What might an expository mathematical wiki be like?
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
http://www.dpmms.cam.ac.uk/~wtg10/2cultures.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
[...] to theorem-proving techniques—will almost certainly exist in the near future. Remarkably, my earlier post on this idea led to an offer of technical help that will be enough to turn it from a fantasy into a reality. And [...]
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 [...]