A History in Sum

Steve Nadis,Shing-Tung Yau

• Editorial: Harvard University Press
• Año: 2013
• ISBN:
  9780674727892
• Categoría: Mathematics
• Categoría: History & Philosophy of Mathematics

In the twentieth century, American mathematicians began to make critical advances in a field previously dominated by Europeans. Harvard's mathematics department was at the center of these developments. A History in Sum is an inviting account of the pioneers who trailblazed a distinctly American tradition of mathematics—in algebraic geometry and topology, complex analysis, number theory, and a host of esoteric subdisciplines that have rarely been written about outside of journal articles or advanced textbooks. The heady mathematical concepts that emerged, and the men and women who shaped them, are described here in lively, accessible prose.The story begins in 1825, when a precocious sixteen-year-old freshman, Benjamin Peirce, arrived at the College. He would become the first American to produce original mathematics—an ambition frowned upon in an era when professors largely limited themselves to teaching. Peirce's successors—William Fogg Osgood and Maxime Bôcher—undertook the task of transforming the math department into a world-class research center, attracting to the faculty such luminaries as George David Birkhoff. Birkhoff produced a dazzling body of work, while training a generation of innovators—students like Marston Morse and Hassler Whitney, who forged novel pathways in topology and other areas. Influential figures from around the world soon flocked to Harvard, some overcoming great challenges to pursue their elected calling. A History in Sum elucidates the contributions of these extraordinary minds and makes clear why the history of the Harvard mathematics department is an essential part of the history of mathematics in America and beyond.

1

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BENJAMIN PEIRCE AND THE SCIENCE OF “NECESSARY CONCLUSIONS”

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Benjamin Peirce came to Harvard at the age of sixteen and essentially never left, all the while clinging to the heretical notion that mathematicians ought to do original mathematics, which is to say, they should prove new theorems and solve problems that have never been solved before. That attitude, sadly, was not part of the orthodoxy at Harvard, nor was it embraced at practically any institution of higher learning in the United States. At Harvard and elsewhere, the emphasis was on teaching math and learning math but not on doing math. This approach never sat well with Peirce, who was unable, or unwilling, to be just a passive recipient of mathematical doctrine. He felt, and rightfully so, that he had something more to contribute to the field than just being a good reader and expositor. Consequently, he was driven to advance mathematical knowledge and disseminate his findings, even though the university he worked for did not share his enthusiasm for research or mathematics journals. (The “publish or perish” ethic, evidently, had not yet taken hold.)

When Peirce was just twenty-three years old, newly installed as a tutor at Harvard, he published a proof about perfect numbers: positive integers that are equal to the sum of all of their factors, including 1. (Six, for instance, is a perfect number: its factors, 3, 2, and 1, add up to 6. Twenty-eight is another example: 28 = 14 + 7 + 4 + 2 + 1.) All the perfect numbers known at that time—and still to this day—were even. Peirce wondered whether odd perfect numbers might exist, and his proof, which is discussed later in this chapter, placed some constraints on their existence. Despite the fact that this work turned out to be more than fifty years ahead of its time, it did not garner international acclaim—or any notice, for that matter—mainly because the leading European scholars did not take American mathematics journals seriously, nor did they expect them to publish anything of note. Nevertheless, Peirce’s accomplishment did signal, to anyone who might have been paying attention, that a new era of mathematics was starting at Harvard—one that the school’s administration could not suppress, even though it did nothing to encourage Peirce in this direction.

Peirce had, however, received strong encouragement from Nathaniel Bowditch, who was considered one of the preeminent mathematicians in the United States. Bowditch helped cultivate Peirce’s interest in “real,” cutting-edge mathematics, and had Bowditch made a different career decision, he might have played an even more direct role in his protégée’s education. In 1806, Harvard offered Bowditch the prestigious Hollis Chair of Mathematics and Natural Philosophy. Bowditch turned down that offer, just as he turned down subsequent offers from West Point and the University of Virginia. But he did not entirely turn his back on Harvard; he later served as a fellow to the Harvard Corporation during a term that overlapped with Peirce’s years there as a student, tutor, and faculty member.

As leading mathematicians go, Bowditch was something of an anomaly. He was almost entirely self-educated; he had never gone to college, nor did he attend high school. Instead, he left school at the age of ten to join the workforce, assisting his father in the cooper trade, making barrels, casks, and other wooden vessels. He helped his father for two years and then joined the shipping industry. After voyaging to distant places like Sumatra and the Philippines, he returned to Massachusetts where he entered the insurance business, while resuming his mathematical studies on the side. Although his exposure to formal education was brief, he had learned enough math on his own to know that a university could never offer him as much money as he came to earn in his job as president of the Essex Fire and Marine Insurance Company.

Bowditch nevertheless continued to pursue his interest in mathematics, focusing on celestial mechanics—the branch of astronomy that involves the motions of stars, planets, and other celestial objects. By 1806, the year Bowditch was recruited by Harvard, he had read all four volumes of Pierre-Simon Laplace’s treatise Mécanique Céleste. (The fifth volume came out in 1825.) Bowditch, in fact, did a good deal more than just read it; he set about the task of translating the first four volumes of Laplace’s great work. His efforts went beyond mere translation—no mean task in itself—and included a detailed commentary that helped bring Laplace within the grasp of American astronomers and mathematicians, who, for the most part, had not been able to understand his treatise before. Bowditch not only brought Laplace’s work up to date but also filled in many steps that the original author had omitted. “I never came across one of Laplace’s ‘thus it plainly appears’ without feeling sure that I have hours of hard work before me to fill up the chasm and find out and show how it plainly appears,” Bowditch said.¹ The French mathematician Adrien-Marie Legendre praised Bowditch’s efforts: “Your work is not merely a translation with a commentary; I regard it as a new edition, augmented and improved, and such a one as might have come from the hands of the author himself if he had consulted his true interest, that is, if he had been solicitously studious of being clear.”²

Peirce, who was born in Salem, Massachusetts, in 1809, would probably have met Bowditch eventually, given Peirce’s manifest talent in mathematics and Bowditch’s growing reputation in the field. But they met earlier than they might have otherwise because Peirce went to a grammar school in Salem where he was a classmate and friend of Henry Ingersoll Bowditch, Nathaniel’s son. The story has it that Henry showed Peirce a mathematical problem that his father had been working on. Peirce uncovered an error, which the son brought to his father’s attention. “Bring me the boy who corrects my mathematics,” Bowditch reportedly said, and their relationship blossomed from there.³

Bowditch moved from Salem to Boston in 1823. Two years later, the sixteen-year-old Peirce moved to nearby Cambridge to enter Harvard, following in the footsteps of his father, Benjamin Peirce Sr., who attended the college and later worked as the school librarian and historian. By the time the younger Peirce arrived on campus, he already had a mentor—not some street-smart upperclassman, but Bowditch himself, who was then a nationally known figure. Hard at work on his Laplace translation at the time, Bowditch enlisted the keen eye and proofreading services of the young Peirce. The improvements suggested by Peirce were reportedly “numerous.”⁴ The first volume of Bowditch’s translation was published in 1829, the year that Peirce graduated from Harvard. The other three volumes were published in 1832, 1834, and 1839, respectively. (Independently, a separate translation of Laplace’s work came out in 1831. That book, titled The Mechanism of the Heavens, was written by Mary Somerville, a British woman who, like Bowditch, had mostly taught herself mathematics and endeavored to make Laplace accessible. Her book, too, went beyond a mere translation, containing detailed explanations that put his treatise into more familiar language.)⁵

Peirce continued to review Bowditch’s manuscripts during his tenure as a Harvard professor. “Whenever one hundred and twenty pages were printed, Dr. Bowditch had them bound in a pamphlet form and sent them to Professor Peirce, who, in this manner, read the work for the first time,” wrote Nathaniel Ingersoll Bowditch, another of Nathaniel Bowditch’s sons, in a memoir about his father. “He returned the pages with the list of errata, which were then corrected with a pen or otherwise in every copy of the whole edition.”⁶

In this way, Peirce was exposed from an early age to mathematics more advanced than could be found in any American curriculum—writings that other undergraduates simply were not privy to. Scholars have speculated that the excitement of reading and mastering Laplace’s work may have drawn Peirce to mathematical research. It is evident that Laplace’s writings made a deep impression on him. Decades later, in the pre-Civil War era, a student told Peirce that he risked incarceration for helping to rescue a runaway slave; the only consolation about being locked up in prison, the student said, was that he would finally have time to read Laplace’s magnum opus. “In that case, I sincerely wish you may be,” Peirce quipped.⁷

Peirce had, of course, an even deeper reverence for his mentor than he did for Laplace. Bowditch, in turn, was convinced that his young charge would go far, claiming that, as an undergraduate, Peirce already knew more mathematics than John Farrar, who then held the Hollis professorship.⁸ Peirce returned the favor decades later, calling Bowditch the “father of American geometry” in a treatise he wrote on analytical mechanics that was dedicated to his mentor.⁹ Before long, a similar term, “father of American mathematics,” was applied to Peirce (by the British mathematician Arthur Cayley, among others). Through the force of his personality and the originality of his work, Peirce came to be known as the leading American mathematician of his generation and, more generally, as the initiator of mathematical research at American universities.¹⁰

On that score, Peirce faced little competition. Before he entered the scene, no one thought that “mathematical research was one of the things for which a mathematical department existed,” Harvard mathematician Julian Coolidge wrote in 1924. It was certainly not a job prerequisite since there were not nearly as many people qualified to conduct high-level research, or inclined to do so, as there were available teaching slots. “Today it is commonplace in all the leading universities,” Coolidge added. “Peirce stood alone—a mountain peak whose absolute height might be hard to measure, but which towered above all the surrounding country.”¹¹

Despite the abilities Peirce exhibited at an early age, it was not obvious that he would have the opportunity to attain the aforementioned heights. After receiving his bachelor’s degree from Harvard in 1829, Peirce had essentially no options for advanced studies of mathematics in the United States, because no Ph.D. programs in math existed at the time. One could go to Europe—Göttingen, Germany, was a popular destination for mathematically inclined young Americans—but this was not a realistic possibility for Peirce, mainly for financial reasons. It appears that his family could not afford the luxury of sending him to school abroad; instead, he had to start earning a living soon after graduation.

He taught for two years at Round Hill School, a preparatory school in Northampton, Massachusetts, before returning to Harvard in 1831 to work as a tutor. But with Farrar, the Hollis chair, away in Europe at the time, Peirce was immediately placed at the head of the department. For health reasons, Farrar never resumed his full duties. Peirce continued to run the department, first as University Professor of Mathematics and Natural Philosophy, starting in 1833, and later as the Perkins Professor of Mathematics and Astronomy, starting in 1842. He retained the Perkins chair until he died in 1880—almost fifty years after joining the Harvard faculty.

Within months of his original appointment, Peirce submitted his aforementioned proof on perfect numbers to the New York Mathematical Diary, one of many journals to which he contributed, whereby he had gained a growing reputation as a talent to be reckoned with.¹² Peirce took the position that people needed to solve actual mathematical problems in order to earn the title of mathematician. “We are too prone to consider the mere reader of mathematics as a mathematician, whereas he does not much more deserve the name than the reader of poetry deserves that of poet,” wrote Peirce, by way of promoting Mathematical Miscellany, a journal that he contributed to frequently and of which he eventually (though briefly) became editor.¹³

His 1832 paper on perfect numbers concerned a topic that had attracted attention since antiquity. Euclid proved in the Elements, which he wrote around 300 B.C., that if 2ⁿ − 1 is a prime number, then 2ⁿ⁻¹(2ⁿ − 1) is a perfect number. Roughly 2,000 years later, Leonhard Euler proved that every even perfect number must be of this form. “But I have never seen it satisfactorily demonstrated that this form includes all perfect numbers,” Peirce wro...

Índice

• Cover
• Half Title
• Title Page
• Copyright
• Dedication
• Contents
• Preface
• Epigraph
• Prologue: The Early Days—A “Colledge” Riseth in the Cowyards
• 1. Benjamin Peirce and the Science of “Necessary Conclusions”
• 2. Osgood, Bôcher, and the Great Awakening in American Mathematics
• 3. The Dynamical Presence of George David Birkhoff
• 4. Analysis and Algebra Meet Topology: Marston Morse, Hassler Whitney, and Saunders Mac Lane
• 5. Analysis Most Complex: Lars Ahlfors Gives Function Theory a Geometric Spin
• 6. The War and Its Aftermath: Andrew Gleason, George Mackey, and an Assignation in Hilbert Space
• 7. The Europeans: Oscar Zariski, Richard Brauer, and Raoul Bott
• Epilogue: Numbers and Beyond
• Photographs
• Notes
• Index