that was that intense and that advanced. The phrase I say to people just to get it across is that, among other things, by the second semester we were discussing the differential geometry of Banach manifolds. That’s usually when their eyes bug out, because most people don’t start talking about Banach manifolds until their second year of graduate school”.
Starting with 75 students, the class quickly melted down to 20 by the end of the second semester. Of that 20, says Harbater, “only 10 really knew what they were doing”. Of that 10, 8 would go on to become future mathematics professors, 1 would go on to teach physics.
“The other one”, emphasizes Harbater, “was Richard Stallman”.
Seth Breidbart, a fellow Math 55 classmate, remembers Stallman distinguishing himself from his peers even then.
“He was a stickler in some very strange ways”, says Breidbart. “There is a standard technique in math which everybody does wrong. It’s an abuse of notation where you have to define a function for something and what you do is you define a function and then you prove that it’s well defined. Except the first time he did and presented it, he defined a relation and proved that it’s a function. It’s the exact same proof, but he used the correct terminology, which no one else did. That’s just the way he was”.
It was in Math 55 that Richard Stallman began to cultivate a reputation for brilliance. Breidbart agrees, but Chess, whose competitive streak refused to yield, says the realization that Stallman might be the best mathematician in the class didn’t set in until the next year. “It was during a class on Real Analysis, which I took with Richard the next year”, says Chess, now a math professor at Hunter College. “I actually remember in a proof about complex valued measures that Richard came up with an idea that was basically a metaphor from the calculus of variations. It was the first time I ever saw somebody solve a problem in a brilliantly original way”.
Chess makes no bones about it: watching Stallman’s solution unfold on the chalkboard was a devastating blow. As a kid who’d always taken pride in being the smartest mathematician the room, it was like catching a glimpse of his own mortality. Years later, as Chess slowly came to accept the professional rank of a good-but-not- great mathematician, he had Stallman’s sophomore-year proof to look back on as a taunting early indicator.
“That’s the thing about mathematics”, says Chess. “You don’t have to be a first-rank mathematician to recognize first-rate mathematical talent. I could tell I was up there, but I could also tell I wasn’t at the first rank. If Richard had chosen to be a mathematician, he would have been a first-rank mathematician”.
For Stallman, success in the classroom was balanced by the same lack of success in the social arena. Even as other members of the math mafia gathered to take on the Math 55 problem sets, Stallman preferred to work alone. The same went for living arrangements. On the housing application for Harvard, Stallman clearly spelled out his preferences. “I said I preferred an invisible, inaudible, intangible roommate”, he says. In a rare stroke of bureaucratic foresight, Harvard’s housing office accepted the request, giving Stallman a one-room single for his freshman year.
Breidbart, the only math-mafia member to share a dorm with Stallman that freshman year, says Stallman slowly but surely learned how to interact with other students. He recalls how other dorm mates, impressed by Stallman’s logical acumen, began welcoming his input whenever an intellectual debate broke out in the dining club or dorm commons.
“We had the usual bull sessions about solving the world’s problems or what would be the result of something”, recalls Breidbart. “Say somebody discovers an immortality serum. What do you do? What are the political results? If you give it to everybody, the world gets overcrowded and everybody dies. If you limit it, if you say everyone who’s alive now can have it but their children can’t, then you end up with an underclass of people without it. Richard was just better able than most to see the unforeseen circumstances of any decision”.
Stallman remembers the discussions vividly. “I was always in favor of immortality”, he says. “I was shocked that most people regarded immortality as a bad thing. How else would we be able to see what the world is like 200 years from now?”
Although a first-rank mathematician and first-rate debater, Stallman shied away from clear-cut competitive events that might have sealed his brilliant reputation. Near the end of freshman year at Harvard, Breidbart recalls how Stallman conspicuously ducked the Putnam exam, a prestigious test open to math students throughout the U.S. and Canada. In addition to giving students a chance to measure their knowledge in relation to their peers, the Putnam served as a chief recruiting tool for academic math departments. According to campus legend, the top scorer automatically qualified for a graduate fellowship at any school of his choice, including Harvard.
Like Math 55, the Putnam was a brutal test of merit. A six-hour exam in two parts, it seemed explicitly designed to separate the wheat from the chaff. Breidbart, a veteran of both the Science Honors Program and Math 55, describes it as easily the most difficult test he ever took. “Just to give you an idea of how difficult it was”, says Breidbart, “the top score was a 120, and my score the first year was in the 30s. That score was still good enough to place me 101st in the country”.
Surprised that Stallman, the best student in the class, had passed on the test, Breidbart says he and a fellow classmate cornered him in the dining common and demanded an explanation. “He said he was afraid of not doing well”, Breidbart recalls.
Breidbart and the friend quickly wrote down a few problems from memory and gave them to Stallman. “He solved all of them”, Breidbart says, “leading me to conclude that by not doing well, he either meant coming in second or getting something wrong”.
Stallman remembers the episode a bit differently. “I remember that they did bring me the questions and it’s possible that I solved one of them, but I’m pretty sure I didn’t solve them all”, he says. Nevertheless, Stallman agrees with Breidbart’s recollection that fear was the primary reason for not taking the test. Despite a demonstrated willingness to point out the intellectual weaknesses of his peers and professors in the classroom, Stallman hated the notion of head-to-head competition.
“It’s the same reason I never liked chess”, says Stallman. “Whenever I’d play, I would become so consumed by the fear of making a single mistake that I would start making stupid mistakes very early in the game. The fear became a self-fulfilling prophecy”.
Whether such fears ultimately prompted Stallman to shy away from a mathematical career is a moot issue. By the end of his freshman year at Harvard, Stallman had other interests pulling him away from the field. Computer programming, a latent fascination throughout Stallman’s high-school years, was becoming a full-fledged passion. Where other math students sought occasional refuge in art and history classes, Stallman sought it in the computer-science laboratory.
For Stallman, the first taste of real computer programming at the IBM New York Scientific Center had triggered a desire to learn more. “Toward the end of my first year at Harvard school, I started to have enough courage to go visit computer labs and see what they had. I’d ask them if they had extra copies of any manuals that I could read”.
