In my experience, Math.SE has a large number of very good mathematics teachers at a variety of levels: some of the regular, reputed users are really skilled at explaining topics within their [sometimes broad] range.
A lot of the answers focus on general methodologies, explanations, generalizations, etc. as opposed to just solving a given problem. This is very valuable knowledge.
Because of these two points, I thought that Math.SE would be a great place to create the definitive mathematics textbook whose primary characteristics would be:
- be accessible at all levels: Equipped with just my brain and the English language, I should be able to start right from the beginning (okay, so 'beginning' might actually be algebra, but that's a separate issue). Equipped further with whatever prerequisites for a topic, I should be able to learn that topic.
- be replete with examples: No matter what topic it is, Math.SE has had a large number of questions and probably a few questions with illustrative answers which would be helpful case-studies for teaching the topic. Further, being a dynamic textbook, you can add/substitute examples as needed (e.g. when someone posts a brilliant question/answer).
- be a useful reference: With appropriate moderation and peer-review, this textbook could be used as an easy-to-cite and accurate reference which would, of course, be backed by references of its own. There would be links between pages for the often-found yet not-so-well-known connections between distant areas of mathematics, for optional reading.
- be free to use (vs. print textbooks/journals), complete (vs. Wikipedia and to some extent, ProofWiki), dynamically community-driven (vs. print textbooks/journals and MathWorld) and rigorous (vs. Wikipedia) but not terse or hard to understand (vs. MathWorld): Of course the parentheses represent my personal opinion, but the adjectives should hold for the proposed textbook nonetheless. Most importantly, the mathematical knowledge of humanity should be freely accessible to humanity (with an Internet connection) and should keep getting better through constant content generation, peer-review, and editing.
In my view, this would be the perfect ultimate textbook; in fact, I didn't realize that my dream for a central math repository and Math.SE could actually be so compatible until a few minutes ago. What do you think?
(inspired by this question and a comment on an answer by @TBongers: "... I'm inclined to think that the mathematics version of the program is just a well-written textbook.")