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Danny Calegari

In high school I took part in many math competitions; the hardest (and therefore to my teenage mind the only ones that counted) were the competitions relating to the international mathematical olympiad (IMO). In Melbourne there was a program of competition and training that culminated in a nine hour exam spread over two days to determine the makeup of the Australian IMO team. I remember very well the first time I took the exam. It was 1987, and the IMO was to be held that year in Cuba. As I sat at my carrel in the Morris library, I took the February sunlight for fortune smiling on me, inspiration spilled liberally from my Pelikano steel nib fountain pen, and I went home at the end of the second day in a blur of fatigue and self-congratulation. Six weeks later a pregnant manila envelope arrived in the mail. From its girth alone I knew I had aced the exam and won my rightful spot on the team. Before even opening the envelope I could see myself in the green woolen team blazer with the Australian coat of arms embroidered on the breast pocket, and by the time I found a letter opener I was shaking hands with Fidel Castro.

The envelope contained…sixty-odd sheets of loose-leaf paper, no invitation, no cover letter: my exam papers, bloodstained with question marks, lines through paragraphs, squiggles of uncomprehension, Xs and Os. My stomach fell. I blushed. In a fraction of a second I rewrote or recolored dozens of memories and fantasies from the recent past and future, and became intensely conscious of and embarassed by my vanity and foolishness. What I now find remarkable was the speed and scope of my transformation; the analogy that comes to mind is being struck by a speeding car.

So how does one deal with disappointment? Freud, in Civilization and Its Discontents identifies three typical measures:

powerful deflections, which cause us to make light of our misery; substitutive satisfactions, which diminish it; and intoxicating substances, which make us insensible to it.

If these are the typical responses, are there any others? Before trying to answer this it might be helpful first to articulate what disappointment actually is, and then to ask what it’s for. Evidently, disappointment is a form of mental suffering. It is so unpleasant that we can experience it in a host of physiological dimensions. Profound mental suffering involves a complex array of interactions between any number of processes and subsystems, both conscious and unconscious, involving both the brain and the limbic system. The suffering that arises from disappointment is that that accompanies disruption: disappointment causes a certain kind of shake-up or realignment of our worldview and self-image and consequently of our priorities; this disruption can be so great that we sometimes emerge from it a very different person.

According to certain schools of cognitive science (e.g., Minsky’s Society of Mind model) the idea of a “self” as a unified, indivisible entity is an oversimplification; rather (they suggest) a self is an uneasy federation of simpler subsystems (sometimes termed “agents”) with their own local goals and interests, which are frequently in competition with one another. Under ordinary circumstances stability is achieved by a complicated system of temporary alliances, detentes, three-way standoffs, and so forth. Our subjective sense of the unified self is—in itself!—also a source of stability. Sometimes a dramatic change in (real or perceived) external circumstances—an unforseen event, an unpleasant discovery—can lead to a cascade of disruptions to this order. This is the mechanism of disappointment, and why it is so painful; it is both a crisis and an opportunity—in Homer Simpson’s inspired terminology, a crisitunity.

Disappointment measures in pain the gap between reality and what we want the world to be. Disappointment matters. It matters because we don’t actually live in the real world. We live in our heads, in a mental world of assumptions, recollections, anticipations, desires and conjectures. And even when we do meet reality, it’s a mistake to think that what our senses feed us is objective, unfiltered, unsorted. Rather we operate according to an interrogative protocol—we ask the world questions to confirm what we already “know” (or, more accurately: hope), and only when we get an unpleasant surprise do we take a closer look. As Proust says,

The real voyage of discovery consists not in seeking new landscapes, but in having new eyes. Happiness is beneficial for the body, but it is grief that develops the powers of the mind.

In his famous paper, “How not to prove the Poincaré Conjecture,” John Stallings writes about an adventure in his mathematical life, how he discovered a proof of the Poincaré Conjecture but later found it to be mistaken. He goes so far as to describe this episode as a “sin;” but the sin was not in the mistake per se, rather it was his resistance to recognizing it as such. He writes,

There are two points about this incorrect proof worthy of note …(t)he second…is that I was unable to find flaws in my ‘proof’ for quite a while, even though the error is very obvious. It was a psychological problem, a blindness, an excitement, an inhibition of reasoning by an underlying fear of being wrong. Techniques leading to the abandonment of such inhibitions should be cultivated by every honest mathematician.

Few mathematicians are as honest or as generous as Stallings in sharing their own stories of disappointment. This is because disappointment often comes wrapped in shame, because our goals are inextricable from our personal and social attachments and relationships. Convention and social norms dictate that any display of human weakness or failing is “unprofessional.” We’re not supposed to admit it when we feel stupid, or underappreciated, or jealous, or that we cared so much about something that when it didn’t go our way we felt shattered. Techniques leading to the abandonment of such inhibitions should be cultivated by every honest mathematician.

Disappointment takes many forms; a partial list from my personal history includes:


being un- or under-acknowledged in a colleague’s paper or talk;


being scooped;


missing out on a job/prize/conference invitation;


having a prospective student work with someone else;


having a potential advisor turn me down as a student;


having a promising line of attack on a problem fail to pan out;


discovering an error in an amazing proof;


having a paper go unread or a book go unreviewed;


seeing an admired senior colleague behave badly;


realizing that I haven’t lived up to my own standards of behavior;


discovering that success, when it came, was not all I hoped it would be.

The last one, perhaps, deserves elaboration. Some acute disappointments in my career were the result of getting what I thought I wanted: a paper in a fancy journal; a job offer; tenure; an invitation to talk at a fancy conference. I don’t mean to diminish the value of such things at all, or the challenges (personal or structural) many people must overcome to achieve them; much about the way such “rewards” are distributed in academic culture is unfair, often in systematic ways, and it should be the goal of all of us to point this out and work to change it wherever we can. I also don’t mean to suggest that success has been joyless; the opposite is true. Nevertheless it is the case that sometimes when we get what we think we want, we discover that these things weren’t what we thought they were, and (more importantly) that we are not who we though we were. When disappointment accompanies success it is worth paying special attention to. If we get what we want but it doesn’t bring us fulfillment, then what’s really going on? In my experience, it has only been at the point of my posing this question that I have acquired insight, and the agency to really change things or come to terms with them.

It took a month of pain after the manila envelope arrived before curiosity got the better of me and I opened it again. And a remarkable thing happened. The exam pages: my answers, the blots, the corrections, the red ink, the comments, were exactly as before. But time and some strange alchemy of which disappointment itself was the catalyst had altered their meaning. An actual human being had taken the time to read my work and share valuable feedback with me. My annotated exam was no longer a certificate of failure, it was a how-to manual: it was about how to prove an inequality by leveraging the convexity of a cleverly chosen auxiliary function, or how to recast a geometric figure in terms of complex numbers and understand it with algebra. These math problems weren’t “problems” at all: they were windows into mathematics itself. And the manila envelope wasn’t a slap in the face, it was a gift; but to see it as a gift I had to see it with new eyes. I never got to Cuba, but I’d taken my first steps on a longer and far more interesting and rewarding journey that continues to this day.


I would like to thank Kathryn Kruse for her extensive feedback on and advice about an early draft of this essay.


Photo of Danny Calegari is courtesy of Danny Calegari.