A folk saying, a body of research, and what a biology lab can teach mathematicians about doing their best work.
Jim Rohn, the American entrepreneur and motivational speaker, had a saying he returned to often: “You are the average of the five people you spend the most time with.” It is the kind of aphorism that sounds almost too neat, the sort of thing you find stitched on a pillow or printed on a motivational poster, easy to nod at and forget.
Shane Parrish, writing in his newsletter on Farnam Street, put the same idea in more mechanistic terms: “Being around successful people shifts your ambition because extraordinary success feels attainable. And when success feels attainable, you lock in and work longer, harder, and with more focus.”
Parrish’s version is not just a restatement. It is an explanation, it tells you why the average matters. Proximity to achievement raises your mental benchmark for what is normal, and that recalibration changes behavior. The question worth asking is whether any of this is actually true, or whether it is simply a pleasing story we tell ourselves.
What the Research Actually Shows
It turns out the evidence is fairly solid, though with important nuance. The core claim, that social environment shapes behavior, ambition, and outcomes more than people intuitively credit, is well-supported across several distinct lines of research.
Social contagion of behavior. Nicholas Christakis and James Fowler’s landmark work using the Framingham Heart Study showed that behaviors like smoking, obesity, and even happiness spread through social networks up to three degrees of separation, friends of friends of friends. This was not explained by genetics or shared environment. It was genuine social transmission.
Income and peer effects. Raj Chetty’s research on economic mobility found that neighborhood, and specifically, the people around you during development, has measurable, lasting effects on future earnings. Children who moved to higher-opportunity areas earned meaningfully more as adults, even after controlling for other factors.
Aspirational peer effects in education. Studies on college roommates consistently find that being paired with a higher-achieving roommate raises GPA, study hours, and stated ambition. The effect is asymmetric: the lower performer rises more than the higher performer falls.
Reference point theory. Behavioral economics shows that people calibrate effort relative to what they perceive as normal. Your “reference class,” the people you implicitly compare yourself to, directly shapes what you consider worth striving for.
The mechanism Parrish describes maps cleanly onto this last finding. Success feels attainable when you can observe it at close range. That felt attainability is not mere optimism, it is a recalibration of your reference class, and it changes what you are willing to work toward.
The caveats are real, too. Some people respond to high achievers with discouragement rather than inspiration, particularly when the gap feels unbridgeable, psychologists call this the contrast effect. Selection bias is hard to untangle: ambitious people seek ambitious peers, so some of the correlation is self-selection. And the “five people” framing, while memorable, is too simple. Network effects diffuse across many more contacts than five, and they operate at multiple degrees of removal.
But the direction of the evidence is clear: the people around you shape you more than most people suppose.
The Lab Model
If you want to observe this principle working in a deliberately constructed institutional form, look at how a well-run experimental science laboratory operates.
In a typical research lab, in biology, chemistry, neuroscience or physics, a principal investigator (PI) maintains a coherent multi-year research program. Each student and postdoc owns a distinct project, but all the projects are pieces of the same larger scientific vision. Lab meetings rotate through members presenting progress, obstacles, and new data. The group critiques together, and this is where something interesting happens: a researcher working on one protein problem regularly spots the flaw in a colleague’s membrane experiment, or suggests a technique from their own work that happens to apply next door. The cross-pollination is nearly automatic, because everyone shares enough common language and scientific culture to engage with each other’s work, but not so much overlap that they are competing for the same result.
The PI’s role is as much curator and connector as director: seeing how the pieces fit, redirecting when someone is stuck, noticing when two students’ work has unexpected convergence. The whole, demonstrably, exceeds the sum of its parts.
The best research environments are not collections of isolated individuals who happen to share an address. They are deliberately structured communities of practice, where culture, standards, and tacit knowledge are transmitted continuously and almost invisibly.
What does this culture transmission actually consist of? Much of what makes an excellent researcher in any field cannot be written down in a textbook. It is taste, knowing which problems are worth pursuing. It is intuition about which approaches are promising and which are dead ends. It is a calibrated sense of what “good work” looks like in this particular field. These things spread almost entirely through proximity. Being around exceptional researchers essentially downloads their aesthetic judgment over time.
The Problem with Direct Translation to Mathematics
Research mathematics has some unusual properties that create genuine tension when you try to import the lab model directly.
The most important is that mathematical results are often binary in a way that experimental science results are not. Either you proved it or you didn’t. This makes certain peer effects cut differently. In a biology lab, multiple people can make simultaneous progress on related problems and all benefit. In mathematics, if two people are working on the same theorem, one of them is going to be scooped, and the awareness of that possibility distorts behavior. Competitive proximity can push researchers toward incremental publishable results rather than swinging at hard questions.
There is also the question of cognitive immersion. Some mathematical work requires a kind of prolonged, solitary focus that social environments can genuinely interrupt. The archetype of the isolated mathematical genius, Ramanujan working alone in Madras, Andrew Wiles spending seven years in secrecy on Fermat’s Last Theorem, Grigori Perelman withdrawing entirely from the mathematical community, is not purely myth. Some problems require a depth of concentration that is hard to sustain in a socially dense environment.
And close peer groups can generate intellectual monocultures. When everyone around you attacks problems the same way, you unconsciously inherit their blind spots. Some of the most significant mathematical breakthroughs have come from outsiders, or from researchers working in partial isolation, who were not anchored to the dominant framework.
The Sweet Spot: Same Area, Different Problems
Most of this tension dissolves with one structural adjustment: build peer groups around a shared area rather than shared problems.
Working in the same general field, say, harmonic analysis, without working on identical questions preserves everything valuable about peer proximity while eliminating most of its costs. The benefits of ambient ambition transfer perfectly: students absorb the culture and standards of the field, calibrate their effort against genuine excellence, and develop taste by observing what a good result actually looks like in this corner of mathematics. The main costs of close proximity, scooping anxiety, convergent thinking, groupthink on approach, are dramatically reduced because everyone has their own problem and their own ownership.
There is also a subtler benefit: cross-pollination without collision. A technique one student develops for one operator problem may turn out to be exactly what another student needed for an interpolation question. Neither would have found it working in isolation. In a shared-area group, this kind of transfer becomes almost automatic, it is how a lot of mathematical progress actually happens, tools migrating across subproblems within a field.
This is, not coincidentally, what made certain mathematical schools historically productive. The Chicago school, the tradition of Soviet mathematics, the Bourbaki collective in France, in each case, what bound the group together was a shared aesthetic and a shared technical language, not identical problems. Individual ownership of questions; collective ownership of culture.
Running It in Practice
The structure that emerges from this reasoning looks, in practice, something like a modified version of the lab meeting model applied to pure mathematics. A research group organized around a common area, meeting regularly, with each member presenting their own work to the others. The group shares enough common technique to engage substantively with each other’s problems, to ask useful questions, to suggest approaches, to recognize when someone is stuck in a direction that won’t work. The advisor’s role is to curate a portfolio of problems that are close enough for genuine cross-pollination, far enough apart to preserve psychological ownership, and united by enough common structure that the group has real value to each member.
Several features make this work well in practice:
- Weekly presentations normalize the struggle. Seeing a colleague make incremental progress, or hearing them describe a week of failed approaches, calibrates your own sense of what research pace looks like and makes the process feel less abnormal when it is slow.
- A common technical language makes the group useful. Students need enough shared background that they can genuinely engage with each other’s work, not merely nod politely through presentations they cannot follow.
- A supportive rather than competitive culture changes the incentive structure. When group members are genuinely invested in each other’s success, celebrating each other’s results, helping each other past obstacles, the distorting effects of competition largely disappear.
- Advisor-curated problem proximity is the design choice that makes everything else work. Getting the distance between problems right, close enough for cross-pollination, far enough for independence, is the core structural challenge.
The advisor is, in this framing, doing something analogous to what the PI does in a lab: maintaining the overarching research program, curating the problem portfolio, and serving as the connector who sees how the pieces fit together. The difference is that mathematical projects are more self-contained than experimental ones, so the interdependence has to be cultivated intellectually rather than built into the work structurally.
What If the Environment Doesn’t Exist?
Not every motivated student finds themselves inside a well-structured research group. What then?
The honest answer is that the right peer environment is more constructible than it appears, and the tools available now make it more constructible than ever before.
The most underutilized intervention is simply organizing a reading group yourself. Pick a paper at the edge of your current knowledge, recruit two or three serious people, and commit to meeting weekly. The topic matters less than the discipline. The act of organizing it is itself valuable, you become a de facto center of intellectual gravity, and the people who show up consistently self-select into your real peer group. A single other motivated person is enough to start.
Online seminars, which proliferated after 2020, now offer genuine access to the culture of active research in most mathematical areas. Attending a weekly seminar in harmonic analysis, even just listening regularly, immerses you in the ongoing conversation of the field in a way that was simply unavailable to students at less-connected institutions a generation ago. The harmonic analysis community, like most active areas of mathematics, runs several open online seminars.
Direct contact with researchers is more effective than most students believe. A genuine, specific, carefully composed email to a researcher whose work you have read closely, asking a real question about it, gets a response more often than people expect. Mathematicians are, on the whole, accessible to serious students. One or two such relationships can provide much of what a local peer group would.
The deeper principle is this: the students who navigate impoverished institutional environments best tend to be proactive in a very specific way. They treat building their intellectual community as part of the work itself, not as something separate from it. The peer environment does not have to be found, it can be built. And building it is, itself, a form of intellectual seriousness.
Jim Rohn’s aphorism is too simple, but it is pointing at something real. The people around you calibrate your sense of what is possible, and that calibration shapes everything downstream, how hard you work, how long you persist, how ambitiously you aim. The lab model institutionalizes this insight deliberately, and the best mathematical research groups do something structurally similar: shared culture, shared language, individual problems, collective standards. The folk wisdom and the research are, on this question, in unusual agreement.