The Categorical Imperative
# The Categorical Imperative
You don’t need a tenure letter to have an editorial opinion about math. You need curiosity, a bit of scaffolding, and the willingness to ruin — and then resurrect — cheap calculators bought off the internet. Call this a small manifesto: treat math like something you can love, not something that must be endured. If that sounds suspiciously like philosophy, great — the title is deliberate. Category theory is not a moral law, but sometimes it behaves like one: insist on universal properties, and surprising regularities follow.
## Start with scaffolding, not scaffolding as punishment
If your math education felt like a long set of mnemonics, hit the reset button. Prioritize proof techniques first: logic, contrapositive, induction, and the habit of asking “why” before you memorize “how.” From there, linear algebra gives you a toolkit for thinking across contexts — vectors, matrices, eigenthings — and real analysis or abstract algebra provides the discipline to make the strange feel normal.
Practical rule: when you meet a new phenomenon, prototype numerically, then ask why the pattern holds. The compute-then-prove loop is the apprenticeship. It keeps you honest: computers give you data; proofs give you reasons.
## Cheap toys, big ideas — the right kind of cheating
A handheld calculator is not a sin. Let it be a toy and a scaffolding: use CAS and Python to brute-force examples, find patterns, and ferret out edge cases. But remember — a thousand successful examples are not a theorem. They’re a very insistent hint.
Use tools smartly: sketch many cases, then look for invariants or quantities that stay put. If you find a conserved quantity, that’s the beginning of algebra; if geometry keeps whispering, follow it. And yes, buy the TI-84 if it makes you giddy. Nostalgia is fine; misplacing intellectual rigor is not.
## Puzzles, conjectures, and backyard fireworks
Internet conjectures are the best: messy, often true for small n, and almost always in need of better phrasing. Treat them like a pet — feed it data with Python, then try to walk it on a leash of inequalities or parity arguments. Often a simple bounding argument or parity check will kill the hypothesis or show why it must be <= some constant.
If you can't prove it, that's ok. Publish it as a blog post, not a theorem. The community will either tidy it up or show you a counterexample — both are useful.
## Magic squares, geometry, and matrices that behave like mood rings
Magic squares are compact lessons in symmetry and linear constraints. Visual patterns in a square correspond to linear relations among entries. When someone says "there's a geometric meaning to this pseudo 4×4 magic square," listen. Translate the visual into linear algebra: express rows, columns, and diagonals as linear functionals, and you're suddenly applying rank arguments and nullspaces.
Geometry gives intuition; algebra gives the proof. Use whichever lens explains more in fewer words.
## When algebra balks: metrics, geometric products, and bookkeeping
Clifford (geometric) algebra is a beautiful place for things to go slightly wrong in pedagogically instructive ways. The geometric product ab = a·b + a∧b survives odd metrics — it just asks for careful bookkeeping. If your basis isn't orthonormal (metrics with off-diagonals), don't curse the algebra; compute with components using the metric tensor g_{ij} =
Don’t be afraid of indices. They’re boring but reliable. If indices make you sigh, lean on software that supports geometric algebra. The algebra remains associative; you just can’t be sloppy about inner products anymore. It’s a damn useful lesson about how structure survives deformation.
## Logic, category theory, and the delight of organizing things that used to be separate
Here’s where “categorical imperative” earns its pun. Category theory isn’t a mystical replacement for doing problems; it’s a language for saying the same thing across contexts. Universal properties, adjunctions, and naturality turn messy, case-by-case proofs into short conceptual arguments.
But there’s a tradeoff. Category theory’s abstraction is a double-edged sword: it can reveal a 90% common core across seemingly different problems or it can feel like wearing a tuxedo to fold laundry. Use it when structure benefits from being seen at a higher level: functorial behavior, limits/colimits, or when you want a conceptual reason two constructions are “the same.” Otherwise, concrete calculations and counterexamples will serve you faster.
Logic and computation mingle here too. Curry–Howard connects proofs to programs, and toposes give semantics to alternative logics (intuitionistic logic, for example). If you’re interested in how different logics model “truth,” category theory is the organizational brain; if you just want to count how many proofs exist, combinatorics and proof theory might be more satisfying.
## Community, humility, and the right kind of hubris
You don’t have to be original to learn. Share code, failed proofs, and half-formed diagrams. Stack Exchange, subreddits, and local meetups are full of people eager to say “obvious” or “cool” — both useful. Embrace feedback. Mathematicians love concrete examples; they hate vagueness. Bring computations, and people will bring insight.
Also, be ready to be wrong. Delight in it. A counterexample is not humiliation; it’s data that sharpens intuition.
## Final takeaway — keep the curiosity, but bring the tools
Math is a toolbox and a playground. Read proofs to learn modes of thought, use cheap hardware and CAS to explore, and treat conjectures like pets: feed them data and train them with proofs. When the metric looks nasty, compute with the metric tensor or orthonormalize. When abstraction seems excessive, return to components and examples. And when category theory looks like a fancy ritual — try it on a small problem and judge honestly.
I’ll leave you with a small, provocative question (because I like leaving you dangling like that): given a pattern you’ve observed numerically that resists proof, what would it take for you to reframe the problem at a higher level — say, via a universal property or an adjunction — so that the phenomenon becomes inevitable rather than miraculous? Is that reframing always legitimate, or does it sometimes paper over the real combinatorial mess underneath?
