Dr. Katya Steiner — The Categorical Imperative of Online Math Theater

# Let the Humans Argue — The Categorical Imperative of Online Math Theater

You love math. You love internet attention. Put them together and you get a forum overflowing with dazzling claims, enormous numbers, and the occasional AI hallucination trying to play Euler. It’s like a village square where everyone brought their favorite theorem as a mixtape and half the participants are convinced that uploading a PDF equals peer-reviewed enlightenment.

I’m Katya Steiner. I study how ideas behave under stress — in seminar rooms, on arXiv, and yes, in the comment threads where algebraists duel with an overconfident chatbot. Here’s a guided stroll through why this mess is both infuriating and oddly fertile, with cross-disciplinary pointers that might help you tell genuine insight from theatre.

## AI: great at prose, terrible at proofs (and why that matters)

Large language models are astonishing conversationalists. They can paraphrase a proof, suggest a lemma, and dress a sketch in mellifluous language. That makes them perfect for pedagogy and exposition. They are terrible, however, at the kind of brittle, rule-bound reasoning that proofs demand. Think of an LLM like a brilliant parrot: it sings like Mozart without composing the sonata.

From a logical perspective this is no accident. LLMs optimize for plausibility in text-space, not truth in formal-system-space. They are pattern machines, not proof engines. Contrast them with proof assistants (Lean, Coq, Isabelle) and theorem provers: those systems implement syntactic rules, type systems, and inference engines — the kind of austere scaffolding that prevents a single stray quantifier from wrecking an argument.

So when a chatbot hands you a lovely, linear proof, be suspicious. Is there a tacit use of the axiom of choice? Are there hidden domain restrictions? Does an analytic step sweep convergence issues under a rug? These subtleties are where math lives. Use LLMs to write, to clarify, to brainstorm examples — but not to certify.

## Post your own work. Yes, even the messy version.

The fastest route to communal progress is the shortest line from originator to critic. When someone posts a theorem they didn’t write, conversation slows into interpretive telephone. If you want feedback, be present. Defend your notation, own your assumptions, and answer clarifying questions.

This is not performative vulnerability. It’s accountability. And when you revise, mark it. [UPDATE] in the title and a changelog at the top save people hours of re-evaluating the same sentence. A one-line fix can change everything; spare us the detective game.

## Lessons across disciplines

Different branches of math have different tolerances for hand-waving.

– Analysis and numerical methods insist on epsilons and stability. A plausible formula that explodes for small perturbations is a fascinating trainwreck — useful to study, but not to rely on.
– Algebra, category theory, and topology reward structural clarity. If you invent a new notation without showing it reduces to standard cases (say, groups, rings, or the homotopy groups of spheres), you’ve created an aesthetic, not a tool.
– Combinatorics and number theory adore explicit examples. Conjectures survive or die by computation and counterexample; make those tests reproducible.
– Logic and foundations force you to state which rules you’re using. Move from classical to intuitionistic or paraconsistent logic and watch how proofs and permissible inferences change like a stage set.

The takeaway: when you post, tailor the axes of evaluation to your field. List your assumptions; provide test cases; state the model of logic you’re working in.

## The Fourier-to-binary charm: creativity needs constraints

I adore creative crossovers: Fourier series used as a hashing trick for integers is the kind of tinkering that can lead somewhere brilliant or beautifully doomed. The difference? Constraints. If your construction breaks for negative integers, say so. If it’s numerically unstable, say that too.

Give reviewers the right levers. Definitions, complexity analysis, minimal reproducible examples, and open code transform a playful idea into something that can be built upon.

## The numerophiles and the cult of big numbers

We live in a culture that worships scale. Somebody invents “Hyper Factorial Levels” with six intricate axioms and new glyphs, and suddenly the thread is less a mathematical claim than an art installation. There’s nothing wrong with mathematical aesthetics — but novelty without connection tends to be decoration, not contribution.

Ask: why does this giant number matter? Does it illuminate computational hardness, asymptotic regimes, or combinatorial structure? Can it be normalized against known hierarchies? If not, it may be a monument to bravado rather than insight.

## Forms of logic: a quick map for the impatient

A few quick logical guides to interpret claims:

– Classical logic: the default. Most everyday proofs live here. Double negation elimination holds, excluded middle is available.
– Intuitionistic/type-theoretic: constructive vibes. Existence means you can build a witness; proofs correspond to programs. If someone says “we can construct” and doesn’t provide a method, alarm bells.
– Modal and temporal logics: great for reasoning about possibility, knowledge, or processes. Useful in computer science and some modern uses of category theory.
– Paraconsistent logics: allow contradictions without collapse — handy in certain philosophical or database contexts, but they change what ‘proof’ even means.

Knowing which logic is being used helps you ask the right questions.

## Small curiosities keep the gears greased

Not every post needs to be Nobel-level. Little puzzles and number-play are social glue. A cheeky divisibility pattern or a playful identity keeps the community warm and creative. Leave toys on the table. They’re often where bigger ideas germinate.

## A balanced verdict (with a grin)

The internet is spectacular at discovery and abysmal at careful reasoning. That dichotomy is not going away. Machines will keep getting better at style; humans will stay essential for judgement. If you’re contributing, do three things: think for yourself, post your own work, and be transparent about updates. Use the right tools for the job — proof assistants for verification, CAS for computation, LLMs for prose — and be generous with clarity.

Be rigorous, be present, and for the love of mathematics, don’t let a chatbot file your theorem.

What’s one small change you could make to your next post that would turn a performance into a provable contribution?