HISTORY OF MATHEMATICS Volume 2
A Century of Mathematics in America
Part II Edited by Peter Duren
with the assistance of Richard A. Askey Uta C. Merzbach
American Mathematical Society. Providence, Rhode Island Library of
Congress Cataloging-in-Publication Data A century of mathematics in
America. Part II. (History of mathematics; v. I-) 1. Mathematics-United
States-History-20th century. I. Askey, Richard, 1933 II. Duren, Peter
L., 1935- . III. Merzbach, Uta C., 1933 QA27.U5C46 1988 510.973 88-22155
ISBN 0-8218-0124-4 (v.1) ISBN 0-8218-0130-9 (v.2)
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0 Contents Preface ix Mathematics at American Universities
HARVARD
Mathematics at Harvard, 1836-1944
3 Garrett Birkhoff The Scientific Work of Maxime Bocher [Reprint]
59 G. D. Birkhoff William Fogg Osgood [Reprint]
79 J. L. Walsh YALE Mathematics and Yale in the Nineteen Twenties
87 Harold Dorwart The Scientific Style of Josiah Willard Gibbs [Reprint]
99 Martin J. Klein The Department of Mathematics [Reprint]
121 P. F. Smith CHICAGO Mathematics at the University of Chicago:
A Brief History
127 Saunders Mac Lane Eliakim Hastings Moore and the Founding of a
Mathematical
Community in America, 1892- 1902 [Reprint]
155 Karen H. Parshall Graduate Student at Chicago in the '20s [Reprint]
177 William L. Duren, Jr. Reminiscences of Mathematics at Chicago
[Reprint]
183 Marshall H. Stone The Stone Age of Mathematics on the Midway
[Reprint]
191 Felix E. Browder PRINCETON The Emergence of Princeton as a World
Center for Mathematical Research, 1896-1939 [Reprint]
195 William Aspray
Garrett Birkhoff has had a lifelong connection with Harvard
mathematics. He was an infant when his father, the famous mathematician
G.D. Birkhoff, joined the Harvard faculty. He has had a long academic
career at Harvard: A.B. in 1932, Society of Fellows in 1933-1936,
and a faculty appointment from 1936 until his retirement in 1981. His
research has ranged widely through algebra, lattice theory, hydrodynamics,
differential equations, scientific computing, and history of mathematics.
Among his many publications are books on lattice theory and hydrodynamics,
and the pioneering textbook 'A Survey of Modern Algebra', written jointly
with S. Mac Lane. He has served as president of SIAM and is a member of
the National Academy of Sciences.
Mathematics at Harvard, 1836-1944
GARRETT BIRKHOFF
0. OUTLINE
As my contribution to the history of mathematics in America, I decided
to write a connected account of mathematical activity at Harvard from
1836 (Harvard's bicentennial) to the present day. During that time, many
mathematicians at Harvard have tried to respond constructively to the
challenges and opportunities confronting them in a rapidly changing world.
This essay reviews what might be called the indigenous period,
lasting through World War II, during which most members of the Harvard
mathematical faculty had also studied there. Indeed, as will be explained
in 1-3 below, mathematical activity at Harvard was dominated by Benjamin
Peirce and his students in the first half of this period.
Then, from 1890 until around 1920, while our country was becoming a great
power economically, basic mathematical research of high quality, mostly
in traditional areas of analysis and theoretical celestial mechanics,
was carried on by several faculty members. This is the theme of 4-7.
Finally, I will review some mathematical developments at Harvard in
the quarter-century 1920-44, during which mathematics flourished there
(and at Princeton) as well as anywhere in the world.
Whereas 1-13 of my account are based on reading and hearsay, much of 14-20
reviews events since 1928, when I entered Harvard as a freshman, and
expresses my own first-hand impressions, mellowed by time. Throughout,
I will pay attention not only to "core" mathematics, but also to
"applied" mathematics, including mathematical physics, mathematical
logic, statistics, and computer science. I will also round out the
picture by giving occasional glimpses into aspects of the contemporary
scientific and human environment which have influenced "mathematics at
Harvard". Profound thanks are due to Clark Elliott and I. Bernard Cohen
at Harvard, and to Uta Merzbach and the other editors of this volume,
for many valuable suggestions and criticisms of earlier drafts.
1. BENJAMIN PEIRCE
In 1836, mathematics at Harvard was about to undergo a major transition.
For a century, all Harvard College students had been introduced to the
infinitesimal calculus and the elements of physics and astronomy by
the Hollis Professor of Mathematics and Natural Philosophy. Since 1806,
the Hollis Professor had been John Farrar (1779-1853), who had accepted
the job after it had been declined by Nathaniel Bowditch (1773-1838),'
a native of Salem.
Bowditch's connection with mathematics at Harvard was truly unique.
Having had to leave school at the age of 10 to help his father as a
cooper, Bowditch had been almost entirely self-educated. After teaching
himself Latin and reading Newton's Principia, he sailed on ships as
supercargo on four round trips to the East Indies. He then published
the most widely used book on the science of navigation, The New American
Practical Navigator, before becoming an executive actuary for a series
of insurance companies.
After awarding Bowditch an honorary M.A. in 1802, Harvard offered him
the Hollis Professorship of Mathematics and Natural Philosophy in 1806.
Imagine Harvard offering a professorship today to someone who had never
gone to high school or college! Though greatly honored, Bowditch declined
because he could not raise his growing family properly on the salary
offered ($ 1200/yr.), and remained an actuarial executive. A prominent
member of Boston's American Academy of Arts and Sciences, he did however
stay active in Harvard affairs.
In American scientific circles, Bowditch became most famous through his
translation of Laplace's Mecanique Celeste, with copious notes explaining
many sketchy derivations in the original. He did most of the work on
this about the same time that Robert Adrain showed that Laplace's value
of 1/338 for the earth's eccentricity (b -a)/a should be 1/316. Bowditch
decided (correctly) that it is more nearly l/300.
Bowditch's scientific interests were shared by a much younger Salem
native, Benjamin Peirce (1809-1880). Peirce had become friendly at
the Salem Private Grammar School with Nathaniel's son, Henry Ingersoll
Bowditch, who acquainted his father with Benjamin's skill in and love
for mathematics, and Peirce is reputed to have discussed mathematics
and its applications with Nathaniel Bowditch from his boyhood. In 1823,
Benjamin's father became Harvard's librarian; in 1826, Bowditch became
a member of the Harvard Corporation (its governing board). By then,
the Peirce and Bowditch sons were fellow students in Harvard College,
and their families had moved to Boston.
There Peirce's chief mentor was Farrar. For more than 20 years, Farrar
had been steadily improving the quality of instruction in mathematics,
physics, and astronomy by making translations of outstanding 18th century
French textbooks available under the title "Natural Philosophy for the
Students at Cambridge in New England". Actually, his own undergraduate
thesis of 1803 had contained a calculation of the solar eclipse which
would be visible in New England in 1814. Although very few Harvard
seniors could do this today, it was not an unusual feat at that time.
By 1829, Nathaniel Bowditch had become affluent enough to undertake the
final editing and publication of Laplace's Mecanique Celeste at his own
expense, and young Peirce was enthusiastically assisting him in this
task. Peirce must have found it even more exhilarating to participate
in criticizing Laplace's masterpiece than to predict a future eclipse!
In 1831, Peirce was made tutor in mathematics at Harvard College,
and in 1833 he was appointed University Professor of Mathematics and
Natural Philosophy. As a result, two of the nine members of the 1836
Harvard College faculty bore almost identical titles. In the same year,
he married Sara Mills of Northampton, whose father Elijah Hunt Mills had
been a U.S. senator [DAB]. They had five children, of whom two would
have an important influence on mathematics at Harvard, as we shall see.
By 1835, still only 26, Peirce had authored seven booklets of Harvard
course notes, ranging from "plane geometry" to "mechanics and astronomy"
Moreover Farrar, whose health was failing, had engaged another able
recent student, Joseph Lovering (1813-92), to share the teaching load as
instructor. In 1836, Farrar resigned because of poor health, and Lovering
succeeded to his professorship two years later. For the next 44 years,
Benjamin Peirce and Joseph Lovering would cooperate as Harvard's senior
professors of mathematics, astronomy, and physics.
Nathaniel Bowditch died in 1838, and his place on the Harvard Corporation
was taken by John A. Lowell (1798-1881), a wealthy textile industry
financier. By coincidence, in 1836 (Harvard's bicentennial year) his
cousin John Lowell, Jr. had left $250,000 to endow a series of public
lectures, with John A. as sole trustee of the Lowell Institute which
would pay for them. Lowell (A.B. 1815) must have also studied with
Farrar. Moreover by another
coincidence, having entered Harvard at 13, John A. Lowell had lived
as a freshman in the house of President Kirkland, whose resignation in
1829 after a deficit of $4,000 in a budget of $30,000, student unrest,
and a slight stroke had been due to pressure from Bowditch!6 As a final
coincidence, by 1836 Lowell's tutor Edward Everett had become governor
of Massachusetts, in which capacity he inaugurated the Lowell Lectures
in 1840.
In 1842, Peirce was named Perkins Professor of Astronomy and Mathematics,
a newly endowed professorship. By that time, qualified Harvard students
devoted two years of study to Peirce's book Curves and Functions,
for which he had prepared notes. Ambitious seniors might progress to
Poisson's Mecanique Analytique, which he would replace in 1855 with
his own textbook, A System of Analytical Mechanics. This was fittingly
dedicated to "My master in science, NATHANIEL BOWDITCH, the father of
American Geometry".
Meanwhile, Lovering was becoming famous locally as a teacher of physics
and a scholar. His course on "electricity and magnetism" had advanced
well beyond Farrar, and over the years, he would give no less than nine
series of Lowell Lectures [NAS #6, 327-44].
In 1842-3, Peirce and Lovering also founded a quarterly journal called
Cambridge Miscellany of Mathematics, Physics, and Astronomy, but it
did not attract enough subscribers to continue after four issues. They
also taught Thomas Hill '43, who was awarded the Scott Medal by the
Franklin Institute for an astronomical instrument he invented as
an undergraduate. He later wrote two mathematical textbooks while a
clergyman, and became Harvard's president from 1862 to 1868 [DAB 20,
547-8].
2. PEIRCE REACHES OUT
During his lifetime Peirce was without question the leading American
mathematical astronomer. In 1844, perhaps partly stimulated by a brilliant
comet in 1843, "The Harvard Observatory was founded on its present site
by a public subscription, filled largely by the merchant shipowners of
Boston" [Mor, p. 292]. For decades, William C. Bond (1789-1859) had
been advising Harvard about observational astronomy, and he may well
have helped Hill with his instrument. In any event, Bond finally became
the salaried director of the new Observatory, where he was succeeded
by his son George (Harvard '45). Peirce and Lovering both collaborated
effectively with William Bond in interpreting data.' Pursuing further
the methods he had learned from Laplace's Mecanique Celeste, Peirce also
analyzed critically Leverrier's successful prediction of the new planet
Neptune, first observed in 1846.
Meanwhile, our government was beginning to play an important role in
promoting science. At about this time, the Secretary of the Navy appointed
Lieutenant (later Admiral) Charles Henry Davis head of the Nautical
Almanac Office. Like Peirce, Davis had gone to Harvard and married a
daughter of Senator Mills. Because of Peirce and the Harvard Observatory,
he decided to locate the Nautical Almanac Office in Cambridge, where
Peirce served from 1849 to 1867 as Consulting Astronomer, supervising for
several years the preparation (in Cambridge) of the American Ephemeris
and Nautical Almanac, our main government publication on astronomy. This
activity attracted to Cambridge such outstanding experts in celestial
mechanics as Simon Newcomb (1835-1909) and G.W. Hill (1838-1914), who
later became the third and fourth presidents of the American Mathematical
Society.' Others attracted there were John M. van Vieck, an early AMS
vice-president, and John D. Runkle, founder of a short-lived but important
Mathematical Monthly.
Similarly, Alexander Dallas Bache [DAB 1, 461-2], after getting the
Smithsonian Institution organized in 1846 with Joseph Henry as its first
head [EB 20, 698-700], became head of the U.S. Coast Survey. Bache
appointed Peirce's former student B. A. Gould'ˇ as head of the Coast
Survey's longitude office and Gould, who was Harvard president Josiah
Quincy's son-in-law, decided to locate his headquarters at the Harvard
Observatory and also use Peirce as scientific adviser. Peirce acted in
this capacity from 1852 to 1874, aided by Lovering's selfless cooperation,
succeeding Bache as superintendent of the Coast Survey for the last
eight of these years." In both of these roles, Peirce showed that he
could not only apply mathematics very effectively (see [Pei, p. 12]);
he was also a creative organizer and persuasive promoter.
Indeed, from the 1840s on, Peirce was reaching out in many directions.
Thus, he became president of the newly founded American Association
for the Advancement of Science in 1853. He was also active locally in
Harvard's Lawrence Scientific School (LSS) during its early years.
The Lawrence Scientific School. The LSS is best understood as an early
attempt to promote graduate education in pure and applied science,
including mathematics. Established when Harvard's president was Edward
Everett, John A. Lowell's former tutor, and its treasurer was Lowell's
then affluent business associate Samuel Eliot, the LSS was named
for Abbott Lawrence, another New England textile magnate who had been
persuaded to give $50,000 to make its establishment possible in 1847. By
then, the cloudy academic concepts of "natural philosophy" and "natural
history" were becoming articulated into more clearly defined "sciences"
such as astronomy, geology, physics, chemistry, zoology, and botany.
Correspondingly, the LSS was broadly conceived as a center where college
graduates and other qualified aspirants could receive advanced instruction
in these sciences and engineering. Its early scientifically minded
graduates included not only several notable "applied" mathematicians
such as Newcomb and Runkle, but also the classmates Edward Pickering and
John Trowbridge. John Trowbridge would later join and ultimately replace
Joseph Lovering as Harvard's chief physicist, while Pickering would become
director of the Harvard Observatory and an eminent astrophysicist. Most
important for mathematics at Harvard, Trowbridge would have as his
"first research student in magnetism" B. 0. Peirce.
The leading "pure" scientists on the L.SS faculty were Peirce, the
botanist Asa Gray, the anatomists Jeifries Wyman, and the German-educated
Swiss naturalist Louis Agassiz (1807- 73), already internationally famous
when he joined the faculty of the LSS as professor of zoology and geology
in 1847.
The original broad conception of the Lawrence Scientific School was best
exemplified by Agassiz. Like Benjamin Peirce, Agassiz expected students
to think for themselves; but unlike Peirce, he was a brilliant lecturer,
who soon "stole the show" from Gray and Peirce at the LSS. Although he
did not influence mathematics directly, we shall see that three of his
students indirectly influenced the mathematical sciences at Harvard: his
son Alexander (1835-1912); the palaeontologist and geologist Nathaniel
Shaler; and the eminent psychologist and philosopher William James.
The Lazzaroni. Like Alexander Bache, Louis Agassiz and Benjamin Peirce
were members of an influential group of eminent American scientists who,
calling themselves the "Lazzaroni", tried to promote research. However,
very few tuition-paying LSS students had research ambitions. The majority
of them studied civil engineering under Henry Lawrence Eustis ('38), who
had taught at West Point [Mor, p. 414] before coming to Harvard. Most
of the rest studied chemistry, until 1863 under Eben Horsford, whose
very "applied" interests made it appropriate to assign his Rumford
Professorship to the LSS.
Peirce's students. Peirce had many outstanding students. Among them may be
included Thomas Hill, B. A. Gould, John Runkle, and Simon Newcomb. Partly
through his roles in the Coast Survey, the Harvard Observatory, and the
Nautical Almanac, he was responsible for making Harvard our nation's
leading research center in the mathematical sciences in the years
1845- 65.
Although most of his students "apprehended imperfectly what Professor
Peirce was saying", he also was "a very inspiring and stimulating teacher"
for those eager to learn. We know this from the vivid account of his
teaching style [Pei, pp. 1-4] written by Charles William Eliot (1834-
1926), seventy years after taking (from 1849 to 1853) the courses in
mathematics and physics taught by Peirce and Lovering. The same courses
were also taken by Eliot's classmate, Benjamin's eldest son James
Mills Peirce (1834-1906). After graduating, these two classmates taught
mathematics together from 1854 to 1858, collaborating in a daring and
timely educational reform. At the time, Harvard students were examined
orally by state- appointed overseers whose duty it was to make sure that
standards were being maintained. "Offended by the dubious expertness
and obvious absenteeism of the Overseers, the young tutors [Eliot and
J. M. Peircel obtained permission to substitute written examinations,
which they graded themselves" [HH, p. 15].
After 1858, J. M. Peirce tried his hand at the ministry, while Eliot
increasingly concentrated his efforts on chemistry (his favorite subject)
and university administration. Josiah Parsons Cooke ('48), the largely
self-taught Erving Professor of Chemistry, had shared chemicals with Eliot
when the latter was still an undergraduate [HH, p. 11]. Then in 1858,
these two friends successfully proposed a new course in chemistry [HH,
p. 16], in which students performed laboratory exercises "probably for
the first time". Eliot demonstrated remarkable administrative skill;
at one point, he was acting dean of the LSS and in charge of the
chemistry laboratory. In these roles, he proposed a thorough revision
in the program for the S.B. in chemistry, based on a "firm grounding in
chemical and mathematical fundamentals". He then served with Dean Eustis
of that School and Louis Agassiz on a committee appointed to revamp
the school's curriculum as a whole [HH, pp. 23-251. However Eliot's
zeal for order and discipline antagonized the more informal Agassiz,
and Eliot's reformist ideas were rejected. After losing out to the more
research-oriented "Lazzarone" Wolcott Gibbs in the competition for the
Rumford professorship [HH, pp. 25-27], in spite of J. A. Lowell's support
of his candidacy, Eliot left Harvard (with his family) for two years of
study in France and Germany.
MIT. Academic job opportunities in "applied" science in the Boston area
were improved by the founding there of the Massachusetts Institute of
Technology (MIT). Since the mid-1840s, Henry and then his brother William
Rogers, "one of those accomplished general scientists who matured before
the age of specialization", had been lobbying with J. A. Lowell and others
for the benefits of polytechnical education, and in 1862 William Rogers
became its first president. Peirce's student John Runkle joined its new
faculty in 1865 as professor of mathematics and analytical mechanics,
and later became the second president of MIT. '2 Recognizing Eliot's many
skills, Rogers soon also invited him to go to MIT, and Eliot accepted
[HH, pp. 34-7]. There Eliot wrote with F. H. Storer, an LSS graduate
and his earlier collaborator and hiking companion, a landmark Manual of
Inorganic Chemistry.
3. PEIRcE's GOLDEN YEARS
When the Civil War broke out in 1861, J. M. Peirce was a minister in
Charleston, South Carolina. Benjamin promptly had James made an assistant
professor of mathematics at Harvard, to help him carry the teaching
load. Benjamin's brilliant but undisciplined younger brother Charles
Sanders Peirce (1839-1914), after being tutored at home, had graduated
from Harvard two years earlier without distinction. Nevertheless,
Benjamin secured for him a position in the U.S. Coast Survey exempting
Charles from military service. As we shall see, this benign nepotism
proved to be very fruitful.
A year later, his former student Thomas Hill became Harvard's president,
having briefly (and reluctantly) been president of Antioch College. At
Harvard, Hill promoted the "elective system", encouraging students
to decide
between various courses of study. He also initiated series of university
lectures which, like the LSS, constituted a step toward the provision
of graduate instruction. Peirce participated in this effort most years
during the 1860s, lecturing on abstruse mathematics with religious fervor.
From the 1850s on, Peirce had largely freed himself from the drudgery
of teaching algebra, geometry, and trigonometry. Moreover, whereas
Lovering's courses continued to be required, Peirce's were all optional
(electives), and were taken by relatively few students. Peirce was
frequently "lecturing on his favorite subject, Hamilton's new calculus of
quaternions" [Pei, p. 6], to W. E. Byerly ('71) among others. In 1873
Byerly was persuaded by Peirce to write a doctoral thesis on "The Heat
of the Sun". In it, he calculated the total energy of the sun, under
the assumption (common at the time) that this was gravitational. 13
The calculation only required using the calculus and elementary
thermodynamics. Nevertheless, Byerly became Harvard's first Ph.D.,
and a pillar of Harvard's teaching staff until his retirement in 1912!
Benjamin Peirce had long been interested in Hamilton's quaternions a ±
b L ± ci ± dic; moreover Chapter X of his Analytical Mechanics (1855)
contained a masterful chapter on 'functional determinants' of n x
n matrices."
During the Civil War, Benjamin and Charles became interested in
generalizations of quaternions to linear associative algebras. After
lecturing several times to his fellow members of the National Academy
of Sciences on this subject, Benjamin published his main results in
1870 in a privately printed paper.'5 This contained the now classic
"Peirce decomposition"
x - exe + ex(l-e) -f- (l-e)xe ± (l-e)x(1-e)
with respect to any "idenipotent" e satisfying ee = e.
An 1881 sequel, published posthumously by J. J. Sylvester in the newly
founded American Journal of Mathematics (4, 97-229), contained numerous
addenda by Charles. Most important was Appendix III, where Charles
proved that the only division algebras of finite order over the real
field R are R itself, the complex field C, and the real quaternions.
This very fundamental theorem had been proved just three years earlier
by the German mathematician Frobenius. '
Charles Peirce also worked with his father in improving the scientific
instrumentation of the U.S. "Coast Survey", which by 1880 was surveying
the entire United States! But most relevant to mathematics at Harvard, and
most distinguished, was his unpaid role as a logician and philosopher. An
active member of the Metaphysical Club presided over by Chauncey Wright
('52), another of Benjamin's ex-students who made his living as a computer
for the Nautical Almanac, Charles gave a brilliant talk there proposing
a new philosophical doctrine of "pragmatism". He also published a series
of highly original papers on the then new algebra of relations. Although
surely "Imperfectly apprehended" by most of his contemporaries, these
contributions earned him election in 1877 to the National Academy of
Sciences, which had been founded 12 years earlier by the Lazzaroni.
In his last years, increasingly absorbed with quaternions, Benjamin
Peirce's unique teaching personality influenced other notable
students. These included Harvard's future president, Abbott Lawrence
Lowell, and his brilliant brother, the astronomer Percival. They were
grandsons of both Abbott Lawrence and John A. Lowell! To appreciate
the situation, one must realize that the two grandfathers of the Lowell
brothers were John A. Lowell, the trustee of the Lowell Institute, and
Abbott Lawrence, for whom Harvard's Lawrence Scientific School (see 2)
had been named. From 1869 to 1933, the presidents of Harvard would be
former students of Benjamin Peirce!
Two others were B. 0. Peirce, a distant cousin of Benjamin's who
would succeed Lovering (and Farrar) as Hollis Professor of Mathematics
and Natural Philosophy; and Arnold Chace, later chancellor of Brown
University. Peirce's lectures inspired both A. L. Lowell and Chace to
publish papers in which "quaternions" (now called vectors) were applied
to geometry. Moreover both men would describe in [Pei] fifty years
later, as would Byerly, how Peirce influenced their thinking. Still
others were W. E. Story, who went on to get a Ph.D. at Leipzig,
and W. I. Stringham (see [S-pG]). Story became president of the 1893
International Mathematical Congress in Chicago, and himself supervised
12 doctoral theses including that of Solomon Lefschetz [CMA, p. 2011.
Benjamin Peirce's funeral must have been a very impressive affair. His
pallbearers fittingly included Harvard's president C. W. Eliot and
ex-president Thomas Hill; Simon Newcomb, J. J. Sylvester, and Joseph
Lovering; his famous fellow students and lifelong medical friends Henry
Bowditch and Oliver Wendell Holmes (the "Autocrat of the Breakfast
Table"); and the new superintendent of the Coast Survey, C. P. Patterson.
4. ELIOT TAKES HOLD
When Thomas Hill resigned from the presidency of Harvard in 1868, the
Corporation (with J. A. Lowell in the lead) recommended that Eliot
be his successor, and the overseers were persuaded to accept their
nomination in 1869. After becoming president, Eliot immediately tried
(unsuccessfully) to merge MIT with the LSS [Mor, p. 4181.18 At the same
time, he tried to build up the series of university lectures, inaugurated
by his predecessor, into a viable graduate school. It is interesting
that for the course in philosophy his choice of lecturers included Ralph
Waldo Emerson and Charles Sanders Peirce; in 1870-71 thirty-five courses
of university lectures were offered; but the scheme "failed hopelessly"
[Mor, p. 453].
To remedy the situation, Eliot then created a graduate department, with
his classmate James Mills Peirce as secretary of its guiding academic
council. This was authorized to give M.A. and Ph.D. degrees, such as those
given to Byerly and Trowbridge. At the same time, Eliot transferred his
former rival Wolcott Gibbs from being dean of the LSS to the physics
department, and chemistry from the LSS to Harvard College, where his
former mentor Josiah Parsons Cooke was in charge. A year earlier, Eliot
had proposed an elementary course in chemistry, to be taught partly in
the laboratory. This gradually became immensely popular, partly because
of its emphasis on the chemistry of such familiar phenomena as photography
[Mor, p. 260].
By 1886, all members of the LSS scientific faculty had transferred to
Harvard College. Moreover undergraduates wanting to study engineering had
no incentive for enrolling in the LSS rather than Harvard College. As a
result of this, and competition from MIT and other institutions, there
was a steady decline in the LSS enrollment, until only 14 students
enrolled in 1886! Its four-year programme in "mathematics, physics,
and astronomy", inherited from the days of Benjamin Peirce, had had
no takers at all for many years, and was wisely replaced in 1888 by a
programme in electrical engineering.
Meanwhile, the new graduate department itself was struggling. After
1876, the Johns Hopkins University attracted many of the best graduate
students. Two of them would later be prominent members of the Harvard
faculty: Edwin Hall in physics and Josiah Royce in philosophy. During
its entire lifetime (1872-90), the graduate department awarded only
five doctorates in mathematics, including those of Byerly and F. N. Cole
(the latter earned in Germany, see ¤5).
To remedy the situation, Eliot made a second administrative reorganization
in 1890. From it, the graduate department emerged as the graduate school;
the LSS engineering faculty joined the Harvard College faculty in a new
Faculty of Arts and Sciences. J. M. Peirce's title was changed from
secretary of the graduate department to dean of the Graduate School of
Arts and Sciences, and the economist Charles F. Dunbar ('51) was made
dean of the Faculty of Arts and Sciences.
Additional programmes of study were introduced into the LSS, and Nathaniel
Shaler made its new dean. This was a brilliant choice: the enrollment in
the LSS increased to over 500 by 1900, and Shaler's own Geology 4 became
one of Harvard College's most popular courses. Shaler was also allowed
to assume management of the mining companies of his friend and neighbor,
the aging inventor and mining tycoon Gordon McKay [HH, pp. 213-15],
Shaler persuaded McKay in 1903 to bequeath his fortune to the school,
where it supports the bulk of Harvard's program in "applied" mathematics
to this day!
Mention should also be made of the appointment in 1902 of the
selfeducated, British-born scientist Arthur E. Kennelly (1851-1939). Joint
discoverer of the upper altitude "Kennelly- Heaviside layer" which
reflects radio waves, Kennelly had been Edison's assistant for 13
years and president of the American Institute of Electric Engineers. In
his thoughtful biography of Kennelly [NAS 22: 83-1191, Vannevar Bush
describes how Kennelly's career spanned the entire development of
electrical engineering to 1939. Bush and my father [GDB III, 734-8]
both emphasize that Kennelly revolutionized the mathematical theory of
alternating current (a.c.) circuits by utilizing the complex exponential
function. Curiously, this major application is still rarely explained
in mathematics courses in our country, at Harvard or elsewhere!
For further information about the changes I have outlined, and
other interpretations of the conflicting philosophies of scientific
education which motivated them, I refer you to [HH] and especially
[Love]. The latter document was written by James Lee Love, who taught
mathematics under the auspices of the Lawrence Scientific School from
1890 to 1906, when the LSS was renamed the Graduate School of Applied
Science. Officially affiliated with Harvard until 1911, Love returned
to Burlington, North Carolina, in 1918 to become president of the
Gastonia Cotton Manufacturing Company. Reorganized as Burlington Mills,
this became one of our largest textile companies. During these years,
Love donated $50,000 to the William Byerly Book Fund.
5. A DECADE OF TRANSITION
When Benjamin Peirce died, his son James had been ably assisting him
in teaching Harvard undergraduates for more than 20 years. Byerly had
joined them in 1876. At the time, B. 0. Peirce was still studying
physics and mathematics with John Trowbridge and Benjamin Peirce, but
he became an instructor in mathematics in 1881, assistant professor of
mathematics and physics in 1884, and Hollis Professor of Mathematics and
Natural Philosophy (succeeding Lovering) in 1888. In the decade following
Benjamin Peirce's death, the triumvirate consisting of J. M. Peirce
(1833-1906), W. E. Byerly (1850-1934), and B. 0. Peirce (1855-1913)
would be Harvard's principal mathematics teachers.
As a mathematician, J. M. Peirce has been aptly described as an
"understudy" to his more creative father [Mor, p. 249]. However, "to
no one, excepting always President Eliot, [was] the Graduate School so
indebted" for "the promotion of graduate instruction" [Mor, p. 455].
Moreover his teaching, unlike that of his father, seems to have been
popular and easily comprehended. In the 1880s, he and Byerly began giving
in alternate years Harvard's first higher geometry course (Mathematics 3)
with the title "Modern methods in geometry - determinants". Otherwise,
his advanced teaching covered mainly topics of algebra and geometry in
which Benjamin and C. S. Peirce had done research, such as "quaternions",
"linear associative algebra", and "the algebra of logic".
While James Peirce was administering graduate degrees at Harvard as
secretary of the academic council, Byerly was cooperating most effectively
in making mathematics courses better understood by undergraduates. His
Differential Calculus (1879), his Integral Calculus (1881), and his
revised and abridged edition of Chauvenet's Geometry (1887), presumably
the text for Math. 3, were widely adopted in other American colleges
and universities.19
In 1883-4, Byerly and B. 0. Peirce introduced a truly innovative course
in mathematical physics (or "applied mathematics") which has been taught
at Harvard in suitably modified form ever since. Half of this course
(taught by Byerly) dealt with the expansion of "arbitrary functions"
in Fourier Series and Spherical Harmonics, this last being the title
of a book he wrote in 1893. The other half treated potential theory,
and Peirce wrote for it a book, Newtonian Potential Function, published
in three editions (1884, 1893, 1902). Like Byerly's other books, they
were among the most influential and advanced American texts of their time.
B. 0. Peirce was an able and scholarly, if traditional, mathematical
physicist. A brilliant undergraduate physics major, his "masterly"
later physical research was mostly empirical. Although it was highly
respected for its thoroughness, and Peirce became president of the
American Physical Society in 1913, it lay in "the unexciting fields of
magnetism and the thermal conduction of non-metallic substances". His
main mathematical legacy consisted in his text for Mathematics 10,
and his Table of Integrals..., originally written as a supplement to
Byerly's Integral Calculus. This was still being used at Harvard when I
was an undergraduate, but such tables may soon be superseded by packages
of carefully written, debugged, and documented computer programs like
Macsyma.
In short, Harvard's three professors of mathematics regarded their
profession as that of teaching reasonably advanced mathematics in an
understandable way. Their success in this can be judged by the quality
of their students, who included M. W. Haskell, Arthur Gordon Webster,
who became president of the American Physical Society in 1903, Frank N.
Cole, W. F. Osgood, and Maxime Bocher. In 1888, when the AMS was
founded, two of them had inherited the titles of Benjamin Peirce and
Lovering; only Byerly had the simple title "professor of mathematics".
C. S. Peirce. When his father died, C. S. Peirce (1839-1914) was at the
zenith of his professional career. From 1879 to 1884, he was a lecturer
at Johns Hopkins as well as a well- paid and highly respected employee
of the Coast Survey (cf. [SMA, pp. 13-20]). While there, he discovered
the fundamental connection between Boolean algebra and what are today
called "partially ordered sets" (cf. American J. Math. 3 (1880), 15-57),
thus foreshadowing the "Dualgruppen" of Dedekind ("VerbŠnde" or lattices
in today's terminology). Unfortunately, in describing this connection,
he erroneously claimed that the distributive law a(b V c) ab V ac
necessarily relates least upper bounds x V y and greatest lower bounds xy.
Indeed, the 1880s were a disastrous decade for C. S. Peirce. His lectures
at the Johns Hopkins Graduate School were not popular; his personality
was eccentric; and his appointment there was not renewed after Sylvester
returned to England. He also lost his job with the Coast Survey soon after
1890. Although he continued to influence philosophy at Harvard (see ¤8),
he never again held a job with any kind of tenure. An early member of the
New York Mathematical Society, his brilliant turns of speech continued
to enliven its meetings [CMA, pp. 15-16], but he was not taken seriously.
6. Osgood AND BOCHER
By 1888, when the American Mathematical Society (AMS) was founded (in
New York), a new era in mathematics at Harvard was dawning. Frank Nelson
Cole (Harvard '82) had returned three years earlier after "two years
under Klein at Leipzig" [Arc, p. 100). "Aglow with enthusiasm, he gave
courses in modern higher algebra, and in the theory of functions of a
complex variable, geometrically treated, as in Klein's famous course of
lectures at Leipzig." His "truly inspiring" lectures were attended by two
undergraduates, W. F. Osgood (1864-1943) and Maxime Bocher (1867-1918),
"as well as by nearly all members of the department," including Professors
J. M. Peirce, B. 0. Peirce, and W. E. Byerly.
After graduating, Osgood and Bocher followed Cole's example and went to
Germany to study with Felix Klein, who had by then moved to Goettingen.23
After earning Ph.D. degrees (Osgood in 1890, Bocher with especial
distinction in 1891), both men joined the expanding Harvard staff
as instructors for three years. Inspired by the example of Gottingen
under Klein, they spearheaded a revolution in mathematics at Harvard,
where they continued to serve as assistant professors for another decade
before becoming full professors (Osgood in 1903, Bocher in 1904). All
this took place in the heyday of the Eliot regime, under the benevolent
but mathematically nominal leadership of the two Peirces and Byerly.
The most conspicuous feature of the revolution resulting from the
appointments of Bocher and Osgood was a sudden increase in research
activity. By 1900, Osgood had published 21 papers (six in German), while
Bocher had published 30 in addition to a book 'On the series expansions
of potential theory', and a survey article on "Boundary value problems of
ordinary differential equations" for Klein's burgeoning Enzyklopedie der
Mathematischen Wissenschaften, both in German. Moreover Bocher and Yale's
James Pierpont had given the first AMS Colloquium Lectures in 1896, to
an audience of 13 while Osgood and A. G. Webster (a Lawrence Scientific
School alumnus) had given the second, in 1898. Similar revolutions had
taken place in the 1890s at other leading American universities. Most
important of these was at the newly founded University of Chicago, where
the chairman of its mathematics department, E. H. Moore, was inspiring a
series of Ph.D. candidates [LAM, 3]. Under the leadership of H. B. Fine,
who had been stimulated by Sylvester's student G. B. Halsted, Princeton
would blossom somewhat later. Meanwhile, Cole had become a professor at
Columbia, secretary of the AMS, and editor of its Bulletin (cf. [Arc,
Ch. V]). The Cole prize in algebra is named for him.
Thus it was most appropriate for Osgood, Bocher, and Pierpont to cooperate
with E. H. Moore (1862-1932) of Chicago in making the promotion of
mathematical research the central concern of the AMS. Feeling "the great
need of a journal in which original investigations might be published"
[Arc, p. 56], these men succeeded in establishing the Transactions
Amer. Math. Soc. [Arc, Ch. V]. From 1900 on, this new periodical
supplemented the American Journal of Mathematics, complete control
over which Simon Newcomb was unwilling to relinquish. The Annals of
Mathematics was meanwhile being published at Harvard from 1899 to 1911,
with Bocher as chief editor. Primarily designed for "graduate students
who are not yet in a position to read the more technical journals",
this also "contained some articles ... suitable for undergraduates."
Harvard continued to educate many mathematically talented students
during the years 1890- 1905, including most notably J. L. Coolidge,
E. V. Huntington, and E. B. Wilson, all for four years; and for shorter
periods E.R.
Hedrick ('97-'99), Oswald Veblen ('99-'00), and G. D. Birkhoff ('03-'05)
At the same time, there was a great improvement in the quality and
quantity of advanced courses designed "primarily for graduate students",
but taken also by a few outstanding undergraduates. By 1905, the
tradition of Benjamin Peirce had finally been supplanted by new courses
stressing new concepts, mostly imported from Germany and Paris; in 1906
J. M. Peirce died.
By that time, Harvard's graduate enrollment had increased mightily. From
28 students in 1872, when Eliot had appointed J. M. Peirce secretary of
his new "graduate department", it had grown to 250 when Peirce resigned
as dean of Harvard's "graduate school", to become dean of the entire
Faculty of Arts and Sciences. A key transition had occurred in 1890,
when the graduate "department" was renamed a "school", and the Harvard
catalog first divided all courses into three tiers: "primarily for
undergraduates", "for undergraduates and graduates", and "primarily for
graduates", as it still does.
The Peirces and Byerly had explained to their students many of the meth-
ods of Fourier, Poisson, Dirichiet, Hamilton, and Thomson and Tait's Prin-
ciples of Natural Philosophy (1867). However, they had largely ignored the
advances in rigor due to Cauchy, Riemann, and Weierstass. For example,
Byerly's Integral Calculus of 1881 still defined a definite integral
vaguely as "the limit of a sum of infinitesimals", although Cauchy-
Moigno's Lecons de Calcul Integral had already defined integrals as limits
of sums E f(x).x, and sketched a proof of the fundamental theorem of
the calculus in 1844, while in 1883 volume 2 of Jordan's Cours d'Analyse
would even define uniform continuity.
The key graduate course (Mathematics 13) on functions of a
complex variable became modernized gradually. Under J. M. Peirce,
it had been a modest course based on Briot and Bouquet's Fonctions
Elliptiques. In 1891-92, Osgood followed this with a more specialized
course on elliptic functions as such, and the next year with another
treating abelian integrals, while Bocher gave a course on "functions
defined by differential equations", in the spirit of Poincare. Then,
from 1893 to 1899, Bocher developed Mathematics 13 into the basic
full course on complex analysis that it would remain for the next
half-century, introducing students to many ideas of Cauchy, Riemann,
and Weierstrass. Then, beginning in 1895, he and Osgood supplemented
Mathematics 13 with a half-course on "infinite series and products"
(Mathematics 12) which treated uniform convergence. By 1896, Osgood had
written a pamphlet Introduction to Infinite Series covering its contents.
In his moving account of "The life and services of Maxime Bocher" (Bull.
Amer. Math. Soc. 25 (1919), 337-50) Osgood has described Bocher's lucid
lecture style, and how much BOcher contributed to his own masterly
treatise Funktionentheorie (1907), which became the standard advanced
text on the subject on both sides of the Atlantic. (Weaker souls, whose
mathematical sophistication or German was not up to this level, could
settle for GoursatHedrick.) Osgood's other authoritative articles on
complex function theory, written for the Enzyklopadie der Mat hematischen
Wissenschaften and as Colloquium Lectures,24 established him as America's
leading figure in classical complex analysis.
On a more elementary level, Osgood wrote several widely used textbooks
beginning with an Introduction to Infinite Series (1897). Ten years later,
his Differential and Integral Calculus appeared, with acknowledgement of
its debt to Professors B. 0. Peirce and Byerly. There one finds stated,
for the first time in a Harvard textbook, a (partial) "fundamental theorem
of the calculus". These were followed by his Plane and Solid Analytic
Geometry with W. C. Graustein (1921), his Introduction to the Calculus
(1922), and his Advanced Calculus (1925), the last three of which were
standard fare for Harvard undergraduates until around 1940. Osgood also
served for many years on national and international commissions for the
teaching of mathematics.
Less systematic than Osgood, BOcher was more inspiring as a lecturer
and thesis adviser. As an analyst, his main work concerned expansions
in SturmLiouville series (including Fourier series) associated with the
partial differential equations of mathematical physics (after "separating
variables"). His Introduction to the Study of integral Equations
(1909, 1914) and his Lecons sur les MŽthodes de Sturm ... (1913-14) were
influential pioneer monographs. Like Bocher's papers which preceded them,
they established clearly and rigorously by classical methods24dl precise
interpretations of many basic formulas concerned with potential theory
and orthogonal expansions (Mathematics 1 Oa and Mathematics 1 Ob).
Several of Bother's Ph.D. students had very distinguished careers, most
notable among them being G. C. Evans, who in the 1930s would pilot the
mathematics department of the University of California at Berkeley to
the level of preeminence that it has maintained ever since. Others were
D. R. Curtiss (Northwestern University), Tomlinson Fort (Georgia Tech),
and L. R. Ford (Rice Institute).
By 1900, the presence of Osgood, Bocher, Byerly, and B. 0. Peirce had
made Harvard very strong in analysis. Moreover this strength was
increased in 1898 by the addition to its faculty of Charles Leonard
Bouton (18601922), who had just written a Ph.D. thesis with Sophus Lie.
25 However, it was clear that advanced instruction in other areas of
mathematics, mostly given before 1900 by J. M. Peirce, needed to be
rejuvenated by new ideas.
The first major step in building up a balanced curriculum was taken
by Bocher. In the 1890s, he had given with Byerly in alternate years
Harvard's first higher geometry course (Mathematics 3) with the title
"Modern methods in geometry - determinants". Then, in 1902- 3, he
inaugurated a new version of Mathematics 3, entitled "Modern geometry
and modern algebra", with a very different outline leading up to "the
fundamental conceptions in the theory of invariants." The algebraic
component of this course matured into Bocher's book, Introduction
to Higher Algebra (1907), in which ¤26 on "sets, systems, and groups"
expresses modern algebraic ideas. This book would introduce a generation
of American students to linear algebra, polynomial algebra, and the theory
of elementary divisors. But to build higher courses on this foundation,
without losing strength in analysis, would require new faculty members.
7. COOLIDGE AND HUNTINGTON
Harvard's course offerings in higher geometry were revitalized in the
first decades of this century by the addition to its faculty of Julian
Lowell Coolidge (1873-1954). After graduating from Harvard (summa cum
laude) and Balliol College in Oxford, Coolidge taught for three years
at the Groton School before returning to Harvard. At Groton, he began
a lifelong friendship with Franklin Roosevelt, which illustrates his
concern with the human side of education (see ¤ 15). Indeed, somewhat like
his great-great-- grandfather Thomas Jefferson, our "most mathematical
president", Coolidge was unusually many-sided.26
From 1900 on, Coolidge gave in rotation a series of lively and informative
graduate courses on such topics as the geometry of position, non-Euclidean
geometry, algebraic plane curves, and line geometry. After he had
spent two years (1902-4) in Europe and written a Ph.D. thesis under
the guidance of Eduard Study and Corrado Segre, these courses became
more authoritative. In time, the contents of four of them would be
published as books on NonEuclidean Geometry (1909), The Circle and the
Sphere (1916), The Geometry of the Complex Domain (1924), and Algebraic
Plane Curves (1931).
In 1909-10, Coolidge also initiated a half-course on probability
(Mathematics 9), whose contents were expanded into his readable and
timely Introduction to Mathematical Probability (1925), soon translated
into German (Teubner, 1927). Coolidge's informal and lively expository
style is well illustrated by his 1909 paper on "The Gambler's Ruin".27
This concludes by reminding the reader of "the disagreeable effect on
most of humanity of anything which refers, even in the slightest degree,
to mathematical reasoning or calculation." The preceding books were
all published by the Clarendon Press in Oxford, as would be his later
historical books (see ¤19). These later books reflect an interest that
began showing itself in the 1920s, when he wrote thoughtful accounts of
the history of mathematics at Harvard such as [JLC}
and [Mor, Ch. XV] which have helped me greatly in preparing this paper.
A vivid lecturer himself, Coolidge always viewed research and scholarly
publication as the last of four major responsibilities of a university
faculty member. In his words [JLC, p. 355], these responsibilities were:
1. To inject the elements of mathematical knowledge into a large number
of frequently ill informed pupils, the numbers running up to 500 each
year. Mathematical knowledge for these people has come to mean more and
more the calculus.
2. To provide a large body of instruction in the standard topics for a
College degree in mathematics. In practice this is the one of the four
which it is hardest to maintain.
3. To prepare a number of really advanced students to take the doctor's
degree, and become university teachers and productive scholars. The
number of these men slowly increased [at Harvard] from one in two or
three years, to three or four a year.
4. To contribute fruitfully to mathematical science by individual
research. Coolidge's sprightly wit and his leadership as an educator
led to his election as president of the Mathematical Association (MAA)
of America in the mid1920s, during which he also headed a successful
fund drive of the American Mathematical Society [Arc, pp. 30-32].
An important Harvard contemporary of Coolidge was Edward Vermilye
Huntington (1874-1952). After completing graduate studies on the founda-
tions of mathematics in Germany, he began a long career of down-to-earth
teaching, at first under the auspices of the Lawrence Scientific
School. Concurrently, he quickly established a national reputation for
clear thinking by definitive research papers on postulate systems for
groups, fields, and Boolean algebra. These are classics, as is his lucid
monograph on The Continuum and Other Types of Serial Order (Harvard
University Press 1906; 2d ed., 1917).
From 1907-8 on, he gave biennially a course (Mathematics 27) on
"Fundamental Concepts of Mathematics", cross-listed by the philosophy
deptartment (see the end of 8), which introduced students to abstract
mathematics. He also became coauthor in 1911 (with Dickson, Veblen,
Bliss, and others) of the thought-provoking survey Fundamental Concepts
of Modern Mathematics (J. W. Young, ed.); 2d ed. 1916. This survey
still introduced mathematics concentrators to 20th century axiomatic
mathematics when I began teaching, 25 years later. It is interesting to
compare this book with Bocher's address on "The Fundamental Conceptions
and Methods of Mathematics" (Bull. Amer. Math. Soc. 11 (1904), 11-35),
and with 26 of his Introduction to Higher Algebra.
In the 1920s, Huntington broadened his interests. Four years after
making "mathematics and statistics" the subject of his retiring
presidential address to the MAA (Amer. Math. Monthly 26 (1919), 421-35),
he began teaching statistics in Harvard's Faculty of Arts and Sciences.
Offered initially in 1923 as a replacement to a course on interpolation
and approximation given earlier (primarily for actuaries) by Bocher and
L. R. Ford, it was given biennially from 1928 on as a companion to the
course on probability for which Coolidge wrote his book.
Finally, as a related sideline, he invented in 1921 a method of
proportions for calculating how many representatives in the U.S. Congress
each state is entitled to, on the basis of its population.28 This method
successfully avoids the "Alabama paradox" and the "population paradox"
that had flawed the methods previously in use. Adopted by Congress in
1943, it has been used successfully by our government ever since.
8. PASSING ON THE TORCH
As I tried to explain in ¤5, the mathematics courses above freshman
level offered at Harvard in the 1870s and 1880s could be classified into
two main groups: (i) courses on the calculus and its applications in the
tradition of Benjamin Peirce's texts (including his Analytical Mechanics),
designed to make books on classical mathematical physics (Poisson,
Fourier, Maxwell) readable, and (ii) courses on topics in algebra and
geometry related to the later research of Benjamin and C. S. Peirce.
Broadly speaking, Byerly and B. 0. Peirce revitalized the courses in
the first group with their new Mathematics 10, while J. M. Peirce made
comprehensible those of the second. It was primarily J. M. Peirce's
courses that Coolidge and Huntington replaced, giving them new content
and new emphases.
The first major change in the mathematics courses at Harvard initiated
by Bocher and Osgood concerned Mathematics 13 and its new sequels, and
these changes bear a clear imprint of the ideas of Riemann, Weierstrass,
and Felix Klein, who had "passed the torch" to his enthusiastic young
American students. We have already discussed this change in ¤6.
The emphasis on "the theory of invariants" in Bocher's revitalized
Mathematics 3 and his Introduction to Higher Algebra (cf. ¤6) also
reflects Felix Klein's influence, while the emphasis on "elementary
divisors" clearly stems from Weierstrass. It is much harder to trace
the evolution of ideas about the foundations of mathematics. In ¤ 11 of
his article in the Ann. of Math. 6 (1905), 151-89, Huntington clearly
anticipated the modern concepts of relational structure and algebraic
structure, as defined by Bourbaki, far more clearly than Bocher had in his
1904 article on "The Fundamental Conceptions and Methods of Mathematics",
and probably influenced ¤26 of Bocher's Introduction to Higher Algebra.
However, it would be hard to establish clearly the influence of this
pioneer work. Indeed, although supremely important for human culture, the
evolution of basic ideas is nearly impossible to trace reliably, because
each new recipient of an idea tends to modify it before "passing it on".
C. S. Peirce, conclusion. This principle is illustrated by the evolution
of two major ideas of C. S. Peirce: his philosophical concept of
"pragmatism", and his ideas about the algebra of logic. Both of these
ideas were transmitted at Harvard primarily through members of its
philosophy department, as we shall see.
The idea of pragmatism was apparently first suggested in a brilliant
philosophical lecture given by C. S. Peirce at Chauncey Wright's
Metaphysical Club in the 1870s. In this lecture, Peirce claimed that the
human mind created ideas in order to consider the effects of pursuing
different courses of action. This lecture deeply impressed William James
(1842-1910), whose 1895 Principles of Psychology was a major landmark in
that subject [EB 12, 1863-5]. During our Civil War, James had studied
anatomy at the Lawrence Scientific School and Harvard Medical School,
inspired by Jeifries Wyman and Louis Agassiz. After spending the years
1872-76 as an instructor in physiology at Harvard College, and twenty
more years in preparing his famous book, James turned to philosophy
and religion.
In 1906, James finally applied Peirce's idea to a broad range of
philosophical problems in his Lowell Lectures on "Pragmatism...",
published in book form. In turn, James' lectures and writings on
psychology and "pragmatism" strongly influenced John Dewey (1859-1952),
whose philosophy dominated the teaching of elementary mathematics in
our country during the first half of this century [EB 7, 346-7]. It is
significant that the last three chapters of Bertrand Russell's History
of Western Philosophy are devoted to William James, John Dewey, and the
"philosophy of logical analysis" underlying mathematics, as Russell
saw it.
Peirce's concern with logic overlapped that of Huntington with
postulate theory. Actually, C. S. Peirce was a visiting lecturer
in philosophy at Harvard and a Lowell lecturer on logic in Boston in
1903, and Huntington's article on the "algebra of logic" in the Trans.
Amer. Math. Soc. 5 (1904), 288-309, contains a deferential reference
to Peirce's 1880 article on the same subject, and a letter from Peirce
which totally misrepresents the facts, and shows how far he had slipped
since 1881. The facts are as follows.
Never analyzed critically at Harvard, Peirce's pioneer papers on
the algebra of relations and his 1881 article basing Boolean algebra
on the concept of partial order inspired the German logician Ernst
Schršder. First in his Operationskreis des Logikkalkuls, and then in his
three volume Algebra der Logik (1890-95), Schršder made a systematic study
of Peirce's papers. In turn, these books stimulated Richard Dedekind to
investigate the concept of a "Dualgruppe" (lattice; see ¤ 16), in two
pioneer papers which were ignored at the time.
Although Huntington did impart to Harvard students many of the other
fundamental concepts of Dedekind, Cantor, Peano and Hubert, transmitting
them in his course Mathematics 27 and to readers of the books cited in ¤7,
he paid little attention, if any, to this work of Schršder and Dedekind.
Indeed, it was primarily through Josiah Royce that the ideas of C. S.
Peirce had any influence at Harvard. Royce, whose interests were
many-sided, made logic the central theme of his courses. In turn,
he influenced H. M. Sheffer (A.B. '05) and C. I. Lewis (A.B. '06),
two distinguished logicians who wrote Ph.D. theses with Royce and later
became members of the Harvard philosophy department (see ¤12).
Royce also influenced Norbert Wiener, who wrote a Ph.D. thesis comparing
SchrOder's algebra of relations with that of Whitehead and Russell
at Harvard in 1913, and later became one of America's most famous
mathematicians. Indeed, an examination of the first 332 pages of Wiener's
Collected Works (MIT Press, 1976) shows that until 1920 he felt primarily
affiliated with Harvard's philosophy department.
9. FROM ELIOT TO LOWELL
As the preceding discussion indicates, great advances were made at Harvard
in mathematical teaching and research during Eliot's tenure as president
(1869-1909). However, besides many ambitious mathematical courses,
Harvard also offered in 1900 a number of very popular 'gut' courses. After
30 years of President Eliot's unstructured "free elective" system, it
became possible to get an A.B. from Harvard in three years with relatively
little effort. Moreover, whereas athletic excellence was greatly admired
by students, scholastic excellence was not. Someone who worked hard at
his studies might be called a "greasy grind", and a social cleavage had
developed between "the men who studied and those who played-.29
Abbott Lawrence Lowell, who himself became the world's leading authority
on British government without attending graduate school,3ˇ had in 1887
drawn attention "to the importance of making the undergraduate work out
a rational system of choosing his electives ... [with] the benefit of
the experience of the faculty" [Low, p. 11]. Fifteen years later, he
spearheaded in 1901-2 a faculty committee whose purpose was to reinstate
intellectual achievement as the main objective of undergraduate education
([Yeo, Ch. V], [Mor, xlv-xlvi}). After six more years of continuing
faculty discussions in which Osgood and Bocher were both active [Yeo,
pp. 77-78], and many votes, Eliot appointed in 1908 a committee selected
by Lowell "to consider how the tests for rank and scholarly distinction
in Harvard College can be made a more generally recognized measure of
intellectual power" [Yeo, p. 80]. In 1909 Lowell succeeded Eliot as
president at the age of 52.
In his inaugural address [Mor, pp. lxxix-lxxxviii], Lowell outlined
his plan of concentration and distribution, stating that a college
graduate should "know a little of everything and something well" [Low,
p. 40]. Having in mind the examples of Oxford and Cambridge Universities,
he also proposed creating residential halls (at first for freshmen)
to foster social integration. I shall discuss the fruition of these and
other educational reforms of Lowell's in 12 below. His ideas have been
expressed very clearly by himself and by Henry Yeomans,3' his colleague
in the government department and frequent companion in later life. For
the moment, I shall describe only some major changes in undergraduate
mathematics at Harvard which he encouraged, that took place during the
years 1906-29.
Calculus instruction. During its lifetime (1847-1906), the Lawrence
Scientific School had shared in the teaching of elementary mathematics
at Harvard. In 1910, during its transition into a graduate school of
engineering (completed in 1919), this responsibility was turned over to
the mathematics department, doubling the latter's elementary teaching
load. At the time, "nine-tenths of all living [Harvard] graduates who
took an interest in mathematics at college got their inspiration from
Mathematics C," which then covered only analytic geometry through the
conic sections.
This seemed deplorable to Lowell, who knew that the calculus, its
extensions to differential equations, differential geometry, and
function theory, and its applications to celestial mechanics, physics,
and engineering, had dominated the development of mathematics ever since
1675. Aware of this domination, he sometimes identified the phonetic
alphabet, the Hindu- Arabic decimal notation for numbers, symbolic
algebra, and the calculus, as the four most impressive inventions of
the human mind.
Lowell soon persuaded the faculty to require each undergraduate to take
for "distribution" at least one course in mathematics or philosophy,
presumably to develop power in abstract thinking. Through the visiting
committee of the Harvard mathematics department (see below), he
also encouraged devoting substantial time in Mathematics C to the
calculus. Within a decade, "half of the Freshman course was devoted
to the subject [of the calculus], and in 1922 the Faculty of Arts and
Sciences, through the President's deciding vote, passed a motion that no
mathematics course where the calculus was not taught would be counted
for distribution" [Mor, p. 255]. This change was followed by steadily
increasing emphasis (at Harvard) on the calculus and its applications,
until "In 1925-26, 327 young men, just out of secondary school, were
receiving a half-year of instruction in the differential calculus"
[Mor, p. 255].
Visiting Committees. Since 1890, the activities of each Harvard department
have been reviewed by a benevolent visiting committee, which reports
triennially to the board of overseers. Beginning in 1906, Lowell's
brother-inlaw William Lowell Putnam played a leading role on the visiting
committee of the mathematics department, and in 1912, Lowell invited
George Emlen Roosevelt, a first cousin of Franklin Delano Roosevelt,
to join it as well.
Both men had been outstanding mathematics students, and their 1913
report with George Leverett and Philip Stockton contained "the important
suggestion that the bulk of freshmen be taught in small sections" [Mor,
p. 254].
This new plan allowed an increasing number of able graduate students in
mathematics to be self- supporting by teaching elementary courses (based
on Osgood's texts). For example, during the years 1927-40, S. S. Cairns,
G. A. Hedlund, G. Baley Price, C. B. Morrey, T. F. Cope, J. S. Frame,
D. C. Lewis, Sumner Myers, J. H. Curtiss, Walter Leighton, Arthur Sard,
John W. Calkin, Ralph Boas, Herbert Robbins, R. F. Clippinger, Lynn
Loomis, Philip Whitman, and Maurice Hems served in this role. At the
same time, a few outstanding new Ph.D.'s were invited to participate in
Harvard's research environment by becoming Benjamin Peirce instructors.
Among these, one may mention John Gergen, W. Seidel, Magnus Hestenes,
Saunders Mac Lane, Ho!brook MacNeille, Everett Pitcher, Israel Halperin,
John Green, Leon Alaoglu,
and W. J. Pettis in the decade preceding World War II.
Besides giving benign and wise advice, the visiting committees of
the mathematics department established and financed for many decades
a departmental library, where for at least seventy years the bulk of
reading in advanced mathematics has taken place. Among the many grateful
users of this library should be recorded George Yale Sosnow. More than
60 years after studying mathematics in it around 1920, he left $300,000
in his will to endow its expansion and permanent maintenance.
10. GEORGE DAVID BIRKHOFF
A major influence on mathematics at Harvard from 1912 until his death
was my father, George David Birkhoff (1884-1944). His personality and
mathematical work have been masterfully analyzed by Marston Morse in [GDB,
Vol. I, xxiii-lvii], reprinted from Bull. Amer. Math. Soc. 52 (1946),
357-91. Moreover I have already sketched some more personal aspects of
his career in [LAM, ¤7 and ¤¤14-15]. Therefore, I will concentrate here
on his roles at Harvard.
When my father entered Harvard as a junior in 1903, he had already
been thinking creatively about geometry and number theory for nearly
a decade. According to his friend, H. S. Vandiver [Van, p. 272] "he
rediscovered the lunes of Hippocrates when he was ten years old". In this
connection, I still recall him showing my sister and me how to draw them
with a compass (see Fig. 1) when I was about nine, joining the tips of
these lunes with a regular hexagon, and mentioning that with ingenuity,
one could construct regular pentagons by analogous methods. By age 15,
he had solved the problem (proposed in the Amer. Math. Monthly) of
proving that any triangle with two equal angle bisectors is isosceles.
Before entering Harvard, he had proved (with Vandiver) that every integer
a" - b" (n > 2) except 63 = 26 - 16 has a prime divisor p which does
not divide ak_bk for any proper divisor k of n. He had also reduced the
question of the existence of solutions of xmyn±ymz±zmxz 0 (m, n not both
even) to the Fermat problem of finding nontrivial solutions of ut ± Vt
± wt = 0, where t = m2-mn±n2. Indeed, he had already begun his career
as a research mathematician when he entered the University of Chicago
in 1902. There he soon began a lifelong friendship with Oswald Veblen,
a graduate student who had received an A.B. from Harvard (his second)
two years earlier.33
I have outlined in [LAM, ¤7] some high points of my father's career during
the final "formative years" in Cambridge, Chicago, Madison, and Princeton
that preceded his return to Cambridge. He himself has described with
feeling, in [GDB, vol. III, pp. 274-5], his intellectual debt to E. H.
Moore, Boiza, and Bocher, thanking Bocher "for his suggestions, for his
remarkable critical insight, and his unfailing interest in the often
crude mathematical ideas which 1 presented". It was presumably under
the stimulus of Bocher (and perhaps Osgood) that he wrote his first
substantial paper (Trans. Amer. Math. Soc. 7 (1906), 107-36), entitled
"General mean value and remainder theorems". The questions raised and
partially answered in this are still the subject of active research.34
Moreover his 1907 Ph.D. thesis, on expansion theorems generalizing
Sturm- Liouville series, was also stimulated by Bocher's ideas about
such expansions, at least as much as by those of his thesis adviser,
E. H. Moore, about integral equations.
Return to Harvard. As Veblen has written [GDB, p. xvii], my father's
return in 1912 as a faculty member to Harvard, "the most stable academic
environment then available in this country," marked "the end of the
formative period of his career". He had just become internationally famous
for his proof of Poincare's last geometric theorem. Moreover Bocher had
devoted much of his invited address that summer at the International
Mathematical Congress in Cambridge, England, to explaining the importance
and depth of my father's work on boundary value problems for ordinary
differential equations. Equally remarkable, my father had been chosen
to review for the Bull. Amer. Math. Soc. (17, pp. 14-28) the "New Haven
Colloquium Lectures" given by his official thesis supervisor, E. H.
Moore, and Moore's distinguished Chicago colleagues E. J. Wilczynski
and Max Mason.
It is therefore not surprising that, in his first year as a Harvard
assistant professor, he and Osgood led a seminar in analysis for research
students, or that he remained one of the two leaders of this seminar until
1921. By that time, it "centered around those branches of analysis which
are related to mathematical physics". This statement reflected interest in
the theory of relativity (see ¤11). It may seem more surprising that the
reports of the visiting committee of 1912 and 1913 took no note of this
unique addition to Harvard's faculty, until one remembers that their main
concern was with the mathematical education of typical undergraduates!
1912 as a milestone. By coincidence, 1912 also bisects the time interval
from 1836 to 1988, and so is a half-way mark in this narrative. It
can also be viewed as a milestone marking the transition from primary
emphasis on mathematical education at Harvard to primary emphasis
on research. Since Byerly retired and B. 0. Peirce died in 1913, it
also marks the end of Benjamin Peirce's influence on mathematics at
Harvard. Finally, since I was one year old at the time, it serves as a
convenient reminder that all the changes that I will recall took place
during two human life spans.
During the next two decades, G. D. Birkhoff would supervise the Ph.D.
theses of a remarkable series of graduate students. These included Joseph
Slepian (inventor of the magnetron), Marston Morse, H. J. Ettlinger,
J. L. Walsh, R. E. Langer, Carl Garabedian (father of Paul), D.
V. Widder, H. W. Brinkmann, Bernard Koopman, Marshall Stone, C. B.
Morrey, D. C. Lewis, G. Baley Price, and Hassler Whitney. Four of them
(Morse, Walsh, Stone, and Morrey) would become AMS presidents.
In retrospect, my father's role in bringing topology to Harvard (as
Veblen did to Princeton), at a time just after L. E. J. Brouwer had
proved some of its most basic theorems rigorously, seems to me especially
remarkable. So does his early introduction to Harvard of functional
analysis, through his 1922 paper with 0. D. Kellogg on "Invariant points
in function space", his probable influence on Stone and Koopman, and
his "pointwise ergodic theorem" of 1931. But deepest was probably his
creative research on the dynamical systems of celestial mechanics. It
was to present this research that he was made AMS colloquium lecturer in
1920, and to honor it that he was awarded the first Bocher prize in 1922.
It is interesting to consider my father's related work on celestial
mechanics as a continuation of the tradition of Bowditch and Benjamin
Peirce, which was carried on by Hill and Newcomb, and after them by
E. W. Brown at Yale. Brown became a president of the AMS, and my father
was happy to teach his course on celestial mechanics one year in the early
1920s, and to coauthor with him, Henry Norris Russell, and A. 0. Lorchner
a Natural Research Council Bulletin (#4) on "Celestial Mechanics". This
document provides a very readable account of the status of the theory
from an astronomical standpoint as of 1922, including the impact of
Henri Poincare's Methodes nouvelles de la Mecanique celeste.
Although my father's lectures were not always perfectly organized or
models of clarity, his contagious enthusiasm for new mathematical ideas
stimulated students at all levels to enjoy thinking mathematically. He
also enjoyed considering all kinds of situations and phenomena from a
mathematical standpoint, an aspect of his scientific personality that
I shall take up next.
11. MATHEMATICAL PHYSICS
Among research mathematicians, my father will be longest remembered
for his contributions to the theory of dynamical systems (including
his ergodic theorem), and his work on linear ordinary differential and
difference equations. These were admirably reviewed by Marston Morse in
[GDB, I, pp. xv-xlix; Bull. Amer. Math. Soc. 52, 357-83], and it would
make little sense for me to discuss them further here. At Harvard,
however, there were very few who could appreciate these deep researches,
and so from 1920 on, my father's ideas about mathematical physics and
the philosophy of science aroused much more interest. These were also the
themes of his invited addresses at plenary sessions of the International
Mathematical Congresses of 1928 and 1936, and of most of his public
lectures. Accordingly, I shall concentrate below on these aspects of
his work (cf. Parts V and VI of Morse's review).
Relativity. Of all my father's "outside" interests, the most
durable concerned Einstein's special and general theories of
relativity. Unfortunately, it is also this interest that has been least
reliably analyzed. Thus Morse's review suggests that it began in 1922,
whereas in fact his 1911 review of Poincare's Gottingen lectures concludes
with a discussion of "the new mechanics" of Einstein's special theory of
relativity (cf. [GDB, III, pp. 193-4] and Bull. Amer. Math. Soc. 17,
pp. 193-4). Moreover, he had touched on these theories and discussed "The
significance of dynamics for general scientific thought" at length in his
1920 colloquium lectures,36 before initiating in 1921-22 an "intermediate
level" course on "space, time, and relativity" (Mathematics 16) having
second-year calculus as its only prerequisite. He promptly wrote (with the
cooperation of Rudolph Langer) a text for this course, entitled Relativity
and Modern Physics (Harvard University Press, 1923, 1927). In 1922, he
also gave a series of public Lowell lectures on relativity. Two years
later, he gave a similar series at U.C.L.A. (then called "the Southern
Branch of the University of California"), and edited them into a book
entitled The Origin, Nature, and Influence of Relativity (Macmillan,
1925). It was not until 1927 that he finally published in book form his
deep AMS colloquium lectures, in a book Dynamical Systems, which omitted
many of these topics which he had presented orally seven years earlier.
Bridgman, Kemble, van Vleck. My father's interest in relativity and the
philosophy of science was shared by his friend and contemporary Percy
W. Bridgman (1882-196 1). (Bridgman's notes of 1903-4 on B. 0. Peirce's
Mathematics 10 are still in the Harvard archives, and it seems likely
that my father attended the same lectures.) Bridgman would get the Nobel
prize 25 years later for his ingenious experiments on the "physics of
high pressure", his own research specialty, but in the 1920s he amused
himself by writing the classic book on Dimensional Analysis (1922,
1931), by giving a half-course on "electron theory and relativity", and
writing a thought-provoking book on The Logic of Modern Physics (1927).
The central philosophical idea of this book, that concepts should be
examined operationally, in terms of how they relate to actual experiments,
is reminiscent of the pragmatism of William James and C. S. Peirce.
In 1916, Bridgman had supervised a doctoral thesis on "Infra-red
absorption spectra" by Edwin C. Kemble which (as was required by the
physics department at that time) included a report on experiments made to
confirm its theoretical conclusions. Five years later, Kemble supervised
the thesis of John H. van Vleck (1899-1980), grandson of Benjamin Peirce's
student John M. van Vleck and son of the twelfth AMS president E. B. van
Vleck. This thesis, entitled "A critical study of possible models of
the Helium atom", is a case study of the unsatisfactory state of quantum
mechanics at that time.
Quantum mechanics. However, in 1926, Schrodinger's equations finally
provided satisfactory mathematical foundations for nonrelativistic mechan-
ics, shifting the main focus of mathematical physics from relativity to
atomic physics. In that same year, my father began trying to correlate his
relativistic concept of an elastic "perfect fluid", having a "disturbance
velocity equal to that of light at all densities" [GDB, II, 737-63 and
876-861, with the spectrum of monatomic hydrogen, usually derived from
Schrodinger's non-relativistic wave equation. Although this work was
awarded an AAAS prize in 1927, of greater permanent value was probably
his later use of the theory of asymptotic series to reinterpret the
WKB-approximations of quantum mechanics, which yield classical particle
mechanics in the limiting case of very short wave length (ibid., pp.
837-56). Related ideas about quantum mechanics also constituted the
theme of his address at the 1936 International Congress in Oslo [GDB
II, 857-75].
In the meantime, Kemble had taught me most of what I know about quantum
mechanics. Far more important, he had just about completed his 1937 book,
The Fundamental Principles of Quantum Mechanics. The preface of this
book mentions his "distress" at "the tendency to gloss over the numerous
mathematical uncertainties and pitfalls which abound in the subject",
and his own "consistent emphasis on the operational point of view".
Like Kemble, van Vieck (Harvard Ph.D., 1922) made non-relativistic quan-
tum mechanics his main analytical tool; but unlike Kemble, he attached
little importance to its mathematical rigor. Instead, he applied it so
effectively to models of magnetism that he was awarded a Nobel prize
around 1970. As a junior fellow in 1934, I audited his half-course
(Mathematics 39) on "group theory and quantum mechanics", and was startled
by his use of the convenient assumption that every matrix is similar to
a diagonal matrix.* The courses (Mathematics 40) on the "differential
equations of wave mechanics" given in alternate years through 1940 by
my father, must have had a very different flavor.
12. PHILOSOPHY; MATHEMATICAL LOGIC
From his philosophical analysis of the concepts of space and time,
my father also gradually developed radical ideas about how high
school geometry should be taught. His public lectures on relativity
had included (in Chapter II, on "the nature of space and time") a
system of eight postulates for plane geometry, of which the first two
concern measurement. They assert that length and angle are measurable
quantities (magnitudes, or real numbers), measurable by "ruler and
protractor". Whereas Euclid had devoted his axioms to properties of such
"quantities", my father saw no good reason why high school students
should not use them freely.
A decade later, he proposed a reduced system of four postulates for
plane geometry, including besides these measurement postulates only two:
the existence of a unique straight line through any two points, and the
proportionality of the lengths of the sides of any two triangles ABC and
A'B'C' having equal corresponding interior angles. His presentation to
the National Council of Teachers of Mathematics two years earlier had
included a fifth postulate: that "All straight angles have the same
measure, l80ˇ." This presentation was coauthored by Ralph Beatley of
Harvard's Graduate School of Education, and their ideas expanded into
an innovative textbook on Basic Geometry (Scott, Foresman, 1940, 1941).
Less innovative analogous texts on high-school physics and chemistry,
coauthored by N. Henry Black of Harvard's Education School with Harvey
*Of course, every finite group of complex matrices is similar to a
group of unitary matrices Davis and James Conant,38 respectively, had
been widely adopted. However, perhaps because it came out just before
World War II, the book by G. D. Birkhoff and Beatley never achieved
comparable success.
Expanded from 4 (or 5) postulates to 23, and from 293 pages to 578,
G. D. Birkhoffs idea of allowing high-school students to assume that real
numbers express measurements of distance and angles was developed by
E. E. Moise and F. L. Downs, Jr. into a commercially successful text
Geometry (Addison-Wesley, 1964).
Aesthetic Measure. According to Veblen, my father "was already speculating
on the possibility of a mathematical theory of music, and indeed of art
in general, when he was in Princeton" (in 1909-12). At the core of his
speculations was the formula
(1) M=f(O/C), O=>O, C=>C3,
where the 0, are pleasing, suitably weighted elements of order, the C
suitably weighted elements of complexity, intended to express the effort
required to "take in" the given art object, and M is the resulting
aesthetic measure (or "value"). Attempts were made to quantify (1) by
David Prall at Harvard and others, through psychological measurements;
those interested in aesthetics should read G. D. Birkhoffs book Aesthetic
Measure (Harvard University Press, 1933). See also his papers reprinted
in [0DB, III, pp. 288-307, 32034, 382-536, and 755- 838], the first of
which constitutes his invited address at the 1928 International Congress
in Bologna.
Of my father's last five papers (##l99-203 in [GDB, vol. iii, p. 897]),
one is concerned with quaternions and refers to Benjamin and C. S. Peirce;
a second with axioms for one-dimensional "geometries"; and a third
with generalizing Boolean algebra. His enthusiasm for analyzing basic
mathematical structures and recognizing their interrelations never
flagged.
Like the relativistic theory of gravitation in flat space-time which
was his dominant interest in the last years of his life (see ¤20),
these speculative contributions are less highly appreciated by most
professional mathematicians today than his technical work on dynamical
systems. However, they made him more interesting to the undergraduates
in his classes, his tutees, and his colleagues on the Harvard faculty. In
particular, they contributed substantially to his popularity as dean of
the faculty, and to the high esteem in which he was held by President
Lowell and the Putnam family.39 They must have also influenced his
election as president of the American Association for the Advancement
of Science.
A. N. Whitehead. My father's ventures into mathematical physics, the
foundations of geometry, and mathematical aesthetics were comparable
to the ventures into relativity, the foundations of mathematics, and
mathematical logic of A. N. Whitehead, who joined Harvard's philosophy
department in 1924. The Whiteheads lived two floors above my parents at
984 Memorial Drive, and were very congenial with them.
The situation had changed greatly since 1910, when Josiah Royce was the
only Harvard philosopher who found technical mathematics interesting, and
(perhaps because of William James) Harvard's courses in psychology were
given under the auspices of the philosophy department. In the 1920s and
1930s, not only Whitehead, but also C. I. Lewis (author of the Survey of
Symbolic Logic) and H. M. Sheffer of the philosophy department (cf. ¤8)
were important mathematical logicians. Moreover Huntington's course
Mathematics 27 on "Fundamental concepts..." (cf. ¤7) was cross-listed for
credit in philosophy, and there was even a joint field of concentration
in mathematics and philosophy.
In the 1920s, mathematical logic was a bridge connecting mathematics
and philosophy, making the former seem more human and the latter more
substantial. Whitehead and Russell's monumental Principia Mathematica
was considered in the English-speaking world to have revolutionized the
foundations of mathematics, reducing its principles to rules governing
the mechanical manipulation of symbols. In particular, its claim to have
made axioms "either unnecessary or demonstrable" was widely accepted by
both mathematicians and philosophers.40
In the following decade, Gšdel and Turing would revolutionize ideas about
the role and significance of mathematical logic; the Association for
Symbolic Logic would be formed; and the subject would gradually become
detached from the rest of mathematics, concentrating more and more on
its own internal problems. However, the addition of W.V. Quine to the
Harvard philosophical faculty, and the presence in Cambridge of Alfred
Tarski for several years, continued to stimulate fruitful interchanges
of ideas until long after World War II.
13. POSTWAR RECRUITMENT
The retirement of Byerly in 1913 and the death of B. 0. Peirce in 1914,
together with the deaths of Bocher and G. M. Green, and the departure of
Dunham Jackson after six years as secretary in 1919,41 created a serious
void in Harvard mathematics. This void was filled slowly, at first
(in 1920) by Oliver D. Kellogg (1878-1932), and William C. Graustein
(1897-1942), who had earned Ph.D.'s in Germany before the war with
Hilbert and Study, respectively. Then came Joseph L. Walsh (1895-1973)
in 1921, and (after H. W. Brinkmann in 1925) H. Marston Morse (1892-1977)
in 1926. Both Walsh's and Morse's Ph.D. theses had been supervised by
my father. 42 Like Osgood, Bocher, Coolidge, Huntington, and Dunham
Jackson, Graustein (A.B. 1910) and Walsh (S.B. 1916) had been Harvard
undergraduates.
Kellogg immediately modernized and infused new life into Mathematics 1 Oa
("potential theory"), took on the teaching of Mathematics 4 (mechanics),
and joined my father in running the seminar in analysis. The 1921-22
department pamphlet announced that in that seminar, "the topics assigned
will centre about those branches of analysis which are related to
mathematical physics". This statement was repeated for two more years,
during the first of which Kellogg and Einar Hille (as B.P. Instructor)
directed the seminar, a fact which confirms my impression, described
in ¤12, that in those years it was relativity theory and not "dynamical
systems" that seemed most exciting at Harvard.
Graustein was an extremely clear lecturer and writer. His and Osgood's
Analytic Geometry, and his texts for Mathematics 3 (Introduction to Higher
Geometry, 1930) and Mathematics 22 (Differential Geometry, 1935), were
models of careful exposition. Combined with Coolidge's lively lectures
and more informal texts on special topics, they made geometry second
only to analysis in popularity at Harvard during the years 1920- 36.
Walsh concentrated on analysis. In 1924-5, he expanded Osgood's halfcourse
Mathematics 12 on infinite series, which had remained static for 30 years,
into a full course on "functions of a real variable" which included the
Lebesgue integral. He also soon invented "Walsh functions",43 and became
an authority on the approximation of complex and harmonic functions. His
interest in this area may have been stimulated by Dunham Jackson, who
had done distinguished work in approximation theory fifteen years earlier
(see Trans. AMS 12). Most striking was Walsh's result that, in any bounded
simply connected domain with boundary C, every harmonic function is the
limit of a sequence of harmonic polynomials which converges uniformly on
any closed set interior to C (Bull. Amer. Math. Soc. 35 (1929), 499-544).
Brinkmann came from Stanford, where H. F. Blichfeldt had interested him
in group representations. A year's post-doctoral stay in Gottingen with
Emmy Noether had not converted him to the axiomatic approach. A brilliant
and versatile lecturer, his graduate courses were mostly on algebra and
number theory, in which he interested J. S. Frame and Joel Brenner, see
[Bre]. However, he also gave a course on "mathematical methods of the
quantum theory" with Marshall Stone in 1929-30.
Morse applied variational and topological ideas related to those
of my father (and of Poincare before him). Just before he came to
Harvard, he had derived the celebrated derived Morse inequalities
(Trans. Amer. Math. Soc. 27 (1925), 345-96). The main fruit of his
Harvard years was his 1934 Colloquium volume, Calculus of Variations in
the Large. The foreword of this volume describes admirably its connection
with earlier ideas and results of Poincare, Bocher, my father, and my
father's Ph.D. student Ettlinger.
14. UNDERGRADUATE MATHEMATICS COURSES
Most of my own undergraduate courses in mathematics were taught by
these relatively new members of the Harvard staff, and by two other
thesis students of my father: H. W. Brinkmann and Hassler Whitney, who
joined the Harvard mathematics staff in the later 1920s (Whitney, Simon
Newcomb's grandson, as a graduate student). It may be of interest to
record my own youthful impressions of their teaching and writing styles.45
In this connection, I should repeat that whereas my description of
mathematical developments at Harvard before 1928 has been based largely on
reading, hearsay, and reflection, from then on it will be based primarily
on my own impressions during fifteen years of slowly increasing maturity.
Shortly after joining my parents in Paris in the summer of 1928, my father
ordered me to "learn the calculus" from a second-hand French text which
he picked up in a bookstall along the Seine. Later that summer, after
explaining to me Fermat's "method of infinite descent", he challenged me
to prove that there were no (least) positive integers satisfying x4 ± y4
= z4. After making substantial progress, I lost heart, and felt ashamed
when he showed me how to complete the proof in two or three more steps.
The next fall, I was fortunate in being taught second-year calculus as
a freshman by Morse and Whitney. Their lectures made the theory of the
calculus interesting and intuitively clear; especially fascinating to
me was their construction of a twice-differentiable function U(x, y) for
which U, $ Ufl,. The daily exercises from Osgood's text gave the needed
manipulative skill in problem solving. Likewise, the clarity of Osgood
and Graustein made it easy and pleasant to learn from their Analytic
Geometry, in "tutorial" reading (see ¤15), not only the reduction of
conics to canonical form, but also the theory of determinants.
I learned the essentials of analytic mechanics (Mathematics 4) from
Kellogg concurrently. In his lectures Kellogg explained how to reduce
systems of forces to canonical form, and derived the conservation laws
for systems of particles acting on each other by equal and opposite
"internal" forces. His presentation of Newton's solution of the two-body
problem opened my eyes to the beauty and logic of celestial mechanics,
and reinforced my interest in the calculus and the elementary theory of
differential equations. In an unsolicited course paper, I also tried
my hand at applying conservation laws to deduce the effect of spin on
the bouncing of a tennis ball (I had played tennis with Kellogg, for
several years a next door neighbor), as a function of its coefficient of
(Coulomb) friction and its "coefficient of restitution". GARRETT BIRKHOFF
was also delighted to learn the mathematical explanation of the "center
of percussion" of a baseball bat.
That spring my father gave me a short informal lecture on the crucial
difference between pointwise and uniform convergence of a sequence of
functions, and then challenged me to prove that any uniform limit of a
sequence of continuous functions is continuous. After I wrote out a proof
(in two or three hours), he seemed satisfied. In any event, he encouraged
me to take as a sophomore the graduate course on functions of a complex
variable (Mathematics 13) from Walsh, omitting Mathematics 12. This was
only 15 months after I had begun learning the calculus.
This was surely my most inspiring course. Walsh had a dramatic way
of presenting delicate proofs, lowering his voice more and more as he
approached the key point, which he would make in a whisper. Each week we
were assigned theorems to prove as homework. As I was to learn decades
later, our correctors were J. S. Frame, who became a distinguished
mathematician, and Harry Blackmun, now a justice on the U. S. Supreme
Court. They did their job most ably, conscientiously checking my homemade
proofs, which often differed from those of the rest of the class. What
a privilege it was!
Concurrently, I took advanced calculus (Mathematics 5) from Brinkmann,
who drilled a large class on triple integration, the beta and gamma
functions and many other topics. He made concise and elegant formula
derivations into an art form, leaving little room for student initiative.
Osgood's Advanced Calculus supplemented Brinkmann's lectures admirably,
by including an explanation of how to express the antiderivative f R(x,
J5)dx of any rational function of x and the square root of a quadratic
function Q(x) in elementary terms, good introductions to the wave and
Laplace equations, etc. Through Brinkmann's lectures and Osgood's book,
I acquired a deep respect for the power of the calculus, which I have
always enjoyed trying to transmit to students.
My junior year, I took half-courses on the calculus of variations
(Mathematics 15) from Morse, on differential geometry (Mathematics 22a)
from Graustein, and on ordinary differential equations (Mathematics 32)
from my father. Morse's imaginative presentation again made me conscious
of many subtleties, especially the sufficient conditions required to
prove (from considerations of 'fields of extremals') that solutions of
the Euler-Lagrange equations are actually maxima or minima. Graustein,
on the other hand, explained details of proofs so carefully that there
was little left for students to think about by themselves. I preferred
my father's lecture style, which included a digression on the three
'crucial effects' of the general theory of relativity, and a challenge to
classify qualitatively the solutions of the autonomous DE x = F(x). (He
did not suggest using the Poincare phase plane.)
He mentioned in class the fact (first proved by Picard) that one cannot
reduce to quadratures the solution of
(14.1) U" ± p(x)U' ± q(x)U = 0.
Not knowing anything then about solvable groups or Lie groups, I was
skeptical and wasted many hours in trying to find a formula for solving
(14.1) by quadratures.
Finally, in my senior year, I took a half-course on potential theory
(Mathematics lOa) with Kellogg. There I found the concept of a harmonic
function and Green's theorems exciting, but was bored by elaborate
formulas for expanding functions in Legendre polynomials. I also took
one on "analysis situs" (combinatorial topology) with Morse, which was
an unmitigated joy, however, especially because of its classification of
bounded surfaces, proofs of the topological invariance of Betti numbers,
etc. The reduction of rectangular matrices of Os and 1 s to canonical
form under row equivalence was another stimulating experience.
15. HARVARD UNDERGRADUATE EDUCATION: 1928-42
My own undergraduate career was strongly influenced by the philosophy
of education developed by President Lowell. Lowell was an "elitist",
who believed that excellence was fostered by competition, and best
developed through a combination of drill, periodic written examinations,
oral discussions with experts, and the writing of original essays of
variable length. He also believed that breadth should be balanced by
depth, and that originality was a precious gift which could not be
taught. His primary educational aim was to foster intellectual and
human development through his system of concentration and distribution
of courses, tutorial discussions, "general examinations", and senior
honors theses. I believe that he wanted Harvard to train public-spirited
leaders with clear vision, who could think bard, straight, and deep.
During eleven academic years, 1927-38, 1 slept in a dormitory, ate most
meals with students in dining halls (from 1936 to 1938 as a tutor),
usually participated in athletics during the afternoon, and studied in
the evening. From 1929 on, my primary aim was to achieve excellence as a
mathematician, and I think the Harvard educational environment of those
years was ideal for that purpose also. After 1931, 1 continued to think
about mathematics during summers, if somewhat less systematically.
I think I have already said enough about individual mathematics courses
at Harvard. All my teachers impressed me as trying hard to communicate
to a mix of students a mature view of the subject they were teaching
and (equally important) as being themselves deeply interested in it.
Of even greater value was the encouragement I got in tutorial to do
guided reading and 'creative'
thinking about mathematics and a few of its applications. These efforts
were tactfully monitored by leading mathematicians, who were surely
conscious of my limitations and slowly decreasing immaturity, and
communicated their evaluations to my father.
My first tutorial assignment was to learn about (real) linear algebra and
solid analytic geometry as a freshman by reading the book by Osgood and
Graustein. In Walsh's Mathematics 13, I spent the spring reading period
of my sophomore year on a much more advanced topic: figuring out how to
reconstruct any doubly periodic function without essential singularities
from the array of its poles. In a junior course by G. W. Pierce on
"Electric oscillations and electric waves", I wrote a term paper on the
refraction and reflection of electromagnetic waves by a plane interface
separating two media having different dielectric constants and magnetic
permeabilities, and presented my results as one of the speakers at a
physics seminar the following fall. I surely learned more from giving
my talk than the audience did from hearing it!
From the middle of that year on, my main tutorial efforts were devoted
to planning and writing a senior honors thesis, for which endeavor
approximately one-fourth of my time was officially left free. My tutorial
reading for this began with Hausdorlfs Mengenlehre (first ed.) and
de la VallŽe-Poussin's beautiful Cours d'Analyse Infinit`simale, from
which I learned the foundations of set-theoretic topology and the theory
of the Lebesgue integral, respectively. In retrospect, I can see that
this reading and my father's oral examination on "uniform convergence"
essentially covered the content of Mathematics 12 on "functions of a
real variable" (cf. [Tex, p. 16]). My acquaintance with general topology
was broadened by reading FrŽchet's Thesis (1906), which introduced me
to function spaces, and his book Les Espaces Abstraits. It was also
deepened by reading the fundamental papers of Urysohn, Alexandroff,
Niemytski, and Tychonoff (Math. Ann., vols. 92-95). I was fascinated
by Caratheodory's paper "on the linear measure of sets" and Hausdorlfs
fractional-dimensional measure, so brilliantly applied to fractals by
Benoit Mandeibrot in recent decades. This reading was guided and monitored
by Marston Morse; like all faculty members, his official duties included
talking with each of his 'tutees' for about an hour every two weeks.
By that time, Lowell's ambition of establishing "houses" at Harvard
similar to the Colleges of Oxford and Cambridge had also come to fruition,
and I became a member of Lowell House, of which Coolidge was the dedicated
"master". Brinkmann was a resident tutor in mathematics, and J. S. Frame
a resident graduate student. My mathematical tutorials with Morse
were supplemented by occasional casual chats on a variety of subjects
with these and other friendly tutors, as well as (naturally) with my
father. The Coolidges tried to set a tone of good manners by entertaining
suitably clad undergraduates in their tastefully furnished residence.
Hours of study in Lowell House were relieved by lighter moments. One of
these involved a humorous letter from President Roosevelt to Coolidge,
which ended ".- do you remember your first day's class at Groton? You
stood up at the blackboard - announced to the class that a straight
line is the shortest distance between two points - and then tried to
draw one. All I can say is that I, too, have never been able to draw a
straight line. I am sure you shared my joy when Einstein proved there
ain't no such thing as a straight line!"
As a senior in Lowell House, I wrote a rambling 80 page thesis centering
around what would today be called multisets (but which I called
"counted point-sets"), such as might arise from a parametrically
defined rectifiable curve x(s) allowed to recross itself any number
of times. Not taking the hint from the fact that pencilled comments by
the official thesis reader ended on page 41, 1 submitted all 80 pages
for publication in the AMS Transactions, and was shocked when a kindly
letter from J. D. Tamarkin explained why it could not be accepted!46
In the comfortable and well-stocked Lowell House library, I became
acquainted with the difficulty of defining "probability" rigorously. But
above all, in the one room departmental library funded by the visiting
committee, I discovered Miller, Blichfeldt and Dickson's book on finite
groups, and soon became fascinated by the problem of determining all
groups of given finite order. There I also saw Klein's EnzyklopŠdie det
Mathernatisehen Wissenschaflen with its awe-inspiring multivolume review
of mathematics as a whole. After finishing my honors thesis, which touched
on fractionaldimensional measure, I decided to see what was known about
it. To my horror, I found everything I knew compressed into two pages,
in which a large fraction of the space was devoted to references! Although
profoundly impressed, I decided not to allow myself to be overawed.
Among nearly contemporary Harvard undergraduates, I suspect that Joseph
Doob, Arthur Sard, Joel Brenner, Angus Taylor, and Herbert Robbins
were profiting similarly from their Harvard undergraduate education;
human minds are at their most receptive during the years from 17 to
22. Although expert professorial guidance is doubtless most beneficial
when given to students planning an academic career, and conditions today
are very different from those of the 1930s, I think it would be hard to
improve on my mathematical education!47 It prepared me well for a year
as a research student at Cambridge University (see 16), after which I
was ready to carry on three years of free research in Harvard's Society
of Fellows (see ¤1 7).
As a junior fellow, I ate regularly in Lowell House for three more
years with undergraduates, a handful of resident Law School students,
and tutors. I then participated actively for two more years in dormitory
life, as faculty instructor and senior tutor of Lowell House, trying
to live up to the ideals of intellectual communication from which I had
myself benefited so much.
In retrospect, although I had pleasant human relations with my
prewar undergraduate tutees, I fear I gave some of them an overdose of
mathematical ideology. They decided (no doubt rightly) that mathematics
as I presented it was simply not their 'dish of tea'! As senior tutor,
I was more popular for being otherwise a normal and gregarious human
being, and top man on the Lowell House squash team (#1), than for being
inspiring mathematically.
Thesis topics. To be a good mathematics thesis adviser at any level, one
should be acquainted with a variety of interesting possible thesis topics,
and the mathematical thinking processes of a variety of students. At
Harvard, a substantial fraction of theses in the 1930s dealt with such
simple topics as relating the vibrating string and Fourier series to
musical scales and harmony; there was (and is) a Wister Prize for
excellence in "mathematics and music". My tutee Russell ("Rusty")
Greenhood, later a financial officer at the Massachusetts General
Hospital, got a prize for his thesis on "The x2 test and goodness of fit",
a statistical topic about which I knew nothing. He may have discussed
his thesis with Huntington, but all students were encouraged to work
independently. Generally speaking, prospective research mathematicians
chose advanced thesis topics in very pure mathematics. Thus Harry Pollard
(A.B. '40, Ph.D. '42) wrote an impressive undergraduate thesis on the
Riemann zeta function, which may be found in the Harvard archives.
16. HARVARD, YALE, AND OXBRIDGE
Harvard and Yale have often been considered as American (New
England?) counterparts of Cambridge and Oxford universities in "old"
England. Actually, it was John Harvard of Emmanuel College, Cambridge,
who gave to Harvard its first endowment. More relevant to this account,
the House Plan at Harvard and the College Plan at Yale (both endowed
by Yale's Edward Harkness) were modelled on the educational traditions
that had (in 1930) been evolving at "Oxbridge" for centuries. Moreover,
since the time of Newton, Cambridge had been one of the world's greatest
centers of mathematics and physics, and I formed as a senior the ambition
of becoming a graduate student there.
Fortunately for me, Lady Julia Henry endowed in 1932 four choice
fellowships, one to be awarded by each of these four universities,
to support a year's study across the ocean. President Lowell in person
interviewed the candidates applying at Harvard, of whom I was one. He
asked me two questions: (i) was I more interested in theoretical or
applied mathematics? (ii) since most candidates seemed to want to go to
Cambridge, would I accept a fellowship at Oxford? Thinking that being
a theorist sounded more distinguished than being a problem-solver,
I replied that my interests were theoretical. Moreover I knew of no
famous mathematicians or physicists at Oxford, and stated that I would
try to find another means of getting to Cambridge.
As a research student interested in quantum mechanics, I attended Dirac's
lectures and was given R. H. Fowler as adviser during my first term. Like
Widder two years later [CMA, p. 82], but far less mature, I also attended
the brilliant lectures given by Hardy in each of three terms, and sampled
several other lecture courses. When I first met Hardy, he asked me how
my father was progressing with his theory of esthetics. I told him with
pride that my father's book Aesthetic Measure had just appeared. His
only comment was: "Good! Now he can get back to real mathematics". I
was shocked by his lack of appreciation!
The Julia Henry Fellow from Yale was the mathematician Marshall Hall,
who has since done outstanding work in combinatorial theory. We compared
impressions concerning the system of Tripos Examinations used at Cambridge
to rank students, for which Cambridge students were prepared by their
tutors. We agreed that Cambridge students were better trained than we,
but thought that the paces they were put through took much of the bloom
off their originality!
My course with E. C. Kemble at Harvard had left me with the mistaken
impression that quantum mechanics was concerned with solving the
Schrodinger equation in a physical universe containing only atomic
nuclei and electrons. Dirac's lectures were much more speculative, and
it was not until I heard Carl Anderson lecture on the newly discovered
positron in the spring of 1933 that I realized that Dirac's lectures
were concerned with a much broader concept of quantum mechanics than
that postulated by Schrodinger's equations.
In the meantime, I had decided to concentrate on finite group theory,
and was transferred to Philip Hall as adviser. By that spring, I had
rediscovered lattices (the "Dualgruppen" of Dedekind; see ¤8), which had
also been independently rediscovered a few years earlier by Fritz Klein
who called them "VerbŠnde". Recognizing their widespread occurrence in
"modern algebra" and point-set topology, I wrote a paper giving "a number
of interesting applications" of what I called "lattice theory", and wrote
my father about them. He mentioned my results to Oystein Ore at Yale,
who had taught algebra to both Marshall Hall and Saunders Mac Lane. Ore
immediately recalled Dedekind's prior work, and soon a major renaissance
of the subject was under way. This has been ably described by H. Mehrtens
in his book, Die Entstehung der VerbŠnde, cf. also [GB, Part I].
In retrospect, I think that I was very lucky that Emmy Noether, Artin, and
other leading German algebraists had not taken up Dedekind's "Dualgruppe"
concept before 1932. As it was, by 1934 Ore had rediscovered the idea of
C. S. Peirce (see 8), of defining lattices as partially ordered sets, and
by 1935 he had done a far more professional job than I in applying them
to determine the structure of algebras - and especially that of "groups
with operators" (e.g., vector spaces, rings, and modules). However, by
that time (in continuing correspondence with Philip Hall) I had applied
lattices to projective geometry, Whitney's "matroids", the logic of
quantum mechanics (with von Neumann), and set-theoretic topology, as well
as to what is now called universal algebra, so that my self-confidence
was never shattered!
17. THE SOCIETY OF FELLOWS
Our modern Ph.D. degree requirements were originally designed in
Germany to train young scholars in the art of advancing knowledge. The
German emphasis was on discipline, and Ph.D. advisers might well use
candidates as assistants to further their own research. Having never
"earned" a Ph.D. by serving as a research apprentice himself, Lowell
was always skeptical of its value for the very best minds, somewhat
as William James once decried "The Ph.D. Octopus". Throughout his
academic career, Lowell kept trying to imagine the most stimulating
and congenial environment in which a select group of the most able and
original recent college graduates could be free to develop their own
ideas [Yeo, Ch. XXXII]. In his last decade as Harvard's president, he
discussed what this environment should be with the physiologist L. J.
Henderson and the mathematician-turned-philosopher A. N. Whitehead,
among others.
As successful models for such a select group, these very innovative
men analyzed the traditions of the prize fellows of Trinity and Kings
Colleges at Cambridge University, of All Souls College at Oxford, and
of the Fondation Thiers in Paris. They decided that a group of about
24 young men (a natural social unit), appointed for a three year term
(with possible reappointment for a second term), dining once a week with
mature creative scholars called senior fellows, and lunching together as
a group twice a week, would provide a good environment. The only other
stated requirement was negative: "not to be a candidate for a degree"
while a junior fellow.
The proper attitude of such a junior fellow was defined in the following
noble "Hippocratic Oath of the Scholar" [S0F, p. 31], read each year
before the first dinner:
'You have been selected as a member of this Society for your personal
prospect of serious achievement in your chosen field, and your promise
of notable contributions to knowledge and thought. That promise you must
redeem with your whole intellectual and moral force.
You will practice the virtues, and avoid the snares, of the scholar. You
will be courteous to your elders who have explored to the point from
which you may advance; and helpful to your juniors who will progress
farther by reason of your labors. Your aim will be knowledge and wisdom,
not the reflected glamour of fame. You will not accept credit that is
due to another, or harbor jealousy of an explorer who is more fortunate.
You will seek not a near, but a distant, objective, and you will not
be satisfied with what you may have done. All that you may achieve or
discover you will regard as a fragment of a larger pattern, which from
his separate approach every true scholar is striving to descry.
To these things, in joining the Society of Fellows, you dedicate
yourself.'
Some months later, we were informed frankly that if one out of every
four of us had an outstanding career, the senior fellows would feel that
their enterprise had been very successful.
Like all institutions, Harvard's Society of Fellows has changed with the
times. Thus junior fellows may now be women, and may use their work to
fulfill departmental Ph.D. requirements. But the ceremony of reading
the preceding statement to new junior fellows at their first dinner in
the Society's rooms has not changed.
As a junior fellow, I was so absorbed in developing my own ideas and in
exploring the literature relating to them (especially abstract algebra,
settheoretic topology, and Banach spaces), that I attended only two
Harvard courses or seminars. I had studied in 1932-33 Stone's famous
Linear Transformations in Hi/bert Space, one of the three books that
established functional analysis (the study of operators on "function
spaces") as a major area of mathematics.48 Moreover, Whitney was rapidly
becoming famous as a topologist with highly original ideas. Therefore,
I audited Stone's course (Mathematics 12) on the theory of real functions,
which he ran as a seminar, in 1933-34, and I participated actively in
Whitney's seminar on topological groups in 1935-36.
I also attended the weekly colloquia. At an early one of these, Stone
announced his theorem that every Boolean algebra is isomorphic to a
field of sets. Having proved the previous spring that every distributive
lattice was isomorphic to a ring of sets, I became quite excited. He went
on to prove much deeper results in the next few years, while I kept on
exploring the mathematical literature for other examples of lattices.
There were five mathematical junior fellows during the years 1933-44:
John Oxtoby, Stan Ulam, Lynn Loomis, Creighton Buck, and myself. In
addition, the mathematical logician W. V. Quine was (like me) among
the first six selected, as was the noted psychologist B. F. Skinner.
Like many other junior fellows, the last four of the six just named
joined the Harvard faculty, where their influence would be felt for
decades. But that is another story!
Ulam and Oxtoby. Instead, I will take up here the accomplishments of
Ulam and Oxtoby through 1944. Most important was their proof that, in
the sense of (Baire) category theory, almost every measure-preserving
homeomorphism of any "regularly connected" polyhedron of dimension r 2 is
metrically transitive. As they observed in their paper,49 "the effect of
the ergodic theorem was to replace the ergodic hypothesis (of Ehrenfest)
by the hypothesis of metric transitivity (of Birkhoff)". Philosophically,
therefore, they in effect showed that Hamiltonian systems should
almost surely satisfy the ergodic theorem. This constituted a notable
modern extension of the tradition of Lagrange, Laplace, Poincare, and
G. D. Birkhoff.
During World War H, like von Neumann (but full-time), Ulam worked at Los
Alamos. There he is credited as having conceived, independently of Edward
Teller, the basic idea underlying the H- bomb developed some years later.
Two other junior fellows of the same vintage who applied mathematics
to important physical problems after leaving Harvard were John Bardeen
and James Fisk. After joining the Bell Telephone Laboratories in 1938,
Bardeen went on to win two Nobel prizes. Fisk became briefly director
of research of the Atomic Energy Commission after the war, and finally
vice president in charge of research at the Bell Telephone Labs. I hope
that these few examples will suggest the wisdom and timeliness of the
plan worked out by Lowell, Whitehead, Henderson, and others, and endowed
by Lowell's own fortune. Of the first fifty junior fellows, no less
than six (Bardeen, Fisk, W. V. Quine, Paul Samuelson, B. F. Skinner,
and E. Bright Wilson) have received honorary degrees from Harvard!
The Putnam Competition. The aim of Lowell and his brother-in-law William
Lowell Putnam, to restore undergraduate admiration for intellectual
excellence (see ¤7), was given a permanent national impetus in 1938 with
the administration of the first Putnam Competition by the Mathematical
Association of America. For a description of the establishment of this
competition, in which George D. Birkhoff played a major role, and its
subsequent history to 1965, I refer you to the Amer. Math. Monthly 72
(1965), 469-83. Of.the five prewar Putnam Fellowship winners, Irving
Kaplansky is current director of the NSF funded Mathematical Sciences
Research Institute in Berkeley, after a long career as a leading American
algebraist; he and Andrew Gleason have recently been presidents of the
AMS; while Richard Arens and Harvey Cohn have also had distinguished and
productive research careers. All of them except Gleason (who joined the
U.S. Navy as a code breaker in 1942) contributed through their teaching
to the mathematical vitality of Harvard in the years 1938- 44!
18. FOUR NOTABLE MEETINGS
I shall now turn to some impressions of the moods of, and Harvard's
participation in, four notable meetings that took place in the late
1930s: the International Topological Congress in Moscow in 1935; the
International Mathematical Congress in Oslo and Harvard's Tercentenary
in 1936; and the Semicentennial meeting of the AMS in 1938.
Lefschetz was a major organizer of the 1935 Congress in Moscow. He, von
Neumann, Alexander and Tucker went to it from Princeton; Hassler Whitney,
Marshall Stone, David Widder (informally) and I from Harvard. Whitney's
paper [CAM, pp. 97-118] describes the fruitfulness for topology of this
Congress, an event which Widder also mentions [CAM, p. 82]. For me,
it provided a marvellous opportunity to get first-hand impressions of
the thinking of many mathematicians whose work I admired, above all
Kolmogoroff,
but also Alexandroff and Pontrjagin.
Widder, Stone, and I met in Helsinki, just before the Congress, whence
we took a wood-fired train to Leningrad. There we were greeted at the
station by L. Kantorovich and an official Cadillac. By protocol, he took
a street-car to his home, where he had invited us for tea, while we were
driven there in the Cadillac. I was astonished! I would have been even
more astonished had I realized that within two years I would be studying
the work of Kantorovich on vector lattices (and that of Freudenthal,
also at the Congress); that 20 years later I would be admiring his book
with V.1. Krylov on Approximation Methods of Higher Analysis; or that
in about 30 years he would get a Nobel prize for inventing the simplex
method of linear programming, discovered independently by George Dantzig
in our country somewhat later-50
Marshall Stone, infinitely more worldly wise than I, reported privately
that evening Kantorovich's disaffection with the Stalin regime. I was
astonished for the third time, having assumed that all well-placed Soviet
citizens supported their government. Many of my other naive suppositions
were corrected in Moscow.
For example, when I expressed to Kolmogoroff my admiration for his
Grundbegrzjfe der Wahrscheinlichkeitsrechnung, he remarked that he
considered it only an introduction to Khinchine's deeper Asymptotische
Gesetze der Wahrscheinlichkeitsrechnung. The algebraist Kurosh and I
made a limited exchange of opinions in German, and I also met at the
Congress I. Gelfand, who would get an honorary degree at Harvard 50
years later! Above all, I was impressed by the crowding and poverty
I saw in Moscow (the famine had just ended a year earlier), and the
inaccessibility of government officials behind the Kremlin walls.
At the International Mathematical Congress in Oslo a year later, I was
dazzled by the depth and erudition of the invited speakers, and the
panorama of fascinating areas of research that their talks opened up. I
was permitted to present three short talks (Marcel Riesz gave four!),
and there seemed to be an adequate supply of listeners for all the talks
presented. Paul Erdoes gave one talk, and he must have been the only
speaker who did not wear a necktie!
Naturally, I was pleased that the two Fields medallists (Lars Ahifors and
Jesse Douglas) were both from Cambridge, Massachusetts, and delighted that
the 1940 International Congress was scheduled to be held at Harvard, with
my father as Honorary President! I was also impressed by the efficient
organization for the Zen tralblatt of reviews of mathematical papers
displayed by Otto Neugebauer (cf. [LAM, ¤21]). This convinced me of
the desirability of transplanting his reviewing system to AMS auspices,
if funds could be found to cover the initial cost. Of course, this was
accomplished three years later.
On both my 1935 and 1936 trips to Europe, I stopped off in Hamburg to
see Artin in Hamburg. In 1935, I also stopped off in Berlin to meet
Erhard Schmidt and my future colleague Richard Brauer and his brother
Alfred. Near Hamburg in 1936, the constant drone of military airplanes
made me suddenly very conscious of the menace of Hitler's campaign
of rearmament!
The serene atmosphere of Harvard's Tercentenary celebration that
September was a welcome contrast, and I naturally went to the invited
mathematical lectures. Among them, Hardy's famous lecture on Ramanujan
was most popular.5' It did not bother me that the technical content of
the others was over my head, and I dare say over the heads of the vast
majority of the large audiences present!
The summer meeting of the AMS was held at Harvard in conjunction
with this Tercentenary; its description in the Bull. Amer. Math. Soc.
(42,
761-76) states that: "Among the more than one thousand persons attending
the meetings ..., approximately eight hundred registered, of whom 443
are members of the Society". What a contrast with the Harvard of John
Farrar and Nathaniel Bowditch, a hundred years earlier!
A fourth notable mathematical meeting celebrated the Golden Jubilee of
the AMS at Columbia University in September, 1938. It was to celebrate
this anniversary that R. C. Archibald wrote the historical review [Arc]
on which I have drawn so heavily, here and in [LAM], and that my father
surveyed "Fifty years of American mathematics" from his contemporary
standpoint.
The meeting honored Thomas Scott Fiske of Columbia, who had by then
attended 164 of the 352 AMS meetings that had taken place. (Of these
352 meetings, 221 had been held at Columbia.) A review of the occasion
was published in the Bull. Amer. Math. Soc. 45 (1939), 1-51, including
Fiske's reminiscence that, in the early days of the AMS, C. S. Peirce was
"equally brilliant, whether under the influence of liquor or otherwise,
and his company was prized ... so be was never dropped ... even though
he was unable to pay his dues."
19. ANOTHER DECADE OF TRANSITION
In ¤12 and ¤13, I recalled the mathematical activity in physics and
philosophy at Harvard through 1940. I shall now give some impressions
of the main themes of research and teaching of the Harvard mathematics
department from 1930 through 1943.
During these years, it was above all G. D. Birkhoff who acted as a magnet
attracting graduate students to Harvard. After getting an honorary degree
from Harvard in 1933, he served as dean of the faculty under President
Conant from 1934 to 1938, meanwhile being showered with honorary degrees
and elected a member of the newly founded Pontifical Academy. He directed
the theses of C. B. Morrey, D. C. Lewis, G. Baley Price, Hassler Whitney,
and 12 other Harvard Ph.D.'s after 1930. In 1935, he wrote with Magnus
Hestenes an important series of papers on natural isoperimetric conditions
in the calculus of variations, and throughout the 1930s he wrote highly
original sequels to his earlier papers on dynamical systems, the four
color theorem, etc., while continuing to lecture to varied audiences
also on relativity, his ideas about quantum mechanics, and his philosophy
of science.
Meanwhile, Walsh and Widder pursued their special areas of research in
classical analysis, Walsh publishing many papers as well as a monograph on
"Approximation by polynomials in the complex domain" in the tradition
of Montel, Widder his well-known Laplace Transform. Variety within
classical analysis and its applications was provided at Harvard by Walsh
and Widder. For example, Joseph Doob and Lynn Loomis wrote theses with
Walsh, while Ralph Boas and Harry Pollard wrote theses with Widder during
these years. While Ahlfors was there (from 1935 to 1938), Harvard's
national leadership in classical analysis was even more pronounced,
being further strengthened by the presence of Wiener in neighboring MIT.
Coolidge, Graustein, and Huntington continued to give well-attended
courses on geometry and axiomatic foundations, keeping these subjects
very much alive at Harvard. In particular, Coolidge gave a series of
Lowell lectures on the history of geometry, while Graustein published
occasional papers on differential geometry, and served as editor of the
Transactions Amer. Math. Soc. from 1936 until his death in 1941. In
his role of associate dean,
Graustein also worked out a detailed "Graustein plan" which metered
skillfully the tenure positions available in each department of the
Faculty of Arts
and Sciences, aimed at achieving a roughly uniform age distribution.
Former Presidents of the Society at Harvard University, September 1936
Left to right: White, Fiske, Bliss, Osgood, Coble, Dickson, and Birkhoff.
Moreover, every department member performed capably and conscientiously
his teaching and tutorial duties, undergraduate honors being "based on
the quality of the student's work in his courses, on his thesis, and
on the general examination" (the latter a less sophisticated version of
the Cambridge Tripos).
New trends. However, this seeming emphasis on classical mathematics was
deceptive. By 1935, Kellogg had died, Osgood had retired, and Morse had
gone to the Institute for Advanced Study at Princeton. Their places
were taken by Marshall Stone, Hassler Whitney, Saunders Mac Lane,
and myself. (I recall that like Walsh and Widder, Stone and Whitney
were Ph.D. students of G. D. Birkhoff.) Stone, already famous as a
functional analyst, was concentrating on Boolean algebra and its relation
to topology. Whitney was founding the theories of differentiable manifolds
and sphere bundles [CMA, pp. 109-1171. Mac Lane was exhibiting great
versatility and expository skill in papers on algebra and graph theory.
Before 1936, when I became a faculty instructor after attending all
the four "notable meetings" described in 17, I had never taught a
class. I realized that my survival at Harvard depended on my success
in interesting freshmen in the calculus, and was most grateful for the
common sense advice given by Ralph Beatley regarding pitfalls to be
avoided. "Teach the student, not just the subject", and "face the class,
not the backboard" were two of his aphorisms. All new instructors were
"visited" by experienced teachers, who reported candidly on what they
witnessed at department meetings, usually with humor. I was visited
by Coolidge, and became so unnerved that I splintered a pointer while
sliding a blackboard down. I survived the test, and became a colleague
of Stone, Whitney, and Mac Lane. Thus, after 1938, the four youngest
members of the Harvard mathematical faculty were primarily interested
in functional analysis, topology, and abstract algebra. In addition,
Quine had introduced a new full graduate course in mathematical logic
(Mathematics 19). This treated general "deductive systems", thus going
far beyond Huntington's half-course on "fundamental concepts".
I am happy to say that Stone (Harvard '22), Whitney, and Mac Lane are
still active, while both Widder and Beatley (Harvard '13) are in good
health. Stone recently managed the AMS conference honoring von Neumann,
while Whitney, Mac Lane, and Widder are fellow contributors to the series
of volumes in which this report is being published.
David Widder and I were put in charge of the Harvard Colloquium in
the years 1936-40. My father and Norbert Wiener usually sat side
by side in the front row, and made lively comments on almost every
lecture. C. R. Adams and Tamarkin often drove up from Brown to attend
the colloquium, bringing graduate students with them. My role included
shopping conscientiously for good cookie bargains for these convivial and
sociable affairs, where tea was served by a faculty wife. Most interesting
for me were the talks by Ore, von Neumann, and Menger on lattice theory,
then my central research interest. In 1938, these three participated in
the first AMS Symposium on lattice theory (see Bull. Amer. Math. Soc. 44
(1938), 793-837), with Stone, Stone's thesis student Holbrook MacNeille,
who would become the first executive director of the AMS, and myself. Two
years later the AMS published the first edition of my book Lattice Theory.
Beginning in 1937-38, Mac Lane and I taught alternately a new
undergraduate full course on algebra (Mathematics 6), which immediately
became very popular. I began the course with sets and ended with groups;
in the second year, my students included Loomis, Mackey, and Philip
Whitman. The next year, Mac Lane began with groups and ended with sets;
his students included Irving Kaplansky. After amicable but sometimes
intense discussions, we settled on the sequence of topics presented
in our Survey of Modern Algebra (Macmillan, 1941). In it and in our
course, we systematically correlated rigorous axiomatic foundations with
elementary applications to number theory,
the theory of equations, geometry, and logic.
20. END OF AN ERA
Meanwhile, war clouds were getting more and more threatening! Germany
and Russia invaded and absorbed Poland in 1939, and the International
Mathematical Congress scheduled to be held at Harvard was postponed
indefinitely. After the fall of France in the spring of 1940,
Germany's invasion of Russia, and Pearl Harbor, it became clear that our
country would have to devote all its strength to winning a war against
totalitarian tyranny.
It was clear to me that our war effort was unlikely to be helped by any
of the beautiful ideas about "modern" algebra, topology, and functional
analysis that had fascinated me since 1932, and so from 1942 until the war
ended, I concentrated my research efforts on more relevant topics. Most
interesting of these scientifically was trying to predict the underwater
trajectories of airlaunched torpedoes, a problem on which I worked with
Norman Levinson and Lynn Loomis, a study in which my father also took
an interest. I believe that our work freed naval research workers in
the Bureau of Ordnance to concentrate on more urgent and immediate tasks.
George D. Birkhoff. During these years, my father continued to think
about natural philosophy, much as Simon Newcomb and C. S. Peirce
had. He lectured on a broad range of topics at the Rice Institute,
and also in South America and Mexico, where he and my mother were good
will ambassadors cooperating in Nelson Rockefeller's effort to promote
hemispheric solidarity against Hitler.
My father finally succeeded in constructing a relativistic model of
gravitation which was invariant under the Lorentz group, yet predicted
the "three crucial effects" whose explanation had previously required
Einstein's general theory of relativity. Because it assumed Minkowski's
four-dimensional flat
space-time, the model also accommodated electromagnetic phenomena
such as the relativistic motion of particles in electron and proton
accelerators.52
The exploration of this theory and other ideas he had talked about
provided an important stimulus to the development of the National
University of Mexico into a significant research center. The honorary
degree that I received there in 1955, as well as my honorary membership
in the Academy of Sciences in Lima, were in large part tributes to his
influence on the two oldest universities in the Western Hemisphere.
The department pamphlet of 1942-43. In spite of the war, the pamphlet
of the Harvard mathematics department for 1942-43 gives the illusion
of a balance of mathematical activities that had been fairly constant
for nearly a decade. Although President Conant had gone to Washington
to run the National Defense Research Council with Vannevar Bush, he had
left intact the plan of undergraduate education worked out by Lowell.
Perhaps suggestive of future trends, Beatley was in charge of three
sections of freshman calculus, Chuck (C. E.) Rickart (then a B.P.) of
two; only Whitney's and Mac Lane's sections were taught by tenured
research faculty members. Stone, Kaplansky, and I taught second-year
calculus; of the three of us, Kaplansky was the most popular teacher.
Advanced calculus was taught by Whitney and my father, geometry by
Coolidge, and undergraduate algebra (Mathematics 6) by Ed Hewitt. Real and
complex analysis (our main introductory graduate courses) were taught by
Loomis and Widder, respectively; ordinary differential equations (a full
course) by my father; and mechanics by van Vleck. Graustein had died,
but differential geometry was taught by Kaplansky; topology was taught
by Mac Lane. Widder's student Harry Pollard and I taught Mathematics
lOb and 1 Oa, respectively.
Applied mathematics. The only "applied" touch visible in this 1942-
43 pamphlet was my changed wording for the description of Mathematics 10a:
I announced that it would treat "the computation of [potential] fields
in special cases of importance in physics and airfoil theory", and that
"In 1942-43, analogous problems for compressible non-viscous flow will
also be treated, and emphasis ... put on airfoil theory and air resistance
to bullets". Also, two courses in "mechanics" were listed: Mathematics
4 to be taught by van Vleck, and Mathematics 8 by Kemble. Actually, van
Vieck and Dean Westergard of the Engineering School had agreed with me
that we should teach Mathematics 4 (= Engineering Science 6) in rotation.
When my turn came, John Tate (in naval uniform) was in the class.
Moreover, appreciation for "applied" mathematics as such was reviving
in the Harvard Engineering School, with whose faculty I was getting
acquainted as part of my "continuing education". Though they did not
worry about Weierstrassian rigor, let alone Cantorian set theory or
symbolic logic, Richard von Mises and my friend Howard Emmons knew
infinitely more about real flows around airfoils than I. Associated with
von Mises were his coauthor Philipp Frank, by then primarily interested
in the philosophy of science, and Stefan Bergman of "kernel function"
fame, as well as Hilda Geiringer von Mises at Wheaton and Will Prager at
Brown. After emigrating together from Berlin to Istanbul to escape Hitler,
all of these distinguished mathematicians had come to New England, 54
greatly enhancing its role in Continuum Mechanics, including especially
the mathematical analysis of fluid motions, elastic vibrations, and
plastic deformations.
But most important for the post-war era, the Gordon McKay bequest
of 1903, which Nathaniel Shaler had labored so hard to secure for
Harvard, was about to become available. In addition, the 1940 bequest of
$125,000, given by Professor A. E. Kennelly because "the great subject
of mathematics applied to electric engineering, together with its study
and teaching, have throughout my life been an inspiration in my work",
was being used to pay the salary of Howard Aiken, while he worked at
IBM on the development of a programmable computer. Harvard was getting
ready for the dawn of the computer age!
NOTES
Further Supplementary Notes and references for this essay, identified
by letters, will be deposited in the Harvard Archives.
1 For Peirce's career and influence, see [Pei] and [DAB 14, 393-7].
2 See John Pickering's Eulogy of Nathaniel Bowditch, Little Brown, 1868;
[DAB 2, 496-8]; [EB 4, p. 31], and [Bow, vol. 1, pp. 1-165].
3 See p. 69 of Pickering's Eulogy. The accepted value today is (a -
b)/a = 1/297.
4 Benjamin Peirce senior also wrote a notable history of Harvard,
recording the many benefactions made to it before the American Revolution.
5 See [Cat, 1835].
6 See [TCH, p. 220] and [Qui]. Kirkland was succeeded by Josiah Quincy,
who would be followed in 1846 by Edward Everett.
7 For Lovering's scientific biography, by B. 0. Peirce, see [NAS 2:
327-44]. He was president of the American Academy from 1880 to 1892.
8 For William Bond's biography, see [DAB 2, 434-5]. His son George
succeeded him as director of the Harvard Observatory. For more
information, see The Harvard College Observatory: the first four
directorships, 1839-1919, by Bessie Z. Jones and Lyle G. Boyd, Harvard
University Press, 1971.
9 See Simon Newcomb's autobiography, Reminiscences of an Astronomer for
colorful details about his life, and [DAB 13, 452-5] for a biographical
survey. Hill's first substantial paper was published in Runkle's
Mathematical Monthly. For his later work, see [NAS 8: 275-309], by
E.W. Brown, and [DAB 9, 32-3].
10 See [DAB 7, 447-9] for biographies of Gould (Harvard '44), who founded
the Astronomical Journal, and his father of the same name.
11 [DAB 14, 393-7]. As superintendent, he received $4000/yr, which must
have doubled his salary.
12 Runkle was MIT President from 1870 to 1878.
13 For the model used, see Newcomb's Popular Astronomy, 5th ed., Part IV,
Ch. III. Until nuclear energy was discovered, the source of the sun's
energy was a mystery. W. E. Story was Byerly's classmate.
14 These are associated with systems of linear DE's of the form dx1/dt =
15 Hamilton had discovered quaternions in 1843, while Cayley's famous
paper on matrices was published in 1853.
16 U.S. government employees helped to prepare Peirce's manuscript for
lithographing.
17 Crelle's J. fuer Math. 84 (1878), 1-68.
18 See [HH, p. 42], Eliot's article on "The New Education" in the Atlantic
Monthly 23 (1869), expresses Eliot's opinions before he became president;
his inaugural address is reprinted in [Mor, pp. lix-lxxviii].
19 When MIT was Boston Tech., by Samuel C. Prescott, MIT Press, 1954.
20 Byerly was also active in promoting Radcliffe (Harvard's "Female
Annex"), where Byerly Hall is named for him; see [DAB, Suppl.,
pp. 145-61. Elizabeth Cary Agassiz was its president. For the Radcliffe
story, see [HH, pp. 193-7].
21 [S-G, p. 69]. Oliver Wendell Holmes Sr. wittily observed that
professorial chairs in "astronomy and mathematics" and "geology and
zoology", like those of Louis Agassiz and his classmate Benjamin Peirce,
should be called "settees, not chairs".
22 "See 7 (pp. 32-4) of my article in [Tar, pp. 25-78], and pp. 293-5 of
my father's article in [AMS, pp. 270-315], reprinted in [GDB, vol. iii,
pp. 60552]. A biography of Osgood by J. L. Walsh will be included in
this volume. For "The Scientific Work of Maxime BOcher", see my father's
article in the Bull. AMS 25 (1919), 197-215, reprinted in [GDB, vol. iii,
pp. 227- 45].
23 For Klein's great influence on American mathematics, see the Index of
[Arc]; also [Tar, pp. 30-32], and ¤10 of my article with M. K. Bennett
in Wm. Aspray and Philip Kitcher (eds.), History and Philosophy of Modern
Mathematics, University of Minnesota Press, 1988.
24 Bull. Amer. Math. Soc. 5 (1898), 59-87, and vol. 7 of the AMS
Colloquium Publications (1914).
25 See Bouton's Obituary in Bull. Amer. Math. Soc. 28 (1922), 123-4.
26 For an appreciative account of Coolidge's career, see the Obituary
by D. J. Struik in the Amer. Math. Monthly 62 (1955), 669-82. Ref. 60
there to a biography of Graustein by Coolidge seems not to exist.
27 Ann. of Math. 10 (1909), 181-92.
28 Senate Document # 304 (41 pp.), U.S. Printing Office, 1940. See
also EVH in Quart. Amer. Statist. Assn. (1921), 859-70, and
Trans. Amer. Math. Soc. 30 (1928), 85-110.
29 [Yeo, p. 67]. Owen Wister's book Philosophy Four gives an amusing
description of the "Zeitgeist" at Harvard in those years.
30 Lowell's The Government of England and (his friend) James Bryce's Amer-
ican Commonwealth were the leading books on these two important subjects.
See [Yeo, p. 111]. Lord Bryce, when British ambassador to the United
States, gave Lowell's manuscript a helpful critical reading.
311n the two volumes [Low] and [Yeo].
32Cf. [LAM, ¤12]. For many years, the Putnams graciously hosted dinner
meetings of the visiting committee, to which all the members of the
mathematics department were invited.
33See [GDB, pp. xv-xxi] for Veblen's recollections and appraisal of my
father's work. The grandson of a Norwegian immigrant, Veblen had graduated
at 18 from the University of Iowa before going to Harvard. See [Arc,
pp. 206-18], for biographies of Veblen and my father. MATHEMATICS AT
HARVARD, 1836-1944 55
34 See G. G. Lorentz, K. Jetter, and S. D. Riemenschneider, Birkhoff
Interpolation, Addison-Wesley, 1983.
35 This classic is currently being republished by the American Physical
Society in translated form, prefaced by an excellent historical
introduction by Daniel Goroff.
36 be outline of these (Bull. Amer. Math. Soc. 27, 67-69) includes
many topics of general interest that were not included in the printed
volume. These include from the first lecture: (7) methods of computation
and their validity, (8) relativistic dynamics, and (9) dissipative
systems. The last lecture was entitled "The significance of dynamical
systems for general scientific theory", and dealt with (1) the dynamical
model in physics, (2) modern cosmogony and dynamics, (3) dynamics and
biological thought, and (4) dynamics and philosophical speculation. My
father's interest in relativity presumably dates from a course he took
with A. A. Michelson at Chicago around 1900; see his review "Books on
relativity", Bull. Amer. Math. Soc. 28 (1922), 213-21.
37 Cf. [GDB, III, pp. 365-81], reprinted from the Ann. of Math. 33
(1932), 329-45, and the Fifth Yearbook (1930) of the NCTM.
38 Harvey Davis, after teaching mathematics (as a graduate student),
physics, and engineering [Mor, p. 430] at Harvard, became president of
the Stevens Institute of Technology. Conant, of course, was President
Lowell's successor at Harvard.
39 Mrs. William Lowell Putnam lent her summer home to the Birkhoffs
during the summer of 1927; see also 17
40 See [Whi], in which pp. 125-65 contain an essay by Quine on "Whitehead
and the rise of modern logic".
41 Dunbam Jackson had been Secretary of the Division since 1913. Other
losses were: the differential geometer Gabriel Marcus Green (cf. Bull.
Amer. Math. Soc. 26, pp. 1-13), and Leonard Bouton (in 1921).
42 Actually, Walsh had asked Osgood to supervise his thesis, but Osgood
declined. Like Coolidge and Huntington ('95), Graustein ('10) and Walsh
('16) had both been Harvard undergraduates.
43 Closely related to Haar functions, these would prove very useful for
signal processings in the 1970s.
44 For a charming description of Morse and his contributions, see Raoul
Bott, Bull. Amer. Math. Soc. (N.S.) 3 (1980), 907-50.
"The majority of students, not being interested in a mathematical career,
presumably had very different impressions.
46 Though original, my ideas were not new. Tamarkin kindly softened
the blow by writing that my paper "showed promise". Six months later,
I published a revised and very condensed paper containing my sharpest
results in Bull. Amer. Math. Soc. 39 (1933), 601-7.
47 In Stone's words [Tex, p. 15], "the Harvard of my student days could
not have offered more opportunity or encouragement to a student eager
for study and learning."
"The others were von Neumann's Mathematische Grundlagen der Quan
tenmechanik and Banach's Th`orie des Operations LinŽaires; cf. Historia
Math. 11 (1984), 258-321.
49 Ann. of Math. 42 (1941), 874-920. See also Ulam's charming Adventures
of a Mathematician (Scribners, 1976) for other aspects of his life.
50 For the story of the independent discoveries of linear programming by
Kantorovich (1939), Frank Hitchcock (1941), T. C. Koopmans ('-j 1944),
and G. B. Dantzig (1946), see Robert Dorfman, Ann. Hist. Comput. 6
(1984),283-95.
51 Published in the Amer. Math. Monthly 44 (1937), 137-55, and as Ch. I
of Hardy's book Ramanujan (Cambridge University Press, 1940).
52 See [GDB, pp. 920-83], and the article by Carlos Graef Fernandez in pp.
167-89 of the AMS Symposium Orbit Theory (G. Birkhoff and R. E. Langer,
eds.), Amer. Math. Soc., 1959.
53 In a very different way, von Mises' book Probability, Statistics and
Truth was a famous contribution to the foundations of probability theory,
which are shaky because sequential frequencies are not countably additive.
54 Minkowski's son-in-law Reinhold RUdenberg had also come from the
University of Berlin to Harvard, while Hans Reissner had come to MIT.
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