# Are mathematical articles on Wikipedia reliable?

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.

• Wikipedia owns. It has some mistakes and some badly written things, but it's not better in books. – Git Gud Apr 7 '14 at 20:57
• 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 7 '14 at 21:01
• 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 7 '14 at 21:03
• 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 7 '14 at 21:06
• @user140943: en.wikipedia.org/wiki/Reliability_of_Wikipedia, and less ironically, news.cnet.com/2100-1038_3-5997332.html and pcworld.com/article/251796/… – Mooing Duck Apr 7 '14 at 22:09

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.

• +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 8 '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 9 '14 at 5:30
• 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
• @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 1 '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 9:12

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.

• Thanks for your commendable efforts. Now get back to the salt mines, answering Math.SE Questions! – hardmath Apr 7 '14 at 23:12
• Sanath Devalapurkar's answer is no longer above yours. – KRyan Apr 8 '14 at 13:53
• Not sure if it's okay to ask, but are you related to G.H. Hardy? – StubbornAtom Aug 16 '16 at 11:05
• @StubbornAtom : Not as far as I know. $\qquad$ – Michael Hardy Aug 16 '16 at 11:23
• Well that's disappointing, but thanks for your time. – StubbornAtom Aug 16 '16 at 11:26

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.

• 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 7 '14 at 21:06
• @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 7 '14 at 22:11
• 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 8 '14 at 10:50
• @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 ncatlab.org/nlab/show/polynomial and ncatlab.org/nlab/show/variable – user43208 Apr 27 '14 at 3:25
• 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 3:28

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)

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.

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.)

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.

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.

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.

• thanks for the edit, but 'i' is on purpose, 'i' is not important – Nikos M. Apr 9 '14 at 13:59

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.