Krdytkyu

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.

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.

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.

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.