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Bourbaki 2.0: Or, is massively collaborative mathematical exposition possible? September 7, 2009

Posted by Akhil Mathew in General, math education.
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Warning: I have very little knowledge about these topics (even less than usual).

The Problem

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.

A Suggestion

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 {{P}} must presuppose no knowledge outside of articles {{P'}} with { {P' < P}} 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.

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Comments»

1. anonymous - September 7, 2009

Two things that already exist come to mind: the Stacks program, , and n-lab, . I haven’t spent much time looking at nLab, but I’ve successfully used Stacks for learning a bunch of algebraic geometry. (Its not just about stacks!)

anonymous - September 7, 2009
Akhil Mathew - September 8, 2009

Thanks for the links! The stacks project is much closer to what I had in mind above than nLab. It seems like a useful resource.

There are of course some differences: the stacks project does not apparently build off from existing plethora of mathematical bits of exposition to amalgamate them into new much longer volumes (though I don’t know how practical this is, given the variance among people’s expository styles). Plus, of course, it’s restricted to algebraic geometry. I’d be very interested to see if this is possible.

2. darfferrara - September 7, 2009

As a fellow self-learner I don’t think that the collaborative method works all that well for learning texts. Texts usually need a vision for presenting a subject that is conducive to learning. Lucky for you, David Eisenbud wrote just a book for you for the subject you mentioned at the beginning of your post. It was specifically written to provide all the commutative algebra that was required for Hartshorne.

Akhil Mathew - September 8, 2009

I actually did eventually learn about derived functors and Cohen-Macaulay rings (including Eisenbud’s source); I just was using that an example of how books tend not to be self-contained.

Incidentally, I think that Eisenbud’s approach of filling in the background for Hartshorne is excellent; I think it’s unfortunate more authors don’t make similar choices.

But about the collaborative method: Why not put someone (or some small team) in charge? What about the Bourbaki model?

3. Harrison - September 8, 2009

I don’t have much of substance to say about the real topic of the post, but I want to sort of point out a few little things.

First of all, it’s not even close to the level of rigor of Bourbaki, but for my money the PCM comes the closest of anything I’ve seen to at least giving a high-level overview of much of modern mathematics. The articles are written by lots of different people, of course (as it would need to be, since nobody knows enough about every branch of math to handle that sort of thing) but it also had a very good editor in Tim Gowers, and I think that (as with polymath) some sort of leadership is necessary for this kind of thing.

There’s one thing that I want to take issue with: You seem to be sort of favoring some sort of linear exposition, where first you do topic A, and this leads naturally to topic B, and so on. But this is unrealistic, and worse, it’s boring! There are some areas where you need to have prerequisite knowledge to really understand a subject, but there are lots of other cases where you can at least initially do things intuitively, backfilling as necessary later on. This is where the nonlinear structure of the web would really have a chance to shine — people don’t have to read rigorous proofs if they don’t want to, but they can still have access to them if they want.

Akhil Mathew - September 8, 2009

“There’s one thing that I want to take issue with: You seem to be sort of favoring some sort of linear exposition, where first you do topic A, and this leads naturally to topic B, and so on. But this is unrealistic, and worse, it’s boring!”

Well, there is an easy solution: just open to a random page, and then look back for any terms you might not already know ;).

I understand your point on inevitable backfilling and have read Qiaochu’s post, but wouldn’t such an approach ultimately make backfilling easier? Besides, the fact that backfilling is inevitable doesn’t imply that one shouldn’t try to minimize it (that is by building up a background, not by avoiding hard topics).

4. Charles Siegel - September 8, 2009

To me, this sort of project does seem best accomplished by wiki. In fact, I set up my own personal wiki on my computer (since crashed for unrelated reasons) to learn a few things: started by putting in a theorem, then linked to all the definitions, wrote up the proof, with links to lemmas (lemmas for just that got subsections, more general stuff got their own pages) and then created a section for immediate corollaries. It worked very well for me in trying to figure out the Deligne-Mumford paper on irreducibility of moduli of curves.

5. Open source textbooks « Delta Epsilons - October 10, 2009

[...] Dunfield, open source triumphalism, stacks project, textbooks trackback Well, it seems that the Bourbaki 2.0 idea I suggested some time back wasn’t entirely absurd: as a commenter pointed out, the Stacks [...]

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