Roadmap¶
What exists, and what is deliberately not written yet. A topic listed here has no page and no example — it is a claim about direction, not a stub.
Written¶
01_Precision — how much of this number is real. Four lessons: exact vs approximate, significant figures, uncertainty propagation, catastrophic cancellation.
02_Measure_Zero — how infinitely many points can take up no room. Six lessons: what measure zero means, countable sets, the Cantor set, the fat Cantor set, the Cantor function, probability zero.
The rest of the precision chapter¶
The four lessons close one argument, but they leave three doors open:
- What a float actually stores — the binary mechanics under lesson 4. The Rust library already has a thorough page on this ↗; linked from catastrophic cancellation and the glossary rather than reproduced here. The open question is whether this library ever needs its own — a math treatment would be about representable sets and rounding as a function, not about one language's floats. Duplicating the Rust page would be the wrong instinct.
- Summation algorithms — Kahan and Neumaier compensated summation, and pairwise summation. The natural sequel to "error accumulates over ten terms": here is how to add a million of them without it. Nothing in any sibling library covers compensated summation, so this one is genuinely open — though the Rust library's letting the compiler reorder a float sum ↗ already owns the adjacent half, that
a + b + cmeans(a + b) + cand reassociating changes the answer. - Interval arithmetic — carrying a lower and upper bound through every operation instead of a value and a sigma. The uncompromising version of this whole chapter. The Rust library got there first (Did the rounding decide it? ↗), and its angle is a good one — because interval error is one-sided, a verdict of decided is a proof. A page here would need a different one, or no page.
The rest of the measure-zero chapter¶
The six lessons close their argument, and leave two doors open:
- Two meanings of small — measure zero is one way to call a set negligible; meagre, built from nowhere dense sets, is another. The fat Cantor set is where they first disagree, and they can disagree completely: the line splits into a meagre set and a set of measure zero. The page would need a program that makes that visible, which is the hard part.
- Sets with no length at all — the Vitali set, which cannot be given a length consistently, and so the reason measure theory has to decide which sets get one. Measure zero sidesteps the question, since it only ever measures intervals. But the Vitali set needs the axiom of choice and no program can build it, so by this library's own rule it may never get a page.
Candidate chapters¶
Not started, and listed in rough order of how likely they are to earn a place:
- Probability — the other discipline built entirely on "how much do you know?", and the natural sequel to uncertainty propagation. Bayes, distributions, and why an interval is a better answer than a number. Probability zero already opens the door from the continuous side.
- Proof — induction, contradiction, construction. The part of mathematics that has nothing to do with computation, included precisely because everything else here does.
- Discrete — counting, graphs, recurrences. Best served by runnable examples of anything in this library.
- Linear algebra — worth doing only with a strong angle. Conditioning of a matrix connects it straight back to chapter 1, which is the angle.
Rules for adding a chapter¶
A chapter earns its place by having an argument, not a syllabus. 01_Precision is four lessons long because that is how many it took to get from "is this just rounding?" to "the textbook quadratic formula is 25% wrong"; it is not four because four is a nice number.
Every lesson needs a program. If an idea cannot be demonstrated by something that runs and prints, it may still be a good idea — but it belongs somewhere other than this library.