Many people who have read "Alice's Adventures in Wonderland" and "Through the Looking-Glass" are aware that the author was a mathematician. Exactly what was his work in mathematics?

by Warren Weaver

Lewis Carroll wasn't he a first-class mathematician too? This is a typical remark when the name of the author of Alice in Wonderland comes up. That Carroll's real name was Charles Lutwidge Dodgson and that his main lifelong interest was mathematics is fairly common knowledge. In fact, among his literary admirers there has long been current a completely false but unstoppable story that Queen Victoria read Alice, liked it, asked for another book by the same author and was sent Dodgson's very special and dry little book on algebraic determinants.

Lewis Carroll was so great a literary genius that we are naturally curious to know the caliber of his work in mathematics. There is a common tendency to consider mathematics so strange, subtle, rigorous, difficult and deep a subject that if a person is a mathematician he is of course a "great mathematician there being, so to speak, no small giants. This is very complimentary, but unfortunately not necessarily true. Carroll produced a considerable volume of writing on many mathematical subjects, from which we may judge the quality of his contributions. What sort of a mathematician, in fact, was he?

The Story of his academic career is quickly told. C. L. Dodgson was born in 1832 near Daresbury in Cheshire. His father was a clergyman, as had been his grandfather, great-grandfather and great-great-grandfather. He went up to Oxford in 1850 after six unhappy years in English "public" schools. At the end of 1852 he was given first class honors in mathematics and was appointed to a "studentship" (what we would call a fellowship) on condition that he remain celibate and proceed to holy orders. He took his B.A. with first class honors in the final mathematical school in 1854 and his M. A. in 1857. In 1855, at the age of 23, he was given a scholarship that paid the princely sum of 20 pounds a year, and was appointed senior student, or don, at Christ Church, and mathematical lecturer in the University. He lived, a bachelor, in college quarters in Tom Quad from 1868 until he died, 66 years old, in 1898. His academic life was enlivened, if you will pardon the expression, by the very Victorian activities of being made a sublibrarian in 1855; being ordained a deacon in 1861, and as a climax, when he was 50, being made curator of the Common Room a sort of club steward.

The even tenor of this secluded life gave him ample time for writing, both as Charles Lutwidge Dodgson and as Lewis Carroll. There must be few authors who wrote so much and are remembered for so little. The standard bibliography of his work lists 256 items printed during his lifetime, and nearly 900 items in all. Of these 16 are books about six written for children and about 10 devoted to mathematics and logic. One has to say "about" because it is hard to tell whether some were intended for children or adults, whether others are mathematics or fantasy. In addition Carroll wrote nearly 200 pamphlets. About 50 related to minor academic quarrels at Christ Church, about 30 to word games, ciphers and the like, and more than 50 to wildly miscellaneous subjects: how to memorize dates, how to bowdlerize Shakespeare for young girls, how to score tennis tournaments, common errors in spelling, rules for reckoning postage, and so on.

Of the 256 items printed during his lifetime, 58 were devoted to mathematics and logic. If we consult these works for an estimate of Carroll's or perhaps we should now say Dodgson's stature as a mathematician, we discover that he was first of all a teacher, earnestly concerned with methods of instruction in elementary subjects. He wrote nearly two dozen texts for students in arithmetic, algebra, plane geometry, trigonometry and analytical geometry.

Dodgson's largest and most serious work on geometry, Euclid and His Modern Rivals, gives us an insight into his approach to mathematics. It shows him as a militant conservative, dedicated to defending Euclid against any modern move to improve or change him in any way. Dodgson sought to prove in this book that Euclid's axioms, definitions, proofs and style simply could not be changed for the better. He even insisted that the order and numbering of Euclid's theorems be preserved. Dodgson skillfully ridiculed contemporary geometers who tried to restate Euclid's parallel axiom, and threw out all their attempts as "simply monstrous." (It is, however, worth noting that in a later book entitled A New Theory of Parallels Dodgson himself sought to replace the classical axiom by one of his own devising.)

Euclid and His Modern Rivals must be classed as amusing, ridiculously opinionated and scientifically unimportant. It reflects nothing of the growing realization among contemporary mathematicians that the axiom of parallels was not a self-evident fact of nature but an arbitrarily adopted and unprovable postulate. Non-Euclidean geometry, with its revolutionary consequences for mathematics and science, was not dreamt of in Dodgson's philosophy.

The bleak impression of the Reverend Dodgson created by his pedagogical works is pleasantly relieved when we turn to his other mathematical writings. He comes closer to the man we know as Lewis Carroll in, for example, a strange little book called Pillow Problems. Here Dodgson presents 72 problems chiefly in algebra, plane geometry and trigonometry all of which he had worked out in bed at night without pencil or paper. Dodgson suffered from insomnia, and while he was careful to point out that mathematics would put no one to sleep, he argued that it would occupy the mind pleasantly and prevent worry. It is characteristic of his severely pious attitude that he offered mathematical thinking, during wakeful hours, as a remedy for "skeptical thoughts, which seem for the moment to uproot the firmest faith ... blasphemous thoughts, which dart unbidden into the most reverent souls .unholy thoughts which torture with their hateful presence the fancy that would fain be pure"

The problems in this book, while elementary, are nevertheless complicated enough to require real skill in concentration and visualization, if one is to solve them in his head. A geometrical problem, reproduced from his manuscript, is shown on page 124. Here is another example:

"On July 1, at 8 a.m. by my watch it was 8 h. 4 m. by my clock. I took the watch to Greenwich, and when it said noon, the true time was 12 h. 5 m. That evening, when the watch said 6 h. the clock said 5 h. 59 m. On July 30, at 9 a.m. by my watch, it was 8 h. 57 m. by my clock. At Greenwich, when the watch said 12 h. 10 m. the true time was 12 h. 5 m. That evening, when the watch said 7 h., the clock said 6 h. 58 m. My watch is only wound up for each journey, and goes uniformly during any one day: the clock is always going, and goes uniformly. How am I to know when it is true noon on July 31?"

Dodgson's solutions of the problems in this collection are generally clever and accurate, but one of them ludicrously exposes the limitations in his mathematical thinking. The problem is: "A bag contains 2 counters, as to which nothing is known except that each is either black (B) or white (W). Ascertain their colors without taking them out of the bag." In his attack on this problem (which as stated cannot actually be solved) he makes two dreadful mistakes. First he assumes, incorrectly, that the statement implies the probabilities of BB, BW and WW (the three possible constitutions of the bag) are 1/4, 1/2, and 1/4 respectively. Then he adds a black ball to the bag, calculates that the probability of now drawing a black ball is 2/3 and makes his second fatal error in concluding that the bag now must contain BBW. His line of reasoning thus leads him to the conclusion that the two original balls were one B and one W! This is good Wonderland, but very amateurish mathematics. It has been pointed out that if one applies Dodgson's argument to a bag containing three unknown balls (black or white), he can come out with the conclusion that it was impossible for there to have been three balls at all.

Dodgson's zest for mathematical puzzles produced a second little book which he called A Tangled Tale. The problems are named "Knots," and Knot I, for example, presents this tangle: Two travelers spend from 3 o'clock until 9 o'clock in walking along a level road, up a hill, back down the hill, along the same road, and home. Their pace on the level is 4 miles per hour, uphill it is 3, and downhill 6 miles per hour. Find the distance walked and, within a half-hour, when they were on the summit.

My collection of Dodgson's manuscripts includes his two favorite puzzles, which he did not publish. One, called "Where Does the Day Begin?", considers the paradox that a man who travels westward around the earth at the same speed as the sun will find that though he started on a Tuesday, when he returns to his starting point the day is now called Wednesday. Where and when did the date change? Dodgson troubled many officials in government offices and telegraph companies with correspondence on this question, which he first posed in 1860. No one could answer it, of course, until the arbitrary International Date Line was established in 1884.

Dodgson's other favorite puzzle, named "The Monkey and Weight," was equally baffling to his contemporaries. A weightless, perfectly flexible rope is hung over a weightless, frictionless pulley. At one end is a monkey, at the other a weight which exactly counterbalances the monkey. The monkey starts to climb. What does the weight do? One of the difficulties in this tricky problem is that it is not well defined. For example, does the monkey jerk the rope? Or does he begin pulling on it very gently, and if so, how does he maintain the pull?

In all of Dodgson's mathematical writing it is evident that he was not an important mathematician. As we have seen, in geometry his ideas were old-fashioned even for his time; in the probability problem cited he failed to grasp the principle of insufficient reason; in algebra he once wrote in his notes: "that 2(x2+y2) is always the sum of two squares seems true but unprovable."It took him some time to recall the fact, familiar to any high-school student of elementary algebra, that (2×2+y2) = (x+y)2+(x'y)2. In calculus his concept of infinitesimals was the wholly confused one that these were queer shadowy quantities which were non-infinite, non-finite and also non-zero! His notes record such logical monstrosities as "infinitesimal unit," "infinity unit," "minimum finite fraction." He did not understand the basic aoncept of the limiting process in the calculus, as is indicated by the remark in his notes: "The notion that because a variable magnitude can be proved as nearly equal to a fixed one as we please, it is therefore equal, is to my mind unsatisfactory, as we only reduce the difference, and never annihilate it."

But before we write Dodgson off as a " mathematician we must consider his contributions to formal logic. Nearly half of his mathematical writings were in this field.

The most important were The Game of Logic, published in 1886, and an expansion of it 10 years later into a longer and somewhat more serious book called Symbolic Logic: Part I, Elementary. In these works Carroll developed the use of a scheme which had first been introduced by the Swiss mathematician Leonhard Euler in 1761. It involves the representation of sets of similar propositions by spatial diagrams together with a symbolic language for translating the diagrams back into verbal statements. The examples he invented for the use of the method were characteristically clever and amusing.

For instance, from the premises:

All dragons are uncanny, All Scotchmen are canny;

he derived the comforting conclusions:

All dragons are not-Scotchmen, All Scotchmen are not-dragons.

Another example of the fun he had with simple logic is the following (de-

duce a conclusion from the given premises):

"It was most absurd of you to offer it! You might have known, if you had any sense, that no old sailors ever like gruel!"

"But I thought, as he was an uncle of yours..."

"An uncle of mine indeed! Stuff!"

"You may call it stuff if you like. All I know is, my uncles are all old men: and they like gruel like anything!"

"Well then, your uncles are" [not sailors}.

Amusing as Carroll's games of logic were, they were neither technically original nor profound. In his formal works on logic, as in geometry, he remained a resolute conservative. The British logician R. B. Braithwaite has pointed out that Carroll "did not accept the doctrine that has done so much to simplify traditional formal logic the interpretation of a universal proposition as not involving the existence of its subject-term." Thus to Carroll the statement "All frogs that jump more than 20 feet croak loudly" necessarily implied the existence of frogs that jump more than 20 feet.

Near the end of his life, however, Carroll did make one formal contribution to logic that has intrigued serious mathematicians. It was a problem containing a paradox which no one has conclusively resolved. A barber shop has three barbers, A, B and C. (1) A is infirm, so if he leaves the shop B has to go with him. (2) All three cannot leave together, since then their shop would be empty. Now with these two premises, let us make an assumption and test its consequences. Let us assume that C goes out. Then it follows that if A goes out, B stays in (by premise 2). But (by premise 1) if A goes out, B goes out too. Thus our assumption that C goes out has led to a conclusion we know to be false. Hence the assumption is false, and hence C can't go out. But this is nonsense, for C obviously can go out without disobeying either of these restrictions. C can in fact go out whenever A stays in. Thus strict reasoning from apparently consistent premises leads to two mutually contradictory conclusions.

Bertrand Russell tried to get around the difficulty by saying that the statement "If A goes out then B must go out" is not contradictory to the statement that "If A goes out then B must stay in." He argues they can both be jointly true on the condition that "A stays in." But this is the same as arguing that there is no disagreement between the statement by one politician that "If the Republicans win, things will improve," and the statement by another politician that "If the Republicans win, things won't improve." Neither politician would be satisfied by a logician's assurance that a Socialist victory would make both their statements true. The paradox has recently been answered in a more complicated and interesting way by two logicians in a paper published in the British Journal Mind. The continuing discussion is evidence of the interest which still attaches to this logical puzzle. As Braith-waite observes, "Carroll was ploughing deeper than he knew. His mind was permeated by an admirable logic which he was unable to bring to full consciousness and explicit criticism. It is this that makes his Symbolic Logic so superficial ... and his casual puzzles so profound."

It would be hard to state better than Braithwaite does in these words the conclusion of the matter. The Reverend Dodgson was a dull, conscientious and capable teacher of elementary mathematics. Lewis Carroll was, in a tantalizingly elusive way, an excellent and unconsciously deep logician. But when he tried to approach logic head on, in a proper professional way, he was only moderately successful. It was when he let logic run loose that he demonstrated his true subtlety and depth. In fact, for a full measure of his stature as a logician we must look into Wonderland.

Alice and her companions have often been quoted in books on logic and philosophy. P. E. B. Jourdain, in his delightful book, The Philosophy of Mr. B*rt-r*nd R*ss*ll, relies heavily on Carroll to demonstrate the key notions of logic. The sampling of Carroll's virtuosity which follows is taken from that book.

Logicians have for ages struggled with "theories of identity." Just when is one justified in saying "X is identical with Y," or "X is the same as Y," or "X is Y"? This matter was entirely clear to Carroll's little friends:

"The day is the same length as anything that is the same length as it." (Sylvie and Bruno)

"Bruno observed that when the Other Professor lost himself, he should shout. 'He'd be sure to hear hisself, 'cause he couldn't be far off.' " (Sylvie and Bruno)

Most logicians and most of the rest of us for that matter have to be very careful about precision in definitions and about the confusing overlap between what words denote and what they connote. But this was not a matter of confusion on the other side of the Look-ing-Glass:

"Tweedledum and Tweedledee were, in many respects, indistinguishable, and Alice, walking along the road, noticed that 'whenever the road divided there were sure to be two finger-posts pointing the same way, one marked "to twee-dledum's house" and the other " THE HOUSE OF TWEEDLEDEE." '

" 'I do believe,' said Alice at last, 'that they live in the same house! . . .' "

" 'When I use a word,' Humpty Dumpty said in rather a scornful tone, 'it means just what I choose it to mean neither more nor less.'

" 'The question is,' said Alice, 'whether you can make words mean different things.'

' 'The question is,' said Humpty Dumpty, 'which is to be master that's all.'"

Extremely subtle and intricate problems in modern mathematical logic hinge on the question whether there exists any such thing as a universal class of all possible things. But even the Gnat had that one figured out. The Gnat had told Alice that the Bread-and-Butter-Fly lives on weak tea with cream in it:

" 'Supposing it couldn't find any?' she suggested.

" 'Then it would die, of course!'

" 'But that must happen very often,' Alice remarked thoughtfully.

" 'It always happens,' said the Gnat." (Through the Looking-Glass)

If existence is difficult to analyze, then nonexistence is perhaps even more elusive, but not to Alice:

" 'I see nobody on the road,' said Alice.

" 'I only wish I had such eyes,' the [White] King remarked in a fretful tone. 'To be able to see Nobody! And at that distance, too! Why, it's as much as I can do to see real people by this light!'" (Through the Looking-Glass)

"... The Dormouse went on, .. .: 'and they drew all manner of things everything that begins with an M.'

" 'Why with an M?' said Alice.

" 'Why not?' said the March Hare.

"Alice was silent.

".... [The Dormouse] went on: 'that begins with an M, such as mouse-traps, and the moon, and memory, and muchness, you know you say things are "much of a muchness did you ever see such a thing as a drawing of a muchness?'

" 'Really, now you ask me,' said Alice, very much confused, 'I don't think ...'

" 'Then you shouldn't talk,' said the Hatter." (Alice's Adventures in Wonderland)

Those interested in the logic of modern science are by no means agreed as to the significance and validity of imaginary experiments, particularly if they involve unrealizable conditions. This is a matter which did not worry the White Queen:

"Alice laughed. 'There's no use trying,' she said; 'one can't believe impossible things.'

" 'I dare say you haven't had much practice,' said the White Queen. 'When I was your age, I always did it for half-an-hour a day. Why, sometimes I've believed as many as six impossible things before breakfast.'" (Through the Looking-Glass)

So rare and so great were Carroll's true talents that we need not be condescending about the shortcomings of his formal mathematical writings. Carroll himself had no pretensions about them, and pronounced his own modest verdict in his diary. The very first entry in this two-volume record, which he wrote down on January 1, 1855, at the age of 23, says: "tried a little mathematics unsuccessfully."