From The Graduate School of Arts and Sciences
At
3.15.24
Paul Massari
Author Interview: Jordan Ellenberg, PhD ’98
Jordan Ellenberg
Jordan Ellenberg, PhD ’98, is a professor of mathematics at the University of Wisconsin-Madison and the author of the best-selling book Shape: The Hidden Geometry of Absolutely Everything. He talks about how geometry can be a way of measuring and explaining the world, how an understanding of its principles helped societies manage the COVID-19 pandemic, and how it can help preserve democracy by preventing incidences of minority rule fostered by gerrymandering.
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What makes geometry different from other branches of mathematics?
I think geometry is somehow the most primal and physical part of mathematics that we do. And maybe some evidence for that is that it’s very old. What we recognize as geometry is certainly thousands of years older than the use of anything we would call formal algebra with variables and things like that. That comes much later in the history of mathematics. In some sense, the more symbolic it is, the less natural it is. (That’s not a value judgment, by the way.) In some sense, the triumph of the human spirit is that we do all kinds of things we’re not naturally built to do by building up this incredible extra cognitive infrastructure that we do in mathematics or many other areas as well. But it does kind of start from geometry.
In the book, you say that “geometry is a form of honesty.” What can it teach us about logic?
I came to that formulation because I was reading about Abraham Lincoln and his interest in geometry. Before he was a politician he was a lawyer, and he was troubled by the fact that every day he was asked to go to court and “prove” his case. And he wondered, what does prove mean? So he read the ancient Greek mathematician Euclid to understand how proofs work.
Now, we think of honesty as not saying things that we know to be false. But Lincoln argued for a much stronger version of honesty, which is not only can you not say things you know to be false, but also you can’t say things that you don’t know to be true. That’s the standard of geometry. So, in the book, I say maybe we should call Lincoln “geometrical Abe” instead of “Honest Abe!”
You devote an entire chapter to “the terrible law of increase.” What is it and how did it help us as a society deal more effectively with the COVID-19 pandemic?
The terrible law of increase is a phrase that I sort of plucked from this parliamentary discussion in the 19th century about a cattle plague that swept across England and killed millions of animals. It’s the idea that it’s not that bad now, but it could get really bad just by the continuation of current trends.
I think that is pretty hard for the intuition to grasp, but I think people in the United States did it in March 2020. We were able to say nobody I know has this now, but from what we know we can see that that situation is not going to persist. It means that the terrible increase is already happening, and it will continue to happen unless trends change.
But “unless trends change” hides a lot, right? Things don’t follow a perfect exponential law in real life. After all, trends do change because people react to what happens. If you hit a tennis ball, and you know physics pretty well, you can predict where the ball is going to go but that doesn’t mean you can predict the outcome of a tennis match.
Finally, algorithms in your state of Wisconsin have enabled minority rule by allowing the drawing of districts in ways that weren’t possible before. How could geometry help political leaders create a less unfair system?
Geometry can help us measure and compare different aspects of district shapes, such as area, perimeter, compactness, and contiguity. For example, one common measure of compactness is called the Polsby-Popper score, which is the ratio of the area of a district to the area of a circle with the same perimeter as the district. The closer this ratio is to one, the more compact the district is; the closer it is to zero, the more sprawling and distorted it is. So geometry can help us quantify how much a district deviates from a simple shape like a circle or a square.
But geometry alone cannot tell us which aspect is more important or fair. Some people may prefer districts that are compact and contiguous, while others may prefer districts that preserve communities of interest or reflect proportional representation.
So geometry can be a useful tool for redistricting. It can help us identify and avoid extreme cases of gerrymandering; it can help us generate and evaluate alternative maps; it can help us communicate and visualize the trade-offs involved in redistricting. But it’s not a substitute for democracy. That is something that we have to decide as a society through an open and transparent process that involves public input and independent oversight.
See the full article here .
Comments are invited and will be appreciated, especially if the reader finds any errors which I can correct.
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The Graduate School of Arts and Sciences is the largest of the twelve graduate schools of Harvard University. Formed in 1872, GSAS is responsible for most of Harvard’s graduate degree programs in the humanities, social sciences, and natural sciences. The school offers Master of Arts (AM), Master of Science (SM), and Doctor of Philosophy (PhD) degrees in approximately 58 disciplines.
Academic programs offered by the Harvard Graduate School of Arts and Sciences have consistently ranked at the top of graduate programs in the United States. The School’s graduates include a diverse set of prominent public figures and academics. The vast majority of Harvard’s Nobel Prize-winning alumni earned a degree at GSAS. In addition to scholars and scientists, GSAS graduates have become U.S. Cabinet Secretaries, Supreme Court Justices, foreign heads of state, and heads of government.
GSAS was formally created as the Graduate Department of Harvard University in 1872 and was renamed the Graduate School of Harvard University in 1890. Women were not allowed to enroll in GSAS until 1962.
The Graduate School of Arts and Sciences forms part of the Faculty of Arts and Sciences (FAS), along with Harvard College, the Harvard John A. Paulson School of Engineering and Applied Sciences, and the Harvard Division of Continuing Education. The dean of the Graduate School of Arts and Sciences, who reports to the dean of the Faculty of Arts and Sciences, is charged with the responsibility of implementing and supervising the policies of the faculty in the area of graduate education. In the administration of academic policy, the dean is guided by the Administrative Board and the Committee on Graduate Education. The dean is assisted by an administrative dean of GSAS, who has day-to-day responsibility for the operations of the school, a dean for admissions and financial aid, and a dean for student affairs. While the GSAS office oversees the processing of applications, financial aid and fellowships, thesis guidelines, and graduate student affairs, the individual departments in FAS retain considerable autonomy in the administration of their respective graduate programs.
The Faculty of Arts and Sciences oversees GSAS and is responsible for setting the conditions of admission, for providing courses of instruction for students, for directing their studies and examining them in their fields of study, for establishing and maintaining the requirements for its degrees and for making recommendations for those degrees to Harvard’s Governing Boards, for laying down regulations for the governance of the school, and for supervising all its affairs. The dean of GSAS is responsible for implementing and supervising the policies of the faculty in the area of graduate education.
In addition to its own master’s and PhD programs, GSAS nominally oversees the PhD programs in Harvard’s professional schools: Harvard Business School, Harvard Divinity School, the Harvard Graduate School of Education, Harvard Medical School, the Harvard T. H. Chan School of Public Health, the Harvard Graduate School of Design, and the John F. Kennedy School of Government.
Harvard University is the oldest institution of higher education in the United States, established in 1636 by vote of the Great and General Court of the Massachusetts Bay Colony. It was named after the College’s first benefactor, the young minister John Harvard of Charlestown, who upon his death in 1638 left his library and half his estate to the institution. A statue of John Harvard stands today in front of University Hall in Harvard Yard, and is perhaps the University’s best known landmark.
Harvard University has 12 degree-granting Schools in addition to the Radcliffe Institute for Advanced Study. The University has grown from nine students with a single master to an enrollment of more than 20,000 degree candidates including undergraduate, graduate, and professional students. There are more than 360,000 living alumni in the U.S. and over 190 other countries.
The Massachusetts colonial legislature, the General Court, authorized Harvard University’s founding. In its early years, Harvard College primarily trained Congregational and Unitarian clergy, although it has never been formally affiliated with any denomination. Its curriculum and student body were gradually secularized during the 18th century, and by the 19th century, Harvard University (US) had emerged as the central cultural establishment among the Boston elite. Following the American Civil War, President Charles William Eliot’s long tenure (1869–1909) transformed the college and affiliated professional schools into a modern research university; Harvard became a founding member of the Association of American Universities in 1900. James B. Conant led the university through the Great Depression and World War II; he liberalized admissions after the war.
The university is composed of ten academic faculties plus the Radcliffe Institute for Advanced Study. Arts and Sciences offers study in a wide range of academic disciplines for undergraduates and for graduates, while the other faculties offer only graduate degrees, mostly professional. Harvard has three main campuses: the 209-acre (85 ha) Cambridge campus centered on Harvard Yard; an adjoining campus immediately across the Charles River in the Allston neighborhood of Boston; and the medical campus in Boston’s Longwood Medical Area. Harvard University’s endowment is valued at $41.9 billion, making it the largest of any academic institution. Endowment income helps enable the undergraduate college to admit students regardless of financial need and provide generous financial aid with no loans The Harvard Library is the world’s largest academic library system, comprising 79 individual libraries holding about 20.4 million items.
Harvard University has more alumni, faculty, and researchers who have won Nobel Prizes and Fields Medals than any other university in the world and more alumni who have been members of the U.S. Congress, MacArthur Fellows, Rhodes Scholars, and Marshall Scholars than any other university in the United States. Its alumni also include U.S. presidents and many living billionaires, the most of any university. Turing Award laureates have been Harvard affiliates. Students and alumni have also won Academy Awards, Pulitzer Prizes, and Olympic medals (many gold), and they have founded many notable companies.
Colonial
Harvard University was established in 1636 by vote of the Great and General Court of the Massachusetts Bay Colony. In 1638, it acquired British North America’s first known printing press. In 1639, it was named Harvard College after deceased clergyman John Harvard, an alumnus of the University of Cambridge (UK) who had left the school £779 and his library of some 400 volumes. The charter creating the Harvard Corporation was granted in 1650.
A 1643 publication gave the school’s purpose as “to advance learning and perpetuate it to posterity, dreading to leave an illiterate ministry to the churches when our present ministers shall lie in the dust.” It trained many Puritan ministers in its early years and offered a classic curriculum based on the English university model—many leaders in the colony had attended the University of Cambridge—but conformed to the tenets of Puritanism. Harvard University has never affiliated with any particular denomination, though many of its earliest graduates went on to become clergymen in Congregational and Unitarian churches.
Increase Mather served as president from 1681 to 1701. In 1708, John Leverett became the first president who was not also a clergyman, marking a turning of the college away from Puritanism and toward intellectual independence.
19th century
In the 19th century, Enlightenment ideas of reason and free will were widespread among Congregational ministers, putting those ministers and their congregations in tension with more traditionalist, Calvinist parties. When Hollis Professor of Divinity David Tappan died in 1803 and President Joseph Willard died a year later, a struggle broke out over their replacements. Henry Ware was elected to the Hollis chair in 1805, and the liberal Samuel Webber was appointed to the presidency two years later, signaling the shift from the dominance of traditional ideas at Harvard to the dominance of liberal, Arminian ideas.
Charles William Eliot, president 1869–1909, eliminated the favored position of Christianity from the curriculum while opening it to student self-direction. Though Eliot was the crucial figure in the secularization of American higher education, he was motivated not by a desire to secularize education but by Transcendentalist Unitarian convictions influenced by William Ellery Channing and Ralph Waldo Emerson.
20th century
In the 20th century, Harvard University’s reputation grew as a burgeoning endowment and prominent professors expanded the university’s scope. Rapid enrollment growth continued as new graduate schools were begun and the undergraduate college expanded. Radcliffe College, established in 1879 as the female counterpart of Harvard College, became one of the most prominent schools for women in the United States. Harvard University became a founding member of the Association of American Universities in 1900.
The student body in the early decades of the century was predominantly “old-stock, high-status Protestants, especially Episcopalians, Congregationalists, and Presbyterians.” A 1923 proposal by President A. Lawrence Lowell that Jews be limited to 15% of undergraduates was rejected, but Lowell did ban blacks from freshman dormitories.
President James B. Conant reinvigorated creative scholarship to guarantee Harvard University’s preeminence among research institutions. He saw higher education as a vehicle of opportunity for the talented rather than an entitlement for the wealthy, so Conant devised programs to identify, recruit, and support talented youth. In 1943, he asked the faculty to make a definitive statement about what general education ought to be, at the secondary as well as at the college level. The resulting Report, published in 1945, was one of the most influential manifestos in 20th century American education.
Between 1945 and 1960, admissions were opened up to bring in a more diverse group of students. No longer drawing mostly from select New England prep schools, the undergraduate college became accessible to striving middle class students from public schools; many more Jews and Catholics were admitted, but few blacks, Hispanics, or Asians. Throughout the rest of the 20th century, Harvard became more diverse.
Harvard University’s graduate schools began admitting women in small numbers in the late 19th century. During World War II, students at Radcliffe College (which since 1879 had been paying Harvard University professors to repeat their lectures for women) began attending Harvard University classes alongside men. Women were first admitted to the medical school in 1945. Since 1971, Harvard University has controlled essentially all aspects of undergraduate admission, instruction, and housing for Radcliffe women. In 1999, Radcliffe was formally merged into Harvard University.
21st century
Drew Gilpin Faust, previously the dean of the Radcliffe Institute for Advanced Study, became Harvard University’s first woman president on July 1, 2007. She was succeeded by Lawrence Bacow on July 1, 2018.
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