This post has its origins in a discussion that arose as a result of a very interesting post of Terence Tao. Both the post and the discussion can be found here . The post outlines a rather general idea, or trick, that can be used in many mathematical situations. With such tricks, it is usually difficult, and in any case not desirable, to formalize them as lemmas: if you try to do so then almost certainly your formal lemma will not apply in all the situations where the trick does. This has the unfortunate consequence that they are relegated to something like “folklore,” transmitted orally (to a lucky few) or rediscovered over and over again (the more usual experience).

Since tricks (or, to make them sound better, general research strategies) are very useful, and since it can be extremely illuminating to have them pointed out — Tao’s post is an excellent example of that — it would be highly desirable to have a lot of such tricks accessible in some convenient format. But what format? I take it as axiomatic that some kind of online resource would be ideal, but there are now many ways of collecting information online — in fact, many more than I know about. So I’d like views of other people.

To get the discussion started, here are a few views of my own. They are mostly criteria that I’d like such a resource to satisfy.

1. It should be a genuine collaborative effort, like Wikipedia.

2. It should have very high levels of quality control.

3. It should not be run in a dictatorial way.

These criteria are of course hard to combine: I have written them in what I consider to be their order of importance. Similar issues have been discussed at the n-category café here , and I recommend that discussion, but I think their needs are slightly different.

I don’t know whether it would be easy to set up, but the kind of idea I think might work is a Wiki where authors have full control over their contributions, but others are encouraged to suggest improvements (as happens in some of the comments on Tao’s blog entry). These suggestions could even be simple ones like, “I got a bit lost in the third paragraph — could you elaborate a bit?”

I think it would also be good to have some kind of certification process, so that the Wiki didn’t fill up with entries that were badly written or aimed at far too sophisticated an audience to be genuinely useful. A first approximation might be that if enough people wrote comments saying how great they thought the entry was then it would become an official “great entry.” But it is not obvious how to implement that in an automatic way (without a super-editor who judges which entries have reached this exalted standard). Alternatively, there could be something like an Amazon star rating, so that you could see which entries were generally thought to be good and ignore the bad ones.

Another issue is how to organize the entries so that people can look them up, or browse, in a reasonably systematic way. This could be particularly difficult for rather general tricks that are not tied to any particular branch of mathematics — a quality that many tricks have.

September 11, 2007 at 3:33 pm |

PlanetMath is a wiki-esque site where the original author keeps control of the article, and site readers can suggest corrections (which the the author must accept or reject, or lost control of the article).

Though, for my 2 cents, I don’t think this system works as well as a normal wiki. People are so much more likely to make constructive contributions than destructive ones to pages on something obscure like mathematics that it tends to worth making it a bit easier for them. I would suggest, rather, giving each page a maintainer (or perhaps a few) who can easily ban people from editing that page, making it easier for people who want to be helpful to participate, but hopefully avoiding nasty edit wars.

September 11, 2007 at 3:56 pm |

That is an interesting point, and probably right, especially if one is trying to get the wiki to grow quickly into something that covers a wide range of topics. I suppose my worry is that there would be no point in starting a mathematical-tricks wiki if it didn’t develop into something that was genuinely distinct from other maths-related wikis. Since, at least initially, people will be familiar with a different style of article, they may be tempted to make edits and contribute articles that are not in the spirit of the site. But perhaps the best way to deal with that is to hit the ground running: commission a fairly large number of initial articles that definitely are in the right spirit, have a front page that lays out very clearly what such an article should be like, and then just trust people to be, as you say, much more likely to be constructive than destructive.

September 11, 2007 at 5:56 pm |

Dear Tim,

A mathematical-tricks wiki would be wonderful! But it is going to be more difficult to set up than other mathematics-oriented wikis. One of the first problems is that of nomenclature: mathematical theorems and objects tend to have standardised names, but mathematical tricks usually don’t; indeed, even mathematicians who use a trick routinely may not even be aware that they are doing so, except at a very non-verbal level.

The nomenclature problem spawns several other related problems. For instance, search. It is easy enough to look up, say, “Banach-Tarski paradox” in Wikipedia, but how does one look up, say, “That trick where you embed the free group inside the group you are working in”, and “the Hilbert’s hotel-type trick for the free group” (which are the two tricks that power the Banach-Tarski paradox)? Unless we have some sort of standardised naming procedure (or at least a robust way to move un-named tricks towards such standardisation) it could be almost as chaotic as what we have right now.

A related problem is redundancy. I work in several different fields of mathematics, and am always struck as to how the same theme seems to crop up independently in so many different areas – but everybody has a different name for that theme, and there is definitely a lot of wheel-reinvention going on. Of course, this is one of the things that a tricks wiki would help out a lot with, but I can imagine there would be a lot of tricks articles from very different perspectives that may eventually need to be merged. Such a merged article could lead to some fantastic new analogies and insights, of course – many of the deepest developments in mathematics have been initiated by such an unexpected “merge” – but it could be quite challenging to pull off technically.

It may be premature to worry too much about these sorts of issues, though – the more important thing is to create some content, which is the more difficult task. At worst, one can start with a lightly organised and narrowly focused list of seed articles and then make a serious effort to reorganise them more efficiently (and in a more scalable fashion) once it becomes clearer what the optimal global structure should be.

September 11, 2007 at 6:13 pm |

Looking back on what I just wrote, it occurs to me that one potentially rather useful thing to do (even if it is not _quite_ what you proposed) would be to systematically take the standard proofs of many well-known or otherwise important results (such as Banach-Tarski) and “deconstruct” them into their component tricks. I have found that even if a given theorem requires 100 pages to prove, the argument can often be decomposed into a mere half-dozen or so “major tricks”, plus a countless number of minor tricks and technical “negotiations” between the major tricks in order to connect everything together properly. But once you know the major players in the proof, you are already more than halfway to understanding the whole proof, and without reading anywhere near as much as 100 pages. (Of course, in many cases, the devil really is in the details, but nevertheless knowing the overall strategy of proof is extremely valuable when trying to read that proof.) I could see a “Proofs modulo details” or “Top-level proofs” wiki being incredibly useful. (One would probably have to restrict attention, though, to those arguments which have already been digested for several decades; with cutting-edge research, trying to tease out a top-level structure may be dangerously premature.)

September 11, 2007 at 6:19 pm |

Sounds like a very good idea to me. With regards to the naming problem: while not a solution, it would be nice to have associated to each trick some results where it is used. This could then be used to ‘reverse-lookup’ a trick. For example, say that you remember a nice trick being used for the Banach-Tarski paradox; then you’d search for Banach-Tarski and up would come a list of tricks that have been tagged as being used in the result. This requires the author(s) to have some knowledge of places where the trick is used, but taggings could of course be done later on by people who stumble across an article and go “Ah, that’s exactly the kind of thing X does in the proof of Y”. There will be issues with different proofs of theorems using different tricks perhaps, but such things could be probably be worked around in a reasonable way.

September 11, 2007 at 6:24 pm |

… which is of course related to Terry’s second post (which I missed) :). Top-level proofs would be very nice, and when you want more information on an idea in a particular part of such a proof, you could perhaps follow a link to an article on a related trick, or find other proofs where such a trick is used.

September 11, 2007 at 7:06 pm |

I think one way of addressing what Dr. Tao was talking about with the tricks not having specific names is to give each trick different topic tags based on what the flavor of the technique is. For instance, ‘freegroup’ could be a tag and if I search it then I get every trick that can be done on or with free groups. A string of tags might be selective enough to narrow down the search for a particular trick. I think it also has the benefit of allowing browsing of closely related tricks.

September 11, 2007 at 8:31 pm |

One thing which would also be extremely useful, but which is somewhat different from “tricks” as usually understood, is a way to express/collect “the way mathematicians think” (this was also suggested by T. Gowers in the earlier discussion), which means often un-rigorous arguments that are used by all specialists but never referred to in print (they are sometimes in lectures). As an example, I learnt very early on during my PhD, as most analytic number theorists must, that in trying to figure out what goes on in an argument, one should do as if “integers are either equal or they are coprime”. (Of course, part of the difficulty is that there are always exceptions to this type of rules).

There is of course a link between this and the idea of giving skeletons of proofs of papers, and indeed this would be a great idea because (in part) of the easy searchability: anyone having identified the original paper as of interest could easily look up whether a “skeleton key” has been already written. Similarly, it could be easy to collect a list of papers/books/surveys which are already published and are particularly suited to the style of explanation desired. (There are certainly a lot that already exists but is not easy to find; a paper may include a very enlightening introduction or sketch of proof without this being apparent from the title, summary, or Math Review…)

In any case, I would be happy to contribute with my modest means to projects of this type. I have also had the occasion to think and work with ideas and objects outside my “main” field, and I have felt that getting to the point where one is able to do it is quite a bit more difficult than it could be nowadays. (I am pretty sure that the same applies to other mathematicians trying to learn or use analytic number theory…)

September 11, 2007 at 9:59 pm |

The OEIS, incidentally, is one mathematical resource where the nomenclature problem is completely solved (each integer sequence uses the first few entries of that sequence as its name), and has been immensely useful for discovering unexpected connections in mathematics (though I myself have only utilised it rarely). Of course, the OEIS naming system is totally inappropriate for what we are discussing here – tags and links seem much better – but maybe there is still something there to learn from.

I also wanted to add one more comment, which is that expositions of proofs of existing results, while important, are only one side of the story. It is just as important to talk about

non-proofs; naive arguments, conjectures, or theories which fail, but for an instructive reason. These failures “identify the enemy” and allow one to truly appreciate the strength of the successes. This, I think, is perhaps the least developed area of mathematics exposition currently; nobody wants to talk about failure.September 11, 2007 at 10:39 pm |

Another thing that occurs to me is that one of the bottlenecks will be people to suggest tricks (or mathematical thought processes, or whatever) in the first place; here, we want the “barrier to entry” to be as low as possible. One wants some sort of discussion forum where people can just pop in and casually say “Here’s a nice trick: …” so that someone later with more time and energy can then create a rudimentary web page for that trick. For instance, in the last hour (while talking to a student), we brought up the tricks “near 1, multiplication looks like addition” and “to prove F(x) is close to F(y), bound the derivative of F” and “As a first approximation, drop all error terms and look at the main term” and “up to constants, + is the same as max”. These are all trivialities, and I wouldn’t want to write, say, a full-length blog post on any of these, but it would be nice to have a venue where one could somehow “dump” these mathematical micro-insights to be processed at some later date.

September 11, 2007 at 11:19 pm |

While I’m dumping these things anyway, here are two more tricks that came up while talking with my student: “as a second approximation, keep the error terms but drop the logarithmic factors”, and “when differentiating a complicated product or quotient, use the log-derivative rather than the derivative”.

September 12, 2007 at 12:04 am |

In due course I’ll respond to several points above, but for now let me just add something to the last two. It’s that a sentence like “near 1, multiplication looks like addition” should not be thought of as enough to specify the trick. What one really needs is an example where it is used (such as proving that an infinite product of terms (1+a_n) converges when the a_n are positive and have a finite sum). I think it might be worth making something like that a formal requirement

(though not necessarily formally enforced: that is, tricks must be explained with reference to an example (or, better still, several examples) and not just described in abstract terms. Even a trivial-seeming trick, when backed up with examples, can be enlightening, as it can say explicitly something that many people may not have done without stopping to think about it.

September 12, 2007 at 10:22 am |

OK here are a few more comments.

The naming problem is obviously a central one. A natural idea for dealing with it that I think is not in the end likely to work is to devise a clever classification system — however one did it there would be too many fuzzy boundaries, overlapping categories, and so on. Nevertheless, as we all know, search methods that initially seem too simple to be satisfactory can work incredibly well: who would have guessed in advance that merely looking for appropriate words and phrases could so often lead to what one wanted in Google? (Of course, that’s also thanks to Google’s eigenvector method, and although such iterative ideas could be wonderful for a site such as the one we are discussing, and lead to the most useful and interesting articles coming up first after lots of recommendations, recommendations of recommenders, etc., the idea of actually implementing them terrifies me.)

To begin with, I’d guess that something quite simple could be pretty effective. For instance, one could think of several different tagging systems: key words, area of mathematics (as far as it can be specified), examples of proofs that use the trick, well-chosen titles of tricks, authors of articles (or at least initiators of articles). Then, for example, one might think, “I keep running against the following difficulty concerning a function f and its Fourier transform F. I think I’ll see if Terence Tao has any harmonic analysis tricks that could help.” Or one could look up all tricks where “Fourier transform” is a tag phrase. I think the site would have to get pretty big before simple methods like that started to give too many results.

Incidentally, Emmanuel’s trick above reminds me of one that I (and many others) use constantly: if you want to prove something about a bounded non-negative function f, then assume that f takes just the values 0 and 1 and modify the proof later. That one has a companion trick that says more or less the opposite: if you want to prove something about functions that take values 0 and 1, generalize the statement to functions that take values in the interval [0,1] and prove that first. When I get time I’ll try to write a proper post on that.

Which brings me back to the notion of “proper post”. Of course, when Terry, Emmanuel and I have mentioned tricks above, we have not tried to explain them in the most useful possible way — we have just mentioned them in passing. But for a site such as this, I think a recommended format would help. Here are some elements that I think should go into a good tricks article.

1. A memorable slogan that describes the trick itself (of which there are several examples above).

2. Very explicitly described examples of applications of the trick.

3. An attempt to describe as precisely as possible the circumstances in which the trick tends to be useful.

I think these are all very important. For example, without 3 the reader may go away thinking “Well, that was very nice, but I somehow can’t imagine noticing that this trick can be applied.”

Various people have suggested other informal aspects of mathematics that could perhaps be presented. Needless to say, I am strongly in favour of these as well. That raises the following question: should they have different sites, or different portions of this site, or should the site be generalized to all sorts of valuable but not conventionally publishable expository material? One could have sections on good ideas that don’t work (as suggested by Terry above), motivations for definitions, tricks, and demystifications of proofs (one of my long-standing preoccupations).

On that last topic, Terry’s discussion of skeleton proofs is closely related to a fantasy I have had for a long time, which could I think become a reality. It’s to present mathematics in such a way that no step in any proof, no definition, nothing at all, is “magic”. Flashes of inspiration are absolutely not allowed — all ideas have to come with their origins.

At first this looks a hopeless task, but here’s how it could work in practice. Suppose I’ve got a proof I want to demystify (I really should illustrate this with an example but just now I haven’t got time — sorry). I begin by writing the proof that A implies B. Now suppose that that proof goes via an intermediate stage, so that A implies C implies B. Then I am done by induction if I can justify thinking of C, since then I have two smaller proofs to explain. So it might be that one person gives a very good explanation of how to reduce the original big difficult problem to a collection of smaller subproblems, which might themselves seem very difficult to the novice but would be considered fairly routine by experts. Then different people could come along and deal with these subproblems separately, breaking them up into subsubproblems, and so on. Since the internet is very good at tree-like structures, this could work very well on a website.

That’s enough for now — work calls.

September 12, 2007 at 11:12 am |

It would be fascinating to see how your ‘tricki wiki’ pans out in terms of the two cultures you describe. Atiyah speaks somewhere of important tricks becoming theorems, and later even theories. I wonder if that career path is more common on his side of the cultural divide. Might we expect less tricks in, say, algebraic topology, since most decent ones there get turned into theory?

September 12, 2007 at 1:06 pm |

David, I quite agree that that is interesting. As I argued in The Two Cultures of Mathematics (sorry, haven’t yet worked out how to do html in replies as opposed to original posts, so can’t give the link — incidentally, yours doesn’t seem to work), the surface appearance of subjects such as combinatorics is misleading, and the real advances take the form of exactly the kind of hard-to-classify problem-solving techniques we are discussing here. For some reason, in other subjects such as algebraic geometry, problem-solving techniques seem to be easier to formalize as lemmas. It’s as though algebraic geometers can build machines that work in the same way every time, whereas combinatorialists build partially specified machines that have to be modified for each use. It would be interesting to point to aspects of the subjects themselves that cause this to be the case: is it just that combinatorics is a comparatively new area, or is there something fundamentally different, such as that combinatorialists study less structured objects? I suspect the latter.

Anyhow, from the point of view of a tricks wiki — I can see that the temptation to call it a tricki will be hard to resist — I wouldn’t want to banish the dominant mathematical culture. But I think even a lemma can count as a trick: it is not of much value until you know how to use it

and lecturers often leave you to work that out for yourself, if you can.

An anecdote (not particularly amusing, but relevant) will illustrate this point. I remember in my first term at Cambridge being supervised with Andy Bailey, whom you will remember too. Bela Bollobas was the supervisor. I got completely stuck on a problem, but AB solved it easily with the help of Zorn’s lemma. In retrospect, given that Zorn’s lemma was needed and I had not digested it at all, it is clear — indeed, rigorously provable — that I was attempting the impossible. Anyhow, I was awestruck at the time that AB had thought to use Zorn’s lemma in this context. Now it would be obvious to me that that was what was required because I know how Zorn’s lemma is used. So it’s a perfect instance of a precise statement that could nevertheless make a very valuable tricks article. Or if you don’t want to call it a trick, it could be a how-to-use-this article.

September 12, 2007 at 2:05 pm |

A little while back, Robert Samal and I began thinking about all the things a well-built wiki for mathematics could be good for. Although we didn’t envision math tricks, we made a list which included open problems, survey articles, “book proofs”, and proof expositions. Recently, we started a website called the Open Problem Garden (http://garden.irmacs.sfu.ca) which is a wiki for open problems. If successful, we hope to expand our content to these other areas.

As Ben Webster suggests, we feel that a true wiki is the best format. While users lose control over what they have submitted, the vast majority of updates to be improvements. Also, every revision is stored, so it is easy to revert to an earlier version, and everyone can see exactly who said what when.

Managing quality without being a dictatorship is a significant challenge, but it is one which has been overcome efficiently by sites like Craigslist and Digg (which handle huge amounts of user-contributed content). If you have a critical mass of users willing to rate content, it is quite easy to just hide the stuff which doesn’t belong.

At the moment, the Open Problem Garden has a dozen posts in the Topology category which are pollution. In a week or so Robert and I will act as dictators to demote them to a less visible area (I think we were hoping this wouldn’t be a problem until we had a bigger community and a voting system). Anyway, one very nice property of our site is that it is very flexible. It is built using an open source content management system called Drupal which has a healthy community of developers. This gives us access to thousands of modules (bits of code) which we can just download, modify to our liking, and then install. Unless you really know exactly what you’ll want down the road, I think such flexibility is essential.

September 12, 2007 at 3:31 pm |

Dear Tim: In order to insert links into comments, there are several ways:

1. Just type the URL. For instance, David’s link should be

Click to access 2cultures.pdf

(and, incidentally, you have the ability to edit that comment to repair the link.)

2. Use HTML code. For instance, < A HREF=”http://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf” > The two cultures of mathematics < /A > should yield The two cultures of mathematics (though if I made a mistake, I have probably just typed some rubbish).

3. Use the wordpress editor to edit the comment. There will be a “link” button for the comment editor which functions similarly to the analogous button on the post editor.

Returning to the tricks wiki concept, I guess while all of the things discussed here would be desirable, it may be better to start with a more focused aim. One thought would be to initially focus specifically on those tricks which are used in constructions and counterexamples (the Banach-Tarski paradox would qualify here). These tend to be rather trick-intensive (rarely does one see a “theory of counterexamples”, instead it is mostly ad hoc). Also, as they are associated with a specific example or construction, they will be easier to attach names to (though of course we shouldn’t constrain ourselves into a naming system that will prevent us from referencing things that aren’t constructions or counterexamples).

I agree that the three ingredients for a tricks article (slogan, examples, range of applicability) are all vital, but the beauty of a wiki is that these ingredients can be added asynchronously. Someone may have an example which is seeking a slogan and a range of applicability, for instance, and would write an incomplete page to be filled in later by that person or by someone else; many other permutations are conceivable. So we could have several partial tricks articles needing improvement, similar to the “stub” articles in wikipedia. If one has templates for these sorts of things, then it is actually rather easy to then automatically collate which articles are in need of more examples, of a catchier slogan, etc.

September 12, 2007 at 3:41 pm |

Terry, I think my problem is that I can’t find a comment editor. For instance, I’m just typing this into a box that says “Leave a Reply” and there doesn’t seem to be anything I can do except type into it. And if I type an address such as http://www.bbc.co.uk it doesn’t come up as a link. I had none of these problems with the original posts, so I suppose there must be another place I haven’t yet discovered that’s better for leaving replies.

I agree with your comment about the three ingredients, as long as it’s kept fairly explicit which ones are thought by the author to have been supplied. So it would be a bit more specific than stubs, in that one would say things like, “This article could do with more examples: can anybody think of some?” and not just, “This article needs to be expanded.” Actually, I see that this is dealt with in your message, so I’m just highlighting its importance.

September 12, 2007 at 3:42 pm |

Oops — it did come up as a link!

September 12, 2007 at 7:31 pm |

Dear Tim,

If you are signed in to your wordpress account, you should see an (Edit) next to every comment, which will bring up the comment editor. If you are not signed in, you can go to http://www.wordpress.com to sign in. (Your browser should be able to automatically remember your user name and password so that you don’t have to do this every time.)

You can also edit existing comments from the Comments menu of your Dashboard. But as far as I know, the only way to actually create a new comment is to use the little box at the bottom of every article, which doesn’t have any fancy editing capabilities. I’ve been looking around for a way to be able to preview comments before posting them, but apparently this has been disabled by wordpress due to some sort of security issue. They have a pretty good track record of implementing various interface improvements over time, though, so perhaps something better will show up at some point.

One final warning: comments with < and > in them tend to get interpreted as HTML and are thus often truncated; this is particularly annoying when trying to write down inequalities 🙂 . You have to use < and > instead.

Coming back to the topic of discussion, one should also be able to allow pages which consist of almost nothing more than links to other web pages (e.g. blog posts, on-line articles, lecture notes, etc.); not all the tricks have to be “in-house”. Though I suppose as time goes by, one would want to develop those pages further with commentary, links to other tricks, and so forth.

September 13, 2007 at 4:42 am |

Terry wrote:

(Of course, in many cases, the devil really is in the details, but nevertheless knowing the overall strategy of proof is extremely valuable when trying to read that proof.) I could see a “Proofs modulo details” or “Top-level proofs” wiki being incredibly useful.Indeed, this is often how proofs in theoretical computer science (especially so in cryptography) are presented in papers, and even more so in talks! Specifically, in a paper, it’s generally considered good practice to begin with a top-level proof before presenting the actual proof, which may still be one that’s modulo details (more so in conference versions, where that’s a page limit).

September 13, 2007 at 4:46 pm |

Hello, I’m a bit embarrassed to show this to mathematicians, since I’m a physicist and often feel I’m taking finely tuned mathematical instruments and beating on things with them like a monkey with a hammer. But I’m personally using a mathematical physics research wiki for my own work, and it might be worth looking at to see an example of how an expository math wiki might look. This is the home page:

http://deferentialgeometry.org

and here’s a page with some math on it, so you’ll be less bored:

http://deferentialgeometry.org/#%5B%5BFuN%20derivative%5D%5D%20Welcome

This site was put together using TiddlyWiki and jsMath, both of which are great. A TiddlyWiki loads all at once, so you can click around it like reading a hyperlinked book once it’s loaded.

Good luck with your efforts. Building a wiki is a lot of fun.

September 13, 2007 at 6:27 pm |

gowers, for your “no magic” style of proof, I am reminded of the Fermat’s Last Theorem blog. Some proofs are hyperlinked back at least five layers.

September 13, 2007 at 7:07 pm |

[…] 13 Sept 07: I appear to have been partially-scooped by a fields medalist. Professor Gowers has initiated a nice discussion on the value of a wiki-based math-0-pedia. (I […]

September 13, 2007 at 9:13 pm |

Dear Tim,

I find your observation that in algebraic geometry “problem-solving

techniques seem to be easier to formalize as lemmas” very interesting.

I think that research areas are heavily influenced by their founders.

Modern algebraic geometry was given rigorous foundations by Serre

and Grothendieck. The philosophy of Grothendieck is well essayed in

NAMS 2004, issues 9-10. E.g., page 1197:

One thing Grothendieck said was that one should never try to prove anything that is not almost obvious. … ”if you don’t see that what you are working on is almost obvious, then you are not ready to work on that yet, …”

So with an ocean of definitions, I guess it is possible to find some more or less obvious relationships between them and call them lemmas. An example of what such intellectual endevours can lead to is on page 1196:

This can be seen in the legend of the so-called “Grothendieck prime”. In

a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”

It is interesting how algebraic geometry is being promoted nowadays.

E.g., an article in Wikipedia says about it: “… some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.”

However, the limits of “technique” that the human brain can handle

are now well known. So, chess can be considered a very simple math game, more technical than conceptual. No human can win anymore with

a desktop chess engine. I watched the last loss (2-4) of the world champion Kramnik with Deep Fritz (2006), a Sicilian. At some moment during the game, grandmaster commentators were hailing Kramnik’s position. A few moves later, he resigned. No analysis was given after the game. The computer’s play was just beyond the grandmasters’ imagination

and book knowledge.

September 13, 2007 at 9:15 pm |

Dear Tim,

I find your observation that in algebraic geometry “problem-solving

techniques seem to be easier to formalize as lemmas” very interesting.

I think that research areas are heavily influenced by their founders.

Modern algebraic geometry was given rigorous foundations by Serre

and Grothendieck. The philosophy of Grothendieck is well essayed in

NAMS 2004, issues 9-10. E.g., page 1197:

One thing Grothendieck said was that one should never try to prove anything that is not almost obvious. … ”if you don’t see that what you are working on is almost obvious, then you are not ready to work on that yet, …”

So with an ocean of definitions, I guess it is possible to find some more or less obvious relationships between them and call them lemmas. An example of what such intellectual endevours can lead to is on page 1196:

This can be seen in the legend of the so-called “Grothendieck prime”. In

a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”

It is interesting how algebraic geometry is being promoted nowadays.

E.g., an article in Wikipedia says about it: “… some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.”

However, the limits of “technique” that the human brain can handle

are now well known. So, chess can be considered a very simple math game, more technical than conceptual. No human can win anymore with

a desktop chess engine. I watched the last loss (2-4) of the world champion Kramnik with Deep Fritz (2006), a Sicilian. At some moment during the game, grandmaster commentators were hailing Kramnik’s position. A few moves later, he resigned. No analysis was given after the game. The computer’s play was just beyond the grandmasters’ imagination

and book knowledge.

September 17, 2007 at 7:03 pm |

[…] a comment on Tim Gowers’ blog, Terry Tao suggests that embedding the free group into has a number of applications. I’m afraid that I […]

September 17, 2007 at 7:11 pm |

A bit of self promotion: I have written a post on “That trick where you embed the free group into a Lie group” Comments are welcome.

November 28, 2007 at 10:15 pm |

Terence

referring to your comment

“A mathematical-tricks wiki would be wonderful! But it is going to be more difficult to set up than other mathematics-oriented wikis. One of the first problems is that of nomenclature: mathematical theorems and objects tend to have standardised names, but mathematical tricks usually don’t; indeed, even mathematicians who use a trick routinely may not even be aware that they are doing so, except at a very non-verbal level.”

This was the same problem faced by people in computer science a while back until a book came out by Erich Gamma et al “Design Patterns: Elements of Reusable Object-Oriented Software”.

Naming software patterns, so what you describe as mathematical tricks could be name on the same token as Mathematical Patterns and be define in similar way as in the Gamma book.

Naming the patterns in a very engaging way that is easy to remember. This will make the demonstrations of theorems once ones knows the mathematical pattern language at another level.

see this

http://en.wikipedia.org/wiki/Software_pattern

for some explanation as to what a software pattern is.

If a way of naming the mathematical pattern is selected it will probably be useful to select names that remind us of the solution similar as it is done in the software pattern book.

December 19, 2007 at 12:12 am |

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December 26, 2007 at 3:43 pm |

I think there’s a clean solution to the naming problem: don’t name the tricks. Instead, just rewrite them into Dijkstra’s calculational style of mathematics (predicate calculus is probably plenty for most cases). Then you can give the tricks rigorous definitions, and point out their use in various places.

Any trick takes the form of an implication or identity, so you should be able to easily build a search system for them based on putting some rough set of structures on both sides of an equality into a search box.

For instance a search ‘P = Q and R’ would return all tricks consisting of decomposing something into two properties, or ‘P = <y \in : P.y : y>’ is basically any embedding.

Such an approach is also valuable because then you can treat tricks as theorems, and make them reflexive manipulations of symbols.

Someone pointed out the design patterns stuff in software design. The last thing you want to do is go this route, because it makes it almost impossible to reason about these structures.

March 9, 2008 at 4:13 pm |

[…] is an old discussion on Gower’s Weblog where Gowers says “Terry’s discussion of skeleton proofs is closely […]

July 30, 2008 at 10:40 am |

[…] my hands, which gives me the chance to devote a bit of attention to other projects, of which the Tricks Wiki is one. So in this post I’m going to discuss a relatively elementary piece of university […]

July 31, 2008 at 1:00 pm |

[…] in combinatorics Here is another article that I hope to develop into an entry on the Tricks Wiki. It concerns the use of linear algebra to solve extremal problems in combinatorics. The method is […]

August 6, 2008 at 10:20 am |

[…] idea of looking at papers online, particularly as the arXiv gains popularity. There is even talk on Tim Gower’s blog of “tricks wiki” for the real lifeblood of mathematics, not just the end product, to be […]

August 12, 2008 at 1:25 pm |

[…] to use Zorn’s lemma I am continuing my series of sample articles for the Tricks Wiki with one that is intended to represent a general class of such articles. It is common practice in […]

August 16, 2008 at 9:25 am |

[…] This post is another sample Tricks Wiki article, which revisits a theme that I treated on my web page. Imre Leader pointed out to me that I […]

August 25, 2008 at 7:49 pm |

[…] (which I believe is now in press), has begun another mathematical initiative, namely a “Tricks Wiki” to act as a repository for mathematical tricks and techniques. Tim has already started […]

August 27, 2008 at 3:49 am |

“Another issue is how to organize the entries so that people can look them up, or browse, in a reasonably systematic way. This could be particularly difficult …”

This problem is really difficult, and I believe that the solution will be a new type of online Bookmark, which appears its prototype in del.icio.us and diigo (still not enough for a math wiki).

Here is some of my thoughts on this problem:

How to organize data by a new type of bookmark?

Bookmark=label+clip+alerts

Here:

I. label view=Groups/Folders view+Tag Clouds view+RoadMap view

II. clip=highlights texts×comments (diigo call comments as sticky notes)

III. alerts=rss reader×calendar reminder×searching alerts(like google alerts)

I will explain the item I. label view in a little detail:

labels (tags) is very important, both for bookmarking the website and organizing the personal files on his computer.

There are some suggestions for the “view pattern” of labels (tags)

I-1 Group view=Classical Folders view+Share Function（determine the sharing authority of members）。

Once users set some labels/tags as the “folders” or even “groups”, define their “hierarchies” structure, the user will be able to see the recursive nested view about these folders type of labels/tags. Yahoo bookmarks now has enable such function, but they do not understand that, “Folders” is just another view of labels/tags, — the same named Folder and Tag are different in Yahoo bookmark. But this is unnessary for our brain. And diigo realize the sharing function of the group, but diigo do not realize groups have the hierarchies structure. And Diigo also donot realize “groups” is just another view of labels, — the same named group and Tag are different in diigo.

Further, such hierarchies structure will satisfy the theory of psychology. That is, once you tags a file with “2.1.1” (for example), then this file will(should) be automatically tagged with the upper tags in the hierarchies, such as “2.1” and “2” . This is the ecnomical philosophy. That is once the user tag the item with subGroups/subfolders, then the item is tagged automatically with the fatherGroups/fatherfolders.

I-2. Tag clouds View=Matrix Pattern for the tags,

Tag clouds is now a very popular type of online bookmark. I suggest that, besides the clouds itself, there are two choices line of tags, first, the horizonal line (on the top) is same as that of temporary “tag clouds”, which interprete the “intersection” function of tags.

Second, another line, the verticle line (maybe lie beside the cloud) could be considered, which will interprete the “union” function of tags. Explicitly speaking, when you click the tags in the clouds, these tags will first appear in the verticle line, and the showing files are

related to the “union” of these tags. i.e. this file maybe belong to the first tags in the verticle line, and that file maybe belong to the second tags in the verticle line.

And then, the horizonal line will show the tags which has the intersection with that “unions”. Once the user click the tags in the horizonal line, the “intersection” function will be applied, just like what we do now.

I-3. RoadMap view of Tags: to indicate the logic structure of tags, especially useful for academic and universities.

August 27, 2008 at 10:52 am |

Besides this new type of online Bookmark system (label+clip+alerts), Another new function compared with the standard wiki, will be valuable and considered – “Book Setup” function, That is:

The site permit and encourage the user organize the items of wiki, to set up his own “book” of Tricks. Then the tricks in this book will show a deeper comprehension, according to the level of its organizer.

Hence, besides the bookmark system, the new-comer has another choice to get familar with the materials of this trick-wiki site, by studying the books of different old users, some of whom will be his teachers in reality (school or university), some of whom will be very talent mathematicians.

That reminds me the Chinese old classical book “Book Of Change” (written time is BC), it is indeed a book of combination of “tricks”, not of math, but of the experience of politics and social science.

Now by a web-site, we can expect many of “Book Of Change” for math can be set up. That would be a really sharp power for the improvement of math.

August 27, 2008 at 12:39 pm |

Many thanks for these suggestions. I don’t understand everything but my much more computer-literate collaborators on this project probably will. At least one of your ideas is one that we are actively thinking about right now — the idea that users themselves should create “higher-level pages” that have links to other pages on the site. In fact, I am in the process of writing some pages like this, to act as samples that will I hope encourage others to write similar pages in areas that I do not know about.

August 31, 2008 at 6:06 am |

A “tricki” sounds like a great idea, but it sounds like more examples are needed, in order to get some idea of what it would look like.

The first thing that came to my mind was the “determinant trick” (so called by Atiyah and Macdonald) that is so useful in commutative algebra.

Combinatorics is so full of tricks that systematic compilation of them has already begun. Nowadays a surprising amount of combinatorics boils down to a single “meta-trick”: Recast your problem so that a certain bag of tricks may apply, and pull out that bag of tricks. For example: generate an integer sequence and look it up in Sloane; or recast the problem probabilistically and pull out Alon and Spencer; or write down a recurrence and pull out Wilf’s generatingfunctionology; or write down a determinant and pull out Krattenthaler’s Advanced Determinant Calculus; or write down a hypergeometric sum and fire up Zeilberger’s Maple packages.

Chern allegedly used to say, “When in doubt, differentiate.” Feynman was fond of evaluating integrals by introducing an extra parameter just so that he could differentiate with respect to that parameter. I’d love to see more examples of Chern’s principle.

Gian-Carlo Rota once said that every mathematician, even Hilbert, has only a few tricks. The beautiful book “Making Transcendence Transparent” by Burger and Tubbs illustrates how this can be. They show how all those amazing theorems in transcendental number theory boil down to the fact that there is no integer between 0 and 1.

These examples make me think that a “tricki” might work best if thought of as an incubator—a place to collect isolated, unnamed, pre-theoretical tricks until enough patterns emerge that a book-length exposition can be created. Inexplicit expertise then turns into explicit knowledge that can be routinely consulted by everybody.

August 31, 2008 at 5:32 pm |

Timothy, I plan a post in the fairly near future (a first draft of it is already written) in which I will explain in some detail what the Tricki is going to be like and what its purpose(s) will be. You raise some issues that I have not yet explicitly addressed, so I’ll add to what I’ve written. In particular, I’ll say something about the relationship between the Tricki and the kinds of books you mention. (It was already very much on my mind that a number of the tricks I feel like writing about are discussed in Alon and Spencer.) For now, your final sentence could be regarded as a one-sentence description of what the Tricki is for.

Incidentally, your idea of a meta-trick is one that we already plan should be a major feature of the Tricki. As well as “first-level articles” that describe specific methods for attacking specific kinds of problems, we will strongly encourage higher-level articles that help you to classify your problem appropriately and then direct you to relevant Tricki pages. I will have much more to say about this in my post on the subject.

Meanwhile, I’d love to know what the determinant trick is that you talk about, and would also like to see a few examples of integrals that can be attacked by Feynman’s method. If you were prepared to write a couple of articles that could be included in the Tricki the moment it is launched, that would be great. And if you are someone else reading this comment and have a trick that might conceivably make a suitable article, then it does make a suitable article. If you haven’t got time to polish it up into the perfect article, write an imperfect one and let others add to it. (I’m hoping that there will be a facility for explaining in what way an article could be improved — e.g. by the addition of more examples, or a clearer discussion of exactly when the trick can be used.)

September 1, 2008 at 8:16 pm |

I have started drafting an article (provisional title: “Smooth sums”), but it is not so easy to arrive at something that sustains comparison with the examples you gave earlier (and the one by Terry on his blog)… But with some effort I’m hoping to get a draft by the end of this week.

September 1, 2008 at 9:53 pm |

Fantastic! And as I say above, an imperfect article is far better than no article, especially given that others will be able to make comments or even directly revise what you write.

September 2, 2008 at 4:16 pm |

The simplest incarnation of the “determinant trick” is in the proof that the algebraic integers, defined as roots of monic polynomials with integer coefficients, are closed under addition and multiplication. One might think at first glance that this is just a matter of chasing definitions, but in fact it’s surprisingly subtle. The trick is to observe that if x and y are algebraic integers, then the ring Z[x,y] is finitely generated over Z as a *module* (not just as a ring), with a basis of elements of the form x^i y^j. So given any w in Z[x,y], multiply w by each basis element in turn and re-expand in terms of the basis to get a system of linear equations. This system asserts that a certain matrix involving w and certain (rational) integers sends a nonzero vector to zero, so we conclude that the determinant is zero. The equation det = 0 yields explicitly the desired monic polynomial equation satisfied by w.

In abstract settings the determinant trick doesn’t show up explicitly so much; instead one appeals to something like Nakayama’s lemma. Or one can simply define “integral over R” to mean finitely generated as a module over R. But if you’re doing explicit computations then the determinant trick is still handy to know.

As for integrals attackable by Feynman’s method, there is an excellent collection of examples in the Wikipedia article on differentiating under the integral sign. However, it’s a little dry, and could benefit from a better presentation, that helps students figure out how to come up with these kinds of arguments themselves. The (sin x)/x example would be a good one to start with. Clearly, the x in the denominator is annoying, so one wants to cancel it out. One might first try differentiating (sin xt)/x with respect to t, but this introduces convergence problems. That’s why one chooses exp(-xt) (sin x)/x, so that you still cancel out the x in the denominator, but now you needn’t worry about convergence (since exp(-xt) goes to zero so fast).

While we’re on the topic, here are two tricks that I would like someone to explain to me. The first trick is the use of partitions of unity. Some basic local-to-global arguments on manifolds can be proved using partitions of unity, while others require more subtle methods such as sheaves and cohomology. When can one expect a partition-of-unity argument to work and why?

The second trick is one I encountered years ago when taking a course in algebraic number theory from Goro Shimura. Shimura frequently used the following lemma. Let F be a number field and let J be its ring of integers. Given any integral ideal A and any fractional ideal X, there exists a nonzero element z in F such that zX is an integral ideal and zX + A = J. This lemma can be used to give quick proofs of a number of results, e.g., that every fractional ideal can be written in the form xJ + yJ for some x and y. However, I never fully understood why this lemma was so useful or when one could expect it to be invoked.

September 6, 2008 at 12:14 am |

I see I arrived a bit late to the party. Nevertheless I hope this comment

gets read somehow 😀

· First thing I thought about the subject: we NEED to get organized! In particular, we need to open a forum about the Tricki: the planification and organization of such a feat can’t be done from zero-level, we must keep adding layers of data and information distribution until we arrive to the Wiki, and it seems to me that the first step should be a forum (blogs were the 0-step).

· About the difficult and fascinating subject of naming and identifying every trick (btw, this should be not a thread, but a whole subforum on the forum!):

a) I suggest we “impose” the official name by being the de facto most authoritative source for math tricks (as it happens with OEIS or Wikipedia). Once we have launched the project and having such skilled, expert, famous people on it, authority will follow fast. People will tend to use the names we give to them just because the rest of the people will want to look at them at our Wiki (here, when I say “we” and “our” I refer to everyone who wants to collaborate, of course!)

b) The name should be composed of (at least) three components:

1) An informal, catchy name.

This would be the best known name for the trick, and would methaphorically refer to its use or main characteristics, or to any circumstances around it or its first apparition, like theorems tend to be named “lately” (hairy ball theorem, sandwich theorem, rising sun lemma, egregius theorem, etc).

The word “TRICK” (or any other to be decide) should be the last name, just to let people know what we are talking about.

An invented example: The “1-is-not-so-trivial” trick -> In your main FORMULA [specialized term to be defined within the WIKI], substitute 1 for any fancy SUBFORMULA that does equal to 1, like sin^2(x)+cos^2(x), or partitions of unity, and REARRANGE the main formula (i.e. separate in “components”, simplify, glue, etc).

2) A technical name within an a priori specified naming rule system (to be defined yet) that already implies some categorization, something like the naming of new biological species. For example: FORMULA SUBSTITUTING+ (where Formularity refers to the fact that the trick is about formulas, Substituting to the obvious fact, and the “+” means that the substitution actually gets more complex, and not less (-)).

Another invented categorization family name could be Equation DifferentiatingN (take your equation and differentiate it N times)

3) A series number. Just to catalogue the trick and make internal database work with it (but also to have it uniquely identified!)

c) To solve the search problem, a multitag system looks like the most reasonable system to me, but the selection of the tags is critical. I think we should have two kinds of tag:

predetermined distinguished tags, and configurably non-distinguished tags. The first ones would be created with the project (always subject to suggestions and changes) and would be things like:

WHERE (in which branchs of math)

IN (in which type of structures)

FROM (hypothesis and elements you need to have for the trick to make sense; i.e. the trick starting point)

TO (the generic result you accomplish by applying the trick)

WITH (“the how”, technical terms that describe key parts of the process)

NAME (informal name)

CATEGORY (formal name),

SN (series number)

AUTHOR (the creator or best-known user of the trick, if it’s known)

REFERENCE (well-kown book, paper or source where it’s used in an important manner)

USES (well-known uses of this trick, theorems where it is important, etc)

The non-distinguished tags would all fall to the same category OTHERS, as a list, just like the tags in YouTube.

Not all the tags need to be filled, and some can be multivalued.

An example:

SN 0000

NAME “The connection trick”

CATEGORY Set Identificating Connected

WHERE Topology, Algebraic Topology

IN Topological spaces, Manifolds

FROM Connected set, subset

TO Subset is Whole Set

WITH Connected Sets Open-Closed Characterization

USES

OTHER Standard, Basic

Did you know what trick it was? The informal description (which I think it should be the first thing you get after the tags) would be “In a connected set C there are no subsets that are open and closed at the same time, other than the empty set and C. Therefore, if you want to prove that your subset D of your connected set S is in fact the whole S, just prove that S is open, closed, and non-empty”

·Other basic tricks I thought just know (please forgive their triviality, I’m just a math student yet):

* Smart application of Hölder inequality

* Partitions of unity

* Linearization in the axioms non-associative algebras

* Zorn’s Lemma and, similarly, Ascending Chain Condition and Maximal Condition for noetherian modules

* If in a Normed Space you want to prove that x is 0, just prove that its norm is 0

* If you want to observe the ideals that contain I, then study A/I. If you want to study the ideals contained in the prime P, study the localization A_P.

* If you are working in a bound that involves exp(x) maybe you just need to work with (1+x). In general, if you are working in a bound that involves an analytic function, it may suffice to work with the first n terms of its Taylor Series (where n=0,1,…,5).

Well, I adore to give more ideas and to get involved with this great project, but we definitely need a forum! (If it isn’t implemented yet!). The other way all this effort will eventually come to waste…

Regards!

Jose Brox

September 7, 2008 at 4:21 pm |

My first trial for an article can now be read in this post.

September 30, 2008 at 7:22 pm |

I would love to see more proofs without dreaded phrases like “note that”, “it follows that”, “clearly”. Also, I distinguish between proofs that verify and proofs that explain. Sometimes, we don’t see the big picture and the best we can do is check the facts. Other times, the proof writer “saw something” and we should be able to see that thing too. After a while, I stopped thinking about proofs as a long trudge to proving “Statement 1.1.6”, but a change to learn cool, potentially useful facts about the object of study (hot games or p-adic metrics on field extensions or van kampen’s theorem or whatever). Often time, the statement being proved may or may not be useful, but the ideas accumulated in the proof are very useful in themselves. Probably by its very definition, insight is hard to put into words. – end rant

October 20, 2008 at 7:02 pm |

Genuine collaboration is voluntary collaboration. Wikipedia is very far from that.

Quality control requires collaboration among responsible contributors who know the subject matter and who respect the knowledge and diverse points of view that other bring to bear on the subject. Wikipedia has nothing to do with that.

The wikioid projects of my acquaintance that approach these goals are Knol, MathWeb, MyWikiBiz, PlanetMath, and ProofWiki.

Jon Awbrey

October 20, 2008 at 7:20 pm |

Let me try those links again:

Knol

MathWeb

MyWikiBiz

PlanetMath

ProofWiki

Jon Awbrey

November 3, 2008 at 5:57 pm |

I am in MSRI for the cofference discrete Rigity. Green will give the first lecture.

I just happen to find a question for that tricki wiki:

Whether is there a common-shared refference system for that tricki wiki? Similar to that of Mathscinet of ams math review

It will be a basic instrument for a mathematical website.

November 16, 2008 at 7:02 pm |

Dear Tim,

Have you considered Wikibooks? The Mathematics bookshelf could certainly accommodate this project, and there’s no setup required.

In your desiderata, the key conflict is between

collaborationandquality control,and these are simplyirreconcilable:2 extremes that work in their way are the anarchy of Wikipedia or the order of a collection of essays, as in PCM.The problem is that the burden of reviewing every change is onerous, and that authors come and go (or have time for a few years, then not after).

I would encourage using a public, freely licensed wiki, like Wikibooks, and

trusting that wikis work– the quality is surprisingly good, because it’s trivial to roll back vandalism, and it’s rare for vandals to care enough to vandalize obscure subjects.You may consider, if you are not familiar with it, the history of Nupedia, which tried such a peer-reviewed system of high quality articles, and got nowhere (rather, not very far). Especially for so varied a topic as “mathematical tricks”, allowing all and sundry to contribute can very much help.

Further, using an existing project (notably Wikipedia & co.) attracts contributors – wouldn’t it be great for Wikipedia entries on various topics to link to the tricks used, within the same project?

To address quality concerns, taking a particular revision and blessing it as “good quality”, as in the

Wikipedia Version 1.0 project or mathematics assessment, works quite well.

So I’d urge you to consider a publicly editable wiki (esp. Wikibooks) – they really work far better than one may expect.

Best,

Nils

November 16, 2008 at 7:19 pm |

Regarding namingAs noted, the naming of tricks is a vexing problem. One solution is to

not worryabout the names!Rather,

linkthe trickto and fromtheapplications. Think of how one identifies unnamed concepts in general: onedescribesthem and givescontext. This allows one tofinda trick: “oh, it’s the trick you use to show that the uniform limit of continuous functions is continuous – let’s look up uniform limit!” (in this case, the epsilon/3 trick) (yes, I’m eliding the assumption of uniform continuity/local compactness). Similarly, if one thinks: “In Theorem Y, one uses the same trick one uses in Theorem X”, one can look up theorem X to link to this same trick.Some form of this is already present in Wikipedia: obstruction theory is both a specific term in algebraic topology and geometric topology, and a general trick for defining invariants (usually cohomological) for various problems – and it’s also v. similar to the Hasse principal and sheaf cohomology, though these tend to go by “local-global” problems, not “obstruction theory”.

Ultimately, while good names are useful, they are just a tag: concepts are defined by their content and their connections.

March 16, 2009 at 7:52 am |

Dear Prof. Gowers,

Sorry for bringing this topic up.

One idea would be to find someone willing to extend mediawiki (maybe there’s such an extension) allowing you to add labels to each trick, plenty of labels. Say you have a trick on an inequality dealing with limits often being used in real analysis (say). Then one could label it as “analysis”, “real analysis”, “limit” and “inequality”.

So if a person isn’t sure about what trick he or she is looking for (or even if there is some sort of trick that could help out with the problem), then he or she could specify some labels to search for and getting a list of tricks related to these labels. And who knows, there might be something there to help out solving the problem!

March 16, 2009 at 8:19 am |

Dear Anonymous,

It’s actually quite a good moment to bring the topic up as the Tricki really is about to appear now. And there are various labelling systems such as the one you propose. More on this soon.

May 10, 2009 at 5:56 pm |

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December 19, 2010 at 9:54 am |

It looks like http://math.stackexchange.com is becoming a useful Wiki-like community, and as of December 2010 the level of quality is high (maybe “very high”).

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