The Categorical Imperative: You Can’t Post a Meme to a Math Department (A Mild Manifesto by Dr. Katya Steiner)

# The Categorical Imperative: You Can’t Post a Meme to a Math Department

If you graduated in the era of mixtapes and dial tones and still think you’re being subversive by posting philosophy memes under a math paper, bless your heart — and allow me thirty seconds of pedantry. Logic is a toolbox, not a catch‑all bin for every clever slogan, existential hot take, or metaphysical fever dream. This is a short, affectionate field guide to what the discipline actually studies, why Kleenex‑sized analogies don’t pass as proofs, and how different branches of math and logic talk past each other (sometimes politely, sometimes like two lovers who both thought the other would bring wine).

## What actually counts as logic (and what doesn’t)

Logic studies relationships among claims. That’s the headline. If you have two or more propositions and you’re asking whether one follows from another under stated assumptions, you’ve wandered into logic territory. Classic logic treats propositions as formal objects, explores their consequences, and builds systems that let us say cleanly when an argument is valid.

What logic usually does not include: single‑claim proclamations (those belong to the domain expert in whatever you’re talking about), cute puzzles that are just party tricks, hardware wiring diagrams, or mystical attempts to reveal “ultimate truth.” Yes, set theory, model theory, proof theory, modal logic, and computability are logic’s knitting needles. No, circuit design, astrology, or performative metaphysics are not. If your content doesn’t fit, there’s probably a subreddit, a department, or a philosopher who’ll happily take it — but don’t be surprised if the logicians politely point you to the right room.

## Two short thought experiments to save arguments at dinner parties

1) The meme: “All the things Harvard teaches” plus “All the things Harvard doesn’t teach” equals everything. This is a set‑theory gag about exhaustive coverage — the union of complementary sets gives the whole universe. In formal terms: for any element x, x ∈ (T ∪ ¬T). It’s literal, totalizing, and deliberately hyperbolic.

2) The filmmaker: “I hate every part of making a film; I hate not making a film even more.” This reads as a preference ordering, not a universal condemnation. It says: for this speaker, not‑making ≻ making (not‑making is worse). It doesn’t quantify over everything else — socks and sunsets remain out of logical scope.

Key vocabulary: quantifiers (∀ for “for all”), predicates (properties like “is a film”), and preference relations (A ≻ B). The meme is totalizing set theory; the quote is comparative psychology. Both are logical moves — just different ones.

## Proof by resonance: clever, occasionally useful, not a metaphysical trump card

“Proof by resonance” — the idea that if something fits a definition so well, and behaves correctly under transformations and over time, then it’s verified — is an attractive synthesis. In mathematics, definitions + structural isomorphism + invariance are powerful. If a candidate object satisfies every defining predicate and maps isomorphically to the canonical model, you’ve got a robust identification.

Why it tempts people outside math: it promises a kind of holistic validation. If an idea “resonates” with your prior intuitions and maps to many corners of your theory, it feels right. But in messy real world domains definitions are partial, context‑sensitive, or plain fuzzy. That’s why resonance is a good heuristic in controlled settings, and a lousy epistemic substitute for falsifiability or rigorous specification in the wild. Claiming resonance replaces empirical or logical tests is a kind of intellectual wishful thinking — charming, sometimes illuminating, but not a free pass.

## Quantum logic: a new tool, not a hostile takeover

Quantum mechanics forced a question: what if the algebra of events doesn’t obey distributivity? Birkhoff and von Neumann’s quantum logic is a formal system reflecting the non‑classical lattice of projection operators on Hilbert space. It’s about measurement outcomes, superposition, and the strange algebra that comes with them.

Does it overthrow classical logic? Hell no. Classical logic remains our everyday workhorse. Quantum logic is a domain‑specific instrument that clarifies interpretational puzzles in physics and demonstrates how altering algebraic assumptions changes consequences. It’s fascinating and deep, but not a universal coup against truth tables.

## Where other branches of math speak up (and sometimes save the party)

– Model theory: tells us about structures that satisfy theories. If your meme claims some universal property, model theory asks whether there exists a structure where the claim holds and whether it holds in all structures.
– Proof theory: cares about the syntactic life of proofs — how do proofs reduce, transform, or normalize? If your “proof” is merely persuasive prose, proof theory will call the bouncer.
– Category theory: here’s where my titular pun pays rent. Categories replace ‘elements’ with objects and morphisms; they emphasize relationships over raw membership. The categorical imperative (sorry, couldn’t resist) is: think in terms of structure‑preserving maps. Category theory is beautiful because it unifies patterns across algebra, topology, computer science, and logic itself (via categorical logic and topos theory).
– Computability/type theory: when your argument depends on procedures — algorithms or constructive content — these fields tell you what can be computed, what can be constructed, and what the program actually proves.

Each of these fields offers a different lens. Sometimes they agree; sometimes they don’t — and that disagreement is where the fun and the real insight lives.

## How to post something and not be that person

Be tidy with definitions. Be explicit about your scope and quantifiers. If you’re using analogies, declare their limits. If you’re appealing to ‘resonance’, say what you mean by the predicates and what empirical or logical checks you’ve done. And yes, if it’s a meme, tag it as such and post it where your audience expects memes.

Logic rewards the pedantic. It’s cheaper than therapy and often just as clarifying.

## Parting note (and an invitation)

There’s a charming instinct to want a single, elegant principle — a categorical imperative of thought — that will settle fuzzy debates, glue together disciplines, and coronate our intuitions. Sometimes we get lucky: a unifying notion from category theory or a clean model‑theoretic classification soothes the soul. More often, the world insists on plural tools and domain‑sensitive judgment.

So here’s my open question for you, dear reader: when you reach for a single explanatory principle (a resonant definition, a favored logic, a unifying metaphor), what’s your checklist for deciding whether you’re discovering structure or inventing it?