It’s My Own Invention

Charles Lutwidge Dodgson (27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, wrote Alice in Wonderland and Through the Looking–Glass, and What Alice Found There. The title of this editorial is a quote by the White Knight who is is a daydreaming inventor, a foolish and awkward man who is highly sentimental, and his cleverness is entirely impractical.

In her essay⁸, Serra Ansay posits that the White Knight’s inventions parodies Victorian technology and Victorian fascination with inventions, and states: “Perhaps Carroll cautions against invention getting out of hand and losing its original purpose: inventing for the sake of inventing rather than facilitating life.” I agree with Ms. Ansay since some of the main inventions of the Industrial Revolution (the period from about 1760 to sometime between 1820 and 1840) include the following.

Automobile
Cotton gin
Electric battery
Gas lighting
Gas turbine
Jacquard loom
Mercury thermometer
Propeller
Sewing machine
Steam engine
Steamboat
The typewriter

Since Mr. Carroll was a mathematician, I suggest that he might use the White Knight also to mock the “inventions” of mathematicians since he lived in a time of great changes in the field of mathematics.

Note what author Deepika Asthana⁷ said about Charles Dodgson: “He was a conservative and passionate mathematician who by most accounts was distressed by all the new theories in mathematics that were being sloshed about. It is believed by many that Alice In Wonderland was, in fact, a mathematical satire and Carroll’s way of belittling and perhaps even maintaining equilibrium in the face of dramatic change. Magic mushrooms, babies turning into pigs, and absurd questions (‘why is a raven like a writing desk?’) were perhaps, all meant to show how pointless and annoying these new theories were.”

Note what Wikipedia¹ records about 19th Century mathematics some of which may have distressed Charles Dodgson.

This century saw the development of the two forms of non-Euclidean geometry, where the parallel postulate of Euclidean geometry no longer holds. The Russian mathematician Nikolai Ivanovich Lobachevsky and his rival, the Hungarian mathematician János Bolyai, independently defined and studied hyperbolic geometry, where uniqueness of parallels no longer holds. In this geometry the sum of angles in a triangle add up to less than 180°. Elliptic g e ometry was developed later in the 19th century by the German mathematician Bernhard Riemann; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developed Riemannian geometry, which unifies and vastly generalizes the three types of geometry.

The 19th century saw the beginning of a great deal of abstract algebra. Hermann Grassmann in Germany gave a first version of vector spaces, William Rowan Hamilton in Ireland developed noncommutative algebra. The British mathematician George Boole devised an algebra that soon evolved into what is now called Boolean algebra, in which the only numbers were 0 and 1. Boolean algebra is the starting point of mathematical logic and has important applications in computer science.

Augustin-Louis Cauchy, Bernhard Riemann, and Karl Weierstrass reformulated the calculus in a more rigorous fashion.

Also, for the first time, the limits of mathematics were explored. Niels Henrik Abel, a Norwegian, and Évariste Galois, a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four (Abel–Ruffini theorem). Other 19th-century mathematicians utilized this in their proofs that straightedge and compass alone are not sufficient to trisect an arbitrary angle, to construct the side of a cube twice the volume of a given cube, nor to construct a square equal in area to a given circle. Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks. On the other hand, the limitation of three dimensions in geometry was surpassed in the 19th century through considerations of parameter space and hypercomplex numbers.

In the later 19th century, Georg Cantor established the first foundations of set theory, which enabled the rigorous treatment of the notion of infinity and has become the common language of nearly all mathematics. Cantor’s set theory, and the rise of mathematical logic in the hands of Peano, L. E. J. Brouwer, David Hilbert, Bertrand Russell, and A.N. Whitehead, initiated a long running debate on the foundations of mathematics.

Math in Alice in Wonderland and Through the Looking–Glass, and What Alice Found There

Quaternions

Alice, angry now at the strange turn of events, leaves the Duchess’s house and wanders into the Mad Hatter’s tea party. This, Bayley surmises, explores the work of the Irish mathematician William Rowan Hamilton, who died in 1865, just after Alice was published. Hamilton’s discovery of quaternions in 1843 was 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. 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 conceptualizing 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.”

As Bayley points out, the parallels between Hamilton’s mathematics and the Mad Hatter’s tea 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 ideas 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 that Dodgson would have found absurd: their multiplication is non-commutative. 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’!”

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.¹

Birthday Paradox

Lewis Carroll, the writer of Alice’s Adventures in Wonderland, was also a mathematician. It has been suggested that that there are many references to mathematical concepts in his story. One example of this occurs in the “unbirthday” party scene. In this scene, many believe that Carroll might have been inspired the Birthday Paradox. The Birthday Paradox – also known as the Birthday Problem – states that in a random group of just 23 people, there is approximately a 50% probability that there will be at least a pair of people sharing the same birthday. The Birthday Paradox, although strange and counterintuitive to most people, can be proven using key mathematical concepts and probabilistic systems. In order to appreciate the solution to the Birthday Paradox, one must have, first and foremost, a solid understanding of the power of compound exponential growth. This concept in itself isn’t intuitive, and is arguably one of the main sources of confusion surrounding this paradox. Aside from exponential growth, another element of confusion lies in the fact that many people fail to consider that a comparison is made between every possible pair of people in the room, rather than fixing one individual and comparing them to the rest of the guests. In the case of the Birthday Paradox, we are essentially analyzing 253 different pairs of people and looking for a birthday match. This high number of pairs observed within a group of as little as 23 people is also highly counterintuitive to most. A sound understanding of the concept of permutations and combinations is required to be able to accept it. Although it is highly unlikely that any individual pair of people will share the same birthday, the likeliness of at least one match can become significant when taking into account higher number of “tries”.²

Number Systems and Bases

Alice herself isn’t the focus of Carroll’s ire, so while she thinks circuitously about mathematics, and make mistakes, she’s mostly the straight man for the characters of Wonderland. She gets us started with mathematical concepts early on in the proceedings, when she’s still shrinking down. She wonders if she can shrink forever, getting smaller and smaller, or if she’ll eventually reach the point of nothingness. Where, exactly, is the mathematical partition between a very small something, and nothing at all?

Later, when she gets bigger and attempts to do math, she gets mixed up. She tries simple multiplication, but comes up with four times five equaling twelve, four times six becoming thirteen, and four times seven turning into fourteen.

In regular math, of course, this doesn’t work. If, however, you mess around with the base systems, things change. We work in base ten, meaning we have zero-through-nine digits, and then when we get to ten we move over and put a one in the next column. Alice was calculating in base ten, but her answers slipped into higher base systems. Four times five is twenty, which in base eighteen is one (1) group of eighteen, and two (2) extra singles, making 12. Four times six is twenty-four, got changed to a base twenty-one system, with one (1) group of twenty-one, plus three (3) extra singles, or 13. Four times seven is twenty-eight, but if you change that to a base twenty-four system, that’s one (1) group of twenty-four, and four (4) extra singles, or 14. When you change the system of measurement, but keep thinking of it as the original standard, you can pile on the numbers and never get anywhere, leaving you as lost as Alice.³

Hidden Math

First, we have to remind ourselves of what was going on in mathematics in the latter half of the nineteenth century, when Dodgson wrote his story. It was a turbulent period for mathematicians, with the subject rapidly becoming more abstract. The discoveries of non-Euclidean geometries, the development of abstract (symbolic) algebra that was not tied to arithmetic or geometry, and the growing acceptance – or at least use – of “imaginary numbers” were just some of the developments that shook the discipline to its core. By all accounts, Dodgson held a very traditionalist view of mathematics, rooted in the axiomatic approach of Euclid’s Elements. (He was not a research mathematician, rather he tutored the subject.) Bayley describes him as a “stubbornly conservative mathematician,” who was dismayed by what he saw as the declining standards of rigor. The new material Dodgson added to the Alice story for publication, she says, was a wicked satire on those new developments.

Perhaps the most obvious example is the Cheshire Cat, which disappears leaving only its grin, an obvious reference – critical in Dodgson’s case – to increasing abstraction in the discipline.

For a more focused example, take the chapter “Advice from a caterpillar.” Alice has fallen down the rabbit hole and eaten a cake that has shrunk her to a height of just 3 inches. The Caterpillar enters, smoking a hookah pipe, and shows Alice a mushroom that can restore her to her proper size. But one side of the mushroom stretches her neck, while another shrinks her torso, so she must eat exactly the right balance to regain her proper size and proportions. Bayley believes this expresses Dodgson’s view of the absurdity of symbolic algebra.

The first clue, she says, may be the pipe. The word “hookah” is of Arabic origin, like “algebra”. More to the point, the original Arabic term for algebra, widely known and used in the mathematical community in Dodgson’s time, was al jebr e al mokabala or “restoration and reduction” – which 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.

Bayley suggests that the overall madness of Wonderland 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 tries to remember her multiplication tables, but they have slipped out of the base-10 number system she is used to.

In the caterpillar scene, 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.

The Caterpillar’s warning, at the end of this scene, is perhaps one of the most telling clues to Dodgson’s conservative mathematics, Bayley suggests. “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 say. But the word “temper” has another meaning of “the proportion in which qualities are mingled.” So the Caterpillar could well be telling Alice to keep her body in proportion – no matter what her size. This may be another reflection of Dodgson’s love of Euclidean geometry, where absolute magnitude doesn’t matter: what’s important is the ratio of one length to another. 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. This is 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 turns into a pig.

According to Bayley, the target of this scene is projective geometry, a subject that involved concepts that Dodgson would have found ridiculous, particularly the “principle of continuity.” Jean-Victor Poncelet, the French mathematician who set out the principle, described 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.”

When Poncelet talked of “figures”, he meant geometric figures, of course, but Dodgson playfully subjects Poncelet’s description to strict logical analysis and takes it to its most extreme conclusion. He 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 realizes 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, Bayley says. 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.

The White Knight

Upon leaving the Lion and Unicorn to their fight, Alice reaches the seventh rank by crossing another brook into the forested territory of the Red Knight, who is intent on capturing the “white pawn”—Alice—until the White Knight comes to her rescue. Escorting her through the forest towards the final brook-crossing, the Knight recites a long poem of his own composition called Haddocks’ Eyes, and repeatedly falls off his horse.

The White Knight saves Alice from his opponent (the Red Knight). He repeatedly falls off his horse and lands on his head, and tells Alice of his inventions, which consists of things such as a pudding with ingredients like blotting paper, an upside down container, and anklets to guard his horse against shark bites. He recites a poem of his own composition, ‘A-Sitting on a Gate’, (but the song’s name is called ‘Haddocks’ Eyes‘) and he and Alice depart.

“Haddocks’ Eyes” is an example used to elaborate on the symbolic status of the concept of “name“: a name as identification marker may be assigned to anything, including another name, thus introducing different levels of symbolization. It has been discussed in several works on logic and philosophy.⁵

References

¹ “Quaternions: Part Of The Hidden Math Behind Alice In Wonderland”. 2011. GA Net Updates. https://gaupdate.wordpress.com/2011/07/26/quaternions-part-of-the-hidden-math-behind-alice-in-wonderland/.

² “The Birthday Paradox (Alice In Wonderland)”. 2016. Math In Popular Media. https://mathinmediablog.wordpress.com/2016/08/23/aliceinwonderland/.

³ “A Math-Free Guide To The Math Of Alice In Wonderland | Article | Abakcus”. 2022. Abakcus. https://abakcus.com/article/a-math-free-guide-to-the-math-of-alice-in-wonderland/.

⁴ “The Hidden Math Behind Alice In Wonderland”. 2022. maa.org. https://www.maa.org/external_archive/devlin/devlin_03_10.html.

⁵ “Haddocks’ Eyes – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Haddocks%27_Eyes.

⁶ “19th Century In Science – Wikipedia”. 2018. en.wikipedia.org. https://en.wikipedia.org/wiki/19th_century_in_science.

⁷ “The Hidden Math Behind Alice In Wonderland | The Curious Reader”. 2018. The Curious Reader. https://www.thecuriousreader.in/features/alice-in-wonderland-math/.

⁸ Ansay, Serra. “Inventions In Alice In Wonderland “. 1995. victorianweb.org. https://www.victorianweb.org/authors/carroll/ansay.html.

Additional Reading

“A Math-Free Guide To The Math Of Alice In Wonderland”. 2022. Gizmodo. https://gizmodo.com/a-math-free-guide-to-the-math-of-alice-in-wonderland-5907235.

“Alice’s Adventures In Wonderland – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Alice%27s_Adventures_in_Wonderland.

“Alice In The Land Of Mathematics: 150 Years Later | OpenMind”. 2015. OpenMind. https://www.bbvaopenmind.com/en/science/mathematics/alice-in-the-land-of-mathematics-150-years-later/.

Bayley, Melanie. “Alice’s Adventures In Algebra: Wonderland Solved | New Scientist”. 2022. Newscientist.Com. https://www.newscientist.com/article/mg20427391-600-alices-adventures-in-algebra-wonderland-solved/.

Danim, Rachel. “Alice In Wonderland: A Satire On Math”. 2021. Medium. https://medium.com/@connectrachelfilms/alice-in-wonderland-a-satire-on-math-a85f6ab6af42.

“Lewis Carroll – Timeline Of Mathematics – Mathigon”. 2022. Mathigon. https://mathigon.org/timeline/carroll.

Charles Lutwidge Dodgson (1832 – 1898) is best know under his pen name Lewis Carroll, as the author of Alice’s Adventures in Wonderland and its sequel Through the Looking-Glass. However, Carroll was also a brilliant mathematician. He always tried to incorporate puzzles and logic into his children’s stories, making them more enjoyable and memorable.

⭐ Leung, Barry. “Math with Alice’s Adventures in Wonderland”. 2022. Medium. https://barryleung.medium.com/math-with-alices-adventures-in-wonderland-425252687929.

The Tea Party scene has taught us one thing about life. Approaching daily life in a non-commutative way is perhaps the way to go. Because if we can like what we get and get what we like at the same time, we will be happy and fulfilled!

“Math – Lewis Carroll Society Of North America”. 2022. lewiscarroll.org. https://www.lewiscarroll.org/carroll/study/math/.

“Pictures From Through The Looking-Glass – Alice-In-Wonderland.Net”. 2022. Alice-In-Wonderland.Net. https://www.alice-in-wonderland.net/resources/pictures/through-the-looking-glass/.

“The Mathematics of Alice in Wonderland”. 2022. massline.org. http://massline.org/ScottH/science/MathOfAliceInWonderland-100308.pdf.

“Through The Looking-Glass Chapter 8: “It’s My Own Invention” Summary & Analysis | sparknotes”. 2022. sparknotes. https://www.sparknotes.com/lit/through-the-looking-glass/section8/.

“Through The Looking-Glass – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Through_the_Looking-Glass.

“What Alice Found There (Part 1) | This Side of the Science Delusion: Lewis Carroll’s Adventures in a Wonderland of Imaginary Numbers”. 2022. The Unexpected Cosmology. https://theunexpectedcosmology.com/what-alice-found-there-this-side-of-the-science-delusion-lewis-carrolls-adventures-in-a-wonderland-of-imaginary-numbers-part-1/.

“What Alice Found There (Part 2) | “Keep Your Temper”—A Reaction to Absurdity”. 2022. The Unexpected Cosmology. https://theunexpectedcosmology.com/what-alice-found-there-this-side-of-the-science-delusion-lewis-carrolls-adventures-in-a-wonderland-of-imaginary-numbers-part-2/.

“What Alice Found There (Part 3) | How the “Principle of Continuity” Transforms a Baby Into a Pig…Naturally”. 2022. The Unexpected Cosmology. https://theunexpectedcosmology.com/what-alice-found-there-part-3-the-principle-of-continuity-implies-a-baby-is-also-a-pignaturally/.

“What Alice Found There (Part 4) | The Fourth Dimension: “Freeing the Intellect from the Shackles of the Actual””. 2022. The Unexpected Cosmology. https://theunexpectedcosmology.com/what-alice-found-there-part-4-the-fourth-dimension-freeing-the-intellect-from-the-shackles-of-the-actual/.

Quite unlike pure time, for a conservative mathematician like Lewis Carroll, the non-Euclidean world of Wonderland was pure madness. Non-commutative algebras contradicted the basic laws of arithmetic, and as we’ve come to see, opened up strange new worlds, where the imagination and abstraction roams free. But as always, Alice said it best.

“Let me think. Was I the same when I got up this morning? … But if I’m not the same, the next question is, who in the world am I? Ah, that’s the great puzzle!”

“White Knight (Through The Looking-Glass) – Wikipedia”. 2014. en.wikipedia.org. https://en.wikipedia.org/wiki/White_Knight_(Through_the_Looking-Glass).

Wilson, Phil. “‘Lewis Carroll In Numberland'”. 2008. plus.maths.org. https://plus.maths.org/content/lewis-carroll-numberland.

Wilson, Robin. Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life. New York: W. W. Norton & Company, Inc., 2008.

Just when we thought we knew everything about Lewis Carroll, here comes this “insightful . . . scholarly . . . serious” (John Butcher, American Scientist) biography that will appeal to Alice fans everywhere. Fascinated by the inner life of Charles Lutwidge Dodgson, Robin Wilson, a Carroll scholar and a noted mathematics professor, has produced this revelatory book—filled with more than one hundred striking and often playful illustrations—that examines the many inspirations and sources for Carroll’s fantastical writings, mathematical and otherwise. As Wilson demonstrates, Carroll made significant contributions to subjects as varied as voting patterns and the design of tennis tournaments, in the process creating large numbers of imaginative recreational puzzles based on mathematical ideas.

Videos

Hidden Math in Alice in Wonderland

Mathematics in Literature: Part 2 – Alice in Wonderland

An analysis about Alice in Wonderland as part of the mathematical discussion happening in Victorian times about the Euclid vs Non-Euclid geometry.

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.