How I Learned the Art of Math [Excerpt]
How I Learned the Art of Math [Excerpt]
How I Learned the Art of Math [Excerpt]
In the spring of 1991, I took an algebraic number theory course from Professor Basil Gordon. Gordon loved the material, and the students in the class could sense his deep devotion to the subject. Gordon’s lectures were inspirational sermons. It was like being at a poetry reading. For him, a theorem was not just some odd mathematical fact. It was a work of art whose aesthetic qualities could be described, as could its place in the ongoing intellectual dialogue of mathematics and the opportunities that it presented for further investigation. Gordon would sometimes compare a theorem to a famous work of art or classic poem. It was not unusual for him to juxtapose the majesty of a theorem of Gauss with the breathtaking beauty of a Michelangelo sculpture. I soon understood that Gordon’s relationship with mathematics was unusual. He viewed himself as an artist whose medium happened to be mathematics. It was clear that he thought about mathematics in a way that was very different from my view, which had always involved performance on exams and the memorization of formulas and proofs. I wanted to know more, and I didn’t have to wait long to get my chance.
It was several weeks into the course, during a lecture about ideal class groups, a subject developed by Gauss a century and a half earlier. Gordon was just finishing the proof of a theorem about prime-order torsion elements in these groups using a method introduced decades earlier by MIT mathematician Nesmith C. Ankeny and Penn State professor Sarvadaman Chowla. It began to dawn on me that there was a much more conceptual proof that made use of elliptic curves. After Gordon completed the proof, I raised my hand and offered my alternative proof, which made use of geometric ideas of Mordell and Weil. Gordon’s response was to ask my classmates to applaud my proof, and he invited me to his office after class.
Author Ken Ono, left, and his mentor Basil Gordon.
Credit: Ken Ono/Springer
I nervously made my way to his office, worried that he would scold me for my presumptuousness. Was he mocking me when he asked the class to applaud? Gordon’s office seemed strangely out of place at UCLA. It could have been the office of an Oxford don or one from the Hogwarts School in the Harry Potter novels. The walls were lined with beautiful barrister bookcases, their contents beckoning from behind hinged glass doors. An enormous ornate oriental rug graced the floor, concealing the weathered 1960s-era floor tiles. The desk overflowed with papers and letters, nearly enveloping a set of antique gold pens.
Gordon’s manner evoked images of a different time and place, perhaps a nineteenth-century English manor. Our discussion was brief. Although the proof I had offered in class was not a new result, he was impressed with my insight. He had been following my career at UCLA, and he told me that he would be honored to be my doctoral advisor. He was thinking about retirement, and he wanted me as his final Ph.D. student. Although I was surprised and puzzled by his offer, I accepted on the spot. That meeting with Gordon marked my birth as a mathematician.
Basil Gordon was indeed a gentleman and a scholar, a polymath who was a direct descendant of the Gordon family of British distillers, producers of Gordon’s gin. He was the step-grandson of the famous American general George Barnett, who served as the major general commandant of the Marine Corps during World War I. I was pleased to learn that we had both grown up in Baltimore. He had attended Baltimore Polytechnic Institute and received his master’s degree in mathematics from Johns Hopkins University in 1953. He earned his doctorate in mathematics and physics from Caltech in 1956 working under the mathematician Tom Apostol and the iconic physicist Richard Feynman. Gordon was drafted into the U.S. Army, where he worked with rocket scientist Wernher von Braun. He was part of the team that worked out the path of the satellite Explorer I so precisely that it remained in orbit for a full dozen years after its launch in 1958. Gordon joined the UCLA faculty in 1959.
Gordon and I developed an unusual routine. Instead of weekly sessions in his office at UCLA, we met at his home in Santa Monica. We rarely began our meetings by diving straightaway into the mathematics. Instead, Gordon might begin our meeting by playing a Chopin nocturne on the piano. Sometimes, he would recite poetry from memory. Gordon had a photographic memory and could effortlessly recite reams of literature. I recall him intoning the first few pages of Melville’s Moby dikk.
To leave the safe familiarity of the shore and sail off into unknown territory, that is what it is like to do mathematics. Gordon was constantly reminding me that our mathematical research, as difficult and as confusing as it can be, is an art form, an exploration, an adventure, something to be appreciated, something to be lived. How could we possibly prove a good theorem if we viewed mathematics as a chore? We weren’t hanging sheetrock, we were creating a masterpiece, cultivated over weeks, months, even years of deep thought and imagination. And so it was music and poetry that set the tone before we began scribbling figures and equations on our yellow pads.
I learned a great deal from Gordon on those Saturday afternoons. We would spend hours huddled in his den, struggling with difficult concepts, trying to break an impasse and find a way to bridge a logical gap in an argument. From time to time, on rare occasions made the sweeter for their rarity, we were rewarded with a breakthrough, an elegant argument, a watertight proof. Those moments of revelation were so awesomely gratifying that we quickly forgot the doubt and despair that can creep into the soul when one has lost one’s way.
From Basil Gordon I learned what it meant to “do mathematics.” When I was a child, I understood as a child, and I thought that math was only about manipulating numbers and “solving for x.” In college, I thought mathematics was about memorizing theorems and their proofs, mastering techniques for carrying out difficult calculations, and solving textbook problems.
Gordon taught me that “doing mathematics” begins with a state of mind that allows you to travel to a place deep inside the subconscious to open body, mind, and spirit to the contemplation of a mathematical idea. Doing mathematics can be a mental voyage to a place where clarity of thought and openness to insight make it possible to see the deeper beauty of a mathematical structure, to enter a world where triumph over a problem depends less on conscious effort than on confidence, creativity, determination, and intellectual rigor.
I didn’t expect that my thesis would be of interest to many mathematicians, but that didn’t bother me. A thesis was the first step in the life of a professional mathematician, not the final magnum opus. Gordon had taught me to love mathematics for its own sake, and that discovery sustained me. My goal was to finish the dissertation, and Gordon assured me that I was well on my way to success. Strangely, I believed him. We were having a wonderful time proving theorems, and proving them for their own sake.
