WHAT MIGHT CAUSE THE CHESHIRE CAT TO VANISH of everything but its grin? An increasing abstraction in mathematical discipline would surely be to blame, naturally. To claim that each of us inhabit Wonderland will come as no surprise to some of you. Perhaps the far more practical question is; are we trying to make sense of ourselves in proportion to the nonsensical world surrounding us, as Alice might—or like the creatures which tormented her tutoring; have we settled in as one of Wonderland’s curiouser and curiouser residents? Hold off on your response, because author Lewis Carroll drew a line of the mathematical nature in the sand, and which side we find ourselves on, based of course on the author’s own account of the Science delusion—particularly the math which makes its fabrications entirely possible—will conveniently answer the question as to where we should stand.
Charles Dodgson, pen name Lewis Carroll, was a stubbornly conservative mathematical lecturer and tutor at Christ Church College in the University of Oxford, tasked with preparing its students to pass examinations in the numbers department. For the record, it was the 19th century. His was a generation of swirling tempests, when new controversial concepts in mathematics, such as imaginary numbers (like the square root of a negative number), symbolic algebra, and projective geometry—characteristic expressions only found in allegorical formalism—were explored and ultimately exploited as groundwork to expand whatever perceived realities its master magicians could conjure.
Throughout earlier centuries—for more than two-thousand years, in fact, Euclid’s Elements had been the personification of rationality. The very adjective “Euclidean” was unnecessary on the mere basis that no other sort of geometry had been conceived. His was grounded in a physical reality and backed by rigorous reasoning. Theorems which might fall under his axioms, once proven, were deemed an absolute truth. In such a world, mathematical objects were conceived as the ideal representation of their physical counterparts, both in universal applicability and the objects themselves. Masterfully applying the geometry of circles, quadrilaterals, parallel lines and trigonometry, Euclid could settle complex arguments using simple, logical steps. And it worked.
These were principles however which Dodgson’s contemporaries were openly straying from. The Euclidean presentation was worn fashion; old news; a tired code of moral conduct—because after all, it was the 19th century, and humanism reigned supreme. Rene Descartes was leading the charge of abstraction within the darkened catacombs of his skull. He introduced geometry of an analytical nature. Post-Cartesian mathematics gave its artists freedom to explore new unseen ideas, so long as these manipulated concepts found a consistent framework of operations. Such calculated techniques of modern mathematics weren’t simply the favored tool for esotericism. New Science had found its art too. Russian Nikolai Lobachevsky (1792-1856), Hungarian János Bolyai (1802-1860), Germans Bernhard Riemann (1826-1866) and Carl Friedrich Gauss (1777-1855) further advanced the post Euclidean abstractions once explored by Descartes, and in the intervening years, an entire fairyland of geometric worlds would explode to consciousness beyond that which Euclidean logic had been intended, mainly in N-dimensions, projective geometry, affine geometry, and finite geometry. And yet all of this would seemingly culminate with Albert Einstein’s theory of general relativity. Space itself, according to Einstein, is not Euclidean.
In short, Wonderland inhabited both inner-space and the fourth-dimension of time. Over a decade after Alice’s famous adventures—1879 to be exact, Dodgson would publish his mathematical treatise, “Euclid and his Modern Rivals,” where-as thirteen contemporary geometry textbooks were exhibited and shown to be of inferior quality or equally—if not better explained—by Euclid. According to Dodgson, post-Cartesian mathematics was a nonsensical land of the delusional mind, where his students were being guided to perversion and led away from an arithmetic which actually described the real world. For a true Euclidean, planes are flat and parallel lines never meet. Yet this is simply not so on a globe. The earth has multiple longitude lines that all meet at the North and South Poles, despite being parallel. Alice’s Adventures in Wonderland was written with an ink-dipped dagger, mocking such reckless concepts which opened up a slippery slope between what one could fathom through the language of algebra and that of geometry at the cost of concrete existence. Wonderland was a warning for the 20th century, particularly what it might become. Depending on what side of the line you’re standing on, you’ll agree with me. Mad results followed.
“I wonder if I shall fall right through the earth! How funny it’ll seem to come out among the people that walk with their heads downward!” – Alice
Indeed, we all inhabit Wonderland. And like Alice, falling down the rabbit hole means leaving the world of Euclidean geometry behind. Goodbye plane Earth. Hello globe logic.
IF WONDERLAND HAD A MAXIM, IT MIGHT READ: reductio ad absurdum, which, when translated from the Latin, means “reduction to absurdity.” In philosophical logic as well as mathematics, it is a form of argument quite familiar with Euclid’s proofs and which attempts to discredit a statement by establishing contention between both parties and showing the absurdity of its denial, which will—if its course is not properly redirected—inevitably lead all participating members to a ridiculous and most impractical conclusion. With that in mind, let me now introduce to you one of Wonderland’s most esteemed residents; the caterpillar whom Alice, being only 3-inches tall at the time, had the displeasure of taking advice from.
He is a 19th-century London math professor, and his name, according to Oxford PhD graduate Melanie Bayley, is none other than Augustus De Morgan. De Morgan was the first British mathematician to lay out a consistent set of rules for symbolic algebra. Morgan’s book, Trigonometry and Double Algebra, first published in 1849, “explained the departure from universal arithmetic,” wrote Bayley, “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.” Bayley added, “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.’” Much like the hookah-pipe which the caterpillar smoked, De Morgan employed algebra’s original Arabic translation in his footnotes, “al jebr e al mokabala,” which means, “restoration and reduction”—but more on that in a moment.
If Dodgson, aka Lewis Carroll, took personal displeasure with symbolic algebra, it’s partly because of the added aggravation he saw in his own students, who had to unlearn a perfectly logical system of real world practicality for something which might easily become perverse in application, as Alice would soon come to find out.
Firstly, consider Alice’s baffling stabs at reality. Already she had grown far too behemoth in size to enter through the Lilliputian door, which in turn led directly into the rose garden. To quote Alice, “Now I’m opening out like the largest telescope that ever was! Good-bye, feet!” After indiscreetly indulging in a mysterious bottle which read: “DRINK ME,” she then became too small to reach the key on the table. An oddly placed cake on the floor made her enormous, while the White Rabbit’s fan shrunk her down to size again. After nearly drowning in her own tears and surviving the political campaign-trail of a Dodo-based Caucus race, another unmarked bottle caused her to swell up again in the Rabbit’s house while, quite conveniently, pebbles thrown in the window turned to cakes and, upon ingestion, was thoroughly waned again.
“Being so many different sizes in a day is very confusing,” she confessed to the caterpillar, having been asked, rather impolitely, who she was, “…at least I know who I was when I got up this morning, but I think I must have been changed several times since then.”
Certainly she has changed—quite inconsolably too; fluctuating anywhere between 9 feet and now 3 inches. Alice functioned best in a rational world, where her multiplication tables and something as simple as poetic grammar might be rightfully recited. Not so here. “Alice,” writes Bayley, “bound by conventional arithmetic where a quantity such as size should be constant, finds this troubling.” The post-Cartesian rules which regulated Wonderland were very confusing indeed, especially for the traditional Euclidean sort such as Carroll. But the caterpillar, who behaves as erratically as the imaginatively skewed world he inhabits—for you see, he has seemingly mushroomed up from nowhere—responds with the very apathy which might lend someone to believe the little girl, and her mocking narrator, are both siding with the wrong side of history. Regarding any concluding notion that Alice is dutifully confused or ebbed of mind, he quips, “It isn’t.”
Lewis Carroll’s views on the madness of this new symbolic algebra might best be explained—reductio ad absurdum—by the mushroom which the hookah-smoking caterpillar instructed her to partake of, seeing as how “one side will make you grow taller, and the other side will make you grow shorter.” Esoteric references aside, this advice was dreadfully confusing for Alice, “trying to make out which were the two sides of it; and as it was perfectly round, she found this a very difficult question.” Perhaps this perplexing circle of rationality relates to Dodgson’s many tutoring sessions. Unfortunately for Alice, she calculated the wrong balance between tall and short, because within a moment her chin collided with her foot.
What happens when Alice eats the other side of the mushroom? Carroll writes: “…she found that her shoulders were nowhere to be found: all she could see, when she looked down, was an immense length of neck, which seemed to rise like a stalk out of a sea of green leaves that lay far below her.”
Suffice to say, Alice might have interpreted the Caterpillars further promulgation of apathy, particularly his advice, to “Keep your temper,” quite differently than Lewis Carroll’s own contemporaries. Among Oxford’s educated, the word “temper” still retained its original definition, essentially meaning: “the proportion in which qualities are mingled.” One might consider tempered metal. And here, when faced with Alice’s difficulty formulating a tempered resolve with the mushroom, we can once more discern Carroll’s own frustration with symbolic algebra. It is a dreadfully poor substitute to the grounded realities of Euclidean geometry.
WHAT HAPPENS WHEN A TERRIBLE COOK DOUSES HER STEW with too much pepper? Well, if it has anything to do with projected geometry, particularly Lewis Carroll’s distrust of it, and the narrative unfolds in Wonderland, we follow such action to its logical conclusion. Let’s observe. Everyone in the room sneezes, except for the Cheshire Cat. But more importantly, the baby turns into a pig, naturally. This certainly doesn’t seem to surprise the Cheshire Cat at all. As Master of Ceremonies to the unfolding madness, his simple and conclusively apathetic reply to a perturbed Alice, we might assume, leads us to the very voice of Carroll himself when he speaks from behind his crescent-shaped grin. Carroll, I mean, the Cheshire Cat concludes:
“I thought it would.”
French mathematician Jean-Victor Poncelet (1788-1867) served as an army lieutenant in the Corps of Military Engineers. Actually, Poncelet took part in Napoleon’s invasion of Russia. That was in 1812 when, at the Battle of Krasnoi, he was left for dead. Despite interrogation by General Mikhail Andreyevich Miloradovich, the mathematician disclosed no private information. It was during his imprisonment over the next two years when he wrote his most notable work, “Traité des propriétés projectives des figures,” which outlined the foundations of projective geometry.
Projective geometry examines the properties of figures that stay the same even when that figure is projected onto another surface. A basic precept is that projective space has more points than Euclidean space for any given dimension. Therefore, geometric transformations are permitted which mutate Euclidean points to its counterpart “extra” points, and vice versa. Accordingly, it is possible to assign meanings to the terms “point” and “line” in such a way that they satisfy the first four postulates but not the parallel postulate. Poncelet describes his theory 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.” At any rate, if Wonderland is a series of ridiculous notions and academic grudges, then Poncelet’s geometry made Carroll’s hit list. It’s a notion that the Euclidean math tutor found ridiculous.
The scene in question plays out like this. Alice enters the house of a Duchess who is doing a terrible job of nursing a baby. It howls inconsolably. The cook is doing a terrible job too. Smoke fills the kitchen. The pepper is causing everyone but the Cheshire Cat to sneeze. Frustrated beyond her own limitations, the Duchess violently tosses the baby through the air so that Alice must catch it, which causes the seven year-old Euclidean to ruminate silently among herself: ‘If I don’t take this child away with me, they’re sure to kill it in a day or two: wouldn’t it be murder to leave it behind?’ This of course comes after the cook begins throwing everything within arm’s reach at the Duchess and her baby. “The fire-irons came first; then followed a shower of saucepans, plates, and dishes.” If Alice were to conclude that everyone in the house is preforming terribly, which she most certainly does, then she would be correct.
It has already been established by this point that Alice has left the world of plane Euclidean geometry behind. Here Projective Geometry is free to perform its theatrics. With Euclid, railroad tracks will never meet. But with projective, at some point in the infinity of our horizon they will. With infinity there is no distance. Parallel lines may be free to wander—so to speak; just as an imaginative mind has room to wander. “The case of two intersecting circles is perhaps the simplest example to consider,” writes Melanie Bayley. “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.”
“The principle of continuity,” Melanie Bayley continues, is “a bizarre concept from projective geometry, which was introduced in the mid-19th century from France.” That would be mathematician Jean-Victor Poncelet. “This principle (now an important aspect of modern topology) involves the idea that one shape can bend and stretch into another, provided it retains the same basic properties — a circle is the same as an ellipse or a parabola (the curve of the Cheshire cat’s grin). Taking the notion to its extreme, what works for a circle should also work for a baby. So, when Alice takes the Duchess’s baby outside, it turns into a pig.”
The reader of Carroll’s work will immediately notice that the baby and the pig essentially keeps its same basic original features, as any theoretical object going through a continuous transformation must. Carroll writes: “Alice caught the baby with some difficulty, as it was a queer-shaped little creature, and held out its arms and legs in all directions, ‘just like a star-fish,’ thought Alice.” Like the Duchess herself, a caricature likely based off of sixteenth-century Flemish artist Quentin Matsys painting of the fourteenth-century Duchess Margaret of Carinthia and Tyrol—she had the reputation of being the ugliest woman in history, and Matsys’ portrait is titled “The Ugly Duchess,” appropriately—Alice notes that the baby is somewhat homely in and of itself. It has a queer shape, turned-up nose and small eyes. Alice only realizes the transformation when its howling sneezes turn to grunts.
What follows between Alice and the Cheshire Cat is perhaps one of the most popular exchanges in the entire history of literature. Wonderland’s Master of Ceremony has witnessed the baby’s untidy transformation into a pig from the branch of a tree, naturally, and the short transaction of words to follow may best describe Carroll’s personal frustration towards the maddening rush of his academic contemporaries away from a solid reality-based Euclidean construct to the abstraction of Post-Cartesian mathematics.
Cat: “Where are you going?”
Alice: “Which way should I go?”
Cat: “That depends on where you are going.”
Alice: “I don’t know.”
Cat: “Then it doesn’t matter which way you go.”
WILLIAM HAMILTON WAS MAD AS A HATTER. Of course, so was a certain Theophilus Carter. The later was a furniture dealer residing near the Oxford of Carroll’s time. The fact that he was actually known by locals as the Mad Hatter, partly due to an obsession with his top hat and because eccentricities became his order of business, is not a coincidence. Carter’s invention of an “alarm clock bed,” for example, which promised to wake the sleeper by tossing him upon the cold, hard floor, was exhibited at the Crystal Palace in 1851. No surprise though, it didn’t catch on. Critics have long explained Theophilus Carter as the reason why Alice’s Hatter is so concerned with time and arousing a sleepy dormouse, not to forget his gratuitous mentions of furniture. Then again, Lewis Carroll didn’t have a bone to pick with Theophilus Carter.
All things considered, it would most certainly be a mistake to overlook the Irish physicist, astronomer, and mathematician William Hamilton. The fourth dimension had become quite the craze in the 19th-century, and Hamilton played his part. Hamilton was looking for ways of extending complex numbers to higher spatial dimensions. The third-dimension failed him. But in working with four dimensions he created ‘quaternions.’ As early as 1843, quaternions had been hailed as an important milestone in abstract algebra, since rotations could be calculated algebraically, and would later be employed within the Kepler Problem and the study of celestial mechanics. Makiko Minow-Pinkney writes, Hinton “believed that with enlarged imaginative powers gained by the practice of visualizing the four-dimensional cubes which he called ‘tesseracts,’ individuals would gain access to true reality (The Question of the Fourth Dimension).”
Hamilton, it seems, was driven by the notion that algebra was the science of time. Specifically, Melanie Bayley writes, he 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.” As a result, he “discovered a four-dimensional manifold of numbers, the ‘quaternions’—usually called hyper complex numbers today,” writes David Booth in the introduction to Rudolf Steiner’s The Fourth Dimension: Sacred Geometry, Alchemy, and Mathematics (1905). “Hamilton did explore the fourth dimension, but still refused to actually accept the notion of a four dimensional space. He carried out his research at a time in which—according to our hypothetically accepted view of Cultural Revolution—man’s consciousness had descended to the greatest degree into matter. Hamilton used three dimensions (the vectors), along with a fourth (the tensor), that were kept separate so that they did not combine into a single four dimensional manifold.”
The fourth dimension certainly wasn’t loved by all. In his critique of Charles Howard Hinton’s book The Fourth Dimension, Bertrand Russell wrote that such impropriety would “stimulate the imagination, and free the intellect from the shackles of the actual,” and that by claiming “our three-dimension world is superficial,” Hinton came across as a “conscientious bigamist.” Carroll also took an offensive position. For the author of Wonderland, “the fourth-dimension,” says Ana Teixeira Pinto, a lecturer at Berlin University of the Arts, “was a case of the hypostatization of language: abstraction taken literally, and set phrases, metaphors, and figures of speech given concrete reality.”
At any rate, Alice finds herself at a tea party, or perhaps it should read “t-party,” with three curious inhabitants of Wonderland: the Hatter, the March Hare, and the Dormouse. Notice that the character Time is absent from the t-party. Actually, the Mad Hatter has had a falling out with Time, and is stuck at 6 o’clock accordingly—and perhaps even eternally. Says the Hatter to Alice: “If you knew Time as well as I do, you wouldn’t talk about wasting it. It’s HIM.”
Writes Melanie Bayley: “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.” The movement around the table can be seen as Hamilton’s unsuccessful attempts at calculating motion, which was limited to rotations on a plane until he added time to the equation. Bayley continues, “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.”
There is an exchange between the Hatter, March Hare, and Alice that goes like this.
“Then you should say what you mean,” the March Hare went on.
“I do,” Alice hastily replied; “at least–at least I mean what I say–that’s the same thing, you know.”
“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’!”
Hamilton’s stimulation of the imagination, specifically his freeing “the intellect from the shackles of the actual,” as Russell would say of Hinton, found its fullest realization in the establishment of Globe Earth—or rather, it’s only proof of existence. And it’s a mathematical one. “The development of abstract mathematical formalisms, notably that of tensor calculus,” writes the astrophysicist John Barrow. “A deep physical insight orchestrated the mathematics of general relativity, but in the years that followed the balance tipped the other way. Einstein’s search for a unified theory was characterized by a fascination with the abstract fomalisms themselves.”
Gerrard Hickson put the general theory of relativity, backed in part by the fourth dimension, to rest in his book, Kings Dethroned. He wrote: “While claiming ‘time’ as a fourth dimension, Einstein explains that ‘by dimension we must understand merely one of four independent quantities which locate an event in space.’ . . . This is to imply that the other three dimensions which are in common use are independent quantities, which is not the case; for length, breadth and thickness are essentially found in combination; they co-exist in each and every physical thing, so that they are related—hence they are not independent quantities. . . . On the contrary, time IS an independent quantity. It is independent of any one, or all, the three proportions of material things, it is not in any way related; and therefore cannot be used as a fourth dimension.”
“When the scene ends,” Bayley adds, “the Hatter and the Hare are trying to put the Dormouse into the teapot….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.” 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!”