TrickiThere are some proofs that keep getting mentioned on the Tricki. One example is the proof of Roth's theorem on arithmetic progressions, which is a very important argument for the Tricki because (i) it illustrates many different "tricks" and (ii) it can be thought of as a starting point for many more.
This situation is likely to recur again and again, and suggests the possibility of a rather nice kind of article. In it, one would present the proof of a theorem, but instead of just writing down the proof in the usual way, one would justify every single step by saying, "Here we are applying the trick that is explained in Tricki article about that trick." In other words, one would show that if you digested the right Tricki articles, then in principle you'd be able to think of pretty serious arguments for yourself.
For instance, the justification for the analytic proof of Roth's theorem could be contained in a list of articles such as To prove existence, count and get a non-zero answer, Turn sets into functions, If your problem can be expressed in terms of convolutions and inner products then take the Fourier transform, etc.
The only drawback I can see is the possible blurring of the main purpose of the Tricki. It would be very important not to end up with people just writing out proofs: the idea would be to present fully justified proofs, with no rabbits pulled out of hats. And I think it would be important to have these articles written in some format that clearly marked them out as different.
A benefit of articles like this is that they would demonstrate how to get deep results from a sequence of tricks, none of which is individually all that difficult. I.e., it would emphasize the fact that most maths problems are not solved by a single "Aha" insight, but rather by a gradual accumulation of such insights.
We could allow "rabbits coming out from hats" if they actually were yet another tricks not implemented on the Tricki yet, presenting them as "empty links" and opening the possibility that someone goes and write them (this links with the "non-existent parents" topic). We may force the author's of this kind of "proof-with-some-rabbits" posts to create a stub for every rabbit.
I think this would be very useful as well. A fun side-effect of it would be that one could go to one of these pages, see the theorem and the list of Tricki ingredients that are involved in the proof, and then see if one has understood each technique well enough to make progress with the proof.
I think you're right that it's probably best to have a different content type for these types of articles to emphasise the difference between them and core Tricki articles. I don't see that making such a distinction would have any drawbacks.
One thing to think about is whether we should have a recommendation for the amount of detail to put into the proofs. Too much detail might obscure the overall outline of the ideas that have been used, though one could alleviate this by having a separate outline section. A benefit of including a lot of detail is that one can then see a series of Tricki articles live 'in action'. Perhaps
[add]s are the solution.I seem to be very late coming to the table. The latest comment is from nearly two weeks ago. The format of the articles is very nice, so I am surprised at the low traffic. Perhaps it will grow as more topics become available.
I understand the impulse of gowers to expand knowledge organically by linking a topic to other relevant topics. The capacity for hypertext is one of the great strengths of the internet. But as someone trying to learn math, I like to see the essential train of thought covered within the article and presented in a sharp profile. If this requires redundancy to refresh the reader's memory, then so be it, because a bit of judicious repetition can be more efficient and effective than sending the reader down a variety of garden paths leading away from the original topic.
I will be so bold as to offer a second suggestion to math authors: Point out the trouble spots where a reader might go wrong. Contrast correct avenues of proof with possible incorrect ideas that might occur to a reader. Don't just send one to the Heine-Borel Theorem. Summarize the gist of the theorem and offer a bit of explanation for why H-B and not some other theorem is just what is needed for the current stage of a proof.
In my experience, the authors of instructional materials in math almost never attempt to anticipate gaps and misconceptions in the readers' understanding. It is only good instructional practice to do so. I would like to see math texts that are much more explicit about the choices at each step of proof. I'm not requesting that obnoxious popular flippancy which some mistake for motivational entertainment; just more discussion of math reasoning and more recognition of the needs of those readers who learn best when a topic is discussed from various angles rather than presented only in a sequence of formulas or terse prose proofs.
Finally, I would like to know who you are, you who established this site and serve as the primary authors. Would you put up an "About Us" page?
The idea of creating articles for proofs on the Tricki is one that had occurred to me before I read this post, and for reasons similar to those which gowers has outlined above. Firstly, I think that, as pointed out, most proofs use several Tricks, and are likely to be repeated on different articles: having proof pages would prevent this inefficiency. Secondly, I think that describing proofs in terms of the Tricks they contain would be a really good way of helping people to learn the proofs: some of gowers's recent blog posts for Cambridge undergraduates have done just this - they have showed that one needs to learn very little in order to reproduce a proof: the little that one has to learn consists, more or less, in the Tricks that make up the formal proof. I realize that the purpose of the Tricki is to teach people about Tricks, and not about proofs, but teaching people about proofs would be a nice collateral, and would not obscure the main purpose of the site. Thirdly, there now (It was started in March 2008) exists a site called proofwiki, whose articles are either expositions of proofs or definitions of mathematical concepts used in the proofs. The proofs given there are very much of the 'rabbits pulled out of hats' variety, and a site which instead gave proofs in terms of the Tricks involved would be more useful for learning (though proofwiki is good as a reference). It could even be possible to synchronize the proof pages on the Tricki to the equivalent pages on the proofwiki, so as to avoid having to find a new way to organize proofs. Certainly, we would not need to create new definition pages.
A list of features which could be added: And I do realize that you haven't been adding any new features to this wonderful site for at least a year, so it's entirely up to you whether you decide to implement these
[TRICK name of Trick]and finish it with[/TRICK]. Everything between the tags would follow very easily from the trick involved: a link to the article on the Trick would appear at the top of the section. Any text occurring outside a pair of[TRICK]tags would not appear in the article (with the exception of general discussion sections etc.) This would force people to write proofs in terms of Tricks.Anyway, I hope very much that you give this your consideration, and I look forward to contributing to the Tricki in the future.
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