Why everything you think you know about success might be wrong
There’s a joke that goes something like this: “Looking at successful careers for career advice is like asking lottery winners for financial planning tips.” The joke is funny because it’s uncomfortably true, and it points to a deeper problem: our understanding of what leads to success is fundamentally corrupted by statistical illusions.
As mathematicians, we pride ourselves on logical rigor. We wouldn’t accept a proof that only examined special cases while ignoring counterexamples. Yet when it comes to understanding our own careers, we routinely commit exactly these logical failures. We study the visible successes, ignore the invisible failures, and draw sweeping conclusions from biased samples.
This post explores how survivorship bias, Berkson’s paradox, and related selection effects create a house of mirrors in academic mathematics, distorting our perception of what matters for success and leading generations of young mathematicians astray.
The Bombers That Didn’t Come Back
During World War II, military engineers faced a problem: how to reinforce bomber aircraft to minimize losses. They carefully mapped bullet holes on returning planes, finding clusters of damage on the wings and fuselage. The obvious solution was to reinforce these areas that seemed to take the most fire.
Then Abraham Wald, a mathematician, pointed out the fatal flaw: they were only examining planes that survived. The planes that didn’t return, the actual failures they were trying to prevent, were likely hit in the areas showing no damage on the survivors. The places without bullet holes were where hits were fatal.
This story has become the canonical example of survivorship bias, but its lessons are rarely applied to careers. We constantly examine successful mathematicians, those who “returned from the mission”, and try to discern patterns in their trajectories. We rarely consider that we’re missing the most important data: the mathematicians who didn’t make it.
The Tyranny of Visible Success
Walk into any mathematics department’s hallway and you’ll see it: the wall of fame. Photos of Fields medalists, Abel Prize winners, faculty who solved famous conjectures. These visible successes become the template against which all mathematical careers are measured.
Consider the standard narrative: brilliant young mathematician shows exceptional promise, makes groundbreaking contribution by age 30, receives prestigious position at top university, continues producing important work. We see this story repeated in biographies, award citations, and colloquium introductions. Terry Tao, Maryam Mirzakhani, Peter Scholze, these luminaries define our image of mathematical success.
But here’s what we don’t see: for every one of these luminaries, there might be hundreds of equally talented mathematicians who:
- Burned out from the pressure of early expectations;
- Chose industry over academia for financial stability;
- Contributed primarily through teaching and mentorship rather than research;
- Made crucial but unrecognized contributions to others’ famous work;
- Left mathematics entirely after years of postdoc positions;
- Work at teaching colleges where research is less visible;
- Had their careers derailed by illness, family obligations, or bad luck.
These invisible mathematicians vastly outnumber the visible ones, yet they’re systematically excluded from our data set when we try to understand mathematical careers. It’s as if we’re trying to understand human physiology by only studying Olympic athletes.
Berkson’s Paradox and the Collaboration Illusion
Here’s a puzzle that plagues many mathematics departments: why does it seem like brilliant mathematicians are often terrible teachers, or that careful, rigorous researchers lack creativity?
The answer lies in Berkson’s paradox, a subtle form of selection bias that creates artificial negative correlations. When we observe mathematicians who are both good enough to be hired and available to work at our institution, we’re looking at a restricted sample. In this sample, different positive qualities can appear negatively correlated even when they’re independent or positively correlated in the general population.
Consider how this plays out in collaboration. You might notice that among your potential collaborators:
- Fast, intuitive thinkers seem less reliable about following through;
- Careful, detail-oriented mathematicians seem less creative;
- Brilliant researchers seem poor at explaining their ideas.
These patterns feel real because, in your selected sample, they are real. But the selection process creates them. The brilliant mathematicians who are also great communicators? They’re too busy, too selective, or already fully committed to other collaborations. You never see them in your pool of potential collaborators.
This same paradox affects how we perceive entire research areas. Among papers getting widespread attention (selected for being both mathematically interesting and accessible to general audiences), technical depth and broad appeal seem uncorrelated. We conclude that popular mathematics must be shallow, or that deep mathematics must be inaccessible. But this correlation is an artifact of our attention constraints, not a fundamental truth about mathematics.
The Publication Iceberg
Published papers are the visible tip of a massive iceberg of mathematical effort. For every elegant proof that appears in a journal, there might be:
- Dozens of failed approaches that took months to exhaust;
- Multiple abandoned versions of the paper;
- Years of developing intuition through examples;
- Countless hours of confused wandering through ideas;
- Near-misses where tiny errors invalidated months of work;
Yet we only see the successes. Reading a published paper, with its clean logical flow from assumption to conclusion, gives no hint of the messy, uncertain process that created it. It’s like watching a magic trick without seeing the practice, the failed attempts, or the mechanical preparations.
This creates a particularly pernicious problem for graduate students and early-career mathematicians. They read these polished papers and wonder why their own work feels so inefficient, so full of false starts and confusion. They don’t realize they’re comparing their behind-the-scenes process to everyone else’s highlight reel.
I once heard a colleague admit that a “simple” lemma in one of his famous papers took three months and four completely wrong approaches to prove. But you’d never know that from the published version, where it appears as a brief, elegant observation. How many graduate students have questioned their mathematical ability because their work doesn’t flow as smoothly as the papers they read?
The Hidden Curriculum of Failure
In mathematics, we have a hidden curriculum that no one teaches: how to fail productively. Because failures are invisible, each generation of mathematicians must rediscover through painful experience what doesn’t work:
- Which proof techniques are seductive but ultimately fruitless;
- Which problems look accessible but are actually intractable;
- Which collaborations seem promising but lead nowhere;
- Which research directions attract attention but lack substance.
Mathematics departments could address this with “failure talks”—seminars where speakers present problems they couldn’t solve and approaches that didn’t work. The response might be the following: relief and recognition from the audience. “I tried that exact approach!” “I wasted six months on something similar!” These sessions could reveal the vast hidden infrastructure of failed attempts that underlies every success.
But such initiatives are rare. More commonly, the failures remain hidden, and each mathematician believes their struggles are unique, a sign of personal inadequacy rather than a universal experience.
The Age Myth and Temporal Selection Bias
Perhaps no myth in mathematics is more damaging than the idea that groundbreaking work must be done young. G.H. Hardy’s famous (and infamous) quote that “mathematics is a young man’s game” has traumatized generations of mathematicians who pass 30 without a major breakthrough.
This myth persists partly due to temporal selection bias. We remember when famous mathematicians made their breakthroughs (often young) but forget the full context:
- Many “young” breakthroughs were actually published years after the work was done;
- Older mathematicians’ contributions often involve mentorship, collaboration, and field-building that’s less visible than theorem-proving;
- The pressure to produce young work causes many to leave mathematics, creating a self-fulfilling prophecy;
- Historical biases in who was allowed to pursue mathematics meant many only got chances when young.
Moreover, we don’t see the age distribution of failed attempts. If 90% of mathematicians attempting major problems in their 20s fail, and 85% attempting them in their 40s fail, the slightly higher success rate when young gets amplified by survivorship bias into an iron law.
Learning from Absence
One of the most powerful tools we have for fighting these biases is learning from absence: studying what’s not there rather than what is. This requires actively collecting data on:
- Mathematicians who left the field (why did they leave?);
- Rejected grant proposals (what were they missing that successful ones had?);
- Abandoned research programs (what made them nonviable?);
- Failed collaborations (what went wrong?);
- Unpublished results (why didn’t they make it to publication?).
For example, when analyzing successful grant proposals, we typically look for common positive features. But studying rejected proposals might reveal that the absence of certain elements (clear broader impacts statements, evidence of completion ability, institutional support letters) matters more than the presence of brilliant ideas.
Similarly, when departments wonder why they can’t retain good junior faculty, they often study what successful departments offer. But examining why mathematicians leave or decline offers might reveal that the absence of particular problems (toxic colleagues, unrealistic teaching loads, poor work-life balance) matters more than the presence of particular perks.
The Practical Art of Unbiased Career Planning
So how do we navigate mathematical careers when our information is so fundamentally biased? Here are concrete strategies:
1. Diversify Your Information Sources
Actively seek stories from:
- Mathematicians at teaching colleges;
- Those who left academia for industry;
- People who changed research areas mid-career;
- Mathematicians who took non-traditional paths;
- Those whose contributions were primarily through teaching or service.
These stories are systematically excluded from standard career narratives but represent the majority of mathematical careers.
2. Document the Full Process
When presenting your work, include the meta-information:
- “This proof took five approaches over six months”;
- “We tried three other collaborations before this one worked”;
- “I nearly gave up on this problem twice”.
This calibrates expectations and helps others understand the true nature of mathematical work.
3. Create Failure Infrastructure
Establish forums for sharing negative results:
- Regular “failure talks” in seminars;
- Online repositories of approaches that didn’t work;
- Explicit discussion of abandoned projects in papers;
- Sharing of rejected grant proposals and papers.
This data is as valuable as successes for understanding what works.
4. Question Selection Mechanisms
Whenever you notice a pattern about successful mathematicians, ask:
- What selection process created this sample?;
- Who’s missing from this data?;
- What would the pattern look like if we included failures?;
- Is this correlation real or created by conditioning?
5. Maintain Process Skepticism
Treat career advice, even from successful mathematicians, with the same skepticism you’d apply to a mathematical proof. Ask:
- Is this advice based on systematic data or personal anecdote?;
- Does survivorship bias explain this pattern?;
- Would this advice apply to the median mathematician, not just the exceptional ones?;
- What evidence would contradict this advice?
The Liberation of Low Expectations
There’s something liberating about understanding these biases. Once you realize that the standard success narrative is based on incomplete data, you’re free to:
- Take longer on problems without feeling inadequate;
- Value contributions beyond breakthrough theorems;
- Build a career that doesn’t match the template;
- Fail frequently without seeing it as a personal flaw;
- Appreciate the hidden work that underlies all mathematics.
The most successful mathematicians aren’t necessarily those who never fail, they might simply be those whose failures we don’t see. The most important mathematical contributions might not be the visible theorems but the invisible mentorship, the unrecognized collaborations, the patient building of mathematical communities.
Conclusion: Seeing the Invisible
As mathematicians, we’re trained to be precise, to question assumptions, to demand rigorous proof. Yet when it comes to understanding our own careers, we abandon these principles, drawing sweeping conclusions from biased samples and incomplete data.
The solution isn’t to ignore successful mathematicians or dismiss all career advice. It’s to apply our mathematical training to the problem: recognize the selection biases, account for missing data, question apparent correlations, and always, always remember the denominator, all those invisible mathematicians whose stories we don’t see.
The next time you hear about a mathematical success story, ask yourself: What am I not seeing? Who didn’t make it this far? What failures preceded this success? The answers to these questions, the invisible data, might be more instructive than the success itself.
Perhaps the best career advice for mathematicians is this: Your struggles are not unique, your failures are not unusual, and your non-linear path is more common than the straight roads you see in the distance. The mathematics community is like a vast theorem whose proof we only see in its final, published form. But the real work, the real community, the real mathematics, lives in all those hidden lemmas, failed approaches, and invisible contributors who never make it to the acknowledgments.
In the end, understanding these biases doesn’t guarantee success. But it does offer something perhaps more valuable: the knowledge that what looks like a clear path to mathematical achievement is really just a statistical mirage, and that your own uncertain journey through mathematics is not a deviation from the norm but the norm itself, hidden though it may be.
Remember: every successful mathematician you admire once sat where you sit, confused by a proof, frustrated by a failed approach, wondering if they belonged in mathematics. The difference isn’t that they avoided these experiences, it’s that their struggles, like those of so many others, have been rendered invisible by the very success that eventually followed.