The Categorical Imperative (or: Stop Letting the Internet Do Your Math)
# The Categorical Imperative (or: Stop Letting the Internet Do Your Math)
Every week some brave soul posts a mathematical fever dream: a gargantuan “new” number, a theory recycled by a friend-of-a-friend, a Fourier-to-binary hash on Zenodo hoping for applause, or the inevitable “Can ChatGPT do my homework?” thread. The comment feed swings between awe, pity, and the kind of pedantry that keeps math departments honest. If you left grad school with an algebra diploma and a sense of humor, this is your field guide to surviving — and improving — the chaos without becoming the person who uploads 300-page PDFs and never replies to comments.
## ChatGPT is a conversational parrot, not a theorem-proving prodigy
Let’s be blunt: ChatGPT writes like it understands math. It does not. It predicts tokens, not the next correct lemma. That shows up as arithmetic slip-ups, leaps in logic that feel convincing, and the cheerfully authoritative tone of someone who skimmed an advanced textbook on a plane ride.
Use large language models for polishing prose, brainstorming metaphors, or translating a proof sketch into something friendlier for human readers. Don’t use them as a substitute for checking algebra, edge cases, or boundary conditions. If you want mechanical certainty, use the right machines: Python’s fractions/decimal for numerics, SageMath or Mathematica for symbolic work, and — for the brave and heavily caffeinated — Lean/Coq/Isabelle for formal verification.
Across disciplines this matters differently. In combinatorics a counterexample can be googled away in an hour; in topology a subtle orientation sign can wreck an entire argument; in category theory a misplaced naturality condition makes everything fall apart. A conversational model will happily spin plausible-sounding nonsense in all three.
## If it’s not yours, don’t post it
There are people whose entire posting strategy is ‘be the middleman for someone else’s idea’. Please don’t. The original author is usually better placed to answer subtleties, explain motivations, and provide provenance — and it’s rude to hide them.
Want attention for someone else’s work? Nudge the author to post it, or post only with explicit permission and clear attribution. The internet doesn’t need more anonymous relay stations.
## Updates are not mind-reading: show a changelog
We’ve all read a 40-page PDF only to discover a single paragraph was rewritten. That invites repeated criticism and burned-out readers. Label updates clearly — [UPDATE] in the title is not optional — and put a concise changelog at the top: what changed, why, and what to focus on. Two lines can save reviewers ten minutes and yourself a dozen identical questions.
## Big numbers are not arguments
Congratulations, you invented HFL (Hyper Factorial Levels) and a notation that looks like it was designed by an enthusiastic LaTeX plugin. Here’s the secret: magnitude without context is decoration, not mathematics.
If your contribution is “this number is huge,” explain why anyone should care. Does it bound complexity classes, solve a natural combinatorial extremum, or give a new asymptotic insight? If not, call it art and be proud, but don’t pretend it settles anything.
## Show your work: code, minimal examples, reproducibility
If you claim a new hash, a novel representation, or an algorithm, give reproducible evidence: a GitHub link, a tiny test suite, and sample outputs. Prefer archives with stable DOIs like Zenodo and include runnable instructions. For analytic claims, include numerical checks and edge cases; for symbolic claims, sketch a proof or point to a formal verification attempt.
A good post makes it trivial for a skeptic to run your code or check the base case of your induction. If that’s too much work for you, it’s too much work for us.
## A small arithmetic palate cleanser
Love little patterns? Try this meme: 987,654,321 ÷ 123,456,789.
Quick trick: 123,456,789 × 8 = 987,654,312. So the quotient is 8 with remainder 9, i.e. 8 + 9/123,456,789 ≈ 8.000000073. Cute, tiny remainder, and a pleasant reminder that sometimes beautiful things are simple.
## How reviewers should respond
If you’re commenting: ask for a minimal reproducible example. Point out where rigor is missing. Don’t be proof police for the ego; target the gaps that matter. If the claim is computational, ask for the code and a sample input/output. If it’s theoretical, ask for the minimal lemma that would make the argument go through.
If you’re moderating: enforce authorship rules, require [UPDATE] tags and changelogs, and demand reproducible artifacts for technical claims. Remove plagiarism and be ready to ban repeat offenders who treat the forum as their personal echo chamber.
## Cross-disciplinary nuances
Different branches of math and logic demand different standards. A numerical conjecture in experimental number theory thrives on computational evidence and heuristics. An alleged proof in set theory or category theory often requires careful checking of definitions and coherence conditions. Computer-assisted results (think long computations or exhaustive search) live or die by their reproducibility and verifiability.
Formal provers add a different axis: they give ironclad certainty but at the cost of time and labor. Not everything needs Lean-level rigor. There’s a spectrum: hand-wavy insight → solid proof sketch with referees’ notes → machine-checked proof. Be honest about where your work sits.
## A compact checklist for posting math online
– State authorship and get consent.
– Use [UPDATE] and a top-of-post changelog when relevant.
– Include LaTeX for clarity and a plain-language summary for the rest of us.
– Attach runnable code and sample inputs/outputs for computational results.
– Use a CAS or formal prover for nontrivial claims; don’t rely only on LLMs.
– Explain why your result matters. If it’s a curiosity, label it as such.
## Takeaway — with a bit of Katya’s cantankerous optimism
The internet will happily host your mathematical curiosities — but nobody owes your post infinite attention. Be generous with clarity: show code, show provenance, and show why your idea matters beyond aesthetics. Use ChatGPT as a stylist, not a referee. Post your own work, sign your name, and if you’ve updated something, don’t make the rest of us play Find-The-Difference. We have day jobs, ennui, and the faint hope that one of these threads will actually teach us something new. Make it worth the scroll.
So here’s the closing provocation — because I like leaving you with a puzzle rather than a verdict: when we balance speed, accessibility, and formal certainty on the internet, what trade-offs are we willing to accept — and which ones are non-negotiable?
