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Elementary Mathematics on Wikipedia

By Adrian Riskin 
Department of Mathematics 
Whittier College
Whittier, CA 90608

Ask any college professor about the accuracy of Wikipedia and they will tell you … well, if they’re a mathematician they may actually tell you that it’s quite accurate. I often recommend that my students look up definitions in Wikipedia and I know that many of my colleagues do as well. In fact I look up definitions on Wikipedia myself. If you want to know, e.g., what a Halin graph is, you could do much worse than the linked article. Yes, it’s semi-literate at best, but it’s informative and not wrong. Furthermore, my colleagues in the humanities and the social sciences are so dead set against Wikipedia that I find myself unable to resist teasing them by remarking on how useful I find it in my professional work.

But you know, I’ve also edited Wikipedia, although very rarely mathematics articles, and found the experience to be toxic and soul-killing and the (non-mathematical) articles mostly worse than useless, even the ones I’ve written myself. I unthinkingly assumed that the difference in quality between the technical mathematical articles and, say, the BLP and POV battlegrounds so familiar to Wikipediocracy readers was due to the calm, logical, sociable nature of mathematicians. But recently it occurred to me that (a) I don’t really read the mathematics articles carefully, but rather just skim through them until I find the bit I need, and (b) I never look at the articles on very basic subjects in mathematics, many of which are among the 500 most frequently viewed articles.

I’m too lazy to pick a technical article on an advanced subject in mathematics and read it carefully, but I did take a look at some of the articles on more basic subjects and was both appalled and relieved. I was appalled for the usual reasons; these articles are a hot mess of error, arrogance, obscurity, and nonsense, and they’re the public face of mathematics on Wikipedia. I was relieved because my preconceptions about the general suckiness of Wikipedia were confirmed. And mathematicians have so much social capital that it’s hard for non-mathematicians to criticize their work (obligatory XKCD cartoon). Thus my duty is clear. I must speak out! 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), 

Fail. Who is the audience for this article? It’s not professional mathematicians, it’s kids doing their homework. What is an eighth grader supposed to make of “usually numbers, but not always”? What else can constants be besides numbers? If you don’t already know, you don’t want me to explain. And seriously, anyone who’s reading this article who doesn’t already understand that sometimes constants aren’t numbers does not need to even have the possibility mentioned. It’s not explainable at this level, so shouldn’t be discussed at all. A good mathematics teacher should tell the truth and nothing but the truth, but never ever ever the whole truth.

using only the operations of addition, subtraction, multiplication, and non-negative integer exponents (which are equivalent to several multiplications by the same value).

It’s true that raising variables to positive exponents is equivalent to “several multiplications by the same value” (although this isn’t the case if the exponent is zero, also a non-negative number). But that’s not the reason why such exponents appear in polynomials. It has no explanatory force here and, like many of the non-sequiturs in the article, is probably an ill-conceived bit of showing off.

However, the division by a constant is allowed, because the multiplicative inverse of a non-zero constant is also a constant.

The division of what by a constant, friends? What they seem to mean is that fractions are included in “usually numbers, but not always.” But then why bring it up at all? The number 1/2 is a perfectly good number, and in this context it makes no sense to describe it as “the division by a constant.” And notice the word “allowed.” A legalistic attitude towards mathematics is one of the true hallmarks of the amateur and/or the Aspie. Furthermore, the reason given, about the multiplicative inverse, is just straight-up wrong. Polynomials divided by non-zero constants are still polynomials because of properties of the real numbers. And they say that “division by a constant is allowed” but fail to say that actually only division by a non-zero constant even makes sense. The badness of this sentence is beyond my ability to parse completely.

Skipping ahead, we find: Polynomial comes from the Greek poly, “many” and medieval Latin binomium, “binomial”.[1][2][3] 

And this is not only wrong, as the Oxford English Dictionary shows, but obviously wrong to anyone with a grasp of the Greek and Latin roots involved. “Poly” does mean “many,” but “bi” means “two.” There are also monomials and trinomials. If the word “binomial” has an influence here it’s as a pattern rather than as a root. But there are three citations, it must be true!

One more item:

Apart from the numbers and expressions representing numbers, polynomials are the simplest class of mathematical expressions. 

There is a great deal of idiocy in this sentence. No one who knows them thinks that numbers are simple. Polynomials are simpler in many ways than bare numbers. For instance, polynomials up to degree three were well understood in many ways as many as 3500 years ago. Numbers not so much. In other contexts polynomials and numbers can be considered as examples of the same thing (elements of algebraic structures, if you care). In fact, numbers can be defined as kinds of polynomials and vice versa. The point here again is that what’s really meant is that one learns about polynomials directly after arithmetic in the junior high school curriculum. This has nothing to do with some imaginary objective ranking of “mathematical expressions,” whatever that means, by complexity. There really is no such thing except in the little minds of undergraduates in the throes of a-little-learning-is-a-dangerous-thing-itis.


And look, the same kind of promotional content that one finds in articles about trivial tech start-ups or small-town politicians can be found here too. In the section of the article entitled “Solving polynomial equations” we find a reasonable enough discussion of the history, leading up to its 19th-century culmination with the work of Galois, and then we find:

It has been shown by Richard Birkeland and Karl Meyr that the roots of any polynomial may be expressed in terms of multivariate hypergeometric functions. Ferdinand von Lindemann and Hiroshi Umemura showed that the roots may also be expressed in terms of Siegel modular functions, generalizations of the theta functions that appear in the theory of elliptic functions. These characterisations of the roots of arbitrary polynomials are generalisations of the methods previously discovered to solve the quintic equation.[citation needed] 

How shall I explain the mismatch in importance here? It’s like reading the article on Los Angeles only to find as much space devoted to some random strip mall as to the Pueblo de Los Angeles. It was almost certainly inserted by the (red-linked) authors of the paper or their students, none of whom are fit to tie up the straps on Galois’s sandals. The eighth grader doesn’t need to know this and can’t understand it anyway. The professional mathematician who doesn’t study polynomials doesn’t need to know it and probably can’t understand it either. No one needs to know it but people who already know it. Why is it in this article?

So what shall we make of the quality of the articles about advanced mathematics on Wikipedia? The fact is that those articles are written by specialists for specialists. They have no place in a general-purpose encyclopedia. Wikipedia continues to be a convenient place for mathematicians to host them, but they are essentially the scientific equivalent of Pokemon articles. Polished and perfect, sure, but meaningless in the context of general knowledge. You have to turn to the short, sweet, accurate, intelligent, tasteful Britannica article on polynomials if you actually want to know what they are.


Image credit:  Flickr/3:19, licensed under Creative Commons Attribution 2.0 Generic

17 comments to Elementary Mathematics on Wikipedia

  • Bielle

    “Yes, it’s semi-literate at best, but it’s informative and not wrong.” I’ve used a similar comment on a student paper. It should be embarrassing about an encyclopedia.

  • Andrew

    It is true that the Britannica article on polynomials is accurate and admirably succinct (although hard to read with that video ad going on beside it), and that the Wikipedia article is of low quality. But there is no malign force at work keeping it from improving – just not enough expert editors. So why don’t you improve it instead of complaining or working on non-mathematical articles?

    • Thomas

      > So why don’t you improve it instead of complaining
      See the bit about editing articles being “toxic and soul-killing”. I am not an expert in many areas but I can correct quite a few errors in those related to my industry. Most of them are immediately reverted because I did not include a citation with my edit. I have even had one where I pointed out in the comment that the content of the article was incorrectly copied and the cited reference states the reverse, so I have corrected it. That was apparently not the way to get things done either and I was reprimanded by a couple of users. I now don’t both to correct errors any more. It sounds like lots of other minor experts experience the same treatment and give up too.

  • Michael Hardy

    Prof. Riskin: Could you be more specific about what you consider wrong with the following passage that you seem to say is “semi-literate at best”?

    QUOTE In graph theory, a Halin graph is a planar graph constructed from a plane embedding of a tree with at least four vertices and with no vertices of degree 2, by connecting all the leaves of the tree (the vertices of degree 1) with a cycle that passes around the tree in the natural cyclic order defined by the embedding of the tree.[1] Halin graphs are named after German mathematician Rudolf Halin, who studied them in 1971,[2] but the cubic Halin graphs had already been studied over a century earlier by Kirkman.[3] END OF QUOTE

  • Andrew

    My HTML skills are a little rusty, so I’ll have to try again: Compare the latest version. They should have a preview option on this site.

  • Thank you for your article. These are very important points that you make. I too have tried to participate in the en wikipedia. (i mainly write math articles for the mk wikipedia where i live.) the collaboration has been mixed at best. I was particularly pleased with the collaboration on e.g. slope and congruence (geometry). Luckily, these topics do not have a higher math definition and so all the persons working on them were looking to create truly useful and readable material. It was indeed beautiful team work. However, any title that has a “higher math definition” (such as linear functions and or linear equations) are guarded by what i term (when i am in a good mood) the higher math police. (i have a ph.d. in theoretical maths and have been working in higher math education for over 35 years.) Any chance of making a useful article here is completely lost. And the problem is not just that these articles are useless for english readers. It is also that they serve as models in the wikipedia translation programs in many other countries. It is indeed a pity.

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  • Martin

    I don’t know, what “short, sweet, accurate, intelligent, tasteful Britannica article on polynomials” Mr Riskin is talking about, but I consider the article currently accessible under the given link terrible and certainly not on par with the quality of the Wikipedia article. To the mentioned “eighth graders” it must be completely incomprehensible, and for any more advanced readers there is not enough substance. On the Wikipedia, that would be called a “Stub”.

    About the Wikipedia article, I agree with only a small portion of the criticism presented.

    I think that several explanations in the Wikipedia article that Mr Riskin declared redundant or misleading, are actually very helpful, especially the part where the exponents are justified. By skipping the definition of exponentiation as iterative multiplication, the polynomials lose their connection to algebras that permit only addition, subtraction, multiplication, which is an essential factor in the importance of polynomials.

    Also, Mr Riskin seems to have misread a perfectly correct sentence. The Wikipedia article never mentioned that “numbers are simple”. It is instead talking about different “classes of mathematical expressions”, and expressions that consist of only a single number are indeed very simple. They are the base cases of the recursive definition of expressions and as such the simplest cases by definition.

    So while there are a indeed some problems that we should criticize Wikipedia for, this pamphlet doesn’t help anyone.

  • Martin

    “But there are three citations, it must be true!”

    Funny comic about that problem: https://xkcd.com/906/

    It’s indeed a problem, unfortunately not only on the Wikipedia. I’ve seen the same effect a lot in science.

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  • Oleg Alexandrov

    I would like to note that the Wikipedia style is that the introduction of any article is meant to be as informal non-technical as possible. If you are looking there for rigorous definitions, it is the wrong place. Please look in the right section below.

    I will argue that the Wikipedia article on polynomials is vastly more detailed, accessible, and informative than the little blurb you see on Britannica.

    I very much disagree with your assertion that articles must be written only by specialists for specialists. The Wikipedia article on polynomials starts with a gentle introduction and does a great job at easing the reader into the concept, and gradually gets deeper, with pointers to very complicated matters about polynomials later on.

    The statement “Apart from the numbers and expressions representing numbers, polynomials are the simplest class of mathematical expressions.” is perfect at comparing polynomials and numbers. Yes, we, professional mathematicians (I have a PhD in math) know that the real numbers are built as equivalence classes of Cauchy sequences of rational numbers who are in turn the smallest field containing the Euclidean domain of integers, all the way to the Peano arithmetic, but these facts are useless to a person who wants to learn, who is the ultimate audience of this article.

    Lastly, your choice of words, like “idiocy” is simply unwise. A nuanced and more balanced review, appreciating the tremendous wealth of information and effort put into this article while noting some perceived deficiencies would have been better than full throated insults.

  • Oleg Alexandrov

    I probably mis-represented your comment about “specialists for specialists”, so you can gloss over that.

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