The Categorical Imperative: How to Relearn Math Without Turning Into a Hermit Genius (Unless You Want To) — Dr. Katya Steiner

# The Categorical Imperative: How to Relearn Math Without Turning Into a Hermit Genius (Unless You Want To)

You want math back in your life. Maybe you woke up at 45 and realized that algebra once felt like a foreign tongue and now it’s an intriguing dialect. Maybe your fingers reflexively reach for a calculator and you want them to remember how to do a little work on their own. Or maybe you stared at (a+b)^3 and felt that old betrayal — the kind that lingers until you redraw the cube and swear never to be intimidated by letters again.

Good news: relearning math is mostly logistics — correct resources, a tiny strategy, fewer excuses. Bad news: occasional actual practice is unavoidable. Also good news: it’s damn freeing when the pieces click.

Why bother? Because math is the language that keeps showing up — in finance, data, code, policy debates, and those smug Twitter threads. Reclaiming it sharpens your thinking, opens practical doors, and gives you access to a way of reasoning that’s both unforgiving and beautifully precise.

—

## From arithmetic to category theory: a tour without the pretension

Math isn’t a monolith. Different subfields cultivate different muscles.

– Arithmetic & algebra: the toolkit. Fluency here is like being able to read street signs. Fractions, manipulations, algebraic patterns — this is where competence lives.
– Calculus & analysis: change and limits. These are the models we use to talk about motion, optimization, and approximation. Learning to think in derivatives and integrals trains you to spot structure in continuous change.
– Linear algebra: vectors and transformations. Once you grok linear maps, a surprising number of problems start to look the same. It’s the discipline where computation meets elegant abstraction.
– Discrete math & combinatorics: counting with attitude. This is where algorithms, graphs, and finite structures play. Very satisfying for problem solvers who like sharp answers.
– Probability & statistics: coping with uncertainty. Back when we thought numbers were always polite, probability taught us otherwise. Crucial for real-world decisions.
– Abstract algebra & category theory: the meta-level. These branches force you to think about systems of systems. Category theory is the “how math speaks about math” — the categorical imperative, if you will.

Each area trains a different cognitive habit: bookkeeping precision, asymptotic thinking, geometric visualization, combinatorial cleverness, probabilistic humility, or structural abstraction. None is inherently superior; dominance depends on what you want to do.

—

## Logic: the skeleton beneath the muscles

If math is bodywork, logic is the skeleton. Propositional logic teaches clarity of statements. Predicate logic adds variables and structure. Modal or constructive logics introduce nuance about necessity, computation, and proof methods.

Why care? Because the same forms of reasoning you practice in math (if A implies B, and B holds under these conditions…) map directly to clear writing, better debugging, and less magical thinking. Also: proof techniques — induction, contradiction, contrapositive, constructive proofs — are little cognitive tools you can borrow for daily life.

Take induction: the principle is tiny and powerful. It’s not just proving number facts — it’s learning to build reliable chains from base cases. That way of thinking helps in planning, debugging, teaching, and even training teams.

—

## Intuition vs rigor (and why you want both)

There’s a perennial fight: intuition (pictures, metaphors, examples) versus rigor (axioms, epsilon-deltas, formal proofs). I’m Team Both.

Intuition gets you emotionally and quickly — the cube visualization for (a+b)^3 is the perfect example. Rigor keeps you honest; it prevents beautiful but false generalizations from turning into embarrassing tweets. Start with intuition, test it with calculation, then ask: can we formalize this? The healthy discipline cycles among these modes.

A warning: some learners lean too hard on tools. WolframAlpha and Python are wonderful labs, but they can become a crutch. Do enough pencil-and-paper work that your brain encodes patterns, then use code to expand and experiment.

—

## Tools, toys, and the social lifeline

My recommended kit looks like this: one good video series (for intuition), one textbook/workbook (for deliberate practice), one interactive tool (Desmos/GeoGebra), and one community (Math StackExchange, AoPS, or a study group).

Why limit it? Choice overload is the enemy. Pick one of each and stick to them until you feel confident.

Communities are underrated. Posting a specific stuck step will get you better help than a vague cry for guidance. Teach a problem back to someone — nothing fixes gaps faster than explaining.

—

## Study habits that don’t require asceticism

– Short, focused sessions (30–60 minutes) beat marathon binges.
– Spaced repetition and mixed practice (hit algebra, then calculus, then statistics) build durable learning.
– Keep a “problem graveyard” — a list of mistakes you can return to weekly.
– Embrace errors: they’re your debugging logs. Analyze why you tripped.

And for the love of practicality: schedule meals and friendships. You can be brilliant and human at the same time.

—

## The philosopher’s nudge: a categorical imperative for learners

If category theory is about morphisms and objects, my imperative for learners is simple: make your learning composable. Build small, verifiable modules of understanding that you can compose into larger results. That way, if one piece fails, you can isolate it quickly and repair it — the mathematical version of modular code.

This approach respects both the structure-sensibility of advanced math and the need for incremental wins that keep you motivated.

—

## The trade-offs (because life isn’t all equations)

You’ll face choices: depth vs breadth, abstract pleasure vs applied utility, solitary practice vs group learning. None of these is universally right. If you need math for work, prioritize utility. If you fall in love with abstraction, allow yourself to wander. If social life matters, schedule it. The alarmingly good news is that math can accommodate all these approaches — it will welcome you back whether you’re pragmatic or romantic.

—

Parting thought: relearning math is not a return to some mythical school-you who aced tests. It’s an invitation to build habits that make thought clearer, curiosity kinder, and problem-solving more deadly efficient — without becoming a hermit genius (unless that’s your fantasy, in which case, fine: schedule walks and vitamin D).

So: what tiny, composable module will you learn this week — and who will you show it to?