Wikiproofs
This is just a brainstorming idea, of course. Whether it would work or be useful in practice is beyond my ability to see. Nevertheless, the idea of a network of articles on results with various proofs may be useful. Some differences between proofs here and on Wikipedia:
- Wikipedia article proofs are always included for importance or instructive reasons. In wikiproofs, however, there would be no such requirements. Every result or proof would be worthy of inclusion by merit of being a result or proof.
- Wikipedia article proofs are seldom linked in a fashion showing their interdepencies and logical relations. This is mostly due to the fact that so few proofs are included in Wikipedia articles. Moreover, the linking showing the logical relations is not really in the spirit of Wikipedia itself. However, such a linking between results or proofs may be of use to working researchers and students.
- There is the issue of notation and what is assumed in a proof (what is known, level of audience, etc.) Obviously, a given result can have multiple proofs, varying by level of audience, what is assumed or known, and level of hand-waving. Moreover, there are several interesting and useful different proofs of the same result; these could be listed together.
- The linking nature of wikiproofs would make what appears clumsy in textbooks perhaps more transparent. Often in textbooks or articles, authors refer to previous results or theorems ("by proposition 3.4", "by the previous corollary", "by definition 2.5", "by reference to other source", etc.) Wikis allow this reference to be direct and transparent, as a quick link could be made to the result or proof used. This would also allow students more freedom to "learn a proof at their own level"; i.e. linking allows students to pursue a line of argument at a level of abstraction or results at which they are familiar or comfortable, whether higher (omitting lots of details) or lower (following links to more basic proofs and definitions which are subsidiary to a proof).