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I know that Wikipedia gets a bad rap, and it seems like some teachers of mine have nothing better to do in class than harp on about the Great Academic Pastime of calling Wikipedia untrustworthy, but let's face it - Wikipedia is probably the single best resource on the internet for getting quick introductions/reviews of new mathematical ideas. I have used it extensively, and I have seen links to its pages provided in hundreds of the questions on this site.

My main concern is: is Wikipedia really unreliable for mathematics? I realize that this may be true in general, but it doesn't seem like something mathematical could be posted "incorrectly" on the site insofar as mathematics is basically true (in an objective sense). I put quite a bit of trust in what I read on that site, and I assume that there will be no falsehoods - is this a justified presumption?

Please note that I personally love Wikipedia. I'm expecting the answer to be "yes" but I just wanted to make sure.

user127096
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user140943
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    Wikipedia owns. It has some mistakes and some badly written things, but it's not better in books. – Git Gud Apr 07 '14 at 20:57
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    I believe that Wikipedia's mathematics articles are substantially more reliable than its non-mathematics articles. The articles to watch out for are the ones about contentious issues or where someone stands to gain by a favorable report. Mathematics articles are not contentious. – MJD Apr 07 '14 at 21:01
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    The issue here is only tenuously connected to mathematics. Some of the factors that make Math.SE a great resource are also working in Wikipedia's favor. – hardmath Apr 07 '14 at 21:03
  • My impression is that Wikipedia is very reliable for mathematical topics. On the other hand, for most topics it seems too much of a smorgasbord to me, both with regard to depth and with regard to the intended audience, to be all that useful for self study, and I find it curious that so many people seem to use it for this purpose (or so it seems from the questions I often see here). – Dave L. Renfro Apr 07 '14 at 21:05
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    Wikipedia is great for learning, but is not a primary source, and so should not be referenced by primary sources in science, in order to avoid mutual confirmation bias (e.g. wiki references paper which references back to wiki) - this is the (crucial and valid) obstacle that professors across all scientific subjects try to warn their students about (but also frequently miscommunicate as "wiki=bad"). In maths, however, there is no risk of creating this sort of self-referential loop - a proof is only a proof if it's valid, and it can't come from nowhere. – Joshua Pepper Apr 07 '14 at 21:06
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    There was a recent study that found encyclopedia britannica to have more mistakes in scientific areas than wikipedia. – recursive recursion Apr 07 '14 at 21:09
  • @recursiverecursion lol that's funny; nice comment. (ps you don't have a link to the study by any chance?) – user140943 Apr 07 '14 at 21:10
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    @user140943: http://en.wikipedia.org/wiki/Reliability_of_Wikipedia, and less ironically, http://news.cnet.com/2100-1038_3-5997332.html and http://www.pcworld.com/article/251796/has_wikipedia_beat_britannica_in_the_encyclopedia_battle_.html – Mooing Duck Apr 07 '14 at 22:09
  • @MooingDuck Thank you, those are interesting articles. – user140943 Apr 07 '14 at 22:15
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    I like to look up mathematics on wikipedia (and mathworld), but I usually either know the result or can easily check the correctness. And contrary to a book, I can correct all the errors forever. – Phira Apr 07 '14 at 23:49
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    But there are many errors in formulas, and even worse, pages for typical first/second year courses are often correct, but very bad. (No examples, obfuscating language, etc.) Beginners' courses have to be structured very well and wikipedia cannot do that. – Phira Apr 07 '14 at 23:51
  • @Phira Can you give some examples of errors? – user140943 Apr 07 '14 at 23:54
  • @user140943 I cannot point to current errors, because I would fix them. But for example, I remember that the formula in Stewart's theorem was wrong at some time. – Phira Apr 08 '14 at 00:12
  • Speaking of links from this site: there are more than 40000 Math.SE posts with links to Wikipedia. And more than 20000 comments. – user127096 Apr 08 '14 at 03:31
  • This answer by Pete L. Clark (to another question related to Wikipedia) might be interesting in this context. – Martin Sleziak Apr 08 '14 at 05:11
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    @JoshuaPepper Unfortunately, I fear that you are giving many professors too much credit: the issue is often not one of miscommunication, but of personal bias. It's not merely that they miscommunicate and students get the impression that "wiki=bad," it's that many professors actually believe "wiki=bad." Which is a shame, because while what you say about Wikipedia not being primary is absolutely true and one should always dig deeper into primary sources for serious research, a well-written Wikipedia article is simultaneously an excellent introduction and an index of relevant primary sources. – KRyan Apr 08 '14 at 13:57
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    they might be generally reliable but are they understandable? unf a case can be made that style is challenging in collaboratively edited documents, and imho many wikipedia math articles often tend to be difficult to follow compared to similar book accts based on minimal exposition and heavy technical/formal emphasis.... & feel they tend not to define math terms well before using them etc.... many misc/prominent issues related to comprehensibility! similar complaints on misc net forums eg reddit – vzn Apr 08 '14 at 18:47
  • see also elementary mathematics on wikipedia by prof Riskin with similar biting criticism – vzn Apr 08 '14 at 18:56
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    I agree with @vzn. In my experience, most articles are either written/edited by non-mathematicians and plain wrong, or else written by mathematicians using the maximum possible generality, and therefore incomprehensible. – Flounderer Apr 08 '14 at 23:01
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    @KRyan: wikis and internet education resources in general are in competition with paid universities professors. As they become good enough there will be less demand for high priced schools and downward pressure on professor jobs. Don't expect someone to talk up their competition... see the book "The Nearly Free University and the Emerging Economy" for more on this topic http://www.oftwominds.com/CHS-books.html – Michaela Light Apr 09 '14 at 05:34
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    If you can read other languages, it may be useful to check what they say. In my experience, Spanish wikipedia is less reliable that the English one (and way less complete), but if they both agree, you are most probably on a good thing. – Davidmh Apr 09 '14 at 08:46
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    I do a patrolling on Wiki and I'm used to roll-back plenty personal insults, offensive or stupid comments. It's pretty common that somebody writes to the middle of an article a word "penis". As long as the math article doesn't contain word "penis", it seem to be treated by administrators, which means, that it should not be completely stupid. – V-X Apr 09 '14 at 09:02
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    Wikipedia has mistakes, books have mistakes. In my experience the bigger issue with using it for learning is that the articles are not easy to understand and often contain inconsistencies (in the order of ideas being presented, in notations, etc.). What it's good for is looking up stuff you already know, or finding references to other sources. But I find it unsuitable for learning about new concepts. Get a good book for that instead, or at least look at a site which is not collaboratively edited. – Szabolcs Apr 10 '14 at 23:04
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    A free-for-all-to-edit article can never be really good at explaining stuff well. That's not the same thing as being correct. To contribute to a correct article, you only need to understand the topic. To contribute to an educational article you need to be very familiar with the article itself and the "storyline" used for explaining. If you think that WP is an excellent resource for learning, then I recommend you take a look at some good books and you'll be amazed ;-) – Szabolcs Apr 10 '14 at 23:13
  • @Szabolcs, absolutely, and in addition the typesetting is awful, there are so many great math books out there: I find a bunch of them are posted by their authors for free, and if you're lucky to have access to electronic sources, nowadays you can download a bunch of properly written books! – PatrickT Apr 12 '14 at 15:28
  • @PatrickT Actually if you register you user, and log in, then you can set math rendering to use MathJax, it's much better quality. :-) I use this all the time because I tend to magnify the text on webpages. – Szabolcs Apr 13 '14 at 00:22
  • Here is a good example of an article with an almost comical amount of useless information: http://en.wikipedia.org/wiki/Beta_distribution – Flounderer May 03 '14 at 00:27
  • Reliable? Perhaps. Well-written, readable and comprehensive? Less so... – user1729 May 23 '14 at 11:46
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    That 2005 study that supposedly found Wikipedia more reliable than Britannica was seriously flawed, and Britannica was rightly furious that it got so much press. The authors handpicked which articles to compare. Sometimes they used the real Britannica article; sometimes they used article from the young student edition. In at least one case they saw that the Wikipedia article was much shorter than Britannica's, cut the Britannica article to comparable length, then po-facedly wrote up that the reviewers complained the Britannica coverage was incomplete. It was really a travesty. – MJD May 23 '14 at 16:24
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    Britannica's response to the study. “One Nature reviewer was sent only the 350-word introduction to Encyclopædia Britannica’s 6,000-word article on lipids.” – MJD May 23 '14 at 16:25
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    I was curious whether Nature had a response to the response, and they did, here and here. (I have no contribution to make regarding which side is in the right; I'm just presenting the documents for others who are curious). – dfan May 23 '14 at 21:28
  • This question in Skeptics.SE might be of help – hrkrshnn Jul 12 '14 at 12:50
  • Enjoy your shiny new badge. You've earned it. :) – Ken Jun 14 '15 at 04:18

10 Answers10

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In my personal experience, I've found Wikipedia tremendously useful and reliable both in my studies and in my research. Rarely are there ever mistakes. Anytime you get information, especially from the internet, you should always check with at least one other source, of course. I usually use wikipedia and another source to make sure they agree, but I don't know if I've ever found an error on Wikipedia. It can especially be very useful to get a broad overview of whatever it is you're researching, and can provide some links to some very reliable sources.

Logan Tatham
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    +1 for "provide some links to some very reliable sources." In fact, I would say that this is, in a nutshell, what makes Wikipedia reliable: you can immediately see where information is coming from (if anywhere) and you can get that material yourself. It is simultaneously a verification that this particular article is serious (most articles on Wikipedia are, but things do fall through the cracks on occasion), as well as a fantastic resource for further research. – KRyan Apr 08 '14 at 13:52
  • The way a wiki gives links to terms used in the article so you can look those wiki articles up too makes learning about a new math subject much easier than a book. I still use books and the wiki is fast way to get started. Often when starting a new subject I don't know not just the main topic but the definitions and related terms and topics... – Michaela Light Apr 09 '14 at 05:30
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    I've seen errors (of all kinds, up to "theorems" which fail in obvious corner cases) in peer-reviewed papers published in highly-reputed journals, many years after the publication with no other hints at the mistake... – vonbrand Apr 10 '14 at 10:13
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    @vonbrand That's an entire can of worms of its own, and there are multiple questions across multiple SE sites regarding this. Sometimes the reason is as simple as "modern math is hard, and we trust certain mathematicians a lot." Sometimes it bites us in the butt. – zibadawa timmy Sep 01 '14 at 20:46
  • In fact, I've used wikipedia to correct an error in a textbook I was reading, before :-D (it disagreed with the other sources I had, too) – galois Sep 27 '15 at 09:12
81

Three days ago ---- Friday the 4th --- the speaker at the weekly probability seminar at the University of Minnesota was Larry Gray, who's been doing research in probability theory since the '70s. He began by saying that when he wanted to learn Markov-chain Monte Carlo methods, he began by going to the principal source of information on all things mathematical: Wikipedia.

I've edited Wikipedia math articles every day since 2002 and other articles as well, having done (I think) around 180,000 edits.

Sanath Devalapurkar's answer posted here is probably approximately correct.

Michael Hardy
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There are a few people on Wikipedia who are knowledgeable about their subjects. For example, in mathematics, concepts like Floer homology are probably edited by actual mathematicians. However, popular concepts like numbers are sure to have some rubbish in it (Disclaimer: I haven't read the article on numbers - that was the first topic that came to my mind with "popular"). Therefore, my answer to your question would be as follows: $$\text{Wikipedia is reliable for math?}=\begin{cases} \text{Most probably yes} & \text{if it's $\geq$ third/fourth year undergrad math} \\ \text{Most probably no} & \text{if it's $\leq$ third/fourth year undergrad math} \end{cases}$$ The requests in the comments have led me to finding this out: From this Wikipedia Page,

Adrian Riskin, a mathematician in Whittier College commented that while highly technical articles may be written by mathematicians for mathematicians, the more general maths topics, such as the article on polynomials are written in a very amateurish fashion with a number of obvious mistakes.

For an example, let me quote Riskin's example:

I’m going to take you through the lead section of the Wikipedia article on polynomials and try to explain some of what’s wrong with it.

In mathematics, a polynomial is an expression constructed from variables (also called indeterminates)

Variables are not the same as indeterminates! Even the linked articles acknowledge as much.

and constants (usually numbers, but not always),

Many more examples can be found in the link above.

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    I've actually found very informative articles on rather elementary subjects, I'd be interested in some specific examples. – Logan Tatham Apr 07 '14 at 21:04
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    Yes I second the request for actual examples of rubbish... although I think your point is a good one. – user140943 Apr 07 '14 at 21:05
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    I must say I disagree with this statement. I've never found anything that's just plain wrong on wikipedia, and if you read the article on numbers, it is actually very thorough and informative. – recursive recursion Apr 07 '14 at 21:06
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    @recursiverecursion, user140943, LoganTatham See my cited example above. –  Apr 07 '14 at 21:12
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    @SanathDevalapurkar Wow I didn't know there was a whole website devoted to critiquing Wikipedia! :) However, I'm not sure those are blatant falsehoods, just loose usages of language that in elementary algebra is not that big of a problem. Also, did you read who commented to Riskin's article at the bottom of that page? ... – user140943 Apr 07 '14 at 22:11
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    The critique on the cited example is unfair, and just shows how difficult it is to serve every possible audience. Polynomials can be in terms of variables or or indeterminates (depending on the precise context) and often it does not matter exactly which point of view is taken. The sentence may not be 100% accurate, but there is no obvious way to improve it without losing out on some counts. Riskin himself says a bit further on: "anyone reading this article who doesn’t already understand ... does not need to even have [it] mentioned"; this applies to the variable-indeterminate distinction too. – Marc van Leeuwen Apr 08 '14 at 10:50
  • @MarcvanLeeuwen Note that this answer is my interpretation of Wikipedia. I admit, I have used Wikipedia to give me research ideas. However, I use Wikipedia only for topics that are higher level. I agree with you on what Riskin has said. However, what about those who are math novices? Those who are just perusing the article due to an interest in math would not be able to place a distinct line between what is true and what isn't. That was the basis of my above interpretation on how correct Wikipedia really is. –  Apr 08 '14 at 15:26
  • In my experience wikipedia math articles on all degree level topic are good. I have read hundreds of them and not seen an issue. I did a math degree and masters in math at Cambridge and still actively study math. – Michaela Light Apr 09 '14 at 04:55
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    One interesting thing about Wikipedia is that if you do find an error, you can just go and correct it. – Michael Hampton Apr 11 '14 at 02:28
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    Mathematicians do use the terms "variable" and "indeterminate" pretty interchangeably, I find, as in "polynomial ring in $n$ variables" or "polynomial ring in $n$ indeterminates". – user43208 Apr 27 '14 at 03:15
  • @user43208 Well, variable is used more often to represent something unknown - indeterminate usually gives me an expression like $\dfrac{\infty}{\infty}$, e.t.c. –  Apr 27 '14 at 03:17
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    @SanathDevalapurkar No, that's just one meaning of "indeterminate" (usually occurring as part of the phrase "indeterminate form" used in elementary calculus). And "variable" as used by mathematicians typically doesn't mean "something unknown", but rather is a formal symbol, typically invoked in the description of a free algebraic structure and the like. See for example http://ncatlab.org/nlab/show/polynomial and http://ncatlab.org/nlab/show/variable – user43208 Apr 27 '14 at 03:25
  • Incidentally, it's "etc.", not e.t.c. – user43208 Apr 27 '14 at 03:25
  • @user43208 I know that - I'm just saying that in common math term usage, the immediate impression (on me) is that which I mentioned above. –  Apr 27 '14 at 03:26
  • @user43208 Also, yeah - it should be etc. (However, etc.=et cetera, so shouldn't we technically be writing et.c.? :-P) –  Apr 27 '14 at 03:27
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    So the point is that Riskin's example is perhaps not a good one, since professional mathematicians do use those terms rather interchangeably -- it's not actually "wrong" as he claims. – user43208 Apr 27 '14 at 03:28
  • @user43208 Hmm, yeah, perhaps... –  Apr 27 '14 at 03:29
  • when you say mathematician do you mean "mathematicians" like yourself? –  Jun 04 '14 at 19:12
  • @SanathDevalapurkar: this is a matter of taste, but you are rather an "internet/youtube mathematician", since you apparantely never contributed anything to mathematics (in the sense of research articles in respected journals). –  Jun 06 '14 at 16:29
  • @Sanath I'd give it a little more time before you proclaim to people that you are a mathematician. Even Paul Halmos titled his automathography "I Want to Be a Mathematician" (emphasis mine), and that attitude sounds about right. :-) – user43208 Nov 19 '14 at 00:53
  • i put my ideas in wikipedia :D but i make sure the equations are right, i can not publish my ideas anywhere :( – Jose Garcia Oct 20 '15 at 18:37
29

To give a personal anecdote.

I was thinking about the complexity of a particular algorithm for a particular type of graph (I'm a Computer Scientist rather than a Mathematician).

I pop onto Wikipedia to find that a particular subproblem of the algorithm is equivalent to a known problem in mathematics. I dig a little more and find that this problem is solved with a particularly good time complexity on 'claw-free graphs', I do some more digging and by clicking on a couple more links I find I can convert my input graphs to claw-free graphs in linear time. With half an hour of broadly unrelated Wikipediage I've found (the main steps of) a proof of worst case time complexity for a problem that had already had 15 papers published on it.

So I would say that not only is Wikipedia pretty good for teaching maths, it's quite astonishingly good for researching it. (Disclaimer, this obviously only is the case if you are sensible about checking references and the like, in this search there were several false starts)

Joe
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Outright errors in articles are rare -- usually the problem with math articles is that they are incomplete, confusing, or disorganized, not that they are wrong. But you do run into the occasional howler (at one point the article on orthogonal matrices claimed that they preserved all inner products -- my attempts to correct this were rebuffed at the time but it appears this has finally been fixed) and definitions and notation are far from standardized, so it doesn't hurt to double-check any fact you have some doubts about.

The usefulness of articles varies widely by topic, even within mathematics. Linear algebra, algebra, and number theory articles, generally speaking, are excellent. So are most analysis articles, though sometimes surprisingly basic information (such as a characterization of what functions are integrable) is difficult to find. Topology is a mixed bag, with long lists of properties and relationships between properties, with no citations or proofs, fairly common. Differential geometry articles could use a lot of work -- they are a hodgepodge of conflicting and inconsistent notation, motivation, and exposition, to the point where I fear the articles on differential forms, covariant derivative, connections, jet bundles, etc. would be unintelligible to those not already familiar with the subject. PDEs are covered thoroughly if you happen to be interested in a particular, named PDE -- the articles on general theory leave something to be desired, however.

user7530
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Joshua Pepper wrote:

Wikipedia is great for learning, but is not a primary source, and so should not be referenced by primary sources in science, in order to avoid mutual confirmation bias

Such a statement might be slightly dangerous. At least in Germany, a written thesis has to be prepared as part of the work for a academic degree, and it has to include a declaration that all used sources have been cited. If you used a tertiary source like wikipedia, you better cite it at the appropriate places, see for example the case of Annette Schavan. Also note that many original research math papers list specific "private communications" with specific other mathematicians in the bibliography among other references.

I like to cite wikipedia in my questions and answers, because this nails down the notion I'm talking about, and makes it clear that it is "well known". Also the provided links to other sources are often really valuable.

When it comes to learning something genuinely new, I found Stanford Encyclopedia of Philosophy orders of magnitude better (for the subjects that it covers). The same is also true for in a slightly different sense for nLab, see for example the explanation of the red herring principle:

The mathematical red herring principle is the principle that in mathematics, a “red herring” need not, in general, be either red or a herring.

Frequently, in fact, it is conversely true that all herrings are red herrings. This often leads to mathematicians speaking of “non-red herrings,” and sometimes even to a redefinition of “herring” to include both the red and non-red versions.

With respect to reliability, it's often hard to notice all the minor and major errors. When I tried to apply some of the information I learned from the wikipedia article on the Dedekind-MacNeille Completion, I was surprised to find that the information wasn't as accurate as it seemed to me when I read it without trying to apply it. I would have to check in detail how much of this misinformation is still present in that article today, but my guess is that most misinformation is still present (even if transformed slightly to make it less wrong.)

Community
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Thomas Klimpel
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There is a huge difference between the reliability of Wikipedia articles based on their topic. The worst reliability problems are found at articles dealing with subjective and often debated topics, like politics, religion, and history. In these articles the reliability of Wikipedia can border on horrible, as there are large groups of people with the motivation to "prove" their side of the issue, and in a subjective topic you can create a heavily biased article without lying and without making up facts, just by selecting your sources: a well sourced paragraph using citations from a major news site tends not to be deleted, so one side of an issue can be over-represented. Fallacies like "someone posted about this event on a right-extremist site, so this fact definitely proves that the event did not happen at all" are also common. I've encountered articles about historical events I witnessed personally, which stated the opposite of what I've seen with my own eyes. If a large ideological group has a lot of people editing that article under a heavy influence of confirmation bias, I can't do anything to correct it.

However, science in general, and mathematics in particular, is a completely different story, as the motivations presented above are largely missing, and the topic is objective enough that there are no major conflicting theories which non-scientifically educated people would have a tendency to fight over.

My experience is that articles about science are very reliable, while articles about politics and similar subjective topics tend to be less reliable.

vsz
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I make heavy use of Wikipedia articles (especially on mathematics and related fields).

They give me the impression that are written by people who understand the subject (but may not be professionals in the field) also they have sources and references which one can use to further investigate a subject (sometimes they state alternative formulations, theories, results, which also helps) .

So all in all, Wikipedia (and related *pedias) (although can be "biased" sometimes, either knowingly or unknowingly), may be the best encyclopedia up to now.

UPDATE: imo, using Wikipedia as a source/reference in research has no use, since Wikipedia (and other pedias) explicitly state that they DON'T DO RESEARCH but provide mainstream content, albeit refined and validated as possible (and it is expected from an encyclopedia, especially one written everyday by many people), however using Wikipedia as reference to subjects not directly related to research (e.g. to point another issue or field, or mainstream information) can be done.

Martin Sleziak
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Nikos M.
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My field is computers, I have found and fixed a number of errors in technical articles to do with various computer subjects.

However I find wikipedia to be an excellent starting place for learning about a topic and much of the information presented is outstanding in it's depth and accuracy. My general impression is that the vast majority of information is usually accurate, perhaps with minor errors.

So, it is a mixed bag as far as accuracy goes, but you yourself and everyone else have the opportunity to improve and correct the errors, which is what makes it a self-correcting system that tends to converge toward a position of accuracy. The self-correcting aspect is the fundamental philosophy underlying wikipedia that enables it to be successful.

If it was seriously inaccurate people would cease to find it useful.

codeslinger
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I vehemently disagree with @vsz’s point about “subjective and often debated topics”. Conflicts around an article are apparently a blight, but really a fortune. If the article’s topic is controversial and the article undergoes edit wars, then an experienced Wikipedia surfer can extract information of much better quality from it; but s/he has to browse edit histories and talk pages. For example, and to stay on topic, there was a conflict about the concept of manifold that left no traces in the current version of the article (but can be found by a wiki archeologist).

On the other hand, articles on calm, “trivial” things, such as logical connectives and inference rules, may be incredibly silly because are mainly edited by guys who don’t know anything but stupid Boolean logic and, consequently, have very poor grasp of these concepts. One of these guys even “proved” Modus ponens ☺

You can also learn how Wikipedes once injected an imbecilic redirect in place of important topic of “square” (2), also because nearby articles were edited by guys who either don’t know anything but integer and real numbers or don’t like to think; note that the article was eventually written (after more than 1 year).

Wikipedia articles, in short, have

  • Advantages: many links to nearby concepts, unexpected relations and analogies (because some authors look on the topic “from outside”).

  • Disadvantages: widespread ignorance, absence of incentives to think, and irresponsibility of users.

Incnis Mrsi
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