Lewis Carroll and algebra

Charles Dodgson was one of the most famous mathematicians of all time. I mean that literally; he was not at all a renowned mathematician, spending his career as lecturer in mathematics at Christ Church, Oxford, but he was famous as Lewis Carroll, the author of Alice’s Adventures in Wonderland and Through the Looking-Glass, nonsense in a style which seems to have a particular appeal to mathematicians.

Once upon a time, I had two books giving explanations and interpretation of Lewis Carroll’s Alice books: The Annotated Alice edited by Martin Gardner (with long footnotes explaining the jokes and obscure references), and Aspects of Alice, edited by Robert Phillips (a volume of critical essays discussing many different features of the books). Somewhere along the way, the second of these has gone missing, though I still have the first. But I have also acquired the Wordsworth edition of The Complete Lewis Carroll, edited by Alexander Woollcott.

In the Christmas edition of New Scientist, there is usually a light-hearted piece about mathematics by Ian Stewart. This year, for a change, we have an article by Melanie Bayley, a DPhil student in Oxford writing a thesis on Victorian literature. She proposes a new interpretation of some of Carroll’s writing.

Several chapters were added to Alice’s Adventures Underground, the first (private) edition of Carroll’s classic, before it appeared as Alice’s Adventures in Wonderland. Bayley claims that these chapters form a satire on modern developments in algebra: Carroll was troubled by such ideas as non-commutativity, and this was his response.

Armed with my copy of The Complete Lewis Carroll (which, sad to say, belies its title by including none of his mathematical work), I decided to test Bayley’s thesis.

It is true that much of Dodgson’s mathematics was conservative and old-fashioned, even in his day. His geometry was that of Euclid; his logic, that of Aristotle.

Traces of this are visible in the items printed in The Complete Lewis Carroll. The editor of that work reproduces an extract from Symbolic Logic, mainly a set of sixty “logic puzzles”, in which conclusions such as “Donkeys are not easy to swallow” or “No engine-driver lives on barley-sugar” are to be derived from sets of up to ten premises by means of classical syllogisms. In one of his polemic pieces, on a new belfry at Christ Church (this might well have been a blog, had the technology existed!), he refers to three forms of the classical syllogism, Barbara, Celarent, and Festino.

Logic has moved on a long way since then. Syllogisms are, in essence, a very strait-jacketed way of deriving a conclusion from a pair of hypotheses; both hypotheses and conclusion must be in one of the four forms “All A is B”, “No A is B”, “Some A is B”, or “Some A is not B”, represented by the letters A, E, I, O respectively (the vowels representing the three components in a valid syllogism are combined into a mnemonic. Thus “Barbara” is the syllogism

All B is C
All A is B
Ergo, All A is C

Thankfully, I was never made to learn the list of mnemonics (though, curiously enough, I first learned about them from one of my sister’s high-school textbooks). An explanation can be found here.

Today logic goes far beyond this. First, the proposition “All A is B” is recognised as being a conflation of two things, implication (“is”) and quantification (“all”). Implication belongs to what is known as Propositional Logic. Even in Carroll’s time, George Boole was developing a very flexible instrument for this, in Boolean algebra (a subject which according to Bayley’s interpretation would be anathema to Dodgson since the variables obey strange laws not satisfied by “ordinary” numbers). With the combination of Boolean algebra and the method of truth tables, all logical relationships between simple or “atomic” propositions can be worked out very easily.

Incidentally, in Boolean algebra (interpreting, as usual, “or” as +, and “and” as ×, we have, in addition to the familiar distributive law

a(b+c) = ab+ac,

which holds for ordinary numbers, also a much less familiar one, namely

a+bc = (a+b)(a+c).

It is easy to see how a conservative mathematician would be troubled by this. (I am troubled too, for a different reason. I think it is not sensible to use + and × for the Boolean operations, since it obscures the symmetry between them.)

There is an alternative view here, which is expressed by Bertrand Russell in the marvellous Logicomix, a fictional account of Russell’s search for logical certainty. At a certain point he says

“Lewis Carroll”, a.k.a. Mr Dodgson, is an expert in Boole’s ideas! The said Boole being the man who made logic as clear as algebra!

Of course this is only an interpretation; I don’t know whether Russell thought this. But it does sit uneasily with Bayley’s view of Carroll.

Quantification is part of a more general process recognising that propositions themselves, like atoms, have internal structure, involving firstly the objects of mathematics (constants, variables, functions, and relations, including the relation of equality), and then new operations such as quantification over specific variables. Perhaps the result looks sufficiently unlike algebra that Dodgson would not have felt threatened by it? I suppose we will never know.

No Dodgsonian geometry is reproduced in The Complete Lewis Carroll, but in some sense there is Carrollian geometry. In another polemic entitled “The Dynamics of a Parti-cle”, he comments on a contemporary Oxford controversy (unspecified by the editor) in a form which parodies Euclid. It begins with definitions, of which the first states:

Plain superficiality is the character of a speech, in which any two points being taken, the speaker is found to lie wholly with regard to those two points

and proceeds through postulates, axioms, propositions, etc. Triangles are referred to, as usual in Euclid, by three letters supposedly representing the vertices. In this case they are undoubtedly the initials of people prominent in the controversy, but again the editor leaves us uninformed.

So are there any clues to Dodgson’s attitude to the mathematics of his day?

A well-known anecdote about him is repeated by the editor. Queen Victoria, it seems, was charmed by Alice’s Adventures in Wonderland, and graciously invited its author to dedicate his next work to her. The Queen was not amused when the next work turned out to be An Elementary Treatise on Determinants.

Now, to a modern mathematician, determinants are intimately associated with matrices; and the algebra of matrices is one which, on Bayley’s thesis, Dodgson would have rejected. (After all, matrix multiplication is non-commutative.) But my understanding is that the theory of determinants pre-dated that of matrices by a considerable time. A determinant, as its name suggests, is a function which gives some information, for example, about whether a system of equations has a unique solution. In algebra, one meets the discriminant of a polynomial, or the resultant of a pair of polynomials, which gives information about multiple or common roots. In calculus, the Jacobean of two functions, or the Wronskian of a family of solutions to a differential equation, play a similar role. All of these are defined as particular determinants.

Without calculating engines, mathematicians are reluctant to solve systems of equations larger than about three, and for these it suffices to have a formula for the determinant, or a mnemonic for calculating it. (In the 3×3 case, imagine the “matrix” tiling the plane, and add the products of terms on the northwest-southeast diagonals, subtract those on the northeast-southwest diagonals. There is no similar formula for larger determinants.) We are kept well away from scary non-commutativity. Indeed, from a modern perspective,

det(AB) = det(A)det(B) = det(B)det(A)= det(BA),

so the non-comutativity has been successfully neutralised!

Bayley spends some time on the subject of quaternions, perhaps the first non-commutative algebra to catch the attention of mathematicians. A quaternion has four coordinates, conveniently regarded as a scalar and a 3-dimensional vector. Quaternion addition and multiplication incorporate all the basic operations with these objects. To add two quaternions, we add the scalar parts and the vector parts. To multiply quaternions, the scalar part of the product is the product of the scalars minus the dot product of the vectors; the vector part is the sum of the vector part of each quaternion multiplied by the scalar part of the others, plus the cross product of the vectors.

Similarly, a quaternion field has both a scalar and a vector part (just like the electromagnetic field); its spatial derivative involves the gradient of the scalar part and the divergence and curl of the vector part.

There is a small historical mystery here, to which I don’t know the answer. Quaternions seem ideally suited for the formalism of general relativity, with the scalar part representing time and the vector part space. Indeed, fifty years earlier, Hamilton had already had the idea of using the scalar part of quaternions to represent time. Bayley quotes the preface of his Lectures on Quaternions in 1853:

It seemed (and still seems) to me natural to connect this extra-spatial unit with the conception of time.

Einstein was happy to use differential geometry in general relativity ten years later. Why not quaternions in special relativity? Or were quaternions dead by then?

Bayley’s thesis is that the Mad Hatter’s tea party satirises quaternions. The Mad Hatter, March Hare and Dormouse represent the three spatial dimensions; Time, who had left in a huff after being “murdered” by the Mad Hatter at the Queen’s concert, was absent, so it was always six o’clock and the tea party continued indefinitely. Quaterions conveniently describe 3-dimensional rotations, which Bayley links to the rotation of the Hatter and the others around the tea-table.

Far-fetched? Who knows? Certainly less far-fetched than some of the interpretations in Aspects of Alice!

None of the above counts as history, of course; but it is fun to speculate, and the fun should not be left entirely to students of literature.