If You Must Invent a Number, At Least Explain It: The Categorical Imperative — Dr. Katya Steiner
# If You Must Invent a Number, At Least Explain It: The Categorical Imperative
You scroll a forum and there it is: a proud post that births a new number system, a mashup of Fourier transforms and binary hashes, or—my personal favorite—the lonely fraction 987654321/123456789, abandoned like a message in a bottle. Welcome to the modern math salon, where enthusiasm often outruns rigor and etiquette is… negotiable.
Call this my categorical imperative for online math: act only according to maxims you’d be willing to make universal for the internet. In plain English — if you insist on inventing a number, explain it, show examples, and don’t make the rest of us do detective work for your weekend hobby.
## Why this matters (and why I’m being a little bitchy about it)
First: amateur math online is a net good. Curious people tinker, hobbyists find beautiful bugs in mainstream assumptions, and occasionally a real idea gets traction. Second: LLMs have made everything sound polished. ChatGPT can spin a convincing narrative of a proof the way an infomercial sells miracle cookware. Both facts together produce an environment that looks smarter than it is — and that’s where etiquette becomes hygiene.
This isn’t snobbery. It’s damage control. When notation is undefined, when updates aren’t labeled, when someone posts 50 pages of LaTeX and disappears, conversations stall, moderators weep, and promising ideas get ignored. There’s a real opportunity cost: time wasted, correct insights buried, and a slow erosion of patience in communities that should be welcoming.
## The categorical imperative, math edition
Borrowing Kant badly: treat every post like it could be the canonical version someone cites. What would that require?
– Define notation. If you invent f† or a symbol that looks like a pilgrim’s staff, state what it means. Ambiguity is the enemy of reuse.
– Give tiny examples. If your new operator produces monstrous values for n=10, show n=1,2,3. Concrete beats grandiose claims.
– Show why it matters. A number defined by a trillion factorials is neat, but why should anyone care? Rate of growth? Decidability consequences? A counterexample to a conjecture? Give a hint of motivation.
– Make results reproducible. Share code, a Sage/Wolfram snippet, or a tiny Lean/Coq file. If your claim depends on symbolic manipulation, say so.
– Label updates. [UPDATE] in the title with a one-line changelog is the internet’s version of a courtesy flush.
If this sounds like “publish like an adult,” good. It is. But it’s also about being generous to the future — to the stranger who finds your thread in three years and wants to replicate your toy experiment.
## Across disciplines: what different corners of math would add
Category theory: please don’t invent ad hoc notation for a functorial idea without saying which category you’re in. Many a beautiful insight becomes unreadable when source and target categories are left implicit. The category-theoretic move is to state objects, morphisms, and universal properties. It’s the social contract: if you want to be abstract, please label the arrows.
Number theory & combinatorics: give small n examples and modular reductions. If your monstrous fraction simplifies (spoiler: 987654321/123456789 = 9/11), say it. Number theory loves counterexamples and cute coincidences; context turns a neat simplification into a teachable moment.
Logic & foundations: specify your metatheory. Is your proof constructive or classical? If you lean on excluded middle or choice, say so. Folks in constructive circles will rightly ask different questions; hiding that choice is a fast way to derail useful critique.
Computability & complexity: if your “new number” encodes an algorithmic process, discuss complexity and computability. Is it computable in polynomial time? Is it definable via a primitive recursive function, or does it need something fancier? These distinctions aren’t arcane — they determine whether others can play with your object.
## ChatGPT: delightful assistant, lousy oracle
LLMs are terrific at making things look finished. Use them to draft, to find wording that doesn’t put readers to sleep, or to generate example computations you then check. Do not use them as mathematical referees. They’ll provide proofs that read well and fail spectacularly under scrutiny. The right tools here are proof assistants (Lean, Coq), CAS engines (Wolfram, PARI/GP), and human collaborators.
A good rule of thumb: if a claim is interesting enough to attract attention, run it through at least one mechanized check or one competent friend.
## How to ask for feedback (and how to give it)
Ask like you want help. Don’t drop a Google Drive link and vanish. Tell readers exactly what you want: correctness, notation advice, examples, or application suggestions. Point out where you suspect a hole. Provide a minimal test case and the expected outcome.
If you’re replying: be kind, be specific. Point to the line where an inference fails. Offer a concrete suggestion or a pointer to a tool that can check the claim. The internet does not need more snark; it needs read receipts and useful nudges.
## Practical checklist before you click Post
– Is this your work? If not, credit the author and link appropriately.
– Did you include at least one worked example and a sanity check?
– Is there reproducible code or a tiny proof-assistant file?
– If you update, did you add [UPDATE] and a one-line changelog?
– Did you state whether your argument is constructive/classical or relies on heavy axioms?
Do that much and you transform irritation into curiosity. People are fascinatingly generous with their time when you make it easy for them.
## Final thought (and yes, an open question)
Math online is a messy, brilliant bazaar. We get bad notation, glorious half-baked ideas, and the occasional ChatGPT-native monstrosity. If we treat our posts like artifacts someone might one day cite — clear, reproducible, and gently annotated — we keep the bazaar lively without making it unbearable.
So here’s the question I leave you with: if you could impose just one non‑negotiable rule for every public math post (one that everyone had to follow), what would it be — and why?
