Warning: I have very little knowledge about these topics (even less than usual).
One of my goals is to learn mathematics independently. I’ve had lots of trouble especially in certain areas such as algebraic geometry, where the preqrequisites are large and interconnected. When reading books nowadays, I frequently come across words I don’t know with (sometimes) recommended supplementary sources. But I can’t really learn the definition of say, a Cohen-Macaulay ring, just from reading Hartshorne’s short blurb or Wikipedia without actually seeing some properties of these rings proved, so I go to the supplementary sources. When I looked up, say, Matsumura’s book on commutative algebra, I then find that I am expected to know what derived functors are to understand depth. Time to find another book!
Other algebraic geometry books are even less encouraging: Grothendieck opens EGA by demanding that the reader be conversant with all of general topology, homological algebra, sheaf theory, and commutative algebra, and cites several and somewhat intimidating long research papers and books. The truly beginning (by which I mean polynomial equations, not EGA!) books on algebraic geometry that I read before don’t help bridge this gap; it is kind of like trying to learn modern number theory from a popular historical account.
In short, the problem is that, in many areas, it’s more or less impossible to find books nowadays that are reasonably self-contained and fully prove everything, yet get to really interesting material. The Bourbaki volumes were a noble attempt to avoid this, but they leave out large parts of fundamentals such as category theory and homological algebra; they don’t even discuss combinatorics. Moreover, they tend to be, in my view at least, somewhat over-pedantic and not always fun to read as a result.
So what to do? Well, I think the Bourbaki approach suggests a potential idea: the power of the group. Whatever the faults of their approach, Bourbaki can rightfully boast in the preface to give “complete proofs” and presuppose, at least in theory, only a certain comfort with mathematical reasoning (and patience); they were a group, so they could write a linear volume series of such great length that few mathematicians (except perhaps Serge Lang) would have managed it. Today, with the Internet, the group could be vastly expanded.
But wait, isn’t this just Wikipedia, or Tim Gowers’s Tricki project? (By the way, I took the title from a post of his.) Not quite. I’m looking for something that could be a textbook. I don’t think either is designed or well-suited to be the kind of thing you sit down with for hours; both are reference guides. What I’m suggesting instead is Bourbaki 2.0: a scheme whereby anyone can contribute to an enormous mathematical book. Perhaps someone with expertise could set up a table of contents spelling out the order of topics to be covered. Then, people could submit mini-articles under the following condition: an article in position must presuppose no knowledge outside of articles with in the (total) ordering.
Moreover, unlike existent approaches such as Wikibooks, one doesn’t even have to start from scratch. There is a wealth of mathematical information available on the web—ranging from course notes to blogs to already existent wikis. Willing authors could submit them as entries, perhaps in a partial form. In the end, if there were conflicts, members could vote to decide them. The success of John Baez’s nLab project suggests that potentially a large amount of material could be compiled.
In the end, ideally the website would automatically produce a catenated file with all the entries, one after another, that would read as a textbook (available freely online).
Oh, and I should say that I’m tossing this out as a purely academic exercise; I certainly lack the knowledge in both mathematics and programming to know whether it is even remotely feasible. So feel free to shred it to pieces in the comments, or explain to me how it doesn’t make sense. I was just curious whether all the mathematical expository efforts that go on could be coordinated to make learning mathematics easier for students, following the Ubuntu philosophy.