I arrived at graduate school from a large state university in the middle of the rankings, where I had been something of an anomaly. I stood out in the mathematics classes, I knew it, and it did me a great deal of good. Within a few weeks of arriving, I had worked out where I stood in the new room. Some of my peers already had master’s degrees. Others came from programs where strong undergraduates routinely took graduate courses. I had taken no graduate courses, and fewer pure mathematics courses than most of them. I was, I concluded, somewhere around the median, or more likely behind it.
I’ve told part of this story before, in The Gap That Closes, as a story about effort: I decided I was behind, and the fix was to work until I wasn’t. There I called that first fear a real skill gap. I’m less sure now. Some of the gap was real. Much of what I felt was the gap amplified by the room. And all of it was a comparison of where we started, which turns out to say very little about where anyone goes.
I’ve tried hard, for most of my career, not to compare myself to other mathematicians. It took me much longer to notice how thoroughly the field does it for me.
Objective about everything but us
Psychologists have argued since the 1950s that people need to evaluate their own abilities, and that when there’s no objective standard to do it with, they compare themselves with other people, usually with people doing better.
Here is what makes mathematics strange ground for this. We have the most objective standard of any field for evaluating claims. A proof is right or it isn’t, and the profession is organized around being able to check. We have almost no objective standard for evaluating mathematicians. Whether you’re good, whether you’re on track, whether your output is normal for your stage and subfield: none of that comes with a proof. So, exactly as predicted, comparison fills the gap. And the checkability of the results lends the comparisons an air of objectivity not earned.
The hierarchy is also unusually public, more than in most fields. Many of us were ranked numerically as teenagers and never quite stopped being ranked. Journal tiers are common knowledge; that’s an Annals paper is a comparison sentence. Prizes come with age and career-stage windows, which turns “how far along am I for my stage?” into a public question. Famous problems are solved or they aren’t. The arXiv announces everyone’s output every morning, in a list, to subfields small enough that everyone knows everyone. And tenure letters routinely ask referees to make the comparison in writing.
Notice what all of these have in common. Every one is an outcome. The field publishes its outcomes, carefully and in public, and keeps its processes almost entirely private. Nobody’s arXiv listing shows the eight stuck months. So nearly every comparison available to you sets your process, which you know in exhausting detail, against someone else’s product, written in definition-theorem-proof, a format designed to erase the path.
The pond
There’s a well-replicated finding in education research called the big-fish-little-pond effect: students of equal ability think less of their own ability when they’re surrounded by stronger students.
I had both ponds. As an undergraduate the effect worked in my favor, and I think it mattered more than I’d have admitted then; standing out is a good way to come to believe you belong in a subject. Then I moved to a stronger pond, and my sense of myself did exactly what the research says it does. Nothing about my mathematics changed on the drive between campuses. The move that most improved my training most damaged my reading of myself, and I responded by working harder, as though the pond were a skill I lacked.
The deeper mistake was in what I compared. Peers with a master’s degree were ahead of me in courses taken. That was true, and it was a statement about position at t = 0. What a research career runs on is closer to a rate: how fast you learn, how you respond to being stuck, what you do with a problem nobody has solved. Two more years of coursework says almost nothing about the derivative. I was comparing levels when the quantity that mattered was the slope.
And some of those peers who looked ahead were very likely, in the same room, looking at someone else and feeling behind. People reliably underestimate how much others are struggling, because struggle is the thing everyone hides. A room full of people each measuring themselves against everyone else’s composure is a room in which nearly everyone can feel below the median at once.
Why it doesn’t help
Many mathematicians don’t just do mathematics; they are it. When you are the mathematics, every comparison is a comparison of you, and the unfavorable ones land harder than the favorable ones lift.
And outcome comparisons carry very little information. A prize, a journal, a job offer is one draw from a noisy random process. Setting your draw beside someone else’s tells you little about either of you. When I’ve compared myself with others, it hasn’t helped. It produced jealousy and envy. It never told me why someone had succeeded where I hadn’t. Much of it looked like randomness, with nothing I could learn from or change. All of the cost, almost none of the information.
My defense has been to remember how much of this is outside anyone’s control. Refereeing has real randomness; excellent work can land on a reviewer at a bad moment. Good work tends to get found eventually. I believe that, and I think it’s mostly accurate. But when I ask whether I give my own successes the same share of randomness I give my setbacks, the honest answer is no. Almost nobody does. Where that matters is which column other people’s successes go in. If theirs are luck and mine are earned, the coping frame has turned into a quieter way of diminishing the work: it’s just a computation; they had a lot of help; nobody really cared about that problem. And even when the frame is accurate, it closes the door on learning. Sometimes there was nothing to learn. Sometimes there was, and the frame hid it.
Comparing correctly
There’s the old line that comparison is the thief of joy, and there’s the advice I’ve heard my whole career: know your reference group. The first says don’t compare. The second says compare correctly. I’ve come down on the side of the second, because in a field with no objective standard for people, not comparing isn’t really on offer. Comparing well is.
The method is a mathematician’s: specify what you’re estimating before you choose the sample. Most bad comparisons happen because nobody asked what question the comparison was meant to answer. There are roughly three legitimate ones.
For calibration (am I on track?), compare with people similar on the things that should predict output: years since the degree, teaching load, the subfield’s publication norms, coauthorship, funding, constraints at home. Compare with that group’s median, not its visible maximum. And where people started in different places, compare slopes, not positions.
For learning (how do they do it?), compare upward freely, even with the best people, but compare process with process. That means asking: How long did it take? What didn’t work first? Mathematicians answer those questions honestly in private far more than in public.
For motivation (what should I aim for?), choose targets one or two steps ahead. A star whose path looks reachable inspires; one whose path doesn’t, deflates. Superstars are for admiring. Near peers are for chasing.
And one filter I arrived at the hard way: if a comparison doesn’t give me something to learn, improve, or change, it isn’t worth the time. Most outcome comparisons fail that test. Most process comparisons pass it. I don’t read other people’s CVs, not out of avoidance so much as declining to sample the noisiest tail available.
For a graduate student, the slope is hard to see because you’re inside it. A cheap instrument: once a semester, pick up something you couldn’t read a year ago, a paper, a chapter, a problem you abandoned, and notice what’s changed. That’s a comparison with the one person whose starting point you know exactly.
For a postdoc, comparison stops being optional. The market ranks you on position, quickly and in public, against a cohort whose CVs you can see, and the CVs you can see are the visible tail of that cohort, not its median. You can’t make the market measure slopes. You can keep the two separate: let the market rank your position, and measure your own progress on the slope.
The other side of the letter
Eventually the request arrives in your inbox: please compare the candidate to others in the field at a similar career stage.
I try to follow it when I can make the comparison meaningful. When I can’t, I’ve sometimes sidestepped it, because a careless comparison doesn’t help the person and can hurt them. I’ve come to think the sidestep has its own cost. Committees read many letters, and an unanswered question can be read as the writer having nothing good to say. The better move is to answer while changing the comparison class: compare to career stage and productivity, normalized for the subfield, without naming names. In an area where papers are long and the publication rate is low, her output over five years compares well with that of strong people at the same stage answers the question. So does saying plainly that someone started behind and has a steep slope, if that’s what you see. The letter writer is often the only person in the process who knows the subfield well enough to normalize, which makes the letter one of the few places where comparison can be done correctly.
There’s also a comparison in every letter that nobody asks for: the writer against themselves at the same stage. It’s worth noticing when you’re making it, and asking whether it’s one you’d defend to the committee.
The same goes for institutions. I know of a school that keeps a list of outside researchers it considers a junior colleague’s peer group and sets the colleague’s output against theirs every year. Chosen well, a list like that is a declared, normalized reference group that holds still. Chosen aspirationally, it’s the visible tail of better-resourced places, written down and applied annually. Someone chose it, and it’s fair to ask who, and how.
What to do with it
Four things, in increasing cost.
Name the question before the comparison. Calibration, learning, or motivation. If you can’t name one, it probably isn’t a comparison you need.
Compare slopes, not positions. Especially early, and especially after moving to a stronger pond. Measure your own slope on purpose, because nobody else will report it.
Run the symmetry check. Whatever share of luck you grant your own setbacks, grant other people’s successes the same, and your own successes too.
If you have standing, choose the reference group out loud. In letters, in annual reviews, in peer lists, and in what you tell students about how your own career actually went, including the part where you were behind at the beginning.
When a letter request is on my screen now, I compare to stage and productivity, and wherever I can, I describe the slope. If I could give that first-year graduate student one sentence, it wouldn’t be that he wasn’t behind. He was, a little, at the start. It would be that behind at the beginning is a statement about position, and that the quantity worth tracking was the slope. Nobody was going to report it to him. He’d have to measure it himself.