## Wednesday Book Club: *Lewis Carroll in Numberland*

**This week’s selection:** *Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life* (2008) by Robin Wilson.

**In brief:** Part biography and part catalogue of Charles Dodgson’s mathematical interests, *Numberland* is a crisp introduction to Dodgson’s professional work outside of the classic literary diversions he penned as Lewis Carroll. Wilson is content to explicate mathematical puzzles and present collections of facts rather than weave them into a story or thesis, but does so admirably enough to produce a fine survey of what captivated the man.

(The Wednesday Book Club is an ongoing initiative of mine to write a book review every week. I invite you to peruse the index. For more on *Lewis Carroll in Numberland*, keep reading below.)

Avid readers of Lewis Carroll are likely familiar with the story of how Queen Victoria received *Alice’s Adventures in Wonderland*. “Send me the next book Mr Carroll produces,” the Queen demanded—hardly expecting that Carroll, who was a lecturer in mathematics by the name of Charles Lutwidge Dodgson, would then present her with a copy of *An Elementary Treatise on Determinants, with Their Application to Simultaneous Linear Equations and Algebraical Geometry*.

Dodgson later denied this story, but the tenor of the anecdote befits his reputation as a master of wordplay who often toyed with miscommunication and the boundary between the figurative and the literal. Examples abound in Robin Wilson’s biography, *Lewis Carroll in Numberland*, most of them drawn not from *Alice* but from Dodgson’s lesser-known works, private correspondence, and mathematical puzzles.

Wilson’s book begins with a sample of scenes from Dodgson’s writings as Lewis Carroll that draw on curiosities in arithmetic, geometry, and logic, but otherwise pays little attention to his literary work. The composition of *Alice* is a story that here occupies no more than two pages. The bulk of *Numberland* considers Dodgson as a pioneer of recreational mathematics; as a consummate scholar and educator who composed riddles for students and family members alike, devised shortcuts for arithmetic and instructional games for visualizing symbolic logic, and dabbled in everything from tennis tournament seeding to schemata for electoral reform.

Half of *Numberland* is straightforward biography, offering a portrait of what life was like for Dodgson as a clergyman’s son, model student, and Oxford scholar. Much of the evidence is drawn from letters and diary entries. Wilson exposes us to Dodgson’s England by way of the mathematical culture of the day, be it in the form of representative examination questions or the debate surrounding whether geometry ought to be taught directly from Euclid’s *Elements* or through new instructional texts. (Dodgson was a staunch advocate of adhering to a classical education in Euclidean geometry, insisting the order and numbering of Euclid’s axioms and propositions were themselves part of standard mathematical literacy. This was a battle he ultimately lost.)

The remainder of the book presents an eclectic sample of problems and other oddities, many of which Wilson leaves for the reader to solve (though solutions are provided in the endnotes). The mathematical content is undemanding and should be accessible to any reader with at least vague memories of middle-school algebra and geometry, though a few of the puzzles will take some thought. Conceptually, there is nothing here more advanced than Dodgson’s condensation method to quickly compute the determinants of large matrices, and Wilson explains it all with clear examples and easy-to-follow diagrams.

This is all that Wilson aspires to do in this book, and he does it well. All the same, the breakneck pace with which he hops from one curiosity to the next, coupled with the wholly expository nature of the text, leaves the impression that we are only receiving a cursory tour of the subject. The narrative frequently tantalizes us with breadcrumbs of fascinating connections only to move on to the next unrelated specimen.

I attribute this to Wilson’s marked distaste for deviating from documented historical fact. While this decision renders *Numberland* a cautious book that dares not synthesize its anecdotes, it may have its roots in what Wilson sees as the grave injustice that present-day revisionism has done to Dodgson’s reputation, notably with respect to his relations with children. As someone who spent a great deal of time with young girls (including Alice Liddell, the inspiration for *Alice’s Adventures in Wonderland*) and a noted amateur photographer who specialized in child subjects, Dodgson’s history raises some eyebrows today. Wilson only broaches the subject once, dismissing any unsavoury suspicions wholesale:

Sadly, much nonsense has been written about Dodgson’s friendships with children. In common with many of his generation, he regarded young children as the embodiment of purity and he delighted in their innocence. His vows of celibacy, which he took very seriously, would have outlawed any inappropriate behaviour, and there has never been a shred of evidence of anything untoward. Subjecting him to a modern ‘analysis’, rather than judging him in the context of his time, is bad history and bad psychology, and often tells us more about the writer than about Dodgson.

I have no reason to doubt that Wilson is correct. Similar controversies persist in fields like Shakespeare scholarship, where modern readings of pederasty and homosexuality in the plays often fail to account for what was a fundamentally different culture of social bonds and normative sexual desire. Psychoanalysis on the whole has proven more relevant to fiction and myth than to real people. But Wilson is unlikely to persuade anyone who believes Dodgson’s relations with children are circumstantially suspect—certainly not by abandoning the subject almost as soon as he brings it up. We all know how moral conservatism and vows of celibacy make for a flimsy defence nowadays.

My point remains that Wilson does little to make arguments and broad connections, even when it is within the scope of the book’s mathematical focus. Personally, I would have liked to see more direct discussion of how Dodgson’s multifarious interests influenced his literary output as Lewis Carroll. The knowledge that Dodgson was an early advocate for proportional representation—specifically, a variant of PR that would replace single-member constituencies with fewer electoral districts consisting of multiple, proportionally distributed representatives—seems to bear directly on the scene of the Caucus-race in *Alice*. Similarly, Dodgson’s advocacy for continued mathematical education through classical texts in polemics like *Euclid and his Modern Rivals* instantly reminds us of his frequent satires of British schooling and begs us to seek out a coherent thread of pedagogical beliefs.

This is not to say that Wilson neglects these connections entirely. One passage in *Numberland* calls attention to the recurring appearances of the number forty-two in the Carroll literature—the most intriguing example being this passage from *Alice*:

I’ll try if I know all the things I used to know. Let me see: four times five is twelve, and four times six is thirteen, and four times seven is—oh dear! I shall never get to twenty at that rate! However, the Multiplication Table doesn’t signify…

Wilson notes that 4 × 5 = 12 in base 18, 4 × 6 = 13 in base 21, and so on as we increment the base in threes up to base 39, where 4 × 12 = 19—after which we reach base 42, where 4 × 13 yields not 20, but 1X (X being the digit in base 42 equivalent to 10 in decimal). In a twisted way, Alice was right.

This reminds me of another story about numbers in amusing books. No mention of the number forty-two goes very far without bringing Douglas Adams to mind, since the number appears in *The Hitchhiker’s Guide to the Galaxy* as the Answer to the Ultimate Question of Life, the Universe, and Everything. What the Question is, we’re not exactly sure, but in *The Restaurant at the End of the Universe*, Arthur Dent draws Scrabble tiles out of a bag and randomly generates the question, “What do you get if you multiply six by nine?” As 6 × 9 = 54, the point is that the universe makes no sense.

It has been observed, however, that 6 × 9 = 42 in base 13. Despite circumstantial oddities like how Arthur’s handmade Scrabble board is 13×13 instead of the standard 15×15, Adams vigorously denied that this was anything but coincidence. “I may be a sorry case,” he said, “but I don’t write jokes in base 13.” Evidently, Lewis Carroll did him one better.

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*The Wind-Up Bird Chronicle*
Well, I certainly didn’t learn geometry using Euclid, but I think Dodgson was correct that the number and ordering of his axioms and postulates, though not his propositions, remains part of mathematical literacy. I’d be pretty surprised at someone who knew what the parallel postulate is, but had never heard that it’s Euclid’s fifth.

Then again, maybe I’m just an old fart.

Anyhow,

A Tangled Taleis online; it’s my favorite collection of Carroll puzzlers. Each month for ten months he published a whimsical short story containing one or more mathematical problems, and the following month he published the answers — with a complete and witty discussion of all the pseudonymous replies he had received. He even stumbles over the problem, some twenty years in advance, of the International Date Line.Thursday, 17 September 2009 at 7:15am

Thanks for the link. His argument for the necessity of an International Date Line is indeed one of the accomplishments discussed in

Lewis Carroll in Numberland, although the only Knot inA Tangled Talereproduced in full is the first.The parallel postulate has some renown as the fifth because its presence or absence makes all the difference between Euclidean and non-Euclidean geometry, but if I were pressed to produce the first four in order

ex tempore, I would probably get it wrong. Regardless, I think Dodgson was just as concerned about preserving the indexing of the derived propositions (like I.47 for the Pythagorean theorem).I find it interesting that some disciplines, especially the maths and sciences, are now taught entirely apart from the texts in which they were introduced; while others, notably philosophy, are still presented as a dialogue where one necessarily studies authors and original texts. As far as I know, up to at least the undergraduate level, biologists don’t teach

The Origin of Speciesand physicists don’t teach the Einstein papers, not even as stepping-stones to our current understanding. They teach the results, along with the methods of obtaining them.I think the heart of the distinction is in what we consider objective knowledge—ideas that exist outside their presentation in a classic document, which we can independently establish through derivation or experiment. We can say that for natural selection, relativity, or the properties of triangles on a plane. We probably can’t say that about Kant.

Thursday, 17 September 2009 at 11:33am

I think you are quite right. We must, of course, except Great Books programs, where students do indeed read Darwin directly (though perhaps not Newton, I don’t know). And of course Stephen Jay Gould and other people who teach the history of science send students to the original documents.

My daughter is reading Plato now, and finding him a great bore. Like the Walrus, I weep for her and deeply sympathize; like the Walrus, I condemn her to intellectual oysterhood nonetheless.

Sunday, 27 September 2009 at 7:41am

It turns out that it’s only me who finds the

Republica great bore; my daughter found it “interesting, but tough”, as Huckleberry Finn says.Monday, 5 October 2009 at 8:53pm