By Adrian Riskin
Department of Mathematics
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)
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”.
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.
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.