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Wikipedia is a widely used resource for mathematics. For example, there are hundreds of mathematics articles that average over 1000 page views per day. Here is a list of the 500 most popular math articles. The number of regular Wikipedia readers is increasing, while the number of editors is decreasing (graphs), which is causing growing concern within the Wikipedia community.

WikiProject Mathematics is relatively active (compared to other WikiProjects, but not compared to MathOverflow!), and there is always the need for more experts who really understand the material. An editor continually faces the tension between (1) providing a lot of advanced material and (2) explaining things well, which generates many productive discussions about how mathematics articles should be written, and which topics should have their own article.

Regardless of the long term concerns raised about whether Wikipedia is capable of being a resource for advanced mathematics (see this closed question), the fact is, people are attempting to learn from Wikipedia's mathematics articles right now. So improvements made to articles today will benefit the readers of tomorrow.


Wikipedia is a very satisfying venue for summarizing topics you know well, and explaining things to other people, due to its large readership. Based on the number of mathematicians at MathOverflow, who are willing to spend time (for free!) answering questions and clarifying subjects for other people, it seems like there is a lot of untapped volunteer potential here. So in the interests of exposing the possible obstacles to joining Wikipedia, I would like to know:

Why don't mathematicians spend more time improving Wikipedia articles?

Recent efforts intended to attract new participants and keep existing ones include the friendly atmosphere of the Teahouse, as well as WikiProject Editor Retention.

In case anyone is interested, my Wikipedia username is User:Mark L MacDonald (which is my real name).

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In my personal experience, it takes quite some self-confidence to make edits which remove someone else's content, and it's usually the wrong people who have this self-confidence. It's easier to add some remarks or fix a typo, and I do think mathematicians do this pretty often (some of the best known MOers are on wikipedia). –  darij grinberg May 22 '13 at 13:49
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The mathematics pages I look at are generally quite good, so I don't feel any particular urge to try and improve them. –  anon May 22 '13 at 14:16
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Hi Mark! I think one important distinction between MO and WP is that with the latter, you never really know if someone has read what you've written, which is discouraging. (Although it doesn't stop us writing papers...) Another is that writing WP articles is very open-ended, which can be daunting. By contrast, it's often quick and easy to give a definitive answer to a question on MO. –  Artie Prendergast-Smith May 22 '13 at 15:33
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For what it's worth, seeing this question made me feel guilty enough to go create a Wikipedia article on a topic not yet covered. –  Daniel Miller May 22 '13 at 20:19
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This question bothers me because regardless of how much mathematicians contribute, you can always ask why mathematicians don't contribute more time, which gives the (false?) impression that mathematicians are not contributing much. I think the mathematical sections of Wikipedia are actually quite good (compared with many others) due to the efforts of many mathematicians. Is there some evidence which says mathematicians don't contribute much? The graphs don't show any dramatic decline, and even if there were a decline, this doesn't mean current levels are low. –  Douglas Zare May 22 '13 at 22:49

13 Answers 13

In general, I find the wikipedia maths pages to be very comprehensive and useful, and I have not found any serious errors. However, they are often not very well organized or clearly explained. I agree that it would be good if more professional mathematicians tried to help with this.

I rewrote the article http://en.wikipedia.org/wiki/Homology_theory a few years ago. I thought about working on some other related ones. However, it seemed to me that it would be best to do a general reorganization, deleting some pages, renaming others and redistributing material in a more coherent way. I did not know how to go about presenting such a proposal to the relevant community, and how to ensure that there was enough consensus that a lot of work would not go to waste, and how to make the changes offline so as not to leave an inconsistent mess while the work was going on. So I did not end up doing anything.

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The best place to make such proposals would be on the talk page of WikiProject Mathematics (en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics ). That page has a lot of eyes on it, so you're likely to get a quick response. Indeed, those sorts of organization-type discussions are very common, and they often end in some kind of agreement. –  Mark M May 22 '13 at 14:07
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@Mark M; WikiProject Mathematics has "eyes" of people who like to talk a lot. If you see how to make wikipedia better, simply do it and check what is the reaction of community. (I think it is called "Be Bold", it is a "rule" in wikipedia). –  Anton Petrunin May 26 '13 at 13:26
    
To a large extend you can follow the "be bold" rule, but some for some things, discussion with others may prove necessary or useful. For the latter sort of thing, the page that Mark M cites is the best place. Some WikiProject talk pages are pretty sleepy, but that one is fairly lively. –  Michael Hardy Jun 4 '13 at 21:03

My feeling towards Wikipedia is that it is great as a quick reference to check facts and results, but it doesn't work as well for explaining well introductory material. The reasons are various.

It is difficult to provide a good explanation for everyone. Often, different intended targets conflict: you can't explain "kernel" to a layman, to an engineer, and to a mathematician at the same time, while making the exposition clear for all of them.

Moreover, often one would like to start from scratch and lay down the whole article from a different point of view; this is diplomatically difficult because it would trash the contributions of the previous editors (good or bad as they are). Yet, for some quickly-evolving topics it might be necessary to rewrite the page from scratch every few years, to reflect the recent advances and the new understanding.

The notation of an article might need changing; however, it's a difficult job to get to agree on "the best notation" for a given topic, even among three co-authors. Edit wars are a concern, and democracy often isn't the best way to solve them.

Pages in other languages are another concern. There is nothing wrong with having a good English page translated in several languages; on the other hand, local writers will tend to start their own new pages independently, even if their quality is worse. Even if someone translates an English page at a certain moment, future changes will get de-synchronized among the different languages.

I feel that many mathematicians would rather write a (part of) a textbook or a review article, and publish it on the internet. Often, too many cook spoil the broth. A good writer working alone can trump many writers working collectively. So, maybe we'd rather need to focus efforts on making good textbooks and lecture notes easy to find on the internet.

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There are also other resources such as Wolfram (mathworld.wolfram.com) with its content published in Eric Weisstein's Concise Encyclopedia of Mathematics which is probably better to use as reference in an academic article than Wikipedia. –  JD Vlok May 22 '13 at 12:10
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@JD Vlok: I'm much more comfortable citing the references in Wikipedia or MathWorld. –  Dustin G. Mixon May 22 '13 at 12:34
    
@Dustin G. Mixon: Fair enough, but there might be instances where no references are given (especially when only MathWorld has a page on a topic). You are then left with using either the webpage or Weisstein's book as reference (or finding another source). –  JD Vlok May 22 '13 at 12:58

Here is my rather personal answer. I like Wikipedia---both in general and the mathematically oriented part, and I would like to collaborate in its improvement and contribute in any way I can. But I don't know where to begin. With MO, when I have free time I can start "at the top" so to speak, and continue on down until I start seeing familiar entries---but in Wikipedia how does one decide what to look for that might benefit from one's inspection and possible editing. Getting over that intimidating problem is what prevents me from even getting started. I think it would help if "we" (i.e., the math community) could set up a website where there was listed mathematical subjects that either someone had spotted as being ignored by Wikipedia, or whose Wikipedia article had been recently edited, or which someone had read and felt would benefit from expansion or improvement. It would be good if such a site was maintained in chronological order by date of entry so one could view it from the top for new stuff as one had time to do so, and it should allow comments and "answers" to each entry to show any progress in fixing the problem. I guess some sort of "stack" is what I have in mind. I would certainly be willing to visit such a site frequently myself and contribute to it as I could.

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There's a bit of the functionality you want implemented: en.wikipedia.org/wiki/Wikipedia:Pages_needing_attention/Mathematics / en.wikipedia.org/wiki/Wikipedia:Pages_needing_attention/… / en.wikipedia.org/wiki/Category:Mathematics_stubs / en.wikipedia.org/wiki/Wikipedia:Requested_articles/Mathematics . I don't know who (and how) decides what should go on those pages, though... –  darij grinberg May 22 '13 at 18:03
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Here is "Yair Minsky" approach to "where to start": Each time a grad student walks into his office and asks a question for which there is no (good) wiki answer, Yair writes a short wiki article. (At least, he used to do so.) –  Misha May 22 '13 at 18:21
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Dick, almost all of what you have suggested is already exists, and it's called WikiProject Mathematics - and we'd like your help! en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics –  Mark M May 22 '13 at 19:25
    
Darij, those lists are automatically compiled, and consist of articles which somebody decided to "tag" with the relevant issue. Once the issue is solved, then you can untag the article, and it disappears from the list. Clearing backlogs is satisfying! :-) –  Mark M May 22 '13 at 19:26
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@Mark M YES! That is a very good approximation of what I was asking for. Thanks for telling me about it. It is now one of my "pinned" tabs and I will try to help with the project. –  Dick Palais May 22 '13 at 21:09

"(...)most academics – despite goodwill to contribute – still perceive major barriers to participation, which typically include a general lack of time to contribute, but also barriers of a technical, social and cultural nature. These encompass the lack of incentives from the perspective of a professional career, the poor recognition of one’s expertise within Wikipedia, the widespread perception of Wikipedia as a non-authoritative source. In combination with the apparent anomaly of collaborative – and often anonymous – authorship and the resulting fluidity of Wikipedia articles, these factors create an environment that significantly differs from the ones experts are accustomed to" Quote from http://blog.wellcome.ac.uk/2011/02/25/wikipedia/

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FYI - The reasons are contained in the second-to-last paragraph. –  Dustin G. Mixon May 22 '13 at 11:18

Here's the math WikiProject's "Current activity" page, which lists each day's new articles, articles for which deletion is proposed, articles needing expert attention, articles needing references, and articles needing various other sorts of work: http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics/Current_activity

Here's the page listing "requested articles"---mathematics articles that do not yet exist: http://en.wikipedia.org/wiki/Wikipedia:Requested_articles/Mathematics

(Inexperienced or non-logged-in users are no longer allowed to create NEW articles, but they can create drafts in the user space, which can be moved by experienced users into the article space.)

DO NOT begin a Wikipedia article by writing "Let $\lbrace T_n\rbrace_{n=1}^\infty$ be a sequence of bounded linear operators on a Banach space $B$." That fails to even hint to the lay reader that the article is not about theology, music, chemistry, banking, politics, etc.

The title of the article should be a singular noun phrase except when there's some special reason to use the plural. One such reason is that the article is about a set of things (in something more like the layman's understanding of "set" than that of the mathematician); for example "Maxwell's equations".

One might begin by writing "In mathematics, an oriented matroid is . . ." Or "In algebra,..." or "In number theory,..." or "In geometry,..." or "In calculus,..." But DON'T start by saying "In functional analysis,...", or "In category theory," or "In topology,....". Again with those, the lay reader is given no reason to suspect that mathematics is what it is about. There's no need for that sort of context-setting phrase if the title of the article is "Mathematical induction" or "Algebraic equation" or "Arithmetic of $p$-adic numbers".

Usually, the word or phrase that is the title of the article should appear in or near the first sentence set in bold, not necessarily verbatim identical to the title (e.g. it may be plural where the title is singular).

Do not begin a biographical article by writing "John Schmon was born in Putford in 1801. His father was a polecat farmer and his mother was a quantum software designer. He attended this school and that school. His older brother died when he was 10....." The reader should be told right at the beginning what John Schmon was notable for, thus: "John Schmon (1801–1998) was a Nevian mathematician who discovered this theorem and that theorem and made fundamental contributions to the understanding of whatever."

Use lower case initial letters in article titles and section headings except where there is a reason to use a capital (e.g. a person's name). The first letter of a section heading is capital except in rare instances where there's a reason to use lower case. Thus a section may be called

Applications of the theorem to population genetics

but should NOT be called

Applications of the Theorem To Population Genetics

If you create a new article, it's a good idea to create redirect pages from alternative names, commonplace misspelling and miscapitalizations, commonplace misnomers, etc. Thus "Schmon's Theorem" (with a capital "T") might redirect to "Schmon's theorem" (the proper title).

One of the easiest mistakes to make in creating a new article is to leave it as an "orphan", i.e. an article to which no other articles link. If you go to the "toolbox" menu and choose "what links here" you can see which other articles link to it. In the relatively early days of Wikipedia (in 2002 or 2003), I created a new article titled "exponential growth". I then used Google to find more than 150 occurrences of "exponential growth" or "grows exponentially" or "growing exponentially" in other Wikipedia articles, and linked them to the new article. Much more recently, in 2011, I found that 60 Wikipedia articles mentioned Gerald J. Toomer, in most cases by citing one of his works, but there was no article about him. I created the links to the new article about him BEFORE creating the article about him. These are "red links": links to an article that does not exist, and are red in color, whereas links to existing articles are blue. One should create appropriate red links in anticipation of the future existence of an article even when one intends never to create it oneself. A red link invites others to click on it and then create the new article.


I've put whatever came to mind instantly into these comments. I could say a lot more if I took more time.

Most of what appears above is codified in Wikipedia's style manuals and guidance pages.

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I just learned a lot about Wikipedia. Makes me wonder if there should be "Wikipedia for mathematicians" somewhere... but maybe it already exists? –  François G. Dorais May 30 '13 at 22:40
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I've thought of writing a "Wikipedia for mathematicians" article and submitting it to the Monthly. Partly it would be a how-to, and partly it would acquaint mathematicians with some of what's there. One probably unprecedented thing is this: en.wikipedia.org/wiki/Lists_of_mathematics_topics Notice that it says "Lists". Plural. It's a list of lists. And going down it, you'll be amazed. Some of the lists are nothing new, and others are themselves probably unprecedented. For example, the "List of exponential topics" or the "List of factorial and binomial topics". –  Michael Hardy May 31 '13 at 17:03
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That's not a bad idea for an article, but I don't think the Monthly would be the best place for it. I would try something like the Notices first. –  Pete L. Clark Jun 27 '13 at 6:13

There are also mathematical models of why the number of editors is decreasing that have nothing to do with a decreasing opinion of Wikipedia or a migration elsewhere. In the beginning, Wikipedia was tabula rasa, and editors could add sections on anything. Addition, multiplication, trigonometry, basic set theory, etc. all needed something, even if it was a very trivial entry. The level of expertise required to contribute was very low. Over time, the articles have indeed improved, and the information content has clearly grown. Those areas with little barrier to entry have been filled out quite extensively.

So, I expect that the data presented in the question is not something to be overly alarmed about. There is much of the trend that should be expected. As the requirements for constructively contributing increase, fewer individuals will be able to add useful information or correct improper or awkwardly worded entries.

I think this is true throughout Wikipedia. The type of editing that is easy these days are more focused on current events. Other data will natural decrease to some convergence and only be edited as cultural changes make different expositions better, a very slow process. Different data would be needed to separate this phenomenon from the other causes for edit rate changes.

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I don't find this at all; the Wikipedia articles I look at daily (many of them in mathematics and technical subjects) could nearly all do with at least copy-editing or improvement of references. It takes about as long to fix a typo as it does to glance at the new mail in one's mailbox, and about as long to correct a sloppy reference as it does to read (but not contribute to) a soft question on MO with a dozen answers. –  András Salamon May 22 '13 at 21:34
    
@András: Notice, I did not say that they could not be improved. That was not my point at all. I said that the barrier to improvement increases with every improvement. Do you think a high-school student could improve the entry on K-Theory? I think there are a number of high-school students who could have contributed originally to getting the Trigonometry entries edited. In the beginning, there were no entries on Trigonometry. Do you see how this can cause contribution rates to decrease over time, unrelated to anything to be worried about? –  ex0du5 May 23 '13 at 23:57

I don't have the time to regularly edit Wikipedia. I've made time for MO lately because asking and answering these questions positively affects my research. Based on my experience writing lecture notes, I find that organizing and presenting math that I already understand really well takes a lot of time compared to the benefit I get from the clearer understanding of the material. For the more specialized math I personally care about, I take the time to write notes on my research blog (like MO, this positively affects my research).

Having said that, if I find an error in the math posted on Wikipedia, I will certainly make an edit for the sake of the community. But this hasn't happened yet. (!)

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Some articles have contributors who are very possessive. It is amazing and impressive that the Wikipedia model works so well. I find it a great first pass on a variety of topics. Often I feel no need to look further. Other times I do. The price of that is that contributing is not worth it unless one is willing to watch the page and decide when to engage and when to let it go.

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One difference between MO and wikipedia is that here you get very quickly answers or remarks on what you have written, and it is then very much more attractive. Going to wikipedia for finding something for oneself, you directly have the answer you were looking for (or dont find it), but really writing new stuff takes times and you don't really obtain a feedback.

I do not say this is a good reason, I just say that it is a reason why people go more often to MO than edit Wikipedia's pages.

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I'd rather say the main difference is that they treat completely different aspects of mathematics. Here the attention is focused on research level mathematics and more elementary or institutional mathematics is often refused, whereas wikipedia mainly treats institutional and encyclopaedic topics, and refuses research topics (en.wikipedia.org/wiki/Wikipedia:No_original_research). I guess this partly explains what you have noticed. –  Pietro Majer Sep 15 '13 at 9:47
    
You are of course right that the level is not the same and that it changes the attractivity. There are also pages on Wikipedia which have however good level in math. –  Jérémy Blanc Sep 15 '13 at 20:02

I will expand on one of Aaron Meyerowitz's remarks.

I am somewhat possessive about what I have written. Indeed one of the hurdles I had to overcome to participate on MathOverflow was to accept that what I submitted could be changed by many other people. I was upset when it happened, and the major thing that kept me from voicing that upset was the FAQ, which said other people could do that. However, most of the edits were sensitive to the thrust of the post, and turned out to be more changes to style than serious changes to content. I have grown to be more comfortable with posting answers on this forum, as my input has mainly been given due respect.

If I were a contributing author to Wikipedia, I would have to overcome a similar hurdle, especially as there would be no consensus as to what is "the" information to present. The posting would not be "mine" anymore.

If Wikipedia had a mechanism for including outside references in a useful fashion, I might be more inclined to offer material for its use. My impression of the current system is that there is a "References" section in the article which includes hyperlinks to other material. My suggestion would be to enhance this so that the link expands into essentially two documents: my version of the article, which I agree to allow Wikipedia to keep a local unedited copy and display repeatedly for their benefit, and a meta document which I agree can be modified in tandem by me and Wikipedia editors whose main purpose is to explain discrepancies, notation changes, and other elements of context to allow the reader to transition between the Wikipedia intro and my version. Wikipedia could then use (or not) my version of the article, the content of which I have control, while maintaing editorial control over their version. Even if I decided that my version was no longer appropriate, I could only petition for its removal, as I had granted Wikipedia the right to use a copy of the unedited version in perpetuity, and I have access to the meta document to say that I think a better version is available elsewhere.

It seems a little more complicated then just providing a hyperlink, but it has the advantage that it could be maintained by Wikipedia, the situation between editor and author is clearly defined and separated further, and the meta document has the flexibility to handle most of the situations that arise. Also, this kind of mechanism would accommodate my sense of possessiveness, and allow me to write things which I could use for my own purposes as well as allow Wikipedia to enhance their collection.

Gerhard "Will Write For Venti Mochas" Paseman, 2013.05.22

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Yes, to contribute to Wikipedia, you have to be comfortable accepting the conditions of Wikipedia's copyright license, which basically implies you are fine with other people mercilessly changing your work. Of course, if what you write is so good that it's hard to improve upon, then it's more likely to stay intact! :-) –  Mark M May 22 '13 at 19:48
    
There is positive part in this, you can improve your writing skills quite a bit. (If it would be more qualified contributors, it would be even better.) –  Anton Petrunin May 26 '13 at 14:02
    
When I create a new Wikipedia article (and I've created hundreds of them), I may feel somewhat disappointed if it doesn't get further edited by others. In recent years, the first thing I do is go to the mathematics WikiProject's talk page and tell them the new article is there and suggest that they work on it. –  Michael Hardy May 30 '13 at 5:26

Replying to Dick Palais: Any edit to Wikipedia no matter how small is good, even if it's just correcting a typo. One thing I've done is to make a small edit to every page I visit. There is always something to be fixed. Once you've done a number of small edits you can do something more substantial. The mathematics reference desk is quite similar to MathOverflow and could be a good place to get started: http://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Mathematics.

Replying to Marc Palm: It is quite easy to see who made edits to a page, just click on the history tab. A user's contributions can be seen by going to a user's page and selecting user contributions on the left.

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I couldn't resist the temptation. –  Douglas Zare May 22 '13 at 22:13
    
@Douglas Zare: Good editing! However, there is still room for small improvements in the punctuation (which I don't have the rep points to do). (1) In line 1, commas are needed after "Wikipedia" and "small", with the comma after "good" preferably deleted. (2) In line 3, a comma should follow "edits". (3) The comma splice in line 6 needs attention, perhaps by replacing "click" with "by clicking". –  John Bentin May 23 '13 at 13:53
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@John Bentin: You should have more than enough reputation points to edit it. You need $100$ to edit a community wiki post. Some commas are a matter of style, not necessity. I think it you add two commas to the first line the sentence has to be reworked. –  Douglas Zare May 23 '13 at 22:03

+1 for question. Hope it will stimulate us to contribute more, it would be great. I feel great respect to those who contribute to Wikipedia. My experience about Wikipedia and its quality is highly positive, probably one of the reasons, is that many many articles are written by one of the Fields Medalists.

Personally I have contributed some material to Wikipedia, the main reason that my contribution is not so big - is lack of time. I feel somewhat ashamed about it, and some excuse which I see for myself - is that I am currently in industry and cannot enjoy academic freedom.

I would also be second on Jérémy Blanc's words: "One difference between MO and wikipedia is that here you get very quickly answers or remarks on what you have written, and it is then very much more attractive..." Wikipedia pages have view-counters, which somehow indicates how useful is what you have done, but it is not clear what exactly is counted ? I guess some views come from robots or other automatic engines. The page about Capelli identity (which is mostly written by me) is visited about 5 times per day according to http://stats.grok.se/en/latest/Capelli%27s_identity Are there really so many people who know what is it ? :))) Although, it is not very important for me, but just having more feedback would be nicer.

Let me also mention that when I was in academia, I did not see big use of Wikipedia for me - when you are deep in one field Wikipedia cannot help you. But when I switched into industry and I often got a new task about things which I know nothing about - then Wikipedia became extremely useful. Moreover I see that for hundreds of engineers Wikipedia is main window to mathematical knowledge.

I am probably not the only person who finds that many math textbooks are quite boring Why do so many textbooks have so much technical detail and so little enlightenment?. In my experience Wikipedia is an exception. So currently my feeling is that If I need to understand some new topic or an idea - I will look to Wikipedia, if it is described there - then it is possible to learn the subj. If something is missing on Wikipedia - no chance to understand an idea in a reasonable amount of time.

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It might have to do with the fact that it is not that easy to track down which persons have contributed to what articles. On MO and any other public lecture notes/ scientific article/blog, it is more directly visible, thus, maybe more attractive.

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Every Wikipedia page has its full date-stamped history available (it is one of the tabs visible on every page), complete with change tracking. In my experience truly anonymous edits tend to be either spam (and quickly reverted) or small fixes by someone who couldn't be bothered to log on; it is therefore usually easy to get a feel for who has made significant contributions. In fact, the sheer volume of Wikipedia change history can be overwhelming. –  András Salamon May 22 '13 at 21:42
    
It's quite easy to track down who did most of the edits, except that many Wikipedians don't divulge their real names. Usually you can exchange messages with them within Wikipedia. (My own user name is my actual name.) –  Michael Hardy May 31 '13 at 19:41

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