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Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life

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Few people know that Charles Dodgson (aka Lewis Carroll) was an accomplished mathematician and logician, who held a lecturership in Maths at Oxford. Throughout his many children's books (the two Alice in Wonderland books being the best known) one can see the hand of a person obsessed with logic, numbers and wordplay.

This book provides some of Lewis Carroll's life history, but the latter half of the book focuses specifically on his life as a mathematician. He developed some famous mathematical puzzles (as given in the book), a much easier way of calculating the determinants of 3x3, 4x4 and 5x5 matrices (explained in the book), and quite an ingenious way of drawing inferences in propositional logic (a diagrammatic method he called the "Game of Logic" as shown in the book).

If you are not that much into puzzles and logic you might get more benefit from buying a plain biography on Lewis Carroll. However, the maths and puzzles are not crucial to the enjoyment of the book, and you can skip any of them without losing much. Also, the answers to the puzzles are all in the back of the book, and it is fun going through it, even if you don't work them out. If you love the quirky writing style of Lewis Carroll's books and also like working out puzzles, you will love this book and get the most out of it.

It is certainly enough for his reputation that Lewis Carroll wrote the two Alice books, whose whimsy will be part of literature (and not just children's literature) for the ages. Carroll never regretted the fame the books gave him, but he might have regretted that the world did not take him more seriously in his day job, that of mathematician. There is, for those who want to look for it, mathematics in the Alice books, but it is distorted and jocular just as is everything else in the books. Alice fans will be happy to learn more about Carroll's mathematical pursuits, and in _Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life_ (Norton) mathematician Robin Wilson has summarized for non-mathematicians the serious mathematical efforts (often leavened with irrepressible wit) of the Reverend Charles Dodgson - to differentiate him from his pen name. Some of the math is daunting; Wilson invites readers to skip portions of it, but any reader will come away with a better understanding of this curious man's interests and the happy way he was able to handle pure mathematics as well as pure fantasy.

Wilson's book is generally chronological, based on Carroll's life which was a fairly dull and conventional Victorian existence, except for his child friends, most (but not all) of them little girls who loved his jokes and stories. Carroll all his life was adept at making puzzles; as a child he designed mazes both on paper and in the snow. Carroll may not have had passions for adults, but he had a passion for Euclid, which in his time was thought the ideal method for teaching reason and logic. He defended Euclid against modern geometry texts in 1879 in _Euclid and his Modern Rivals_; to lighten it, he wrote it as a play in four acts! Carroll's mathematics intruded into his humor and vice versa. He wrote a young friend, "Please give my kindest regards to your mother, & ½ of a kiss to Nellie, & 1/200 of a kiss to Emsie, & 1/2000000 of a kiss to yourself." When he wrote seriously about syllogisms, the premises tended to be absurd, which is all the funnier since it affects the logic not at all:

A prudent man shuns hyaenas.
No banker is imprudent.
Conclusion: No banker fails to shun hyaenas.

He drew Bertrand Russell's admiration for contributions to logic with his hypotheticals, including a funny dialogue between Achilles and the Tortoise that seems to indicate that there is an infinite regress that must occur before we can accept any idea, just like the infinite steps that Zeno's Achilles must make before he can outrace the Tortoise. (This dialogue was happily a starting point in theme and style for _Gödel, Escher, Bach_.) But plenty of Carroll's logical and mathematical musings were serious and practical; Wilson notes that his work on voting systems (voting has unexpected mathematical complexities) was well regarded in the succeeding decades. He also developed a fairer system of tournaments whereby athletes compete and are eliminated until one is the victor; this work was the best of its kind at the time, and no one for sixty more years looked at the problem in such depth.

There are many puzzles in this volume, and the merciful Wilson has provided answers. Carroll called many of them "Pillow Problems", as he solved them in his head, lying down; his capacity for calculation and reasoning without resort to diagrams or equations seems to have been prodigious. It has to be said that _Lewis Carroll in Numberland_ is not nearly so much fun as either of the Alice books (what could be?), but those of us who are devoted to Alice will get much pleasure in learning the often serious, often outlandish mathematical pursuits of her creator.

and go on till you come to the end: then stop."

Excellent advice, of course, but in the case of Charles Dodgson, the story is so enormous, one may never finish. Author of the two famous Alice books, creator of hundreds of early photographs, mathematician with papers on logic, algebra, geometry and the mathematics of voting, author of almost 100,000 letters, and diarist with ten massive collected volumes -- it's amazing that one person could produce such a volume of work.

The Alice books have gone through hundreds of editions over the years, there are collections of his puzzles and scholarly analyses of his works on the mathematics of voting (some of which would cast light on the complexities of Amazon's Ranking systems). Dodgson was even something of a wine expert with a delicious spoof on wine tasting based on experts in tasting jam (see the first Comment).

Robin Wilson has put together an excellent summary of Dodgson's life and some superb extracts from his writing for children. He alludes to his contributions to photography, albeit with very few examples; Dreaming in Pictures: The Photography of Lewis Carroll by Douglas R. Nickel is an excellent source.

The heart of the book is an assessment of Dodgson's work as a mathematician. It's not really necessary to work out all of the problems -- although it is certainly possible to do so with Wilson's help and basic algebra and geometry training. The book moves swiftly, but be warned, some of the problems are so cleverly written, you may find yourself trapped and spend several pleasant minutes, even hours, working your way through to a solution.

Either way, you'll come away from this excellent study with a deeper appreciation of Dodgson's complex imagination and with a real appreciation of Wilson's skill in bringing that imagination to life.

Robert C. Ross 2009

For most people familiar with Lewis Carroll, it is because he was the writer of that classic story, Alice in Wonderland. What's less known is that even before he made his name in literature, he was a mathematician of some prominence, and that this field would creep into his fictional writing.

Actually, Lewis Carroll was a pseudonym for Charles Dodgson, an Oxford educated mathematician of the mid-1800s. He would also teach at Oxford and start to write his stories there, as well as mathematical works. Always eager to please children (including the inspirational Alice), he would become one of the first people to develop recreational mathematics, a field that focuses on some of the more wonderfully entertaining aspects of numbers (particularly the whole numbers).

Robin Wilson's Lewis Carroll in Wonderland serves as a biography of Dodgson/Carroll, focusing on his work in math. The first half or so is more filled with biographical facts; it is in the second half that we get more of the math, most of which requires no higher learning in the field. We get some of the word play, puzzles, logic problems and riddles that were Carroll's forte. Many are interesting, but admittedly, some of the problems that seem presented as logic problems are anything but, coming off more as tricky riddles and leaving the reader feel a little cheated.

If you have an interest in the life of Lewis Carroll, this would probably be a good book to read; on the other hand, if you enjoy recreational mathematics, this book is merely okay. I tend to think of this book more as a biography, so I'll rate it as a good, four-star read, well-written and with plenty of illustrations.

Although Charles Dodgson's literary works are of course very well known, his mathematical work is more obscure. In fact, his mathematics is notable not so much for its importance in the strict development of mathematics but rather for the algorithmic aesthetic that infused all of Dodgson's work. Dodgson was particularly interested in systematizing rules of thought, and he strove to do this in two areas, symbolic logic and linear algebra (notably determinants) that would not really yield to the power of the computer for generations. At the same time, Dodgson's interest in the paradoxes of systematization, in the interplay between algorithm and intuition, informed his fiction work and his work on puzzles as well.

This pleasant, easy-to-read book nicely captures the milieu of Oxford mathematics education in the nineteenth century generally and Charles Dodgson's life in that milieu in particular. Notable is the affection for scholarship for its own sake, and for a much more sedate pace of life, than obtains nowadays.

Even in political science, Dodgson was way ahead of his time in his interest in systematizing voting protocols; although he did not anticipate Arrow's theorem, he at least did some preparatory work.

As a strict mathematician, however, Dodgson did not have the deep technique of the great 19th century mathematicians. His work rarely exceeds very basic mathematics, what would now be learned, perhaps in less detail, in middle school. There is, at least in the book, no calculus, no complex variables, no power series, no abstract algebra, no non-Euclidean geometry, no infinite set theory, not even impossibility proofs. He deals with Euclidean geometry (plane geometry in the examples) and some basic linear algebra.

At the same time, with this limited palette, Dodgson addressed very important problems - the problems he worked on all wound up leading to major, important fields (computational logic, computational linear algebra, and voting theory).

Not only does this book give insight into Dodgson's work and character, it also has some fascinating old exam papers from that time period. Indeed, the book would have been substantially improved had it included more facsimiles of period-authentic examinations and textbooks (as well as, for that matter, a bit more explanation of the somewhat confusing system - to American readers - of exams and matriculation at Oxford). But what it does have is interesting.

An example of a nice puzzle by Carroll is from The Tangled Tale: a traveller walks along level ground and up a hill, then returns the way; leaving at 3:00 PM and returning at 9:00 PM that day; travelling at 3, 4, and 6 mph uphill, level and downhill respectively. How far did he travel and when, within a half-hour, did he reach the top of the hill?

A nice exam problem from Oxford at the time was to solve

(x+sqrt(a^2-x^2))/(x-sqrt(a^2-x^2)) = b ,

where one presumes this is over the reals. There is something relaxing about math before the high-powered modern machinery took hold, everything concrete and finite.

In this short but charming book, the author introduces the reader to Charles Dodgson whose pen name was Lewis Carroll - creator of Alice in Wonderland. Although the book is categorized as a biography, relatively little of Dodgson personal life is presented. Instead, the author has chosen to focus more on his mathematical genius and accomplishments. Nevertheless, one can glimpse what Dodgson's life must have been like through the many quotes from his diary and his letters, as well as through his dedication and passion for thinking up and solving various mathematical problems - several of which are presented in this book. The writing style is clear, friendly, authoritative and engaging. The book's 209 pages of main text include a great many interesting pictures and useful diagrams, thus making this a relatively short read. Because of its many mathematical problems and brain teasers, this book would most likely be loved the most by math buffs.

I liked this book because it is a bio of Lewis Carroll from a mathematical standpoint. Yes, it does touch on his literary career, but is mostly about his development as a mathematician and an Oxford Don.