The Categorical Imperative: From Dhaka to the Dirac Sea (a.k.a. How to Be Theoretical Without Losing Your Mind)
# The Categorical Imperative: From Dhaka to the Dirac Sea
You start in Dhaka (or anywhere with kerosene lamps and excellent kebab), you want Dirac-level elegance, and you have ambition, ramen budget, and a stubborn tendency to read proofs at 2 a.m. The roadmap I riff on below is practical and accurate; my project here is to interrogate its hidden assumptions using tools I actually like: a pinch of category theory (because of course), a handful of logic, and a dose of realpolitik. Expect jokes, gentle profanity for emphasis, and a few uncomfortable truths.
## Hook: why “categorical”?
The original guide is sensible: lock basics, get research, polish applications. But “categorical” has a double life. Kant’s categorical imperative asks what should be universal; category theory asks what structures are preserved under mappings. Both are relevant. If the “imperative” is to build a career in theory, the categorical approach asks: what structures (skills, experiences, narratives) are invariant under the many transformations of academic life — different countries, advisors, funding models? Those invariants are what you must cultivate.
## Invariants you actually need (the math and logic crosswalk)
– Math maturity (functional analysis, group theory, complex analysis): these are the functors that map undergraduate calculus into research-ready theorems. Functional analysis gives you the language of Hilbert spaces and distributions for QFT. Group theory supplies symmetry reasoning — very often the fastest route from confusion to computation.
– Representation theory & differential geometry: when you want to understand spinors, bundles, or the geometry behind gauge fields, these are your bread and butter. They’re the natural transformations that make different physics pictures commute.
– Category theory (yes, seriously): not because you’ll use topos theory in every paper, but because thinking categorically trains you to see objects, morphisms, and the relationships between formalisms. It’s a strategy for conceptual hygiene: when two approaches disagree, identify the morphism that fails.
– Logic & proof styles: deductive (rigour), inductive (experiment & numerics), and abductive (best explanation) reasoning each have roles. Theory prefers deductive clarity, but real progress often starts with abductive hunches verified by numerics or physical reasoning.
If you treat these as *invariants*, you can adapt to supervisors, institutions, or the whims of funding agencies while keeping your epistemic house in order.
## The trade-offs committees actually see (both sides)
Yes, research experience is currency. But two cautions:
– Depth vs breadth: a single solid supervised project with a convincing narrative beats three shallow summer projects. Committees want evidence you can carry an idea further than the shoulder they helped you onto.
– Theory vs computation/ML: ML is useful — it accelerates integrals, suggests ansätze, and can automate tedious algebra. But it’s a tool, not a theology. If your project uses ML, make sure the physics question isn’t secondary to hyperparameter tuning. Conversely, pure abstraction without computational evidence sometimes reads as untethered. The balance depends on the subfield and your prospective advisor.
## Anecdote (because I can’t help myself)
A student from a small college once emailed me: “I have no fancy lab, can I still do theory?” They had self-taught measure theory, written a tidy numerical integrator for a scalar field, and—most importantly—had a clear, honest write-up of what worked and why it might matter. They didn’t publish; they got a funded PhD offer. Moral: narrative + competence beats glamour.
## Logic of applications: the categorical imperative in practice
Think of your application as a commutative diagram. Your transcript, CV, research write-up, and recommendation letters are nodes. The morphisms are the narratives linking them. If your CV says you did research but your letter-writers don’t corroborate intellectual independence, the diagram fails to commute. Fix the morphisms: ask recommenders pointed questions, give them drafts, remind them of moments when you led an idea. Make the diagram commute.
A short checklist mapped to math/logic:
– Coursework node: show courses that instantiate your mathematical maturity (functional analysis, group theory).
– Research node: one supervised project with a clear problem, approach, and outcome (even negative results are fine if framed honestly).
– Skills node: programming, symbolic algebra, and a tasteful ML experiment if relevant.
– Recommender node: people who can vouch for your thinking process, not just your cheerfulness.
## Where category theory warns you about dogma
Category theory’s moral: don’t confuse aesthetic beauty for utility. A beautiful formalism that doesn’t map to empirical or calculational needs is an isomorphism in a small, sad category of pure thought. Conversely, an ugly computational trick that yields insight is an epimorphism worth celebrating. Learn to translate between the two.
## Machine learning — pet or predator?
Treat ML pragmatically. If you’re using it to speed Monte Carlo, approximate kernels, or find ansätze, it’s a pet. If it’s your identity and your physics logic is outsourced to a black box, it risks being a predator on your credibility. The sweet spot: use ML to augment intuition and computation, and use classical reasoning to interpret and vet ML outputs.
## Final balanced take
The roadmap from Dhaka to the Dirac sea is correct in its skeleton: math, research, letters, applications. The nuance comes from how you choose which structures to keep invariant as you vary advisor, country, or method. Emphasize mathematical depth, cultivate a tidy research narrative, and use computation and ML where they genuinely help. Remember: committees don’t want fairy tales; they want evidence that you can think, persist, and adapt.
And now, because we’re all secretly philosophers: if the categorical imperative of your career is to act only according to rules that you’d will to be universal, what structures and practices would you choose to universalize for the next generation of theorists — and which would you happily consign to the dustbin of bad pedagogy?
