Or: Why Your Career Anxieties Are Mathematically Inevitable
Every mathematician knows that moment. You’re staring at your desk, covered with:
- A half-finished proof that could be beautiful if you had six more months
- Unmarked calculus exams from three weeks ago
- A grant proposal demanding “practical applications” for your work on abstract sheaf cohomology
- An email from your department chair about committee assignments
- A text from your partner asking if you’ll make it home for dinner (again)
Which do you tackle first? More fundamentally: How do you build a career that optimizes across all these dimensions?
Here’s the liberating, terrifying truth: You can’t. And I mean that literally, as in, mathematically proven impossible by Kenneth Arrow’s Nobel Prize-winning theorem.
The Impossibility at the Heart of Decision-Making
Arrow’s Impossibility Theorem is one of those results that seems abstract until it punches you in the gut with its real-world implications. Originally formulated for voting systems, it proves that when you have three or more options and multiple criteria for choosing between them, no decision-making system can satisfy all reasonable fairness conditions simultaneously.
Let me translate that from social choice theory to your academic life.
The Academic Trinity That Cannot Be Solved
Consider the three pillars of academic evaluation:
- Research Excellence (those groundbreaking papers)
- Teaching Quality (actually caring about your students)
- Service Contribution (keeping the department running)
Now imagine three colleagues up for promotion:
Dr. A: Published a proof that made the Annals. Students flee her courses. Hasn’t attended a faculty meeting in three years.
Dr. B: Solid, incremental papers. Students worship him. Runs the undergraduate program.
Dr. C: Two decent papers. Good teacher. Chairs five committees and organizes every conference.
Who deserves promotion most?
Here’s where Arrow’s theorem bites: There is no fair, consistent way to rank them. Any system you devise will either:
- Become dictatorial (“only research matters”)
- Create paradoxical cycles (A beats B, B beats C, but somehow C beats A)
- Violate basic logic (adding Dr. D to the mix changes how you rank A versus B)
This isn’t a failure of your department’s promotion committee. This is mathematics.
The Research Paradox: Choosing Your Problems
The impossibility deepens when selecting research problems. You cannot simultaneously optimize for:
Mathematical Elegance: That beautiful connection between modular forms and elliptic curves that speaks to your soul.
Practical Impact: The NSF wants applications. They’re paying the bills.
Career Advancement: You need papers. Lots of them. Before tenure review.
Field Advancement: The important problems are hard. Grothendieck spent years on foundations before proving anything.
Personal Fascination: You got into math because of the beauty, remember?
Take the extreme case: working on the Riemann Hypothesis. Maximum prestige if solved? Absolutely. Publishable progress? Unlikely. Career suicide pre-tenure? Probably. Worth it? Arrow’s theorem says there’s no consistent way to answer.
The Collaboration Triangle of Impossibility
Choosing collaborators presents another Arrow paradox:
Option 1: Famous Senior Mathematician
- Your paper gets attention
- Your contribution gets overshadowed
- Your career gets a boost
- Your confidence gets crushed
Option 2: Peer at Your Level
- Equal credit distribution
- Similar struggle levels
- Less prestige boost
- More genuine partnership
Option 3: Strong Junior Researcher
- You get senior author credit
- You provide mentorship
- Less immediate prestige
- More long-term network building
There’s no ranking system that doesn’t privilege one factor dictatorially. Your career optimization function is mathematically doomed to inconsistency.
The Pure vs. Applied Impossibility
Mathematics departments worldwide struggle with this manifestation of Arrow’s theorem. How do you fairly evaluate:
- The category theorist pushing the boundaries of abstraction
- The numerical analyst developing climate models
- The mathematical biologist collaborating with virologists
- The number theorist following Hardy’s “beautiful uselessness”
Each has different publication rates, funding opportunities, and impact metrics. Any evaluation system either becomes dictatorial (h-index rules all!) or produces paradoxes (pure math is more valuable than applied, applied is more valuable than interdisciplinary, but somehow interdisciplinary is more valuable than pure).
The Time Horizon Tragedy
Perhaps most painfully, your past, present, and future selves are stakeholders with incompatible optimization functions:
Past You (the grad student): “Follow beauty and truth wherever they lead!”
Present You (pre-tenure): “I need papers. Any papers. Now.”
Future You (hoped-for Fields medalist): “Why didn’t you work on harder problems?”
Elderly You (mathematical legacy): “Did any of it matter?”
Spending five years learning schemes and stacks?
- Short-term disaster (no publications)
- Medium-term enabling (opens new areas)
- Long-term essential (for modern algebraic geometry)
- Eternal view unknown (might be superseded by better frameworks)
Arrow’s theorem formalizes what you feel in your bones: These timeline-selves cannot be simultaneously satisfied.
The Liberating Conclusion
Here’s the plot twist that took me years to internalize: Arrow’s Impossibility Theorem isn’t a source of despair. Instead it’s a form of liberation.
That guilt you feel about not being excellent at research AND teaching AND service? Mathematically inevitable.
The anxiety that you’re optimizing for the wrong thing? There is no “right thing” as optimization itself is the problem.
The sense that your colleagues have figured out some secret you haven’t? They’re just unconsciously picking their dictator (could be research, could be teaching, could be service, could be something else) and living with the consequences.
The Mathematician’s Response to Impossibility
We’re mathematicians. We don’t deny theorems; we work with them. So what does it mean to build a career acknowledging Arrow’s impossibility?
1. Embrace Conscious Dictatorship Pick your tyrant explicitly. “For the next three years, publication count is my dictator. I acknowledge this means sacrificing teaching excellence and possibly my sanity.”
2. Time-Box Your Trade-offs “Pre-tenure: career optimization. Post-tenure: mathematical truth. Near retirement: mentoring the next generation.”
3. Accept Cyclical Preferences Monday: “Research is everything.” Wednesday: “My students need me.” Friday: “The department would collapse without service.” This isn’t inconsistency, it’s a necessary sacrifice.
4. Find Your Comparison Class Stop comparing across categories. A category theorist and an applied mathematician are playing different games. Find your reference group and optimize within it.
5. Meta-Game Awareness Play the game while working to change it. Optimize for the current broken system while advocating for better metrics.
The Ultimate Paradox
The final twist? A mathematically literate response to Arrow’s theorem undermines the very idea of optimization that drives academic culture. We’ve proved that the search for the optimal career path is itself suboptimal.
This is either deeply depressing or strangely comforting. I choose comfort.
Your career dissatisfaction isn’t a personal failing, it’s a mathematical certainty. Every choice disappoints some stakeholder, including some version of yourself. That’s not failure; that’s Arrow’s theorem.
So the next time you’re agonizing over whether to:
- Polish that proof or grade those exams
- Chase citations or follow curiosity
- Please your department or your family
- Build a career or build mathematics
Remember: The impossibility is the point. There’s no perfect solution because perfection is mathematically undefined in multi-stakeholder optimization.
Make your choice. Accept the trade-offs. Move forward.
And take solace in this: At least your career anxiety has a Nobel Prize-winning proof behind it.
P.S. The irony isn’t lost on me that I’m writing a blog post about optimal career strategies while proving optimization is impossible. Arrow would appreciate the paradox. Then again, optimizing for Arrow’s appreciation would violate… well, you get it.