Wikipedia and Math

I am a relatively big fan of Wikipedia. It tends to be my go-to source for things I want to know that are of a general nature - for instance, when I finished Bleak House the other day, I read/skimmed the Wikipedia articles on Charles Dickens, Bleak House, and the Chancery court that is such a big feature of that book.

But for professors in many disciplines, Wikipedia is a kind of sore spot, because students will often try to cite it. Not only is it generally inappropriate to cite an encyclopedia in a college class, Wikipedia is extra-suspect since anyone can edit it, and so it may or may not be rife with errors. (Everything in life is full of errors, really, but at least published encyclopedias have editors.)

But in math, people seem to like Wikipedia a lot. Several of my professors have referred to looking up things in Wikipedia themselves before presenting them in class, or to using it in general.

In fact, a few weeks ago, we had a visitor from the NSA who came to talk about careers there. It came up that of course (for security reasons) they don't have Internet access at their workstations there. I asked the woman how they did math without Wikipedia, and she immediately replied, "Oh, we have our own copy of Wikipedia." She didn't seem to find the question bizarre (like if I'd asked, "Oh, how do you do math without Facebook?")

I think there are some legitimate reasons why Wikipedia is different for math than for other subjects.

First of all, I imagine that when, say, history professors read Wikipedia, they find errors that irritate them. (This is probably true of many encyclopedias as well, but I doubt it comes up much that professors read encyclopedias.) You can make a lot of factual errors in history, or you can simply write an article that is unbalanced - that goes into a lot of detail on one small point and completely fails to include other major points. This is especially likely if the topic is controversial.

In math, on the other hand, there are not so many facts. When you look up a math topic in Wikipedia, you want to answer questions like

• How is this thing defined?
• What areas is it used in?
• What are some theorems about it?
• What are the different notations or ways that it is conceptualized?

You might think of definitions as being akin to facts, but actually they're quite flexible. It's true that some things are simply NOT the definition of an algebraic group, for instance, but there are several different definitions to choose from, and you can do math from a lot of different standpoints or bases.

I commented to Ed the other day that, unlike in other fields, in math it's the facts (definitions and axioms) that are matters of taste or opinion, and the conclusions drawn from those facts (theorems, etc.) that are either right or wrong.

The second reason I think Wikipedia is different for math is that, honestly, it's difficult to abuse it. You can't read and understand a Wikipedia math article unless you actually know enough math that any errors are probably not going to be dangerous to you. Is there a proof that is erroneous? You should be able to tell. (Nobody sophisticated enough to read proofs in Wikipedia should be foolish enough to treat any proof as authoritative.)

So if you were going to write a paper about Hausdorff spaces and you looked up the Wikipedia article and started there, it wouldn't really hurt you any. Either the definition in the article would work for you as a starting point in your research or it wouldn't. Once you know generally what's being discussed, you can make up your own definition if you want (though of course if it's not roughly equivalent to a commonly-used one, you'll only confuse your audience by calling it "Hausdorff"). You don't need a source for a mathematical definition, so you're not likely to mistakenly cite Wikipedia.

So, math Wikipedia - all upside, no drawbacks (if you can read it at all).

As a side note, my ability to read Wikipedia articles in math has absolutely skyrocketed since I started grad school. I'm actually starting to get enough background in the various general areas of mathematics for these things to make sense.