The Unapologetic Mathematician

Mathematics for the interested outsider

Deconstructing XKCD

Okay, evidently I need to flex my Critical Theory muscles, atrophied from years of disuse, and bring them to bear on yesterday’s offhand remarks.

To recall, we’re looking at the XKCD comic from Monday, February 18. The title is given as “How It Works”. This is where the ambiguity begins. The phrase “how it works” can either indicate either an observational or a normative description. Observationally, we might catalogue the operations of a certain system. Here, those operations are the ways in which the system works — “how it works” as an entity isolated from the reader. Normatively, we might take our knowledge of a system and give instruction on proper interactions with the system. Such instructions tell how to achieve such results as the author finds worthy — “how it works” to the benefit of the reader, as seen by the author.

The ambiguity is important here because of the different connotations of the two readings. The observational reading is emotionally neutral with respect to its content. The system simply is, and the author renders an image of the system as it is, with no inherent judgement or commentary. On the other hand, the normative statement is inherently an endorsement. The author instructs the reader to interact thusly.

It should be noted here that with slight modifications, the observational mode can be turned to a critical mode. For example, instead of merely describing the human condition as he saw it, Nietzsche entitled his book, Menschliches, Allzumenschliches. In doing so, he explained that he was describing what it was to be “human-like”, but emphasized his disapproval by picking it out as “all too human-like” — something to be escaped rather than merely documented, let alone embraced.

Now, as to the content itself. The comic compares two nearly-identical situations. In each case, two people stand at a chalkboard. In each, the person on the right has just finished writing out the formula

\displaystyle\int x^2=\pi

on the board. In each, the person on the left comments in response. I will refer to the person on the right as the “Writer”, and the person on the left as the “Speaker”.

First, let’s dismiss the details of the mathematical fact. Others have pointed out various flaws. The alt text for the image correctly lists one such possibility — the writers have omitted constants of integration. Another problem is that each image omits the “dx” from the integral. However, these details are actually immaterial to the setup. The expression is not meant as mathematics itself, but as an icon representing mathematics. That is, it acts as a symbol meaning “mathematical work containing a glaring error”. However, the presence of an integral sign picks out the level of the signified work: basic calculus.

In each situation, the Speaker is drawn identically, and generically. The identity is clearly meant to suggest that the two speakers are actually the same person, reacting to two slightly different situations. The difference is all in the Writer.

The author’s style is for “stick figures” with a minimum of recognizable features. However, there are a few standards to his iconography. Most important here is that almost all of the characters are bald, with the exception of a female character. These look identical to the male characters, except inasmuch as they have hair.

The difference in the situations is clearly that in the first, the Writer is male, while in the second the Writer is female. And thus the Speaker’s different reaction is solely a result of the difference in the Writer’s sex. In the first situation, the Speaker says, “Wow, you suck at math.” In the second, he says, “Wow, girls suck at math.”

So we have a significant error in calculus-level mathematics. Nothing about the Writer suggests to the Speaker that this is a one-time error by a normally-competent person. The reaction is not “that’s a mistake”, but “you suck” in both situations, indicating the glaring nature of the error. The Writer, it may be assumed, actually is bad at mathematics.

But then why is someone bad at math at the board anyhow. People with little mathematical skill don’t seek out public fora like chalkboards without provocation. If they must do calculus, it will be hidden on paper, so the numerous false starts and errors can be easily swept under the rug. This identifies the Writer as a student, and the Speaker as someone with enough sway over the Writer to force an appearance at the chalkboard — likely an instructor. Since the Writer is a student with mediocre mathematical ability, it is unlikely that the instruction is taking place in a high school setting. Far more likely, this is at a college, where calculus is often a general requirement.

But despite cultural assumptions, college calculus instructors generally don’t hold their individual students in contempt. We complain about students en masse, but each individual student is to be helped to understand the material. Yes, some instructors don’t fit this mold, but if we are to adopt the observational mode with respect to this comic, we must understand it as speaking generically. There are no identifying features about the Writer or the Speaker (other than the Writer’s sex), and so we cannot understand either of them to be established characters. They are generic placeholders, filling roles to be determined (as we have above) from the context.

And so each situation — with a male Writer or a female — rings false when interpreted generically, as an observation. And yet our prior knowledge of the author tells us that he can’t be meaning this normatively. We are left unsatisfied, with an awkward, ill-contextualized comic. However, if we did not have prior knowledge of the author (as many readers may not) then the awkward contextualization provides reason to read the comic normatively. Either way, the work surely fails to achieve its goals.

February 20, 2008 Posted by | rants | 36 Comments