Alice's adventures in algebra

There's a stunner of an article by Melanie Bayley over on the New Scientist site at the moment. Called "Alice's Adventures in Algebra: Wonderland Solved", it purports to demonstrate that much of Alice's Adventures in Wonderland is not the simple surrealist fantasy we've all believed but a satire of the newfangled mathematical ideas that Carroll/Dodgson, a very conservative mathematician, believed nonsensical. I think Bayley comes very close to proving her case; I say "very close to" because in a couple of instances I have just a suspicion she might be overreaching. Overall, though, I'm convinced.

Here's an appetite-whetter from this longish piece:

Alice, angry now at the strange turn of events, leaves the Duchess's house and wanders into the Mad Hatter's tea party, which explores the work of the Irish mathematician William Rowan Hamilton. Hamilton died in 1865, just after Alice was published, but by this time his discovery of quaternions in 1843 was being hailed as an important milestone in abstract algebra, since they allowed rotations to be calculated algebraically.
 

Just as complex numbers work with two terms, quaternions belong to a number system based on four terms. Hamilton spent years working with three terms - one for each dimension of space - but could only make them rotate in a plane. When he added the fourth, he got the three-dimensional rotation he was looking for, but he had trouble conceptualising what this extra term meant. Like most Victorians, he assumed this term had to mean something, so in the preface to his Lectures on Quaternions of 1853 he added a footnote: "It seemed (and still seems) to me natural to connect this extra-spatial unit with the conception of time." 

Where geometry allowed the exploration of space, Hamilton believed, algebra allowed the investigation of "pure time", a rather esoteric concept he had derived from Immanuel Kant that was meant to be a kind of Platonic ideal of time, distinct from the real time we humans experience. Other mathematicians were polite but cautious about this notion, believing pure time was a step too far.

The parallels between Hamilton's maths and the Hatter's tea party - or perhaps it should read "t-party" - are uncanny. Alice is now at a table with three strange characters: the Hatter, the March Hare and the Dormouse. The character Time, who has fallen out with the Hatter, is absent, and out of pique he won't let the Hatter move the clocks past six.

Reading this scene with Hamilton's maths in mind, the members of the Hatter's tea party represent three terms of a quaternion, in which the all-important fourth term, time, is missing. Without Time, we are told, the characters are stuck at the tea table, constantly moving round to find clean cups and saucers.

Their movement around the table is reminiscent of Hamilton's early attempts to calculate motion, which was limited to rotations in a plane before he added time to the mix. Even when Alice joins the party, she can't stop the Hatter, the Hare and the Dormouse shuffling round the table, because she's not an extra-spatial unit like Time.

The Hatter's nonsensical riddle in this scene - "Why is a raven like a writing desk?" - may more specifically target the theory of pure time. In the realm of pure time, Hamilton claimed, cause and effect are no longer linked, and the madness of the Hatter's unanswerable question may reflect this.

Alice's ensuing attempt to solve the riddle pokes fun at another aspect of quaternions: their multiplication is non-commutative, meaning that x × y is not the same as y × x. Alice's answers are equally non-commutative. When the Hare tells her to "say what she means", she replies that she does, "at least I mean what I say - that's the same thing". "Not the same thing a bit!" says the Hatter. "Why, you might just as well say that 'I see what I eat' is the same thing as 'I eat what I see'!"

It's an idea that must have grated on a conservative mathematician like Dodgson, since non-commutative algebras contradicted the basic laws of arithmetic and opened up a strange new world of mathematics, even more abstract than that of the symbolic algebraists.

When the scene ends, the Hatter and the Hare are trying to put the Dormouse into the teapot. This could be their route to freedom. If they could only lose him, they could exist independently, as a complex number with two terms. Still mad, according to Dodgson, but free from an endless rotation around the table.

I can barely recall any other article that's entranced me so much. Do please go read the whole thing!

one of the things I'll be doing in 2010 (and likely beyond)

One of the lesser known but perhaps more important projects underway in sf at the moment is the preparation of the third edition of The Encyclopedia of Science Fiction, this time under the editorship of Peter Nicholls, John Clute and David Langford. The new edition, which is already approaching twice the wordcount of the second edition (to which I devoted several years of my life), is to be an online publication; I'm not certain of the proposed publication date, and I'm not sure anyone yet is.

What I do know, though, is that a couple of weeks ago I was approached by Clute and Langford with the commission to write the art/illustration strand of the new encyclopedia, and you can bet that -- honoured and flattered like you wouldn't believe -- I accepted with some alacrity. When we were putting together the second edition this was something I could not have done: my knowledge of f/sf art was in those days pretty woeful, despite the efforts of artist close friends like Ron Tiner to enlighten me. Since then, of course, I've had the powerful education of running Paper Tiger for a few years.

So, if you're an extensively published sf artist, expect sometime in the new year to be getting a questionnaire from me; if you're not an artist, well, I'll probably corner you in a bar somewhere and weep on your shoulder about the travails of dealing with artists. "They're not like us, you know . . ."

More seriously, this is going to be a joy. I'm almost nervous of getting started on it.