The Categorical Imperative: From TI-84s to Tensor Tantrums

# The Categorical Imperative: From TI-84s to Tensor Tantrums

You keep a TI-84 in a drawer for the aesthetic. You have strong opinions about Spivak. You can still solve a puzzle at 2 a.m. and feel like you are alive again. Welcome to the club — the curious-but-overworked former student who refuses to let calculus become a distant, exasperated memory.

This is not a manifesto. It is a set of friendly prescriptions, mild parables, and some Kantian wordplay about why the mathematical life after graduation is both a moral duty and a delicious hobby. Consider it the categorical imperative of recreational rigor: act so that your curiosity could be a maxim for all caffeine-fueled problem solvers.

## Pick a backbone, not a buffet

When time is a scarce good, eclecticism becomes an excuse for incompletion. The smarter move is to adopt a tripod: one rigorous text, one intuitive exposition, and one applied or computational companion per subject. That method keeps both your soul and your résumé nourished.

– Proofs & logic: Velleman as ritual practice. Learn how to argue rather than only how to show work.
– Calculus/analysis: Spivak if you like the deep-breath, mathematical-aesthetic route; Stewart or Courant if you need pragmatic tools fast.
– Linear algebra: Axler for clean theory; Strang for matrices that actually behave in the real world.

Augment these with videos from 3Blue1Brown, MIT OCW lectures, and problem forums. Read fewer things well. Skill compounds like interest; attention is the currency.

## Tools: buy, borrow, or stop fetishizing plastic

A non-working TI-84 makes a good paperweight and a perfect ironic prop for your Zoom background. For doing math, the hierarchy is simple:

– Visualization first: Desmos and GeoGebra are free and beautiful.
– Computation next: Python with NumPy/SymPy scales. If you are automating algebra, do it like an adult.
– Emulators or physical graphing calculators if you enjoy tactile rituals or are stuck in legacy exam land.

The categorical choice here is: tools should amplify thought, not mask laziness. If you find yourself repeatedly fixing calculator-battery issues instead of proving lemmas, you have violated the imperative.

## Conjecture-making: entitled, but verify

Everyone has a cute conjecture tucked in the mental glove compartment. They are delightful and dangerous: delightful because they train intuition, dangerous because they encourage overconfidence.

A practical route from ‘I think’ to ‘I prove’:

– Exhaust the low-hanging fruit with a quick script. Computers are savage truth-tellers.
– Search for invariants and extremal cases. Digits, eigenvalues, matrices — usually something is bounded or monotone.
– Use inequalities. AM-GM, Cauchy, and their cousins show up like old friends at a very nerdy reunion.
– Reduce to a finite check plus a bounding argument whenever possible. Human brains love patterns; proofs demand fences.

This is the triage method: let your computer debunk you, then sharpen the cases that survive.

## Playful structures aren’t toy boxes

Magic squares, surprising bijections, and combinatorial puzzles look like party tricks, but they often encode deeper symmetry or algebraic structure. When someone hands you a bizarre 4×4 variant, ask: what is preserved? Sums, inner products, projection properties? Reinterpretation is the mathematician’s sleight of hand.

Sometimes the aesthetic variant leads straight into category theory: what maps to what, and why does structure persist under that mapping? If category theory feels like high-brow Kantianism, that is not an accident — both ask what universals govern particular instances.

## When geometric algebra throws a tantrum

Geometric and Clifford algebras are gorgeous and bitchy. In an orthonormal world life is neat; in a general metric you suddenly owe algebraic rent.

Triage plan:

– Change coordinates. Gram–Schmidt or diagonalize the metric and get back into a friendlier basis.
– Use the defining relations symbolically: e_i e_j + e_j e_i = 2 g_ij. Don’t sneak antisymmetry where the metric says otherwise.
– Matrix representations help. If bookkeeping bores you, let a computer carry the heavy sums.

There is dignity in translating a tricky algebra into a simpler language. If you cannot simplify, at least make the problem readable.

## Two sides of the same coin: beauty vs. utility

There is an academic piety that elevates pure abstraction as the only virtuous math. There is an industrial piety that worships utility. Both are right and both are wrong.

– Beauty matters because it guides. Elegant proofs reveal deeper connections and conserve cognitive energy.
– Utility matters because math that cannot be used risks becoming a cathedral without parishioners.

Your categorical imperative should be to balance: pursue beauty that can be communicated, and utility that can be understood. If you tilt too far into ivory-tower fever or spreadsheet pragmatism you will miss large swaths of delightful math.

## A modest program for the curious-but-overworked

– Pick one backbone per subject and stick to it.
– Automate grunt checks with Python; visualize with Desmos.
– Make conjectures, then immediately try to break them with code.
– Translate uncomfortable algebra into friendlier bases or matrix form.
– Read one deep proof a month and try to explain it out loud to someone (or a plant).

This is not discipline for discipline’s sake. It is a recipe for joy: deliberate learning, small wins, and the slow accretion of competence.

## Final thought

Math after graduation is not a competition or a completion task. It is an ethical stance toward curiosity: you either honor the itch and let it guide you into careful thinking, or you ignore it and watch the itch migrate into Twitter arguments.

So here is my categorical question for you, the reader who still keeps a calculator around for nostalgia: what maxims about your mathematical curiosity would you like everyone to follow, and which one are you least likely to keep?