Actually, I don’t see anything ludicrous about the Amazon example — that seems to be another example of canonicity. And the whole point of what I’m saying about canonicity is that although people link randomly, the distribution of how they link is far from uniform . Rather, random linking biases usually reinforce links to the more easy-to-find, already “canonical” resources, thus making them even more canonical.

As for people hovering over links to read where they point and then determining whether to go there or not — well, not everybody does that . But even if they do, the point I’m trying to make is that when people just link up to a default source such as Amazon or Wikipedia, without it being clear whether it is linked to simply on account of it being a default or on account of their finding it good in the specific instance, some information on how the link choice was made is lost. And this can lead to both humans and bots concluding that the particular specific instance (e.g., the specific Wikipedia entry) has been “recommended” by the linker.

For more about link-by-default, the way I would handle it, if indeed I planned to link to Wikipedia, is to write something like:

Complete metric space(Wikipedia entry)

If there exist good entries at multiple sources, I’d comma-separate them, e.g.,

Complete metric space(Wikipedia entry, Mathworld entry)

On the plus side, it reduces the canonicity or “by-default” appearance. On the minus side, it consumes more space.

I am not sure what “actually good” means here. Do you mean, good relative to other Wikipedia articles, or good relative to other online references, or good relative to all existing references?

It did seem to me that Tao actually checks the Wikipedia entry before linking to it. I think this is evidenced by the fact that in some cases, he links to the Planetmath entries, or other entries, or puts excerpts from books.

To check up on your comment, I decided to look at some Wikipedia entries linked to by Terence Tao from what’s currently his latest blog post (245B, Notes 9). He links to Wikipedia articles on Lebesgue differentiation theorem, complete metric space, dense set, nowhere dense set, Baire category theorem, and many more. Anyway, so the Wikipedia entry on Lebesgue differentiation theorem seems reasonable in itself, and fits in reasonably well with what Tao seems to want for his work. Good one. As for complete metric space — well, here the Wikipedia entry probably does a reasonable job by itself, though it seems to not fill in with Tao’s flow too well. The entry on nowhere dense set is short and not outstanding, but does a reasonable job again for Tao’s flow. Ditto for dense set: the Wikipedia page isn’t what I’d consider great, but it is okay for Tao’s purposes. Baire category theorem on Wikipedia has a lot of useful information, but I’m not too fond of the way it has been presented.

So anyway, I looked up the corresponding definitions on Mathworld. The Mathworld entries for dense and nowhere dense seem shorter, but to me they seem to be better presented and more useful to link to for serious math students. But Mathworld definitely doesn’t win hands down and it is possible that Tao made a conscious choice of the Wikipedia entry after comparing with the Mathworld entry.

A little later down, I see Terence Tao linking to tempered distribution: here at last is some evidence that perhaps Tao isn’t inspecting the outward Wikipedia links. Tao’s link lands up at a disambiguation page, that was last edited in July 2008. Another link that seems extremely fishy is the link to harmonic analysis, where reading the Wikipedia entry doesn’t seem to help directly with understanding the material. On the other hand, when it comes to the Fefferman-Stein decomposition theorem, Tao links to the paper’s MathSciNet page rather than to a (non-existent) Wikipedia entry, and Tao does point his link to Radon-Nikodym derivative when mentioning the Radon-Nikodym theorem. Some of Tao’s later Wikipedia links are also unhelpful to people trying to grasp the material first-off, but may be value addition to people who have understood bulk of the material.

All in all, I don’t know how Tao is choosing Wikipedia entries — whether he is comparing them with other online references, whether he is reading the pages, whether there are many other things for which he chose not to link because of the poor quality of the entry. But the few examples I gave above show that Tao isn’t inspecting each entry very critically — and I’d be surprised if he did, given the number of links he’s given. Whatever the case, readers of his blog (other than me) do seem to get the impression of link-by-default to Wikipedia: for instance, look at the comments section of this blog post where an anonymous commenter suggests a convention indicating links to Wikipedia for all underlined terms in the “print” version of the blog.

[I read a comment somewhere else noting that almost everything Terence Tao linked to on Wikipedia was a good article, while most mathematics articles on Wikipedia weren't very good... the simple explanation is that for the linked topics, Wikipedia was indeed a good resource, which is why it was linked to.]

[1] http://en.wikipedia.org/wiki/Federated_search