From today’s vantage, it comes as a surprise to learn how slow the American academy was to embrace the practice of original mathematical research. It was at Harvard that the pursuit finally gained a foothold in the 19th century, and the history of the university’s mathematics department tracks well with the rising regard for mathematics in America. Steve Nadis and Shing-Tung Yau trace these developments in A History in Sum: 150 Years of Mathematics at Harvard (1825-1975), which they introduce below.
These days, the importance of math is widely recognized. The ongoing revolution in computers and communications devices, to take a familiar example, owes to advances in information theory, solid-state physics, and other disciplines that are, themselves, rooted in mathematics. Our country considers mathematics education a priority (even though existing programs often fall short of our goals). In terms of understanding the world around us, many people realize that the physical laws that govern our universe are—in their clearest form—mathematical statements. As Galileo Galilei once said, “Mathematics is the language with which God has written the universe.”
Isaac Newton’s law of universal gravitation is summed up in a simple mathematical expression that accurately describes the motions of objects in our solar system. Newton’s law has since been subsumed into Albert Einstein’s broader theory of general relativity, which encapsulates the workings of gravity in ten linked equations. String theory is more ambitious still, attempting to explain the particles and forces of nature (including gravity) through the geometry of tiny, hidden six-dimensional spaces. Carnegie Institution researcher Daniel Kelson, who hunts for distant galaxies, claims that his work in astronomy is “100 percent” mathematics. MIT astrophysicist Max Tegmark goes even further, suggesting that the universe we inhabit is, in itself, a “mathematical structure.”
Americans, however, have not always valued mathematics in this way. When Harvard College, the nation’s first institute of higher learning, was founded in the 1630s, “arithmetic and geometry were looked upon … as subjects fit for mechanics rather than men of learning,” the historian Samuel Eliot Morison wrote. In those early days, students did not need to demonstrate any proficiency in mathematics to gain admittance to Harvard, nor did they receive any training in the subject until their third and fourth years of school. The training they did receive was often far behind the times. Algebra, for instance, did not make its way into the Harvard curriculum until the 1720s or 1730s—roughly a century after René Descartes introduced modern algebraic notation. Calculus instruction began in the 1720s as well, more than 50 years after Newton and Leibniz put the discipline into a form that’s recognizable today.
Mathematical work that pushed the field towards new insights and new frontiers was essentially nonexistent in that era at Harvard and, indeed, elsewhere in the nation. Professors’ primary duties were to teach classes and write textbooks. The notion of proving new theorems or working through problems that had never been solved before was not encouraged.
For a couple of centuries, Colonial America and the United States remained a veritable backwater in mathematics—far removed, both geographically and intellectually, from the field’s nerve center in Europe. But that started to change in the 1800s, particularly in the latter part, and this country now sits on top of the mathematics world. While great mathematics is being done all over the globe, no other country can compete with the United States in terms of the quantity, quality, and breadth of activity taking place here.
How did that transformation come about? It may be instructive to look at Harvard—the nation’s oldest university and arguably its most influential one—to see how attitudes towards mathematics, and the culture within the field, have changed over the years and centuries.
Benjamin Peirce, who graduated from Harvard in 1829, was a key player in this tale. He couldn’t afford to go to graduate school in Europe, and no such programs then existed in the United States, so he took a job teaching at a Massachusetts prep school. In 1831, Harvard hired Peirce as a tutor and soon asked the twenty-two-year old to run its math department. A year later, Peirce published a proof about “perfect numbers”—numbers like six that are equal to the sum of their factors (3 + 2 + 1). European mathematicians, who probably never considered looking at the lowly American periodical that Peirce’s paper appeared in, reproduced his result fifty-six years later.
Rather than applauding Peirce, Harvard president Josiah Quincy urged him to focus on writing textbooks. Peirce complied with that request, publishing seven textbooks over the next ten years, while continuing to do original work in mathematics. He saved his greatest accomplishment for last—a treatise on “Linear Associative Algebra” that was not published until 1881, a year after his death. Much of this work was duplicated twenty years later by two German mathematicians who probably did not consider the American Journal of Mathematics worth following.
Although many of Peirce’s achievements did not garner international attention, he nevertheless helped establish a research ethic in mathematics that had been lacking at Harvard and at other institutions of its sort. Two of Peirce’s successors, William Fogg Osgood and Maxim Bôcher, did their doctoral work in Germany and began teaching at Harvard in the early 1890s, training students to make new contributions to their field. George David Birkhoff joined the department as an assistant professor in 1912 with aspirations of becoming the strongest mathematician in the world. A year later, Birkhoff made a name for himself by solving the famous “Geometric problem” posed by Henri Poincaré. He claimed to have worked out the proof in three months, losing thirty pounds in the process. Birkhoff’s career had many highlights—including his celebrated proof of the “ergodic theorem”—as well as some “lowlights.” Perhaps equally important is the fact that he trained 46 Ph.D. students, four of whom went on to become presidents of the American Mathematical Society and three of whom won National Medals of Science.
A mathematical research establishment was starting to take hold in this country, and that trend accelerated in the 1930s and 1940s, when upwards of 150 European mathematicians—most of whom were Jewish—emigrated to the United States. The center of gravity had shifted, and American mathematics has thrived in the years since World War II.
It would be foolish to suggest that this started with a single person or a single school. Nevertheless, Peirce set an important example by insisting—and demonstrating—that mathematicians need to break new ground in their field to earn that name, just as poets need to write poetry rather than just recite it. This philosophy gained momentum as others heeded the call, furthering the process of mathematical creation that is not only beautiful unto itself but is also helping to unlock the secrets of a wondrous universe.