Mathematics

Math-aware maps, the Board, function plots, and the Answer Guard.

When the conversation turns quantitative, Logos stops drawing a mind-map about the problem and starts drawing a map of the work: each state of the expression is a node, each operation an edge with the move written on it — −6 both sides, factor — so the solution reads as a chain. Math is written in real LaTeX everywhere it appears: the chat, the map, the Board, the node panels, the drafts.

A map of the work#

Eleven node types exist for this — givens, unknowns, equations, definitions, transformations, theorems, steps, inferences, verifications, results, and errors — with proofs chaining on implies and a theorem attaching to the step it licenses via justifies.

Four kinds of math moment#

The map also reads why the math is happening, because the right behavior differs:

Learning

You are working a problem to understand it. The Answer Guard engages — guidance toward the next step, never the full worked solution.

Verification

You did the work and want it checked. Logos finds the first step where it diverged, names exactly what went wrong there, and helps you repair that step — it never silently rewrites.

Utility

A quick calculation where teaching would be friction — “what’s 18% of 340”. It just computes it, plainly. No Socratic detour.

Exploration

Understanding a concept rather than solving an assigned problem. It explains, and describes functions so they can be seen.

The goal is never to withhold an answer artificially — only to avoid replacing reasoning you are trying to build. When in doubt between learning and utility, Logos assumes learning if a specific problem is being worked.

The Board#

The Board renders the same map as a worked-through notebook page: equations in a serif math face, operations and asides in a handwritten one, givens tagged given and unknowns tagged find across the top, the chain flowing down the page with hand-drawn arrows, theorems as margin notes beside the step they justify. Mistakes are struck through in rust with the fix written beneath — never erased — the result gets a hand-drawn box, and verified steps a handwritten check.

Every wobble and tilt is seeded from the node’s own id, so the board looks hand-made but never jitters between renders. Watch it work the demo problem — the sign slip goes down, stays down, and the fix arrives beside it:

The function plot#

When the map holds something graphable, the Plot lens draws it — up to four curves with a KaTeX legend. What counts as graphable is deliberately narrow: a single-variable expression with an actual operation in it (a bare x or 42 is not a curve), never an error node (a wrong line does not get drawn), and implicit multiplication like 2x or x sin(x) is understood.

The Answer Guard#

The guard arms on one condition: the work is math and the intent is learning. Verification, utility and exploration are never guarded — those answers are what you asked for.

While armed, it is one shared state across every surface, so the chat cannot withhold an answer that the map, the Board, or a node panel quietly reveals. The chat behaves like an excellent lab instructor, climbing a six-rung ladder one rung at a time — observe, nudge, concept hint, specific hint, one minimal partial step, and only then the full solution. Pushed with “just tell me”, it gives the next-smallest hint and mentions the reveal button; it does not lecture, and it does not cave. On the map and the Board, the result and any verification of it are masked as = ? — the working stays visible, because the working is yours. The plot still draws your curve faithfully — and it never annotates roots, intercepts or extrema with their values, so it can never become the surface that hands over what the chat is withholding.

The bar under the chat says what is happening — “Guiding, not solving — you’re working this one out” — with two buttons: Another hint, and Show solution, which lifts the guard for that conversation and regenerates every surface with the results unmasked. The reveal is per-conversation, remembered, and never priced.