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Proposal: Mathematical Exposition July 28, 2009

Posted by Phi. Isett in Essays, polymath proposals.
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Disclaimer:  My last entry discussed how the use of personal profiles, a bookmarking system and selective provision of statistics can, in combination, help polymath deal with many of the challenges it faces, including the issues of how to provide leadership and promote overall efficiency.  But when I wrote it, I considered only the problem of producing research mathematics en masse, so I do not claim that the same principles necessarily apply to the collaborative production of mathematical exposition.

First of all, I am not suggesting we discontinue working on problems — rather I’m suggesting a (necessarily) smaller side-project.  Polymath should try to write a textbook… or something expository.

There are plenty of reasons to feel initially apprehensive about this idea.  We might imagine a group of authors’ artistic disagreements turning into edit wars.  Would a great mathematician’s insight be lost through this process?  And after all, doesn’t Wikipedia already provide this kind of service?  And look at all the obvious inadequacies there!  Not the ideal way to be learning mathematics at all, is it?  And yet we still try reading it…  Well, even if I don’t, other people do and it ends up consistently at the top of Google’s search results along with some other random papers, which only contributes more to the difficulty of finding good mathematical exposition on the internet.

These reasons to feel apprehension only highlight the importance of finding the solution to the more broad problem: find the best way to compose en masse.  In some ways, it seems to be open, at least for mathematical purposes, and yet the applications clearly extend beyond mathematics.  So I think it’s worth solving (carefully), and here are a few more reasons we have to solve it:  (For the time being, the reader may envision we are trying to write a textbook on some well-understood mathematical subject)

  1. To help the Tricki —  Many applications in the articles in the tricki are to problems commonly solved in textbooks.  And yet this is a bit antisymmetric — isn’t it more often that we see these funny-looking solutions to problems and then afterward spend a great deal of time and energy thinking “What?  Where the heck did that idea come from?” or “I’ll have to remember this technique later for my completely unrelated purposes”, etc.  At the same time, we will find that these collaboratively written expositions may inspire many articles for the Tricki, and give the Tricki good places to which to link for examples.  So in short, this project may have great potential to grow along with the Tricki, and that is something that is (to my knowledge) essentially absent from the math we usually read.
  2. To help the polymath projects –  You may consider the problem of mathematical exposition to be just one important subproblem of how to optimally do Polymath.  This problem appears throughout the course of the project when summaries appear, and at the end of the project when the papers must be written out in full.  I think this subproblem (just like the one considered in my last post), can be isolated, so we might benefit from concentrating on it.
  3. Stretching our abilities –  Somehow, a printed topology textbook always fails to say as much as it wants to.  And, I don’t know about you all, but I am far from comfortable including dynamic media in my posts, so I wouldn’t be able to write a decent online topology book, even if I did understand it very well.  You can probably see I have other things to fix in my technical writing, too, but one way for me to learn is to be able to copy the techniques which appear through optimized collaborative writing.  As we develop these online expository techniques/technologies and make them easier, we can record how they work in a separate wiki.
  4. Saying things in more than one way – When you listen to music in a car, you probably get to choose the volume, the level of bass, the active speakers, and so on.  On the contrary, when you read a textbook (or Wikipedia), you have no knob for adjusting the level of abstraction or technical detail.  Authors of textbooks have spent much energy optimizing these aspects to their liking, and once they have made their choices, you are left to figure out for yourself how to say it in “layman’s terms” or what meaning, if any, lies behind the particular algebraic manipulations.  Grant it, these can often be great exercises, but they can also be frustrating and we will always find good conceptual exercises for ourselves.  Having these different options available, one can then construct one or many different printable books by making these same choices in one of many different ways, but there are advantages to having various approaches available simultaneously.  ( I also think this flexibility is necessary because, for good reason, nobody will agree on exactly one single best way to present something, but we just might be able to have just a bounded number of fundamentally different modes of presentation / points of view)
  5. Possible updating –  Some fantastic books require supplementation simply because they are outdated in maybe a few ways which aren’t fundamental.  When your product is on the internet, you can always update it.
  6. A concentration of fantastic problems –  In my previous essay, I contended that “ratings” were not in the spirit of doing mathematics — that we need, instead, to be more specific when classifying/criticizing posts, but still cannot afford to leave things unclassified.  A collaboratively written textbook, even if it somehow manages to be not so well-written, is more likely to attract the most fantastic problems, especially if we provide ratings and other simple, popular statistics for these problems (which measure difficulty, estimate time commitment, how valuable the lessons we learned, number of people marking it as a “favorite” or whatever seems appropriate).  Of course, there is a delicate issue regarding how and when to store solutions if at all, and we’d have to hope people give the problems an honest shot before rating.  But I find it hard to imagine how a truly massive collaboration can fail to accomplish at least this feat, which by itself is a great thing.

I’ll stop here and hope I convinced everyone that solving the massive exposition problem is worth a shot even though it’s clearly a very difficult problem (but who better than mathematicians and programmers to take it on, eh?).  I think it’s better to try to solve this problem simultaneously with the development of polymath.  To me these times for polymath resemble the times in which the US Constitution was written — the US could have easily started a government without even a Bill of Rights were there not so much rigorous intellectual debate during the formation!  (which is why it’s so great that those leading the polymath movement have chosen to do things this way)

I’ll leave you all with some questions…

  1. How would we determine what to try to write first?
  2. What does a good candidate exposition project look like?
  3. Is there any kind of exposition a massive, internet-based approach seems inherently incapable of doing?  Or does the potential seem unlimited?
  4. Are there any subtle but important issues I have completely ignored?
  5. Should I have called this a polymath proposal or should the collaborative exposition idea be considered separate?

The one comment I think I should make is that the first experimental exposition projects, if enough people are interested, should be very small, with well-defined boundaries, and should help to set a precedent of interaction with the Tricki.  I also think that, much like the research-flavor of polymath (and the US constitution), we would have to make things extremely robust from the very beginning.  For example, if the Tricki were instead an idea which instead came into being during the future, it should have been easy to integrate into whatever expository polymath system we introduce.

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Unrelated… How do you turn off the annoying preview thing for the hyperlinks?  And how do you make it so that you only get a small preview of the entry from my blog’s homepage?  I’m new to this blogging stuff..

Edit ( 30 Aug, 2009 ):

There is one thing which can be accomplished on the internet which might have great applications for exposition: we can keep our writing simple by attaching small links (like footnotes in LaTeX) to statements whose proofs may be obvious to a decently large fraction of readers, or when proving such a statement would disrupt the flow of the prose.  A pop-up could then appear giving a complete proof or two.  This feature would not only help to compress the prose and accelerate reading (people get to read their choice of details), but simultaneously this feature would allow for a more detailed exposition of whatever we decide to write.

Comments»

1. Jacob Steinhardt - August 1, 2009

This is indeed a compelling idea.

Without thinking too hard about it, one possibility would be an expository account of Tate’s Thesis. I believe this to be a good candidate because current texts, even if well-written, seem to suffer from the linear presentation of the ideas. There is so much technical background for an undergraduate or even beginning graduate student (topological groups, topological representations, spectral theory, abstract harmonic analysis, algebraic number theory, and even some class field theory) that it is easy to get lost in the details. Having a nice “big picture” with hyperlinks to gradually more detailed explanations could make the material much easier to absorb, in my opinion.

Phi. Isett - August 1, 2009

Tate’s Thesis does sound like a compelling candidate! A few other grad students and I organized a study group to read through Ramakrishnan’s book Fourier Analysis on Number Fields (which attempts to present the thesis with background material, all-details-included) as well as the actual thesis. Our group actually eventually dissolved and it did have very much to do with a complete lack of a big picture and getting lost in the details. Tate’s thesis itself is widely considered very well-written, so it seems the real demand for prose in this subject lies more in the presentation of background material.

If we wanted to write something expository on a subject which has a formidable background requirement, Tate’s thesis seems pretty much ideal. And in general “subjects with broad and formidable prerequisites” do seem like good projects for massive collaborative writing, since they require so many different kinds of insight and points of view to come together.

What I wonder, though, is whether such a project would be ideal as a first effort. For instance, they might actually be more difficult to write, and therefore require more experience and foresight to be done well. And if, say, very many people were to get involved, we run the risk of writing 48 “books” on very different subjects — although I guess the challenge is really to figure out how to prevent that from happening. So I think it’s really a great idea and I see where you’re coming from; my question is only.. is it too hard for a first effort? I don’t really know. I hope more ideas and interest come in so we can compare and discuss.

(( By the way, Terry Tao wrote a nice expository article on Tate’s proof of the functional equation for zeta, and I think the related part of chapter 3 in Gel’fand, Graev, and Shapiro’s Representation Theory of Automorphic Forms is also quite accessible. ))

2. Jacob Steinhardt - August 2, 2009

I can definitely understand where you are coming from with it being a suboptimal first effort.

Perhaps something in algebraic combinatorics would ideal? There are lots of topics to cover, and most are fairly concrete, but it gets a bit theoretical at points as well. Also there is lots of room for examples, both in terms of figures and in terms of specific applications of more general techniques.

Of course, there is a lot out there already, like Alon’s treatise on Combinatorial Nullstellensatz, so there is the question of how much it would benefit from a polymath approach.

We could also pick some fairly new topic for which it is not even obvious how to organize the existing material, and perhaps a polymath approach will more quickly make it clear how to present it. This has the nice advantage of having a direct impact on current research, and may perhaps even inspire new avenues of inquiry. The disadvantage is that there could be fairly high barriers to entry if we pick the wrong field.


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