6/20/2023 17:16

Mathematical Writing Failures

I get frustrated trying to read anything any mathematician has ever written about anything ever. For example, in Wikipedia it was probably some grad student who wrote this:

 • In hyperbolic geometry, the circumference of a circle of radius r is greater than 2πr.

A natural thought would be, that is interesting; tell me more. But the next words out of this contributor's keyboard are:

 • Let R = 1 - (1 / Sqrt (-K)), where K is the Gaussian curvature of the plane. In hyperbolic geometry, K is negative.

If you are a mathematician and you see an expression like (1 - 1/c), you are trained to think, "Ah, the remaining fraction after 1/c is taken away!" And if you see Sqrt (-K) you suspend your analysis while you wonder, "Is this an imaginary value or is K negative?" Normal people do not react that way.

Normal people see an equation and think it is an algorithm: Do this.

In fact, if you see an undefined expression K and are neither troubled nor flustered it shows you are at least partly mathematicianistic. Any ordinary person would stop at K and say, "I do not know what K is. I think you are providing this material out of order." Which is true!

The article on hyperbolic geometry is written with a presumption of knowledge of Gaussian curvature -- which is absurd approaching the verge of being circular reasoning. Gaussian curvature is not used to define the geometry so there is no reason to presume one reader must know the concept in order to learn about Lobachevsky geometry. The definition is based solely on varying just one of Euclid's postulates. So all that should be presumed is a basic familiarity with Euclid's postulates. That is rare enough in the modern generation; it would be silly to presume more.

If you did need curvature the author really should have written, "That is an interesting fact, and I can quickly show you how it is proven. But first I will need to tell you about Gaussian curvature." Saying that immediately raises the question, "Should we stray from the primary topic to discuss this related but fundamentally extraneous concept right here in the middle of an introduction to hyperbolic geometry?" Probably not. But in typical mathematical writing it is uncommon for the question to even be raised.

In this context I suggest comparing mathematical style, such as it is, with Gogol's famous sentence in "The Overcoat":

 • Even at those hours when the gray Petersburg sky is completely overcast and the whole population of clerks have dined and eaten their fill, each as best he can, according to the salary he receives and his personal tastes; when they are all resting after the scratching of pens and bustle of the office, their own necessary work and other people's, and all the tasks that an overzealous man voluntarily sets himself even beyond what is necessary; when the clerks are hastening to devote what is left of their time to pleasure; some more enterprising are flying to the theater, others to the street to spend their leisure staring at women's hats, some to spend the evening paying compliments to some attractive girl, the star of a little official circle, while some--and this is the most frequent of all--go simply to a fellow clerk's apartment on the third or fourth story, two little rooms with a hall or a kitchen, with some pretensions to style, with a lamp or some such article that has cost many sacrifices of dinners and excursions--at the time when all the clerks are scattered about the apartments of their friends, playing a stormy game of whist, sipping tea out of glasses, eating cheap biscuits, sucking in smoke from long pipes, telling, as the cards are dealt, some scandal that has floated down from higher circles, a pleasure which the Russian do never by any possibility deny himself, or, when there is nothing better to talk about, repeating the everlasting anecdote of the commanding officer who was told that the tail had been cut off the horse on the Falconet monument--in short, even when everyone, was eagerly seeking entertainment, Akaky Akakievich did not indulge in any amusement.

Or almost any page of William Faulkner. But Faulkner, and probably Gogol, had a fine sense of when to leave the reader spinning in a page and a half of sentence and when it is better to stop and write, "He went to the door." I do not see that in math articles.

 • https://en.wikipedia.org/wiki/Lobachevsky_geometry
 • https://en.wikipedia.org/wiki/Periodic_sentence