows who chose to come to Princeton, as compared with other universities, Princeton ranked first in the field of mathematics.
Standard textbooks and authoritative advanced works issued from the pens of Princeton mathematicians and were used in classrooms of colleges and universities throughout America: notably, A College Algebra and Calculus by Fine; Coordinate Geometry by Fine and Thompson; Projective Geometry, a massive two-volume work by Veblen and Young; and Differential Geometry by Eisenhart.
When Einstein landed in New York in 1921, he immediately expressed a desire to lecture at Princeton University because "he felt grateful to the faculty of Princeton, which was the first college to become interested in his work." Einstein was at the time very interested in some of Princeton's worknamely, that of Eisenhart and Veblen on Riemannian geometry and tensor analysiswhich related to the development of his theory of general relativity.
In 1928 President Hibben was informed of a comment made at the Leipzig meeting of the German National Academy of Sciences. A German professor, who had just returned from a study tour of American universities, reported that in his opinion the most valuable contributions in mathematics during the previous decade had come from Princeton.
The fame of the Mathematics Department rested not on its undergraduates, but on its upper echelons of higher learningits faculty, postdoctoral research fellows, and graduate students. Although all the freshmen and many sophomores took mathematics to fulfill strict degree requirements, not more than twelve upperclassmen elected mathematics courses in any one year. One big reason for this paucity of departmental majors was the fact that so many attractive managerial positions in American business were open to Princeton graduates. The number of graduate students, however, was somewhat larger and increased steadily as the renown of the faculty spread. Graduate students in mathematics were drawn from the undergraduate colleges of the entire country and from other nations, only a small fraction of them from the Princeton University undergraduate body. The star exception is Alonzo Church '24 (Ph.D. '27), whose 30-year Princeton career has established him as a world authority in the domain of mathematical logic.
The $3,000,000 Scientific Research Fund established in 1928 was another giant step for Princeton in its growth as a university and a big boost to the cause of mathematical research. When the General Education Board of the Rockefeller Foundation offered to give $1,000,000 towards this fund, provided that the University raise an additional $2,000,000, Fine was put in charge of the Fund Campaign Committee and at once approached his good friend and kindred spirit, Thomas D. Jones '76. Characteristically, Jones' response was more than generous. He began by establishing the Henry Burchard Fine Professorship in Mathematics. He and his niece, Miss Gwethalyn Jones, then endowed three additional advanced professorships, one in chemistry and two in physics, and added $500,000 to the research fund, which was soon completed by other gifts. (Jones had been a follower of Wilson and a Life Trustee of Princeton from 1906 until 1912, when he resigned. Retiring from active law practice in Chicago in 1900, he then served as President of the Mineral Point Zinc Company and Director of the International Harvester Company and the New Jersey Zinc Company.) After establishment of the fund, Fine headed the committee to advise the president concerning its administration and once again was instrumental in bringing a group of outstanding scholars to the Princeton scientific faculties.
At the time of its institution in 1926, the Henry Burchard Fine Professorship was the most distinguished mathematical chair in the country. G. D. Birkhoff (then at Harvard), commenting on the dearth of scientific research positions in America at the time, said: "Most fortunately, for American mathematics, the situation is now changing for the better, and I rejoice that Princeton by instituting its Henry Burchard Fine Professorship has set an example in this direction. Unless I am mistaken, this is only the beginning of a large movement looking toward the permeation of a few of our greatest universities with the true university spirit of productive research."
For several months before a fatal bicycle accident, which occurred in late December, 1928, Fine had spoken of his desire for a mathematics building. Immediately after Fine's death, Mr. Jones and his niece offered to endow a Hall of Mathematics as a memorial.
"Nothing is too good for Harry Fine," said Jones, as plans for the building got under way in magnanimous style. The donors saw to it that Fine Hall was not only a comfortable and functional home for mathematicians, but also an unusually substantial structure. Maintenance of the building and renewal of the furnishings were adequately provided for in the endowment.
The idea built into Fine Hall, said Oswald Veblen at the dedication ceremonies in 1931, is that of a university as a seat of learning and creative scholarship. A university, he went on, should provide centers for people of like intellectual interests, where young and old can gather for mutual encouragement and support and for informal and easy contacts. Each intellectual group should be placed physically so that it becomes automatically conscious not only of itself, but also of its relation to the university as a whole.
Fine Hall today, besides being a unique collegiate structure in many ways, is the materialization of this idea. Connected as it is to one of its giant neighbors, Palmer Physics Laboratory, and overlooking Guyot, the great hall of Biology and Geology, its strategic location dramatizes the role of mathematics as the unifying spirit behind all the sciences. Every afternoon at 4 o'clock tea is served in its large and luxurious common room, which fills up with mathematicians, mathematical physicists, engineers, visitors from Forrestal and the Institute for Advanced Study, research scientists from neighboring cyclotrons and industrial laboratories, and others mathematically inclined. Students and teachers alike relax with cup and saucer. Conversation is lively, and now and then a comprehensible word or phrase can be heard.
"Livability" is another idea that was built into Fine Hall, in keeping with Jone's wish that it be a place mathematicians would be "loath to leave"a dwelling house rather than a stiff classroom building. Faculty offices are designed as studies or living rooms, comfortably furnished, with oak-paneled walls that often hide blackboards or filing cabinets. One of the most distinctive features of the building is a locker room provided with showers, so that members of the Department can use the nearby tennis courts and gymnasium without having to return home to dress. In the immortal words of the Faculty Song, Fine Hall is a splendid "country club for math, where you can even take a bath."
A close look at architectural details in the Common Room and in the equally sumptuous Professors' Room on the other side of the building reveals delightful touches guaranteed to make any mathematician feel at home. Incorporated into the leaded designs of windows in both rooms are the mathematical formulae of Newtonian gravitation, the Einstein theory, the quantum theory, and other esoteric equations. On one side of the fireplace in the Professors' Room is a winsome carving of the predicament of a fly on a one-sided surface. Above the mantel in the Professors' Room is carved the provocative Einstein statement: "God is clever, but he is not malicious" (freely translated: "God is slick, but he ain't mean").
In addition to these two chambers, Fine Hall contains two seminar rooms, two lecture rooms, nine large studies assigned to senior faculty members, and sixteen smaller studies for junior members and advanced research fellows. The entire top floor is given over to the library of mathematics and physics, one of the finest collections to be found anywhere in the world. A fund established by the class of 1880 enables the Mathematics Department to buy important new books as they appear.
Inside the cornerstone of Fine Hall is a lead box containing copies of significant works by Princeton mathematicians. Alongside these treasures are laid the venerable tools of the tradewithout which no mathematical discoveries would ever be madetwo pencils, one piece of chalk, and an eraser.
The man who stepped into the shoes of Henry Burchard Fine resembled his predecessor in many qualities of intellect and character. Luther Pfahler Eisenhart was also an educator, in the broadest sense of the word, and an energetic administrator, as well as a leader of modern mathematical thought. In the early 1930's Eisenhart could frequently be found chatting learnedly with Einstein. Today at age eighty-two, Dean Emeritus, Professor Emeritus Eisenhart commutes to Philadelphia twice a week to perform his duties as executive officer of the American Philosophical Society.
Eisenhart did his undergraduate work at Gettysburg College, where it was rumored that he majored in baseball, accidentally acquiring an interest in mathematics. He received his Ph.D. from Johns Hopkins University. Coming to Princeton as an instructor in 1900, Eisenhart was promoted by Fine to a preceptorship in 1905 and jumped to a full professorship four years later, at age 33. In addition to his post as Chairman of the Mathematics Department from 1928 until his retirement in 1945, Eisenhart served as Dean of the Faculty from 1925-1933 and Dean of the Graduate School from 1933-1945. Largely through his efforts, while Dean of the Faculty, the four-course plan of independent study was instituted at Princeton and started on its path of successful operation.
A prodigious variety of professional activities extended Eisenhart's influence in the educational world far beyond the boundaries of Princeton University. While at the helm of the Princeton Mathematics Department, he also served, at various times, as President of the American Mathematical Society, President of the Association of American Colleges, Chairman of the division of Physical Sciences of the National Research Council, a member of the sScientists' Committee on Loyalty Problems (with Einstein and Veblen), and Vice-President of the National Academy of Sciences. In his book The Educational Process, Einsenhart sets forth the central theme of his life's work: the aim of education is to teach the student how to educate himself.
With the building of Fine Hall, the Mathematics Department at Princeton came of age. The early years of the Eisenhart era were marked by a stream of distinguished visitors brought to Princeton through the resources of the new Scientific Research Fund. In mathematics, the roster included leading lights from many nations, among them G. H.. Hardy and P. A. M. Dirac of Cambridge, P. A. Alexandroff of Moscow University, and Hermann Weyl from the University of Gottingen, Germany. Noted scholars were appointed to the mathematics faculty during this period, including John von Neumann and Eugene P. Wigner. Dr. Einstein, then holding a position in the Berlin Academy of Sciences, was invited to occupy the Jones Professorship of Mathematical Physics, but declined with the remark: "One must not dig up a flowering plant."
The strength of mathematical scholarship at Princeton was underscored by decisions to locate the Institute for Advanced Study in the area, to make its first school one of mathematics, and to invite several members of the Princeton Mathematics Department to form the nucleus of the new effort in postdoctoral education. The Institute's School of Mathematics came into being in October 1933. Its staff of five professors included Veblen, Alexander, and von Neumann, formerly of the University; Einstein, who found this second call to Princeton irresistible, and Hermann Weyl. Since interim headquarters were needed for the infant enterprise, Fine Hall opened its doors and generously provided office space and library facilities. No organic connection existed between the University Mathematics Department and the Institute, but scholarly cooperation between the two immediately became the accepted practice. Students and faculty of either institution were allowed to attend courses or seminars given by the other without paying additional fees. Faculty members of both schools constituted the advanced-level Mathematics Club, which met weekly for presentation of papers. The Department and the Institute shared editorial responsibilities for the Annals of Mathematics, a quarterly journal of world renown.
There followed in the wake of the early Institute elite an impressive international parade of the world's most accomplished mathematicians. The present group of postdoctoral mathematics visitors at the Institute numbers over seventy.
What might have been a death knell turned out to be a shot in the arm. The Department, far from being drained of all vigor by the Institute, continued to grow in power, productivity, and recognition. The added opportunity for association with scholarly leaders acted as a drawing card in bringing bright young students and teachers to the campus. Princeton's stature as a world center for mathematical research was enhanced. Today the Department and the Institute continue to work together as partners, united in the pursuit of mathematical knowledge.
Granting the importance of postdoctoral education, one must not forget that mathematical genius and youth go hand in hand. A mathematician is like a baseball player; he must have vigor and enthusiasm in order to play the game well. When a mere lad of sixteen B. Pascal proved "one of the most beautiful theorems in the whole range of geometry," involving the "mystic hexagram." K. F. Gauss once declared that "such an overwhelming horde of ideas stormed my mind before I was twenty that I could hardly control them and had time to record but a small fraction." Riemann was twenty-one when he recognized in partial differential equations the essential definition of an analytic function of a complex variable. So there you are.
The prize Princeton mathematical prodigy is John Milnor, '51, Princeton Ph.D. of 1954, now an Associate Professor in the Department at the age of 27. When just a freshman, Milnor heard about an unproved conjecture of the Polish topologist, Karol Borsuk, concerning the total curvature of a knotted curve in space. A few days later he brought a written proof to Professor Tucker, with the modest request: "Would you be good enough to point out the flaw in this attempt? I'm sure there is one, but I can't find it." Tucker studied the proof carefully, but could find no flaw, nor could his colleague Fox, nor could Chicago's Professor Chern (from China), then visiting the Institute. Milnor was encouraged to submit his proof for publication, and when he turned in his paper a few months later, it contained a full theory of the curvature of knotted curves, with the proof of the Borsuk conjecture as a mere by-product. This paper, deeper than many an excellent doctoral thesis, was published in the 1950 Annals of Mathematics. Milnor now has a world reputation in topology and other fields, based on more than ten published works of outstanding originality and elegance.
In view of this tendency among mathematicians towards youthful productivity, the Department established the Henry Burchard Fine Instructorships in 1937, appointees to have lighter schedules than regular instructors, some advanced classes, and time for research. These junior chairs were intended to attract strictly first-rate young men to the Department, to give them an opportunity to develop their possibilities to the fullest, and at the same time to provide a stimulus to the rest of the faculty and a natural group from which to draw permanent members of the Department. Among the first Fine Instructors were Norman E. Steenrod and John W. Tukey, both currently Princeton professors.
World War II brought major changes in the mathematical routine at Princeton. Several of the teaching staff took prolonged leaves of absence to engage in defense projects. In addition to the usual classes, the Mathematics Department gave special courses prescribed by the Navy V-12 and the Army Specialized Training Programs. Some members of the faculty joined forces with industrial and governmental experts in emergency instruction programs for war workers in the vicinity. Princeton mathematicians took an active part in statistical and ballistic researches sponsored by the United States government. The Presidential Medal of Merit for outstanding war service was awarded to Wilks of the Department and to Veblen, von Neumann, and Morse of the Institute.
The first international gathering of mathematicians in a "long and terrible decade" took place at Fine Hall in December 1946. In celebration of Princeton's bicentennial year, more than one hundred leaders from nine nations gathered on the campus for a three-day conference on the "Problems of Mathematics." Solomon Lefschetz, who had taken over the reins of the Mathematics Department after Eisenhart's retirement in 1945, served as the conference chairman. Discussions touched on a wide range of topics in the field of advanced mathematics.
Russian-born, French-educated Solomon Lefschetz had joined the Princeton Mathematics Department in 1924. In 1933 he succeeded Oswald Veblen as Henry Burchard Fine Professor of Mathematics. At this crucial time for the Departmentwhen three of its ablest members had been drawn away to the Institute and when Eisenhart had just taken on new duties as Dean of the Graduate SchoolLefschetz exerted forceful leadership and was largely responsible for rebuilding the Department into the strong and flourishing group that it is today. His term as Chairman of the Mathematics Department extended from 1945 until his retirement in 1953.
Professor Emeritus Lefschetz is recognized the world over for his pioneering work in algebraic geometry and topology. Princeton's supremacy in these fields was greatly augmented by the researches he initiated and guided during his three decades on the mathematics faculty. In recognition of his stature as a mathematical leader, Lefschetz has received many high honors during his lifetime, including membership in the American National Academy; honorary degrees from Clark University, the University of Mexico, and the University of Prague, election to the presidency of the American Mathematical Society; and, recently, corresponding membership in the French National Academy in 1954, following Lefschetz' retirement, a three-day research conference was held in his honor, attended by forty scholars from over twenty-five institutions. The topics discussed stemmed from his significant work in algebraic geometry and topology, and the papers delivered were published in a symposium volume by the Princeton University Press. In 1956 Lefschetz received one of the highest mathematical honors in the world, the Antonio Feltrinelli International Prize of Rome's Accademia Nazionale.
Looking at the Princeton scene today, we find the Department of Mathematics of the University carrying forward the great tradition that began over a half century ago. Its excellent reputation rests on the mathematics publications of Princeton University Press, the accomplishments of its distinguished faculty, and the quality and quantity of its past and present Ph.D's. [* This article is devoted to the history and scientific accomplishments of the Department of Mathematics. But it should be remarked that undergraduate teaching is a major activity of the Department. Both terms this year eight sections of freshman mathematics and five sophomore sections have been taught by Department members of professorial rank. Emil Artin, the H. B. Fine Research Professor, conducts a very successful Freshman Honors Course (Math. 105-6) which has been publicized by the Mathematical Association as a model for such courses. Professor William Feller, holder of the Higgins Chair, and Professor Ralph Fox have taught the corresponding sophomore course. Currently twenty-five upperclassmen are enrolled as Departmental Students in Mathematics.]
Recently a Brooklyn high school lad wrote to foreign universities for information comparing missile guidance and bird migration. Moscow University politely informed the boy that he would find just what he needed in a "Princeton publication" in a series entitled the Annals of Mathematics Studies. The Annals Studieswhose volumes contain articles longer than Journal papers but shorter than full-length booksstarted out in 1934 as a collection of mimeographed notes of advanced lectures given at the University and the Institute. Soon the demand for the "Princeton Mathematical Notes" became so great that the University Press took over their publication. Gradually, the Studies were enlarged to include articles by non-Princeton authors. Today this series has thirty-nine volumes to its credit and is ranked with the best of its kind, including the Cambridge Tracts and the French Mémorial Series.
The Annals Studies series is a valuable adjunct to the Annals of Mathematics journal, mentioned earlier, now in its sixty-seventh volume. The third Princeton mathematics publication is the Princeton Mathematics Series, initiated in 1937 to stimulate the publication of full-length advanced mathematics books in America. This series, which has developed very rapidly, is about to bring out its twenty-third volume. All of these publications are edited under the joint auspices of the Department and the Institute.* [* These scholarly series are published by Priinceton University Press, as are the following distinguished selections for amateur mathematicians: Philosophy of Mathematics and Natural Science, by Hermann Weyl (Paperback) $1.95, Symmetry, by Hermann Weyl, $3.75, The Enjoyment of Mathematics, by H. Rademacher & 0. Toeplitz, $4.50, Mathematics and Plausible Reasoning, 2 vols., by G. Polya, $9.00.]
The Mathematics Department at Princeton today is an international one in many respects. Its ranks include natives of nine different lands, including England, Austria, Germany, Poland, Yugoslavia, Hungary, Japan, Canada, and the United States. Seven Princeton mathematics professors received their doctorates from foreign universitiesnamely, Cambridge, Gottingen, Leipzig, Zurich, and Berlin. Each year several members of the teaching staff make extended visits and tours abroad, sometimes on special fellowships or grants for research and study, sometimes to attend important meetings, sometimes as visiting lecturers to other universities. In recent years, thirteen Princeton mathematicians have visited a total of thirteen different countries. Professor Norman E. Steenrod was invited behind the Iron Curtain in June 1956, where he attended the annual meeting of the Russian Mathematical Congress.
As in the past, members of the Princeton mathematics faculty today are being singled out for top academic honors. At various times during the past decade, Professor Donald C. Spencer was a joint recipient of the American Mathematical Society's Bôcher Prizeone of two prizes awarded by this Societyfor a notable research memoir in the field of analysis; Professor K. Kodaira, formerly of the University of Tokyo, was awarded the Fields Medal by the International Mathematical Congress for outstanding contributions to mathematics in the quadrennium 1950-1954; Professor William Feller (whose book An Introduction to Probability Theory and Its Applications is currently the basic modern work in its field) was invited to give the Riesz Lecture of the Institute of Mathematical Statistics; Professor Norman E. Steenrod and Solomon Bochner were asked to deliver the Colloquium Lectures of the American Mathematical Societythe highest research honor of the country's most distinguished mathematical association. Professors Bochner, Steenrod, and Artin are members of the National Academy of Sciences. This year the International Congress of Mathematicians is to be held in Edinburgh. Out of a group of about twenty-one now in the United States who have been invited to deliver addresses, seven are from Princeton: Professors Steenrod, Feller, Moore, and Milnor of the Mathematics Department, Professor F. J. Dyson of the Institute, and Drs. C. D. Papakyriakopoulos and A. Andreotti.
Mathematical research at Princeton is carried on in several different ways. Sometimes it involves one individual working independently along an avenue of particular interest. In other cases, several individuals with a mutual interestincluding graduate students, young teachers, and senior faculty memberswork together on a special mathematical problem. Thirdly, a number of the staff are involved in government-sponsored projects which have been underway on the campus since World War II.
Current government-sponsored research projects include Professor Samuel S. Wilks' Project in Statistics (Office of Naval Research), Professor Solomon Lefschetz' Project on Nonlinear Differential Equations (Office of Naval Research), Professor Albert W. Tucker's Project on Combinatorial Problems related to Logistics (Office of Naval Research), Professor Donald C. Spencer's Project on Boundary Value Problems for Real and Complex Manifolds (Office of Ordnance Research, United States Army), and the Topology Project of Professors John Moore and Norman E. Steenrod (Office of Scientific Research, United States Air Force). In addition, the Department has a relatively new Applied Statistics Project in operation at Forrestal Research Center, led by Professor John W. Tukey, under a grant from the U.S. Army Office of Ordnance Research.
Whenever a national organization establishes a committee on mathematics today, it immediately approaches Princeton for help. More and more, members of the Fine Hall group are being called upon to give of their time, energies, and talents towards the improvement of education, the advancement of science, and the cause of human welfare in this country. Two men extremely active along these lines are Professor Tucker, Departmental Chairman, and Professor Wilks, head of the Section on Mathematical Statistics.
Professor Tuckerwhose undergraduate courses have received great praise from the studentsis so tremendously interested in the problem of mathematics education that he finds it very hard to turn down any jobs in this area. "Leading mathematicians have been so busy developing new ideas," says Tucker, "that they have neglected to keep mathematics teachers informed about the new developments." The revolutionary innovations in mathematics during the twentieth century have led to considerablebut not enoughrevision of the college mathematics curriculum today, but the pre-college curriculum has remained much the same, so that the two are now seriously "out-of-phase."
Aware of the urgent need for reform, Tucker is currently serving as Chairman of the College Entrance Board's Commission on Mathematics, a member of the Committee on the Undergraduate Program of the Mathematical Association of America, and the Committee on Educational Policy of the Mathematics Division of the National Research Council. As if this weren't enough for one man to shoulderin addition to teaching and research and administration of the Mathematics DepartmentTucker is also the mathematics member of the Basic Sciences Program Committee of the Alfred P. Sloan Foundation and mathematics member of the Research Panel of the President's Scientific Advisory Committee.
As Chairman of the College Entrance Board's Commission on Mathematics, Professor Tucker is supervising the long-term nationwide project for complete revision of the high school mathematics curriculum. The goals of this project are the elimination of obsolete materials and the addition of new materials that are a necessary preparation for today's uses of mathematics. The program involves the development of new textbooks, retraining of present teachers, planning special courses for new teachers, development of illustrative material, and conferences to win the support of mathematics teachers.
Another public-spirited Princeton mathematician is Professor Samuel S. Wilks, who has spent much of the last twenty-five years building up the Department's strong Section on Mathematical Statistics. Wilks, too, has been very active in national affairs and is noted for his work in statistical inference and the applications of mathematical statistics in many fields. This year he was elected to chair the Mathematics Division of the National Academy of Sciences' National Research Council. The National Research Council is a private organization of scientists, authorized by the federal government to act as advisor in all matters of scientific and technical interest. The Mathematics Division, which Wilks heads up, serves as a source of interchange between mathematicians, scientists, and educators. Professor Wilks also serves on committees of the Bureau of the Budget, the National Science Foundation, the National Security Agency, and the U.S. Army Office of Research and Development.
The performance of Princeton's mathematics Ph.D.'s offers impressive testimony to the excellence of the graduate training they received while at Fine Hall. The Graduate School of Mathematicsthe third largest one in the University, next to Physics and Chemistrynow averages about forty students a year. A detailed survey, sponsored by the National Science Foundation in 1957, shows that Princeton has produced more mathematics Ph.D.'s since 1935 than any other American university. The survey goes on to show that Princeton's Ph.D.'s outnumber those from other institutions in the two top publication groupingsthat is, more Princeton Ph.D.'s have continued in the field, and more are making substantial contributions to pure research.
The most characteristic feature of the graduate instruction program is a unique opportunity for personal contact and close intellectual association between teachers and students. Secondly, although a student devotes his first two years to a broad training in advanced mathematics, he is given a maximum of freedom to follow his personal bent, and encouraged to undertake investigations on his own as soon as possible. Graduate seminarssometimes conducted jointly with the Institute or with a neighboring industrial research laboratoryafford an unusual opportunity for communication of significant research.
This year twenty of the forty graduate students are here on special fellowshipsfive of these are University sponsored and fifteen granted by outside organizations, such as the National Science Foundation, Bell Telephone, General Electric, the Commonwealth Fund, and the Canadian National Research Council, to name only a few. Approximately five fellows a year come as visitors from nations abroad. Recent statistics show that close to 20% of the National Science Foundation fellows in mathematics choose to do their graduate work at Princeton, rather than at one of forty or fifty other possible institutions of higher learning. This year the Woodrow Wilson graduate fellowships are being offered for the first time to students of mathematics. Over 25% of the recipients so far have listed Princeton as the school of their choice.
Since the turn of the century Princeton's mathematics Ph.D.'s have distinguished themselves in many ways, reflecting great glory on their graduate school. Six now grace the membership rolls of the National Academy of Sciences; one is a fellow of the Royal Society of London. Fourteen are Chairmen of Mathematics Departments in American colleges and universities and one Canadian university. Included in this group are Professors Paul A. Smith of Columbia, Robert J. Walker of Cornell, Stephen C. Kleene of the University of Wisconsin, John G. Kemeny '43 of Dartmouth, and Carl B. Allendoerfer of the University of Washington (Seattle). Three are Deans of Graduate Schoolsnamely, Robert D. Carmichael (Emeritus) of the University of Illinois, Morris S. Knebelman of Washington State College, and Henri F. Bohnenblust of the California Institute of Technology. Several Princeton Ph.D.'s in mathematics have gone on to distinguished academic posts in areas outside their field of graduate study, notably, Frederick C. Mostellar, Professor of Social Relations at Harvard, Paco A. Lagerstrôm, Professor of Aeronautical Engineering at the California Institute of Technology, and Nobel Prize winner John Bardeen, Professor of Physics at the University of Illinois. Many others are now occupying high executive offices in industry, business, and government.
Fifty years of mathematics at Princeton have fulfilled the fondest dreams of Harry Fine and Woodrow Wilson. Fine Hall today is an outstanding international center for mathematical research and a community of scholars serving the nation and the world.
The 6 sketches of parts of the leaded windows of Fine Hall mentioned in the editorial note, plus the Mobius strip illustration, in the order in which they appear scattered in the text:
The Princeton Mathematics Community in the 1930s