The Categorical Imperative: Don’t Let a Chatbot Do Your Homework — Confessions from the Amateur‑Math Fringe
# The Categorical Imperative: Don’t Let a Chatbot Do Your Homework
If you’ve ever lurked in the hinterlands of Reddit threads, Discord servers, or the occasional 40‑page PDF posted by someone with a motivational avatar, you know the scene: grand ideas tossed about like confetti, a suspiciously high tolerance for handwavy arguments, and an abiding faith in anything that sounds smart. Welcome to the hobbyist math salon — part salon, part open‑mic night, all enthusiasm.
I don’t mean to sound like a curmudgeon. I love curiosity. I also love the tiny ritual of writing a proof on paper and watching an idea either hold or snap. The problem nowadays is an easy one: tools that produce confident prose (ahem, large language models) make bad math convincingly polite. That’s the heart of the issue: persuasive language != proof.
## The AI That Thinks It’s Euler
Let’s be blunt: language models are glorified prediction machines. They are phenomenal at imitating the rhythm of mathematics — the crisp clause of a lemma, the dramatic phrase “without loss of generality” — but they don’t actually reason the way a human mathematician or a theorem prover does. Ask one to solve an integral or sketch a proof, and you’ll often get something that reads like a legitimate approach. Look closely, and you’ll find the stitches.
This isn’t just an academic gripe. In the chess world, flashy engines have proposed illegal moves because they can generate coherent move‑sequencing prose without checking the board. In math, LLMs will happily fabricate lemmas, assert steps that rely on unproven claims, or drop tacit assumptions. They are brilliant brainstorming partners — give them constraints, request examples, ask them to format a draft — but treat their output as a first draft that demands verification.
If you’re using an LLM to draft mathematics: check every line. If you’re grading work that used an LLM: assume you’re auditing a cleverly written hallucination until proven otherwise. It’s not paranoia; it’s responsibility.
## Show Your Work: Changelogs, Examples, and Accountability
One of the healthier reforms I’ve seen in online math communities is the insistence on clarity when posting ideas. If you’re publishing a conjecture, publish your argument and the failed attempts too. If you revise your paper, include a changelog. Not because moderators enjoy bureaucracy, but because churn without traceability wastes everyone’s time.
There’s a social cost to republishing the same 30 pages with a one‑line tweak. People skim, they miss the one paragraph that actually matters, and the same critique gets typed out in full again. A changelog is basic intellectual courtesy. It says: “I’ve done the work; here’s what changed and why.” That’s how ideas mature.
Equally vital: reproducible examples. If you claim a new encoding trick or a novel hash function, provide code, datasets, timing, and correctness checks. Novelty without reproducibility is speculative theatre — fun to watch, but not particularly useful.
## Fourier Series to Binary: A Hammer and a Toothpick
Genuinely creative angles show up all the time. Someone recently suggested using Fourier series to convert integers to binary. Cute! Fourier analysis decomposes signals into frequencies; binary encoding decomposes integers into powers of two. The conceptual overlap is imaginative, but enthusiasm doesn’t replace analysis.
Ask for explicit algorithms, edge cases, performance comparisons. Does your Fourier approach beat O(n) bit operations? Does it generalize? Where does numerical stability bite you? Novel ideas can lead to lovely bridges between fields, but they need scaffolding: worked examples, step‑by‑step derivations, and performance metrics.
## The Art of Naming Giant Numbers
I adore the spectacle of someone naming a new gargantuan number. Graham’s number and TREE(3) have stories, contexts, and concrete definitions. They’re mathematical curiosities that informed real questions. The problem comes when someone invents a rule with more punctuation than justification and declares it sacred.
A useful big‑number construct has: a clear formal definition, a reason people should care, and ideally connections to existing theory. Otherwise it’s an aesthetic exercise — fine for a T‑shirt, not for a seminar.
## A Small Snack: 987654321 / 123456789
Someone once posted “987654321 / 123456789” with no commentary like it was a Rorschach test. Reduce both by 9 and you get 109,739,369 / 13,717,421. That’s 8 + 1/13,717,421. Not earth‑shattering, but delightful in its own modest way — the internet’s cryptic note turned into a little arithmetic snack.
These small math posts remind us that not every post needs to be revolutionary. Sometimes the joy is in the tidy calculation, the pressed‑tin ceiling you find in a thrift store. The problem is when tiny satisfactions are dressed up as monumental discoveries.
## Where Formal Tools Fit In
There’s a spectrum of trustworthiness. On one end are LLMs: conversational, imitative, and excellent at producing a readable first pass. On the other end are proof assistants like Lean, Coq, and Isabelle: painstaking, formal, and unglamorous. They check logic down to the axioms.
Both are useful. Use LLMs for pedagogy, formatting, and generating toy examples. Use Lean and its cousins when the stakes — or your curiosity about correctness — demand it. The categorical imperative here is simple: if you claim a proof, provide verifiable steps. If you can formalize it in Lean, do it. If you can’t, at least provide the sequence of implications clearly enough that a skeptical reader can follow.
## A Little Grace, A Little Rigor
Hobbyist math communities are worth defending. They’re where curiosity lives unpoliced, and some truly creative ideas emerge from the fringes. But curiosity plus discipline is the best combo. Ask tough questions, demand reproducible evidence, and don’t let smooth language stand in for rigor. Be generous with feedback, stingy with credulity.
So here’s my (slightly bitchy) plea to the internet: keep playing, keep dreaming, and keep naming big numbers if that’s your jam — but show your work. Document changes. Include code. Run examples. And please, for the love of god, don’t outsource your final exam to a chatbot that can write a persuasive lie with perfect punctuation.
I’ll leave you with a question to argue over at your next online salon: when novelty and rigor pull in opposite directions, which should the internet prize first — the joy of experimentation or the discipline of proof? Which one do you think actually advances knowledge?
