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 x^2=t, from which one can easily generalize to (x-a)^2=t. And then one asks whether there are any other cases, and is quickly led to the idea of completing the square.

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13 Responses to “Very brief Tricki update”

  1. Harrison Says:

    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?

    I feel as if this is a question that the “tricki community” should ask itself collectively. Wikis, of course, work on the same logic as open-source software and prediction markets; not only does pooling everyone’s work or knowledge or bets allow the resources to come together in one place, but the end result tends to approach some sort of optimum over time. A sufficiently robust system, in other words, will take care of these types of problems itself (i.e., by allowing articles to shift focus over time, so that something that starts as a specific example can become more general, and an over-general article can split into several new, more focused, ones).

    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.

    In reference to the general problem of navigation on the Tricki: the TV Tropes Wiki is another collaborative wiki that deals with subjects for which there’s no good naming convention; it divides its pages in several different ways (so that you can get to a given trope page from the page for a series in which it’s used, or from a list of tropes for its medium or genre) and has a lot of index pages, so that if a trope exists it’s possible to find it fairly quickly.

    So, while I’m not sure that “describing the trick” is entirely feasible, it’s certainly possible to reduce ambiguity. In the example you give, our hypothetical user might find the add-a-linear-function trick by first looking at the index on “Reduction Tricks,” tricks for reducing one problem directly to another, and then to “Transformation Tricks,” which involve performing an operation on the object under consideration so that it has a certain desirable property. Or perhaps from “Reduction Tricks” the user could click on “Reduce to special cases” and from there link to the Rolle’s theorem trick.

  2. Noah Snyder Says:

    Isn’t the trick here (both for MVT and the quadratic formula) just “if you have too many parameters try making a linear change of variables to eliminate some of them?”

  3. gowers Says:

    Noah, your comment made me realize that what I really meant to say was that I’d like to find a way to get to tricks by describing the problems they solve rather than the tricks themselves. I’m putting myself in the position of a hypothetical person who is stumped by the apparent complexity of the statement of the mean value theorem and wonders what to do next. Such a person may well not think, “Hmm, I have too many parameters here.” Indeed, it may not be obvious what the difficulty is, beyond, “This looks fairly complicated and I can’t think of anything to do.” And the only way I can think of for dealing with this particular example, which may not be all that bad actually, is to have an index page entitled something like, “I have a complicated looking problem and I can’t get my head round it” with links to many more specific pages with different simplification techniques. If those techniques are given suitable short descriptions, then it could be possible to find the appropriate one reasonably quickly.

    Incidentally, maybe another description of this trick is, “Often it’s enough just to look at zero,” another example being that if you want to prove that you can’t expand a vector in two different ways in terms of some spanning set, then it’s enough to check it at zero, or that to prove that a linear map between Banach spaces is continuous it is enough to prove that it is continuous at zero.

    Harrison, thanks for that interesting comment — I had a look at the TV tropes Wiki. As an experiment I decided to look for a very common trope: the police detective, not the main character, who reveals that he is on the point of retirement, at which point you know that he is doomed to die before the end of the film. I clicked first on “death” and then found a long list of names of tropes. One of them was called “retirony”. I thought it looked promising, and indeed it turned out to be exactly the trope in question. But this felt like a bit of a fluke, as for many of the trope names it was not possible to guess even roughly what the trope was that they were describing. But I definitely agree with your main point, that Tricki users will find solutions to these difficulties. At this stage my priority is to make people aware of the problem, and to urge people not to be satisfied with what I would regard as unsatisfactory solutions to it. In other words, I’m very happy to rely on our collective ingenuity, and that is one of the major ideas of this Tricki, but I want to make it as clear as possible where this ingenuity needs to be applied. Once the site is up and running this will of course be much easier, and I expect there will be surprises in store — some things working better than expected and others less well.

  4. Positive-Knowledge Proofs « in theory Says:

    [...] new, then presenting the proof of an actual theorem may even be besides the point. This gets to the Tricks Wiki, a repository of mathematical techniques which is coming online any day now, and which is [...]

  5. Tom Says:

    about december 14th update: Fantastic news, and very wise to do it in two phases. Looking forward to comment constructively as soon as it’s live!

  6. Yor Naim Says:

    I’m arriving to this discussion a bit late (and from physics), but I think members of the problem-oriented MathLinks community

    http://www.mathlinks.ro/ ,

    might enjoy the Tricki. Apparently, many current, former, and future Olympiad-caliber competitors and problem-solving enthusiasts — not to mention budding mathematicians — use the large and very active MathLinks forums to discuss problems, techniques, etc. Depending on the level of the Tricki, they could benefit from and contribute to it, or perhaps spin off a Junior Tricki.

  7. Michael Lugo Says:

    The naming in the TV tropes wiki, I think, is intended to be more humorous than that in the Tricki.

  8. Yor Naim Says:

    I apologize if I missed it, but is there an answer to an earlier question about making the Tricki software freely available, at least eventually?

    Although the Tricki itself will emphasize “method over matter,” the design or specific features of the Tricki could be adapted to broader educational ends.

    For example, an analogue of arXiv.org with Tricki-style comments and cross-links could serve browser-formatted lecture notes and collaboratively evolving textbooks. Authors would set the policies for community changes to and comments on their own articles, but other works might be almost completely “open,” as with Wikibooks.

    Automatic or relatively painless machine translation between online and conventional formats (*eX, PDF, etc.) might also be useful for converting existing manuscripts, editing articles elsewhere, stripping comments, feeding journals, making hard copies, generating slides for class, etc.

    The Sage developers may be able to help with embedding Sage expressions and plots into Tricki pages. Presumably, live objects would absorb non-trivial server-side resources, but perhaps they could be hosted remotely.

    In a subjunctive mood, I suppose. Much praise to those who toil tirelessly for the Tricki.

  9. gowers Says:

    Apologies — I didn’t get round to answering that. In fact, I’m not really in a position to answer it because I have played no part in the development of the software. So all I can say is that I’ll mention this to Alex and Olof, and also that my own wish is for the Tricki to be very much in the same altruistic spirit as Wikipedia, with people contributing to it because they believe in the concept rather than because they think they will be rewarded for their particular contribution. Possibly that attitude will extend to the software too, especially if the expert-system aspect of it takes off and it becomes more than Wikipedia-type software with a few very nice extra bells and whistles.

    Another thing to say is that the software that Alex and Olof have used to write the Tricki itself is Drupal, which is freely available. So it could in the end be that those wishing to serve broader educational ends would be better off just writing something specially tailored to their needs using Drupal.

  10. Tricks Wiki: Give yourself an epsilon of room « What’s new Says:

    [...] limiting argument, maximum principle | by Terence Tao Today I’d like to discuss (in the Tricks Wiki format) a fundamental trick in “soft” analysis, sometimes known as the “limiting [...]

  11. Anonymous Says:

    I know polymath took most of your “free” time, but what happened to the Tricki? Would you please give another update?

  12. gowers Says:

    It’s now ready. It should go live in two or three weeks’ time. I’ll give a more detailed update soon.

  13. Tricks Wiki: Give yourself an epsilon of room « mathTHÍCHinTOÁNmyHỌCbrain Says:

    [...] I’d like to discuss (in the Tricks Wiki format) a fundamental trick in “soft” analysis, sometimes known as the “limiting argument” [...]

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