Lewis Carroll and my Playing With Universes

I wanted to explain what’s going on that my little toy universes with three kinds of elements changing to one another keep settling down to steady and unchanging distributions of stuff. I can’t figure a way to do that other than to introduce some actual mathematics notation, and I’m aware that people often find that sort of thing off-putting, or terrifying, or at the very least unnerving.

There’s fair reason to: the entire point of notation is to write down a lot of information in a way that’s compact or easy to manipulate. Using it at all assumes that the writer, and the reader, are familiar with enough of the background that they don’t have to have it explained at each reference. To someone who isn’t familiar with the topic, then, the notation looks like symbols written down without context and without explanation. It’s much like wandering into an Internet forum where all the local acronyms are unfamiliar, the in-jokes are heavy on the ground, and for some reason nobody actually spells out Dave Barry’s name in full.

Let me start by looking at the descriptions of my toy universe: it’s made up of a certain amount of hydrogen, a certain amount of iron, and a certain amount of uranium. Since I’m not trying to describe, like, where these elements are or how they assemble into toy stars or anything like that, I can describe everything that I find interesting about this universe with three numbers. I had written those out as “40% hydrogen, 35% iron, 25% uranium”, for example, or “10% hydrogen, 60% iron, 30% uranium”, or whatever the combination happens to be. If I write the elements in the same order each time, though, I don’t really need to add “hydrogen” and “iron” and “uranium” after the numbers, and if I’m always looking at percentages I don’t even need to add the percent symbol. I can just list the numbers and let the “percent hydrogen” or “percent iron” or “percent uranium” be implicit: “40, 35, 25”, for one universe’s distribution, or “10, 60, 30” for another.

Letting the position of where a number is written carry information is a neat and easy way to save effort, and when you notice what’s happening you realize it’s done all the time: it’s how writing the date as “7/27/14” makes any sense, or how a sports scoreboard might compactly describe the course of the game:

0 1 0   1 2 0   0 0 4   8 13 1
2 0 0   4 0 0   0 0 1   7 15 0

To use the notation you need to understand how the position encodes information. “7/27/14” doesn’t make sense unless you know the first number is the month, the second the day within the month, and the third the year in the current century, and that there’s an equally strong convention putting the day within the month first and the month in the year second presents hazards when the information is ambiguous. Reading the box score requires knowing the top row reflects the performance of the visitor’s team, the bottom row the home team, and the first nine columns count the runs by each team in each inning, while the last three columns are the total count of runs, hits, and errors by that row’s team.

When you put together the numbers describing something into a rectangular grid, that’s termed a matrix of numbers. The box score for that imaginary baseball game is obviously one, but it’s also a matrix if I just write the numbers describing my toy universe in a row, or a column:

40
35
25

or

10
60
30

If a matrix has just the one column, it’s often called a vector. If a matrix has the same number of rows as it has columns, it’s called a square matrix. Matrices and vectors are also usually written with either straight brackets or curled parentheses around them, left and right, but that’s annoying to do in HTML so please just pretend.

The matrix as mathematicians know it today got put into a logically rigorous form around 1850 largely by the work of James Joseph Sylvester and Arthur Cayley, leading British mathematicians who also spent time teaching in the United States. Both are fascinating people, Sylvester for his love of poetry and language and for an alleged incident while briefly teaching at the University of Virginia which the MacTutor archive of mathematician biographies, citing L S Feuer, describes so: “A student who had been reading a newspaper in one of Sylvester’s lectures insulted him and Sylvester struck him with a sword stick. The student collapsed in shock and Sylvester believed (wrongly) that he had killed him. He fled to New York where one os his elder brothers was living.” MacTutor goes on to give reasons why this story may be somewhat distorted, although it does suggest one solution to the problem of students watching their phones in class.

Cayley, meanwhile, competes with Leonhard Euler for prolific range in a mathematician. MacTutor cites him as having at least nine hundred published papers, covering pretty much all of modern mathematics, including work that would underlie quantum mechanics and non-Euclidean geometry. He wrote about 250 papers in the fourteen years he was working as a lawyer, which would by itself have made him a prolific mathematician. If you need to bluff your way through a mathematical conversation, saying “Cayley” and following it with any random noun will probably allow you to pass.

MathWorld mentions, to my delight, that Lewis Carroll, in his secret guise as Charles Dodgson, came in to the world of matrices in 1867 with an objection to the very word. In writing about them, Dodgson said, “”I am aware that the word `Matrix’ is already in use to express the very meaning for which I use the word `Block’; but surely the former word means rather the mould, or form, into which algebraical quantities may be introduced, than an actual assemblage of such quantities”. He’s got a fair point, really, but there wasn’t much to be done in 1867 to change the word, and it’s only gotten more entrenched since then.