Harvard University Press Blog

The Rise of American Mathematics

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.

Dimensions in Mathematics, Revealed

In the second book of the Millennium series, The Girl Who Played with Fire, Stieg Larsson’s Lisbeth Salander is devoted to a 1,200 page mathematics text. The book, by one L. C. Parnault, is titled Dimensions in Mathematics, and though Larsson informs readers that it was published by Harvard University Press, the book has been impossible to find. Until now. We’re very excited to announce the long-awaited publication of Parnault’s Dimensions in Mathematics.

Like no work since the Arithmetica of Diophantus two millennia before, L. C. Parnault’s Dimensions in Mathematics presents the fullness of mathematical knowledge attained by man. From Thales to Turing, Pythagoras to Euclid, Archimedes to Newton, the Riemann Hypothesis to Fermat’s Last Theorem, Parnault escorts both serious mathematicians and the non-mathematical mind through the deepest mysteries of mathematics. Along the way he offers the greatest expositions yet of number theory, combinatorial topology, the analytics of complexity, and his own groundbreaking work on spherical astronomy. Dimensions equips even elementary readers with the tools to solve the logical puzzles of the perfect universe that can exist only in the mind of a mathematician.

“Dr. Parnault’s elegant explications of seemingly every extant mathematical concept or quandary make this text as indispensible as any in our field,” says Fields Medal-winning MIT Professor Gerald Lambeau. “His presentation of combinatorial mathematics left me breathless.”

In addition to being a milestone for the field, the publication of Dimensions in Mathematics is a true publishing event, a crowning achievement in our centennial year. We’re extremely proud to finally satisfy the millions of Millennium readers who’ve sought out the book, and are deeply humbled by the experience of working with the legendary Dr. Parnault.

UPDATE: Just April Foolin’. Parnault and his book remain but fiction.

Is Mathematics Not Beautiful?

Since its reincarnation a year or so ago, the NYT Sunday Review section has usually reserved its front page for articles intent on sparking debate or stirring controversy. The pieces often present counter-intuitive arguments graced with hackle-raising headlines. And so witness this week’s above-the-fold inquiry: “Is Algebra Necessary?”, accompanied by an illustration of a trio of poor souls drowning in waves of notation sure to give even MIT’s finest fits.

Andrew Hacker, a political scientist whose most recent book examines “how colleges are wasting our money and failing our kids,” argues in the article that if math is the greatest stumbling block to educational achievement then perhaps we should just remove it. “There are many defenses of algebra and the virtue of learning it,” he writes. “Most of them sound reasonable on first hearing; many of them I once accepted. But the more I examine them, the clearer it seems that they are largely or wholly wrong—unsupported by research or evidence, or based on wishful logic.” The issue as he sees it is that this forced emphasis on the mathematical sequence causes large numbers of students to fail, thus depriving them personally and society generally of the further development of their non-mathematical talents.

In place of our current model of mathematics education, Hacker envisions an alternative wherein the focus shifts to “citizen statistics” so as to “familiarize students with the kinds of numbers that describe and delineate our personal and public lives.” He imagines such an approach would be exciting, and he urges us to do away with a system in which math is but “a huge boulder we make everyone pull, without assessing what all this pain achieves.”

Now, very few people with any exposure to the teaching of math in our schools would disagree with Hacker’s contention that it’s broken. But many who feel passionately about mathematics would surely also note the absence of appreciation from Hacker for anything but math’s most obscure and utilitarian applications. For all his focus on the pain and fear of mathematics, Hacker has little to say about its beauty. He does suggest that we should treat mathematics as a liberal art, “making it as accessible and welcoming as sculpture or ballet,” but in the service of rationalizing its marginalization rather than encouraging its embrace.

Interestingly, much of Hacker’s analysis echoes that of Paul Lockhart, who gained a certain degree of fame for his “Mathematician’s Lament,” a stinging critique of the state of mathematics education. The difference between Hacker and Lockhart is that while Hacker is interested in public policy and the efficient use of resources, Lockhart is full of passion for the creative art of mathematics. He’s not so worried about how broken mathematics education deprives students of the development of their other talents; he’s worried about how it deprives them of math. For Hacker reform will come through cuts, sweeping math aside to join art, music, and athletics on the margins of American public education. For Lockhart the solution is to adopt an approach to mathematical education that frees students from memorization and exposes them to the wonders of what he calls “mathematical reality.”

We’re soon to publish Lockhart’s Measurement, a wonderful book that puts into practice the method of teaching for which Lockhart argued in his lament. We recently spoke with him about the book, and he showed himself to be a living, breathing, laughing embodiment of the deep beauties of math that any call for reform would do well to recognize.

But don’t take our word for it: