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.

About Peter Cameron

I count all the things that need to be counted.
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18 Responses to Lewis Carroll and algebra

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  2. Sam Tarzi says:

    It was Oliver Heaviside who pushed for a description of equations in terms of vectors instead of quaternions, for example re-formulating Maxwell’s equations of electromagnetism in terms of vectors. So perhaps by 1905, quaternions were dead as you say. However, there are quaternionic formulations of Special Relativity.

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  4. Melanie Bayley says:

    A fascinating resume and extension of the article. To follow up on the Boole references: there’s no evidence that Dodgson knew of Boole’s work on logic when he wrote the ‘Alice’ books. Boole’s contribution went almost unnoticed in the nineteenth century and was only formally recognised when Russell and Whitehead examined the philosophy of maths in the early 1900s. A more likely source of inspiration is Augustus De Morgan’s ‘Formal Logic’ (1847). De Morgan’s work was better known than Boole’s in the 1860s and Dodgson’s own texts on symbolic logic (1887, 1895) more closely reflect De Morgan’s approach. Wish I could comment on quaternions and relativity, but it would take several pages!

  5. Contrariwise says:

    Nice theory, but I don’t really think it convinces. Admittedly it isn’t the MOST far-fetched Alice-theory, but that’s not saying very much, sadly 🙂


  6. Paul Raymont lists me as disagreeing with Melanie Bayley’s thesis. I don’t really think that is so; really, like so much of this blog, I was just thinking aloud.

    As to Contrariwise, I don’t see what’s sad about “far-fetched Alice-theories”.

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  8. Contrary to Contrarywise, I must say Bayley’s idea of the Mad Hatter’s tea party being a satire on quaternions clicked before I had even finished reading it. Like most good ideas, it’s so obvious once pointed out.

    Of all the past people I’d like to meet in some parallel universe, Charles Dodgson is easily in the top ten. He must have been a fascinating guy. Had he been alive today, I’ve no doubt he would be organising and coaching the British IMO team!

    As for the vexed question of him having been a kiddie fiddler, in thought if not in deed, I read that a recent biographer had some grounds for claiming he befriended young girls solely as a pretext to become more intimate with their mothers! Sadly his niece, or some relative, burned all his letters after his death (as that was considered almost as scandalous in those days); so I guess we’ll never know.

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  10. jacob says:

    Keeping a fresh blog is challenging and extremely tiresome. You’ve got pulled it off well though.

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  12. Ralph Dratman says:

    While growing up, C. L. Dodgson was accustomed to the company of girls, because he had seven sisters. If he had not enjoyed the company of young girls, we might never have had his wonderful books.

    In many years of reading about Dodgson, I have never seen a suggestion that he harmed, abused or sexualized a single child. Later in life, people who knew him in childhood seem to have thought of him with great affection. That is not a sign of child abuse.

    Every one of us human beings thinks about sex in many different ways. We have no choice. We are constructed with the obligatory powerful reproductive instincts, without which no species can survive for more than one generation. We are also gifted with a large brain which never stops randomly putting ideas together. We cannot get rid of that proclivity either.

    Meanwhile, large numbers of respected religious leaders (and lately, sports coaches) are found to have severely abused children with depressing regularity, while the leadership hierarchy transferred and protected them year after year, decade after decade.

    Dodgson, by contrast, created touching and often brilliant ways of expressing his love for children, particularly girls. In doing so he brought into the world a quantity of exceptionally charming, thought-provoking writing.

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  14. jd2718 says:

    Ten years late to the party, but I recently opened a some old textbooks looking for how they handled matrices – both are College Algebra from the US from the 1920s, and both had full chapters on determinants, with nary a mention of a matrix. (lending a little support to one of your suppositions)

    • Ralph Dratman says:

      How do you write about determinants without writing about matrices? I would have thought you would need a matrix to calculate a determinant, silly me.

      • Good question, and it shows just how far our ways of thought have changed. A determinant was originally a function of n^2 arguments which told you something interesting (e.g. if they were the coefficients of a system of linear equations, whether the equations had a unique solution or not). The idea of a matrix as a representation of an operator on a vector space is more abstract, and did indeed come later. My undergraduate algebra book had a chapter called “determinants of orders 2, 3 and 4”, and in the last case gave simple formulae for calculating them in some special cases.
        An example of this usage is the notion of the resultant of two polynomials, a determinant involving the coefficients which tells you whether or not the two polynomials have a common root.

      • Ralph Dratman says:

        Thanks so much for your reply, Peter. If you wouldn’t mind too much, could you please expand your last paragraph, the one starting with “An example of this usage” — which I find awfully interesting — with a numerical example of how the criterion of two polynomials having a common root is used? Just so I can see the whole process? That would cheer my day a lot!

  15. jd2718 says:

    Consider first the system,
    ax + b = 0, a≠0. (1)
    a’x^2 + b’x + c’ =0, a’≠0. (2)
    consisting of one linear and one quadratic equation. In order that (1) and (2) have a common root, it is necessary and sufficient that the solution x = -b/a which satisfies (1) shall also satisfy (2). This requires that
    a’b^2/a^2 – b’b/a + c’ = 0 (3)
    a’b^2 – abb’ + a^2c’ = 0 (4)

    The relation (4) among the coefficients is the condition that (1) and (2) have a common root.
    The left-hand member of (4) may be put into determinant form as follows: Multiply (1) by x, and the resulting equation in combination with (1) and (2) gives the system
    ax + b = 0 (1)
    ax^2 + bx =0 (5)
    a’x^2 + b’x + c’ =0 (2)

    which should be thought of as linear equations in two unknowns, x and x^2. From Art. 194 [Article 194, the previous section, which is “Systems of equations containing fewer unknowns than equations” – jd] it is necessary that

    [here the author has a 3×3 matrix, bracketed with vertical bars, set equal to 0. I will first attempt Latex, where I am unsure of myself, then in row by row text – jd]

    \begin{bmatrix}  0 & a & b \\ a & b & 0\\ a' & b' & c' \\ \end{bmatrix} = 0     (6)

    {0, a, b
    {a, b, 0
    {a’, b’, c’
    = 0. (6)
    in order that (1), (2), and (5) have a common root. But (6) is merely (4) written in determinant form, as can easily be verified.

    From Introductory College Algebra, H. L. Rietz and A. R. Crathorne, Henry Holt and Company, New York, 1923

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