The Categorical Imperative: Math on the Internet, DIY Theorems, and the Art of Not Ruining a Good Idea

# The Categorical Imperative (of Internet Math)

You know the scene: it is 2 a.m., someone posts “I solved X” in all caps, Reddit ignites, and ten hours later the comments are a mixture of furious brilliance and friendly evisceration. The internet is an excellent place to have late-night mathematical epiphanies — and an equally excellent place to see them die a messy, public death. This is not a takedown piece; consider it a love letter with a stern footnote. If you want your math to be heard, not laughed at, read on.

## The machine that talks like a human is not a mathematician

Okay, this is the least surprising sentence you will see today: large language models are spectacularly persuasive conversationalists and terrible proof checkers. They predict tokens, not truths. That means when you ask for a derivation you’ll often get something that looks exactly like a derivation — until it quietly turns into wishful algebra. This isn’t malice; it’s architecture.

Where you need strict logical chains, bring tools that care about exactness. Proof assistants such as Lean or Coq, and CAS systems like Mathematica or Sage, are built to avoid the sort of creative fudging chatbots indulge in. Use LLMs for intuition, exposition, and polishing—ask them to explain ideas, not to certify them. If an AI supplies a lemma, re-derive it, test edge cases, or formalize it in a trusted system.

## Post your work, or keep it as a thought experiment

Forums operate on an implicit social contract: if you post a theorem, you should be able to answer questions. Outsourcing the author’s voice to a third party turns conversation into telephone; the signal decays, nuance vanishes, and mistakes ossify. If you’re sharing someone else’s draft, get permission and say who will speak for the paper. If the originator can answer comments, the work improves faster.

A corollary: provide a changelog. If your 40-page note gets patched, list the edits at the top. Tag updates in the title. People will love you for it. And if they don’t, well — at least you saved them from re-arguing old versions.

## Where the disciplines collide (and sometimes cuddle)

One of the joys of internet math is the strange cross-pollination. Here’s how some fields illuminate the problem:

– Number theory: spectacular almost-integers and numerical coincidences make for great copy. But an enticing decimal is not a proof. Continued fractions, Diophantine approximation, and transcendence theory are the right ways to drill into why some ratios flirt with integers.

– Analysis & harmonic analysis: ingenious folks sometimes try to use Fourier methods to encode combinatorial objects or design hash functions. That can be brilliant — but show code and failure cases. Fourier transforms can be applied in surprising domains, but the spectral intuitions must be matched with computational experiments.

– Logic & model theory: many internet claims hinge on sloppy quantifiers or hidden set-theoretic assumptions. Is your construction well-founded? Are you equivocating between constructive and classical existence? Spell out your foundations.

– Category theory: the temptation to dress up a heuristic in universal properties is real. Category-theoretic language can clarify, but it can also obfuscate if used as gloss. If you invoke adjoints and monads to sound abstractly wise, also give the concrete instances where those structures illuminate behavior.

– Computability & complexity: an algorithmic claim without asymptotics or complexity analysis is a promise with no warranty. If your method beats something, show runtime, space, and a reproducible benchmark.

## Grand definitions need grand scaffolding

There is a special internet species that invents enormous numbers, gives them names, and then builds a cult around the axioms. Creativity like that is delightful, but it must be anchored. Define connections to classical notions, give computations for small cases, and explain why this object answers a question anyone outside your head cares about. Otherwise it’s a museum piece, admired for craftsmanship but not utility.

## Reproducibility is not optional

If you want constructive critique, make your work runnable. Share code, minimal test cases, and documented failure modes. The community can give excellent feedback if it can reproduce your results in ten minutes and break them in thirty.

## Two examples, briefly

1) Someone uses a neural net to guess a lemma, then proves a theorem relying on that lemma. The theorem collapses if the lemma is false. Moral: separate conjectural heuristics (ML guesses) from certified statements.

2) A hobbyist defines a hierarchy of hyper-factorials and claims they grow “faster than everything.” Great! Now compute the first three values, show an embedding into ordinal arithmetic or fast-growing hierarchies, and demonstrate nontrivial theorems that use them. Otherwise, it’s a beautiful object in search of a motivation.

## Etiquette, or the categorical imperative of online math

– Be the author if you want conversation. If you’re reposting, say who speaks for the work.
– Tag updates and provide a short changelog.
– Use the right tools: proof assistants for rigor, CAS for symbolic work, LLMs for language.
– Include examples, code, and failure modes when proposing new objects or algorithms.
– Stay humble. The internet is merciless but generous — the best projects grew from messy threads.

## A final, friendly reminder

The sweet spot is a community that tolerates half-baked ideas and rewards rigorous polishing. That requires creators who respect readers’ time and a culture that asks for evidence instead of applause.

So here’s the question I leave you with — and please don’t answer it in the comments with a single-sentence “I proved it”: how can we design online spaces and tooling so that late-night mathematical sparks can be shared, critiqued, and matured without turning every thread into a trainwreck? Are proof assistants and reproducible notebooks enough, or do we need social conventions and moderation that value slow verification over fast applause?