Excerpt:
Take the chapter "Advice from a caterpillar", for
example. By this point, Alice has fallen down a rabbit hole and eaten a
cake that has shrunk her to a height of just 3 inches. Enter the
Caterpillar, smoking a hookah pipe, who shows Alice a mushroom that can
restore her to her proper size. The snag, of course, is that one side
of the mushroom stretches her neck, while another shrinks her torso.
She must eat exactly the right balance to regain her proper size and
proportions.
While
some have argued that this scene, with its hookah and "magic mushroom",
is about drugs, I believe it's actually about what Dodgson saw as the
absurdity of symbolic algebra, which severed the link between algebra,
arithmetic and his beloved geometry. Whereas the book's later chapters
contain more specific mathematical analogies, this scene is subtle and
playful, setting the tone for the madness that will follow.
===
Alice's adventures in algebra: Wonderland solved
Critical of the new mathematics (Image: Andrew Hem)
1 more image
What would Lewis Carroll's Alice's Adventures in Wonderland
be without the Cheshire Cat, the trial, the Duchess's baby or the Mad
Hatter's tea party? Look at the original story that the author told
Alice Liddell and her two sisters one day during a boat trip near
Oxford, though, and you'll find that these famous characters and scenes
are missing from the text.
As
I embarked on my DPhil investigating Victorian literature, I wanted to
know what inspired these later additions. The critical literature
focused mainly on Freudian interpretations of the book as a wild
descent into the dark world of the subconscious. There was no detailed
analysis of the added scenes, but from the mass of literary papers, one
stood out: in 1984 Helena Pycior
of the University of Wisconsin-Milwaukee had linked the trial of the
Knave of Hearts with a Victorian book on algebra. Given the author's
day job, it was somewhat surprising to find few other reviews of his
work from a mathematical perspective. Carroll was a pseudonym: his real
name was Charles Dodgson, and he was a mathematician at Christ Church
College, Oxford.
The
19th century was a turbulent time for mathematics, with many new and
controversial concepts, like imaginary numbers, becoming widely
accepted in the mathematical community. Putting Alice's Adventures in Wonderland
in this context, it becomes clear that Dodgson, a stubbornly
conservative mathematician, used some of the missing scenes to satirise
these radical new ideas.
Even
Dodgson's keenest admirers would admit he was a cautious mathematician
who produced little original work. He was, however, a conscientious
tutor, and, above everything, he valued the ancient Greek textbook
Euclid's Elements
as the epitome of mathematical thinking. Broadly speaking, it covered
the geometry of circles, quadrilaterals, parallel lines and some basic
trigonometry. But what's really striking about Elements is its
rigorous reasoning: it starts with a few incontrovertible truths, or
axioms, and builds up complex arguments through simple, logical steps.
Each proposition is stated, proved and finally signed off with QED.
For
centuries, this approach had been seen as the pinnacle of mathematical
and logical reasoning. Yet to Dodgson's dismay, contemporary
mathematicians weren't always as rigorous as Euclid. He dismissed their
writing as "semi-colloquial" and even "semi-logical". Worse still for
Dodgson, this new mathematics departed from the physical reality that
had grounded Euclid's works.
By
now, scholars had started routinely using seemingly nonsensical
concepts such as imaginary numbers - the square root of a negative
number - which don't represent physical quantities in the same way that
whole numbers or fractions do. No Victorian embraced these new concepts
wholeheartedly, and all struggled to find a philosophical framework
that would accommodate them. But they gave mathematicians a freedom to
explore new ideas, and some were prepared to go along with these
strange concepts as long as they were manipulated using a consistent
framework of operations. To Dodgson, though, the new mathematics was
absurd, and while he accepted it might be interesting to an advanced
mathematician, he believed it would be impossible to teach to an
undergraduate.
Outgunned
in the specialist press, Dodgson took his mathematics to his fiction.
Using a technique familiar from Euclid's proofs, reductio ad absurdum,
he picked apart the "semi-logic" of the new abstract mathematics,
mocking its weakness by taking these premises to their logical
conclusions, with mad results. The outcome is Alice's Adventures in Wonderland.
Algebra and hookahs
Take
the chapter "Advice from a caterpillar", for example. By this point,
Alice has fallen down a rabbit hole and eaten a cake that has shrunk
her to a height of just 3 inches. Enter the Caterpillar, smoking a
hookah pipe, who shows Alice a mushroom that can restore her to her
proper size. The snag, of course, is that one side of the mushroom
stretches her neck, while another shrinks her torso. She must eat
exactly the right balance to regain her proper size and proportions.
While
some have argued that this scene, with its hookah and "magic mushroom",
is about drugs, I believe it's actually about what Dodgson saw as the
absurdity of symbolic algebra, which severed the link between algebra,
arithmetic and his beloved geometry. Whereas the book's later chapters
contain more specific mathematical analogies, this scene is subtle and
playful, setting the tone for the madness that will follow.
The
first clue may be in the pipe itself: the word "hookah" is, after all,
of Arabic origin, like "algebra", and it is perhaps striking that Augustus De Morgan,
the first British mathematician to lay out a consistent set of rules
for symbolic algebra, uses the original Arabic translation in Trigonometry and Double Algebra,
which was published in 1849. He calls it "al jebr e al mokabala" or
"restoration and reduction" - which almost exactly describes Alice's
experience. Restoration was what brought Alice to the mushroom: she was
looking for something to eat or drink to "grow to my right size again",
and reduction was what actually happened when she ate some: she shrank
so rapidly that her chin hit her foot.
De
Morgan's work explained the departure from universal arithmetic - where
algebraic symbols stand for specific numbers rooted in a physical
quantity - to that of symbolic algebra, where any "absurd" operations
involving negative and impossible solutions are allowed, provided they
follow an internal logic. Symbolic algebra is essentially what we use
today as a finely honed language for communicating the relations
between mathematical objects, but Victorians viewed algebra very
differently. Even the early attempts at symbolic algebra retained an
indirect relation to physical quantities.
De
Morgan wanted to lose even this loose association with measurement, and
proposed instead that symbolic algebra should be considered as a system
of grammar. "Reduce" algebra from a universal arithmetic to a series of
logical but purely symbolic operations, he said, and you will
eventually be able to "restore" a more profound meaning to the system -
though at this point he was unable to say exactly how.
When Alice loses her temper
The
madness of Wonderland, I believe, reflects Dodgson's views on the
dangers of this new symbolic algebra. Alice has moved from a rational
world to a land where even numbers behave erratically. In the hallway,
she tried to remember her multiplication tables, but they had slipped
out of the base-10 number system we are used to. In the caterpillar
scene, Dodgson's qualms are reflected in the way Alice's height
fluctuates between 9 feet and 3 inches. Alice, bound by conventional
arithmetic where a quantity such as size should be constant, finds this
troubling: "Being so many different sizes in a day is very confusing,"
she complains. "It isn't," replies the Caterpillar, who lives in this
absurd world.
Wonderland's madness reflects Carroll's views on the dangers of the new symbolic algebra
The
Caterpillar's warning, at the end of this scene, is perhaps one of the
most telling clues to Dodgson's conservative mathematics. "Keep your
temper," he announces. Alice presumes he's telling her not to get
angry, but although he has been abrupt he has not been particularly
irritable at this point, so it's a somewhat puzzling thing to announce.
To intellectuals at the time, though, the word "temper" also retained
its original sense of "the proportion in which qualities are mingled",
a meaning that lives on today in phrases such as "justice tempered with
mercy". So the Caterpillar could well be telling Alice to keep her body
in proportion - no matter what her size.
This
may again reflect Dodgson's love of Euclidean geometry, where absolute
magnitude doesn't matter: what's important is the ratio of one length
to another when considering the properties of a triangle, for example.
To survive in Wonderland, Alice must act like a Euclidean geometer,
keeping her ratios constant, even if her size changes.
Of
course, she doesn't. She swallows a piece of mushroom and her neck
grows like a serpent with predictably chaotic results - until she
balances her shape with a piece from the other side of the mushroom.
It's an important precursor to the next chapter, "Pig and pepper",
where Dodgson parodies another type of geometry.
By
this point, Alice has returned to her proper size and shape, but she
shrinks herself down to enter a small house. There she finds the
Duchess in her kitchen nursing her baby, while her Cook adds too much
pepper to the soup, making everyone sneeze except the Cheshire Cat. But
when the Duchess gives the baby to Alice, it somehow turns into a pig.
The
target of this scene is projective geometry, which examines the
properties of figures that stay the same even when the figure is
projected onto another surface - imagine shining an image onto a moving
screen and then tilting the screen through different angles to give a
family of shapes. The field involved various notions that Dodgson would
have found ridiculous, not least of which is the "principle of
continuity".
Jean-Victor Poncelet,
the French mathematician who set out the principle, describes it as
follows: "Let a figure be conceived to undergo a certain continuous
variation, and let some general property concerning it be granted as
true, so long as the variation is confined within certain limits; then
the same property will belong to all the successive states of the
figure."
The
case of two intersecting circles is perhaps the simplest example to
consider. Solve their equations, and you will find that they intersect
at two distinct points. According to the principle of continuity, any
continuous transformation to these circles - moving their centres away
from one another, for example - will preserve the basic property that
they intersect at two points. It's just that when their centres are far
enough apart the solution will involve an imaginary number that can't
be understood physically (see diagram).
Of
course, when Poncelet talks of "figures", he means geometric figures,
but Dodgson playfully subjects Poncelet's "semi-colloquial" argument to
strict logical analysis and takes it to its most extreme conclusion.
What works for a triangle should also work for a baby; if not,
something is wrong with the principle, QED. So Dodgson turns a baby
into a pig through the principle of continuity. Importantly, the baby
retains most of its original features, as any object going through a
continuous transformation must. His limbs are still held out like a
starfish, and he has a queer shape, turned-up nose and small eyes.
Alice only realises he has changed when his sneezes turn to grunts.
The
baby's discomfort with the whole process, and the Duchess's unconcealed
violence, signpost Dodgson's virulent mistrust of "modern" projective
geometry. Everyone in the pig and pepper scene is bad at doing their
job. The Duchess is a bad aristocrat and an appallingly bad mother; the
Cook is a bad cook who lets the kitchen fill with smoke, over-seasons
the soup and eventually throws out her fire irons, pots and plates.
Alice,
angry now at the strange turn of events, leaves the Duchess's house and
wanders into the Mad Hatter's tea party, which explores the work of the
Irish mathematician William Rowan Hamilton. Hamilton died in 1865, just after Alice
was published, but by this time his discovery of quaternions in 1843
was being hailed as an important milestone in abstract algebra, since
they allowed rotations to be calculated algebraically.
Just as complex numbers work with two terms, quaternions belong to a number system based on four terms (see "Imaginary mathematics").
Hamilton spent years working with three terms - one for each dimension
of space - but could only make them rotate in a plane. When he added
the fourth, he got the three-dimensional rotation he was looking for,
but he had trouble conceptualising what this extra term meant. Like
most Victorians, he assumed this term had to mean something, so in the
preface to his Lectures on Quaternions of 1853 he added a
footnote: "It seemed (and still seems) to me natural to connect this
extra-spatial unit with the conception of time."
Where
geometry allowed the exploration of space, Hamilton believed, algebra
allowed the investigation of "pure time", a rather esoteric concept he
had derived from Immanuel Kant that was meant to be a kind of Platonic
ideal of time, distinct from the real time we humans experience. Other
mathematicians were polite but cautious about this notion, believing
pure time was a step too far.
The
parallels between Hamilton's maths and the Hatter's tea party - or
perhaps it should read "t-party" - are uncanny. Alice is now at a table
with three strange characters: the Hatter, the March Hare and the
Dormouse. The character Time, who has fallen out with the Hatter, is
absent, and out of pique he won't let the Hatter move the clocks past
six.
Reading
this scene with Hamilton's maths in mind, the members of the Hatter's
tea party represent three terms of a quaternion, in which the
all-important fourth term, time, is missing. Without Time, we are told,
the characters are stuck at the tea table, constantly moving round to
find clean cups and saucers.
Their
movement around the table is reminiscent of Hamilton's early attempts
to calculate motion, which was limited to rotatations in a plane before
he added time to the mix. Even when Alice joins the party, she can't
stop the Hatter, the Hare and the Dormouse shuffling round the table,
because she's not an extra-spatial unit like Time.
The
Hatter's nonsensical riddle in this scene - "Why is a raven like a
writing desk?" - may more specifically target the theory of pure time.
In the realm of pure time, Hamilton claimed, cause and effect are no
longer linked, and the madness of the Hatter's unanswerable question
may reflect this.
Alice's
ensuing attempt to solve the riddle pokes fun at another aspect of
quaternions: their multiplication is non-commutative, meaning that x × y is not the same as y × x.
Alice's answers are equally non-commutative. When the Hare tells her to
"say what she means", she replies that she does, "at least I mean what
I say - that's the same thing". "Not the same thing a bit!" says the
Hatter. "Why, you might just as well say that 'I see what I eat' is the
same thing as 'I eat what I see'!"
It's
an idea that must have grated on a conservative mathematician like
Dodgson, since non-commutative algebras contradicted the basic laws of
arithmetic and opened up a strange new world of mathematics, even more
abstract than that of the symbolic algebraists.
When
the scene ends, the Hatter and the Hare are trying to put the Dormouse
into the teapot. This could be their route to freedom. If they could
only lose him, they could exist independently, as a complex number with
two terms. Still mad, according to Dodgson, but free from an endless
rotation around the table.
And there Dodgson's satire of his contemporary mathematicians seems to end. What, then, would remain of Alice's Adventures in Wonderland without these analogies? Nothing but Dodgson's original nursery tale, Alice's Adventures Under Ground,
charming but short on characteristic nonsense. Dodgson was most witty
when he was poking fun at something, and only then when the subject
matter got him truly riled. He wrote two uproariously funny pamphlets,
fashioned in the style of mathematical proofs, which ridiculed changes
at the University of Oxford. In comparison, other stories he wrote
besides the Alice books were dull and moralistic.
I would venture that without Dodgson's fierce satire aimed at his colleagues, Alice's Adventures in Wonderland would never have become famous, and Lewis Carroll would not be remembered as the unrivalled master of nonsense fiction.
Imaginary mathematics
The
real numbers, which include fractions and irrational numbers like π
that can nevertheless be represented as a point on a number line, are
only one of many number systems.
Complex
numbers, for example, consist of two terms - a real component and an
"imaginary" component formed of some multiple of the square root of -1,
now represented by the symbol i. They are written in the form a + bi.
The
Victorian mathematician William Rowan Hamilton took this one step
further, adding two more terms to make quaternions, which take the form
a + bi + cj + dk and have their own strange rules of arithmetic.
Melanie
Bayley is a DPhil candidate at the University of Oxford. Her work was
supported by the UK's Arts and Humanities Research Council
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