"One of the deep secrets of life is that all that is really worth the doing is what we do for others." - Lewis Carroll

The Reverend Charles Lutwidge Dodgson, (1832-1898) better known by his pen name, Lewis Carroll, was a writer and mathematician. Carroll is most fondly remembered as the author of *Alice's Adventures in Wonderland* and *Through The Looking Glass.*

Charles Lutwidge Dodgson, was born on January 27th, 1832, in Cheshire, England. His father, Charles Dodgson senior, was the "Perpetual Curate" of the Daresbury church. Charles had two older sisters, but was the oldest son, in a family of eleven children.Charles was a good student--excelling espcially in mathematics--but he stammered and was shy.

In 1851, he matriculated at Christ Church, Oxford. Dodgson received a BA in mathematics in 1855 and an MA in 1857. In 1858 he became a Senior Student, which allowed him to stay on as a teacher for as long as he liked, provided that he became a minister, and remained celibate. He became the Reverend Charles Dodgson.

He remained a bachelor all his life; whether or not he actually remained chaste, was—and continues to be—a subject of rumor and speculation.

The Dean of Christ Church College was Henry George Liddell. Dodgeson became friendly with Liddell's family, occasionally tutoring Liddell's son Henry and picnicking with Liddell's three daughters, Lorina, Alice, and Edith.

On several July afternoons, in 1862, Dodgson, and Dean Liddell's children, and sometimes also a friend of Dodgson's named Duckworth, took rowboat excursions and had picnics. Dodgson amused the girls and his friend with a fanciful story of Alice chasing a white rabbit down a rabbit hole, and having adventures under ground. Alice Liddell asked Dodgson to write up the story and eventually he did. The result was *Alice's Adventures Under Ground*, which he illustrated himself.

He later revised the story, added thee new chapters—including the now famous tea party with the Mad Hatter—and hired well-known illustrator John Tenniel to replace his sketches with professional illustrations.

The result was *Alice's Adventures in Wonderland*, an immediate critical and commercial success.

In order to keep his life as an Oxford don separate from his life as a children's author he published *Alice* under the pseudonym Lewis Carroll.

Seven years later Carroll wrote *Through the Looking Glass*.

*Looking Glass* was written just after Carroll's father had suddenly and unexpectedly died, and the book has a somewhat darker tone than *Alice's Adventures in Wonderland*.

*Looking Glass* contains one of Carroll's most influential works, the poem *Jabberwocky*.

In 1874, when Carroll was forty-two, a single line of a poem popped into his mind while he was taking a walk.

The line was:

For the Snark was a Boojum you see.

Carroll had no idea what it meant, but after thinking about it for a while the rest of the stanza occurred to him:

In the midst of the word he was trying to say,

In the midst of his laughter and glee,

He had softly and suddenly vanished away—

For the Snark was a Boojum you see.

This became the last stanza of the poem *The Hunting of the Snark: An Agony in Eight Fits*, published with illustrations by Henry Holiday.

Carroll presented *Snark* as a nonsense poem for children, but unlike the *Alice* books, or his late *Sylvie and Bruno* novels, *Snark* features no child characters

In the introduction to his annotated version of the poem Martin Gardner wrote:

Although Lewis Carroll thought of *The Hunting of the Snark* as a nonsense ballad for children, it is hard to imagine—in fact one shudders to imagine—a child of today reading and enjoying it.

The Snark is perhaps Carrols best and most profound work. Although he continued to insist that it was purely nonsense, readers have found, and continue to find hidden layers of meaning and enigma within its verses.

Quoting Gardner again:

The Hunting of the Snark is a poem over which an unstable, sensitive soul might very well go mad.

Consider for instance the Bellman's map, presented in Fit the Second:

He had bought a large map representing the sea,

Without the least vestige of land:

And the crew were much pleased when they found it to be

A map they could all understand.

What's the good of Mercator's North Poles and Equators,

Tropics, Zones, and Meridian Lines?

So the Bellman would cry: and the crew would reply

They are merely conventional signs!

Other maps are such shapes, with their islands and capes!

But we've got our brave Captain to thank:

(So the crew would protest) that he's bought us the best--

A perfect and absolute blank!

This is amusing nonsense, but it also hints at deeper meanings and raises questions about several relationships: the relationship between what is actually represented, and what is intended to be represented; the relationship between symbols and meaning; the relationship between symmetry and information.

The importance of this last relationship would emerge in the twentieth century.

Symmetry groups now play a fundamental role in the standard model of particle physics, but questions about this relationship between symmetry and information remain.

The Reverend Charles Dodgson was a competent, but somewhat plodding mathematician. He wrote a textbook on determinants that was not well received; he was a notoriously dull and uninspiring lecturer.

Lewis Carroll, however, was another story. Carroll anticipated several issues in logic, recursion theory, and the philosophy of mathematics, that were not developed until the twentieth century.

Perhaps his greatest contribution was the article *What the Tortoise said to Achilles*, in which he presents his own version of Zeno's famous paradox.

In Carroll's version, Achilles has already beaten the tortoise at the race, and now the two are having a discussion about mathematical logic in Euclid's Elements.

The Tortoise (always capitalized in Carroll's story) asks Achilles to prove that Euclid is correct.

The Tortoise:

"Well, now, let's take a little bit of the argument in that First Proposition -- just two steps, and the conclusion drawn from them. Kindly enter them in your notebook. And in order to refer to them conveniently, let's call them A, B, and Z: --

(A) Things that are equal to the same are equal to each other.

(B) The two sides of this Triangle are things that are equal to the same.

(Z) The two sides of this Triangle are equal to each other.

Readers of Euclid will grant, I suppose, that Z follows logically from A and B, so that any one who accepts A and B as true, must accept Z as true?

Achilles agrees that Z is true but the Tortoise insists that the fact that Z is true if A and B are true must be included as:

(C) If A and B are true, Z must be true. "

"That is my present position," said the Tortoise.

"Then I must ask you to accept C."

"I'll do so," said the Tortoise, "as soon as you've entered it in that note-book of yours. What else have you got in it?"

Achilles seems to have won again, but the Tortoise now demands a "D," an "E," and so on ad infinitum.

Achilles and the Tortoise, raises the problem of the difference between truth and meaning in mathematics.

It is amusing to see some of the responses to Achilles and the Tortoise written by mathematicians. The Tortoise, or Carroll (or both) are often dismissed as simply wrong: the tortoise simply must accept the truth of Z; the paradox disappears if one accepts only the propositional calculus; Z can be reduced to a tautology in a finite number of steps, et cetera. Of course, none of these arguments would have convinced the Tortoise.

The point is that while a mathematical system can be shown to be intrinsically self-consistent, it can never be shown to be intrinsically meaningful: meaning must come from without, in the form of primitive information associated with the terms and symbols of a logical system.

The problem of meaning in mathematics is not intrinsically mathematical but is rather metamathematical. It is about how meaning is expressed in mathematics, not about whether or not a particular mathematical calculus is self-consistent.

Truth can be demonstrated within a certain logical calculus—by showing that a proposition reduces to a tautology, for example—but meaning must come from somewhere else.

As Stephen Cole Kleene wrote in Mathematical Logic in 1967:

If at no stage is an application made outside of formal axiomatics, the whole activity must appear to be futile. We therefore conclude that, if we are not to adopt a mathematical nihilism, formally axiomatized mathematics must not be the whole of mathematics. At some place there must be meaning, truth, and falsity.

But Kleene was born in 1909--eleven years after Carrol died. He developed metamathematic in the 1930s, 40s and 50s. Kleene was also one of the founders of recursion theory, another field anticipated by Carroll in *What the Tortoise said to Achilles*.

It's interesting to note that the Tortoise was taking issue with Euclid, and yet the Revered Charles Dodgson was a confirmed, even reactionary, Euclidean: he wrote a defense of Euclid, including the fifth postulate, and was either uninterested in or actually hostile towards the revolution in non-Euclidean geometry (occurring more-or-less during his lifetime) begun by Gauss, Riemann and Lobachevsky.

And so the dichotomy: on the one hand the Reverend Charles Dodgson, Oxford don, reactionary, dull lecturer in mathematics, clinging to eighteenth century and ancient mathematical principles; on the other hand, Lewis Carroll, creative and insightful genius, questioning all that was old and taken-for-granted in mathematics, anticipating the twentieth and twenty-first century.

Curiouser indeed.

Sources:

http://www.victorianweb.org/authors/carroll/bio1.html

By Jenny Woolf, St. Martin's Press (2010).

What the Tortoise Said to Achilles

Stephen Cole Kleene, Mathematical Logic. Dover, New York (1967).

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- Birthday Date: Monday, 27 January 2014