Very brief Tricki update

December 18th. I suppose there’s good news and bad news. The bad news is that I’m not after all in a position to give a date for the launch, or even the first phase of the launch, of the Tricki. This is because Alex and Olof still have things they need to sort out, some essential and some highly desirable. But that’s also implicitly the good news: that any delay is not due to our being too busy to work on the Tricki but rather to the fact that improvements to it are still being made. And while Alex and Olof sort out the things I have no idea about, I’m continuing to add material, with the result that by the time we do go public it will be possible to get a reasonable idea of what the structure of the site could be like. This will mean that instead of trying to describe that structure in abstract and easily misunderstood terms, I can say the much simpler, “Let’s try to continue what has already been started until it encompasses all of maths,” and others can say, “Well, that’s all very well for the things you’ve written about but it would never work for algebraic geometry,” and the discussion about how best to organize the site can start from a first (or zeroth perhaps) approximation rather than from nothing at all.

I don’t think the remaining problems or unfinished aspects will take all that long for Alex and Olof to deal with. If I turn out to be wrong about that, then I may extract one or two articles and turn them into blog posts.

December 14th. While this blog has been quiet, the Tricki has been rapidly growing — in fact, I’ve been a bit obsessed with it for the last week or so. It’s beginning to take shape, partly because I’ve put in a lot of links to nonexistent articles, just to give an idea of how it might eventually look. We’ll probably go live in two phases: one where it’s read-only, apart from a forum where people can make suggestions based on what they see. And then, if no major changes seem to be necessary — or if they do and we make them — we can open it up properly. I may soon be in a position to give dates for this, as Alex, Olof and I are meeting tomorrow.

Further update added December 7th. I can now say with complete confidence that the Tricki really is happening. I have uploaded quite a bit of content on to it (though of course it’s a minuscule amount compared with what I hope it will eventually contain) and am finding it a joy to use. There are still some aspects of the design that need tweaking: Alex, Olof and I will meet in a few days time to discuss these, and perhaps then we’ll come to a decision about when to go live. I don’t see a strong reason for waiting much longer, but there may be technical problems I don’t know about.

Further update added December 3rd. I am now in the process of adding content to the site. There’s quite a lot to do, but I’ve already moved a couple of sample articles over from this blog and written one or two pages of introduction to the site. I’ll add to this update from time to time, since now I should have a much clearer idea of when we will be up and running.

The original post. This is in answer to Random Student, who spotted that a site that “should be up and running within the next couple of weeks” of October 15th should be up and running. I’ve sort of half learnt my lesson, and will not give an estimate of when it will actually appear, so all I’m going to say here is that it’s quite a bit closer to appearing now than it was then, in the non-trivial sense that quite a bit of work that needed to be done to it has been done. Olof Sisask tells me that, possibly even in the next couple of days, it will be in a state where I can directly add content to it. At that point, he, Alex and I will be able to work on things like instructions to authors, a few more sample articles, and so on. On the negative side, I at least will be pretty busy for the next couple of weeks, so I don’t see myself doing everything I want to do until some time after that. But, like Random Student, I am impatient for the site to exist. And if you’re thinking you might contribute articles, there’s nothing to stop you writing them even now, and a lot to be said for a rapid initial growth of the site, so any time you suddenly realize that something you had thought was hard is not in fact hard, please write something down that will help others get there more quickly, and post it as soon as the site exists.

Meanwhile, here’s something to think about to do with navigation in the site. As I have discussed at length in other posts, it is a serious challenge to find ways of making relevant Tricki articles easy to find once they are on the site. For many potential articles, it is reasonably straightforward to think of ways of achieving this. But Olof raised an example of one that is less easy, and therefore worth thinking about as it may be representative of a large class of such examples. (I haven’t tried to think what it is that makes it difficult or tried to generate similarly difficult examples — that would be a useful thing to do.) His example was the trick that one applies in order to reduce the mean value theorem to Rolle’s theorem. Suppose, that is, that you wanted to prove the mean value theorem, couldn’t see how to do it, were familiar with Rolle’s theorem and its proof, and wanted to turn to the Tricki for help. (This may not be all that likely a scenario, but that doesn’t necessarily lessen its importance, since we’d still like to have navigation tools sufficiently general to apply to it.) How would you arrive at the page that told you to subtract a linear function from yours in order to make the gradient of the chord equal to zero?

The best answer I myself have come up with, which doesn’t feel completely satisfactory, is this. If you can’t see how to solve the problem, your difficulty is not really a difficulty peculiar to the mean value theorem, but rather a much more general difficulty, which is that it doesn’t come naturally to you to apply the following “trick”: if you can’t solve a problem straight away, see if there are interesting special cases of it that you can solve. (Of course, there are a number of similar principles that could be applied here, such as: see if you can simplify the problem with a WLOG or two.) In the case of the mean value theorem, it doesn’t take much to spot that if the end points of the interval are equal, then the formula for the derivative you want in between becomes much simpler, so this is a very natural first case to look at. And at that point, if you are familiar with Rolle’s theorem, you will have solved the special case and could find the Tricki article by using a tag such as “Rolle’s theorem”.

However, I find that unsatisfactory, partly because I’d like a way of finding the trick by describing the trick, and not by using some name of a theorem. I think part of the problem here is that it’s quite hard to say, in full generality, what the trick for reducing the mean value theorem to Rolle’s theorem really is. Is it just a one-hit wonder that does that particular proof for you, or is it a special case of a much more general principle (as I’ve suggested above), or is there an idea of intermediate generality that includes this but that is not a general strategy for solving more or less any problem?

Incidentally, if I were writing a Tricki article on the general trick of finding a simpler special case and then reducing everything else to it (which I may in fact do), then another example would be solving quadratic equations. The case that’s easy to solve is , from which one can easily generalize to . And then one asks whether there are any other cases, and is quickly led to the idea of completing the square.