Mathematical concepts and logic – Mathematician Keith Devlin asserted in the journal of The Mathematical Association of America that Dodgson wrote Alice in Wonderland in its final form as a scathing satire on new modern mathematics that were emerging in the mid-19th century.
“Down the Rabbit-Hole”: Alice’s final size represents the concept of a limit.
“The Pool of Tears”: the multiplication which produces some odd results is the representation of numbers using different bases and positional numeral systems.
“Advice from a Caterpillar”: the Pigeon who asserts that little girls are some kind of serpent, for both little girls and serpents eat eggs represents the substitution of variables.
“A Mad Tea-Party”, the March Hare, the Hatter, and the Dormouse give several examples in which the semantic value of a sentence A is not the same value of the converse of A giving examples of inverse relationship.
Alice ponders what it means when the changing of seats around the circular table places them back at the beginning. This is an observation of addition on the ring of integers modulo N.
The Cheshire cat fades until it disappears entirely, leaving only its wide grin in the air; Alice notes that she has seen a cat without a grin, but never a grin without a cat. This refers to non-Euclidean geometry, abstract algebra, and the beginnings of mathematical logic and can represent the very concept of mathematics and number itself.
A Math-Free Guide to the Math of Alice in Wonderland
Alice in Wonderland got its start as a simple story, told by a mathematics professor to a colleague’s daughter. It’s a strange story that seems to be the result of a drug trip, but is actually a scathing satire of the new-fangled math that the professor was seeing invade his area of study.
Most of us just enjoy the White Rabbit and the hookah-smoking caterpillar. But now you can understand the math in Alice without needing to be a math whiz.
Some people who read Alice in Wonderland find it a whimsical adventure into a world of fun little paradoxes. Other people consider it a creepy march through a world of characters who seem to be set on making life as frustrating as possible as manically as they can. Which side you see might possibly have some bearing on your view of the world. Alice isn’t just fun and games. Charles Dodgson — the real name of Lewis Carroll — added all those paradoxes and puzzles as he was poring over the new math that was springing up in the middle of the 1800s.
Carroll liked good old-fashioned no nonsense algebra and Euclidean geometry — areas of study that could prove things about the natural world. Suddenly math students, and even teachers, were using different mathematical methods to prove things like one and one not equaling two. It seemed to Carroll that they were just being difficult on purpose, so he skewered them in prose.
Alice’s Mathematical Attempts at Control
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.
Alice also has to work on her sense of grace and Euclidean proportion. Alice meets the caterpillar on his mushroom and asks to be made larger. He tells her that one side of the mushroom makes her smaller and one makes her larger, then warns her to “keep her temper.” Temper, in this case, means correct proportions. This is hard when the same object can have exactly opposite effects. This is Carroll’s jab at a new style of math.
For most of us, math has an analog in the real world. That’s the point of it, to use this system of symbols as a translation to figure out problems that would be ungainly if expressed in language. Around Carroll’s time, math became something different. Instead of a translation, it became a language in and of itself, or rather several different languages. If someone set up certain rules for a problem, and then opposite rules for another problem, they could prove opposite conclusions to be true. Each set of rules has opposite effects, but as long as each proof remains true to its internal rules, each is considered correct.
Alice tries to remain in proportion, despite the inconsistency. In the movie she simply grows and shrinks. In the book, at one point her neck grows long like a snake’s, which is even worse for her than being either big or small. She has to figure out how to keep gracefully geometric, having the same proportions at any size, before she can go on.
Hallucinations and Reality
A part that was left out of the movie, and shakes a lot of readers off the book, is an encounter with the Duchess. Alice meets her in her house, eating soup with her baby. When the soup is too peppery, her baby sneezes. What follows is a delightful little poem that starts with the lines, “Speak roughly to your little boy and beat him when he sneezes.” Alice grabs the baby and, when she looks, the baby has retained many of his original features, but has turned into a pig. It is at that point that many people put down the book and quietly go to find some Nancy Drew mysteries.
This section is actually Lewis Carroll ridiculing the work of Jean-Victor Poncelet, who talked about how geometric figures transform. He stated that a geometric figure undergoing a continuous transformation, without any sudden changes or subtractions, will retain certain features. However, it won’t retain those features in a way that can be understood physically. It only manages to keep them on paper, using things like imaginary numbers. The baby-to-pig transformation is Carroll’s commentary on how absurd and grotesque he found that idea. It’s either a baby or a pig, and an infinite series of tiny changes can’t truly make it both at the same time, argues Carroll.
Perhaps the most iconic scene in the Wonderland story is the tea party in the garden. This is where Carroll really starts grinding his axe. William Rowan Hamilton had come up with a new thing called quaternions. This is a sort of coordinate system based on four terms, three that designate place, and one that designated, or so Hamilton decided, time. With these four terms, Hamilton could describe rotation in a three dimensional universe. He could only do this, though, if he added that fourth component. Without it, they rotated in a plane, like the hands of a clock.
Carroll was miffed that someone had appropriated all of time just so they could have a fourth component to allow them to rotate things properly, so in the tea party he took it away. In the movie, no one explains why the Doormouse, the Mad Hatter, and the March Hare are all going in a circle around a table in a perpetual tea time during a perpetual unbirthday. In the book, Time had been the fourth member of their party, but had gotten fed up and walked out. That left the other three to keep going around in circles forever, like an incomplete quaternion. Alice, free of the madness of requiring an extra dimension in order to be of any use, leaves the tea party.
The Lord of Wonderland, much more than the absurd royalty and the sad, manic Mad Hatter, is the Cheshire Cat. He goes where he wants, and does as he pleases, and it’s all because of his grin. At the end of a loopy conversation with Alice, the cat disappears, but the grin remains. Alice remarks that she’s seen a cat without a grin, but she’s never seen a grin without a cat.
This is the heart of the story. Sure, anyone can have a smile on their face, but if people remove the face — the reality of equations — they have to remove the grinning, silly, maddening numbers as well. Wonderland is just fine for the Cat, who has fully embraced and embodied this crazy new math, but what good, Carroll argues, is it to anyone else? The Cat voices Carroll’s opinion. Alice complains that she doesn’t want to “go among mad people.” The Cat tells her that she has no choice: “Oh, you can’t help that. We’re all mad here. I’m mad. You’re mad. . . . You must be, otherwise you wouldn’t have come here.”
Author Lewis Carroll was also a math teacher in Oxford, England, and mathematicians say the Alice books are full of algebraic lessons — such as why a raven is like a writing desk.
That’s the riddle the Mad Hatter asks Alice. And, as Weekend Edition Math Guy Keith Devlin tells NPR’s Jacki Lyden, “That particular scene — and lots of other scenes in Alice in Wonderland — were a reflection on the increasing abstraction that was going on in mathematics in the 19th century.”
Carroll, whose real name was Charles Dodgson, was a very conservative, traditional mathematician, Devlin says, and he didn’t like the changes some were bringing to the discipline of mathematics.
“To him, algebra was all about numbers,” Devlin says. But in the 19th century, people were developing all kinds of bizarre new algebras, where x times y was not equal to y times x.
So why is a raven like a writing desk? Because the new mathematics didn’t make sense to Carroll. “Lots of things that every common-sense person would say are different in this new mathematics turned out to be the same,” Devlin says — a point Carroll found ripe for satire.
Missing Time At The Tea Party
So underneath the madness of the Wonderland tea party lies math — and some snark. Alice finds the Mad Hatter, the March Hare and the Dormouse at their tea party, but Devlin says Carroll deliberately left out one character.
“One of the big developments that was going on at that time … was work by an Irish mathematician called William Hamilton,” Devlin explains. Carroll wasn’t a fan of Hamilton’s work, a new arithmetic called quaternions. “Quaternions were numbers — not to deal with counting things, but to deal with understanding rotations.
“Back in Victorian times, when Hamilton himself was doing this work, he tried to understand his new arithmetic in physical terms,” Devlin says. “He said one of the four terms that was involved in these numbers had to be time. So time was inexplicably, inescapably bound up with these new numbers.”
Yet it’s the Mad Hatter, the March Hare and the Dormouse at the tea party — the character Time is absent. (You can read the chapter here if your memory needs refreshing.)
“What Hamilton said was if you take this time parameter out of these new numbers, then the numbers would just keep rotating around — they won’t go anywhere,” Devlin says. “It was just like the characters rotating round and round the tea party, round and round the table.”
“In fact, when the Hatter and the Hare try to squeeze the Dormouse into the teapot, they’re trying to somehow get away from this complexity — throw away another of the parameters, if you like — so that life can resume as normal.”
Devlin says Carroll’s message is that we “get rid of all of this complexity in the first place, and let’s just go back to the familiar old geometry that we’ve had since Euclid for 2,000 years.”
The hidden math in Alice may come as a surprise to many, but mathematicians have always known Carroll was slipping some numbers into his fiction.
“We knew that Carroll was actually a mathematician,” Devlin says. “Last year, in fact, a scholar in Oxford called Melanie Bayley wrote a complete dissertation analyzing Alice In Wonderland, and she identified a number of mathematical allusions in the story.”
Without Carroll’s secret ingredient, Alice might never have achieved her fame. “The very first version of [the] Alice in Wonderland story — that he wrote for the real Alice — had none of the mathematics,” Devlin says. “He added a lot of new material and it’s all of that new material where you find the mathematical allusions.
“Almost certainly what he did was said, ‘Here’s this cute story that I’ve written for this real Alice. I’m going to take that and I’m going to use it to do this wicked satire of what I think are these crazy, stupid developments in mathematics that are getting us away from the real, solid mathematics that I’ve loved all my life.’ ”
Like James Cameron’s recent blockbuster movie Avatar, Tim Burton’s Alice in Wonderland, released this month, is in 3D. Also like Avatar, I suspect audiences will be uniformly thrilled with the visual spectacle, yet be divided when it comes to the story. Millions loved Cameron’s tale, but personally I (and apparently many others) thought that, although it had all the plot ingredients to have been good, it ended up annoyingly adolescent and cloyingly banal. As for Alice? Well, I’ll let you make up your own mind.
For mathematicians, the real story is not so much whether Burton’s movie will be a hit, rather it’s not often that a mathematical allegory makes it to Hollywood blockbuster status in the first place! So I can’t let the release of Alice go unnoticed in the mathematical literature – to whit Devlin’s Angle. For, as readers of MAA Online will doubtless know, Lewis Carroll was the pen name of the Reverend Charles Lutwidge Dodgson, a mathematician at Christ Church College, Oxford, and most mathematicians are probably aware that elements of the Alice story were inspired by mathematics. (At least, that is the entirely reasonable assumption everyone makes; Dodgson himself provided no commentary to that effect.)
Before I go any further, I should note that the new Alice movie is not based on Lewis Carroll’s original book. (Actually, the memories of the Alice story we all have from our childhood are based on two books, Alice in Wonderland and the later Alice Through the Looking Glass.) Rather, Burton takes as his inspiration a computer game called American McGee’s Alice. In the film, an adult Alice, now a disturbed young woman mourning the death of her parents, returns to the land we are familiar with from Carroll’s original tale, a strange place where animals talk, the Cheshire Cat has a grin, and the Queen of Hearts is wicked. (Or was it the Red Queen? Carroll’s two books had different queens that over the years tend to merge in our memories.)
Though others had looked for political and social allusions in the Alice books, most notably Martin Gardner, whose The Annotated Alice was published in 1960, followed by a sequel More Annotated Alice in 1990, perhaps the first scholar to look in depth for possible mathematical inspirations for Alice was Helena Pycior of the University of Wisconsin-Milwaukee, who in 1984 linked the trial of the Knave of Hearts with a Victorian book on algebra. Now Melanie Bayley, of the University of Oxford in England, has taken the analysis a lot further. She described her findings (well, since we are in the realm of literary interpretation here, I’d better say “her theory”) in an article titled Alice’s adventures in algebra: Wonderland solved, published in New Scientist, 16 December 2009.
Before I relate what Bayley has to say, let me summarize the history of Carroll’s Alice in Wonderland.
In 1862, Dodgson, together with the Reverend Robinson Duckworth, rowed in a boat up the River Thames with three young girls, Lorina Charlotte Liddell, aged 13, Alice Pleasance Liddell, aged 10, and Edith Mary Liddell, aged 8, the daughters of Henry George Liddell, the Vice-Chancellor of Oxford University and Dean of Christ Church College, as well as headmaster of the nearby, private, Westminster School.
The journey started at Folly Bridge near Oxford and ended five miles away in the village of Godstow. As they rowed, Dodgson made up and told the girls a story about a bored little girl named Alice who goes looking for an adventure. The three girls loved it, and Alice Liddell asked Dodgson to write it down for her. Two years later he did just that, and on 26 November 1864 he gave Alice the handwritten manuscript of what he then called “Alice’s Adventures Under Ground,” illustrated by his own drawings.
Most of the story was based on situations and buildings in Oxford and at Christ Church. For example, the “Rabbit Hole” down which Alice descends to begin her adventure symbolized the actual stairs in the back of the college’s main hall.
A year later, Dodgson – now masquerading as Lewis Carroll – published a greatly expanded version under the title “Alice’s Adventures in Wonderland,” with illustrations drawn by John Tenniel. It is in the new material he added, which includes the Cheshire Cat, the trial, the Duchess’s baby, and the Mad Hatter’s tea party, that we find allusions to mathematics. (Tweedledum, Tweedledee, Humpty Dumpty and the Jabberwock appear in the sequel, Alice Through the Looking-Glass.)
The book rapidly became a bestseller, it has never been out of print since it first appeared, and it has been translated into well over 100 languages.
So what does Bayley tell us about the mathematical ideas that Dodgson took inspiration from?
Now we get to the math part
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.
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.
The sting in the tale
Even if you accept Bayley’s suggestions – and obviously I am inclined to do so, at least overall, otherwise I would not have written about her work – you might think the mathematical inspirations for some of the scenes we read in Alice are nothing more than an interesting footnote. Think again, says Bayley. Without those mathematical undercurrents, it is highly unlikely that Dodgson’s book(s) would have achieved lasting, international stardom. His original nursery tale, written for the ten-year-old Alice Liddell, she says, would have been unlikely to attract much attention.
Dodgson was most witty when he was poking fun at something, Bayley explains, and then only 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,” Bayley claims, “Alice’s Adventures in Wonderland would never have become famous, and Lewis Carroll would not be remembered as the unrivalled master of nonsense fiction.”
Put that in your hookah and smoke it.
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.
Down the Rabbit Hole Studi in Alice I, da Marc Edmund Jones
Questa lezione inaugura la seconda metà del quarto anno settanta nella presentazione della filosofia Sabian e inizia una serie di 26 studi in un libro che in poco più di una generazione era diventato immortale ed è una considerazione del primo capitolo di Alice in Wonderland . Le lezioni tratteranno in regola con i dodici capitoli di Wonderland , i dodici capitoli di Looking-Glass e due studi sovrannumerari. Il grande primo principio della saggezza nella filosofia Sabian come rivelato attraverso le avventure di Alice è che tutto nella vita è un sacramento. Nel campo della filosofia tutta la potenza di un’idea in questo concetto di sacramento è spesso poco apprezzato a causa dell’associazione con gli istituti ecclesiastici, ma mentre un sacramento come il segno esteriore e visibile della grazia ricevuto in se stessa, naturalmente avrebbe molto a che fare con la chiesa tuttavia rituale rituale è qualcosa di più grande l’osservanza di riti religiosi e un sacramento comporta quindi molto più che gli elementi nell’Eucaristia. Il principio della vita stessa come un istituzione sacramentale è stato portato fuori a lezione Gabirol Ibn. C’è lo studente vede che tutte le attività dell’essere è resa possibile dalla rituale incastro del cosmo stesso. Tutta la sostanza dei racconti di Alice si rinnova dimostrazione. Nelle lezioni sulla Fairy Tales’s Grimm, le storie rivelano la struttura dell’essere e che sblocca praticamente ogni mistero del funzionamento dell ‘anima, e questo è stato scoperto di essere la diretta conseguenza del fatto che le storie sono state tramandate attraverso il cuore umano o sono state conservate perché erano amati e non è possibile amare se non come rituale. Nulla rimane nella memoria umana, senza alcun vincolo di esperienza, e la gara non può tenere a mano o giù idee che non sono abbastanza in sintonia con la struttura eterno a far parte dell’esperienza razziale. È così che alcune storie muoiono e gli altri vivono e lo stesso principio opera in ogni settore della vita. Nelle lezioni del 1001 Nights, a causa della estrema tenacia con la quale queste storie sono rimasti nel Vicino Oriente coscienza, è stato possibile estrarre da essi il perfetto e completo che la psicologia come base del sistema applicato Sabian si è dimostrato con quasi precisione infallibile. Le avventure di Alice si sarebbero raccomandare la virtù solo della loro sopravvivenza solo, ma vi si aggiungono le considerazioni che danno loro un valore unico nel campo della filosofia. è la chiave per il fondamento di ogni sforzo costruttivo.
È di fondamentale importanza il fatto che sono state scritte da un matematico. La matematica è una scienza divina non a causa delle sue astrazioni, ma a causa di ciò il suo studio non alla coscienza. Un individuo untutored rivolgendosi al nonsense produce non-senso. Anche Lewis Carroll stesso, quando sulla base di un lavoro cosciente di sciocchezze prodotto l’infantile relativamente Caccia allo Snark . Ma il cd assurdità così consentito di emanare spontaneamente da un cervello che si è scanalato secondo rituale cosmico quasi infallibilmente porterà qualcosa di significato cosmico. Questo da solo portato alla sua logica conclusione è la spiegazione di tutti ispirazione sia per mezzo di alcool o più agenzia costruttivo. Non è un caso che il lavoro del tutto spiegabile del rituale cosmico crea paese delle meraviglie, come questo studio. Ci sono fattori occulti, come il fatto che Lewis Carroll ha scritto il libro e non un matematico, ma la spiegazione, piuttosto che essere diversi è semplicemente più dettagliate. Qui per esempio è l’elemento di catalisi occulta o coincidenza di ispirazione nel fatto che il figlio maggiore si chiama Alice Liddell, una parola greca che significa verità, perché l’appoggio del nome con il ragazzo indica allo studente occulto l’indicazione esterna della sua possesso di queste qualità catalitico. è la chiave per il fondamento di ogni sforzo costruttivo.
Il primo capitolo rivela il fattore sacramentale nella foto con la bottiglia di liquido o l’elemento di vino con la descrizione del suo sapore e la torta poco o l’elemento di pane. La bevanda-me e mangia-me delle etichette rappresentano l’idea della costrizione nel processo cosmico o la prima legge di essere astratto, come persistenza in essere. Il bere, o vino come sangue o la coscienza o spirito, è la costrizione della discesa dello spirito nella materia drammatizzata in sempre più piccola mentre il mangiare o il metabolismo attraverso il quale è costretto questione di essere animato progressivamente da forme più alte della vita è la costrizione di ascesa della materia in una piena partecipazione nello spirito drammatizzata dalla crescente sempre più in alto. Questi dettagli sono troppo perfetti per un grido di coincidenza e continuano a sopportare il peso per sempre. Tale è il lupo! lupo! dei materialisti. è la chiave per il fondamento di ogni sforzo costruttivo.
Il raggiungimento di immaginazione nel libro, o il primo grande attesa scientifico, è l’impiego corretto del simbolismo. La funzione dei simboli è quello di dare forma al concetto spirituale. Tutto ciò che serve come simbolo non è essa stessa il simbolo, ma piuttosto il simbolo è idea per cui la forma dà sostanza, oltre a dare sostanza al suo proprio essere. Carroll qui involontariamente anticipato il detto popolare recentemente dalla scienza moderna come tutto ciò che immagina l’uomo può raggiungere. Questo è citato dal latino da Montaigne come un detto dei saggi ed è così lontano da moderna. Tutto risale alla coscienza, come è insegnata nel lavoro Sabian, o interesse, come la psicologia convenzionale e lo stato della scienza, e la base positiva di tutto l’interesse è vivo dell’attività. Così Alice non piaceva il libro ci da riva del fiume il pomeriggio di luglio, perché non ha avuto conversazioni o illustrazioni. politica editoriale moderna riconosce bene in questa sua richiesta di dialogo e illustrazioni ed è curioso notare che qui il genio coincidenti di John Tenniel è sopravvissuta la crescita di una nuova tecnica di arte illustrativa. In realtà, il libro è incompleto senza la sua incomparabile xilografie. La base negativa di interesse è il suo bisogno di francobollo stesso nella vita o per esprimersi. Così Alice in cui doveva parlare. Allo stesso modo ogni persona o gruppo di persone ha a parlare, e la saggezza e la mancanza di saggezza nel concedere o negare la libertà di parola è qui visto. È necessario collegare questa espressione e di interesse per il noto, ed è illuminante vedere Alice costantemente con Dinah come oggetto di conversazione. è la chiave per il fondamento di ogni sforzo costruttivo.
La legge della psicologia applicata o la grande prima idea per la soluzione di problemi personali è messa qui nella tecnica di azione in caduta o guasto. Lo studente deve venire a imparare relatività, in ogni stato di coscienza. Ci si dice che sia speciale provvidenza per ubriachi e pazzi, e questo non è altro che il fatto che l’ubriaco o pazzo rifiuta di permettere la realtà esterna a sconvolgere il senso interiore della realtà. Si può quasi cadere in un buco e scappare letteralmente pregiudizio. Tutto è relativo ed è così che Alice cadendo trova il vasetto di marmellata vuoto mentre in seguito la bottiglia e la scatola di vetro non lo sono. Nella maggior parte dei casi i tentativi di salvare il relitto nella vita sono vuote e inutili. Lo studente deve imparare a disinnesto e lasciarsi andare . Quando Alice è andato a dormire ha trovato il basso, illeso. Il rilassamento è la chiave per il fondamento di tutti gli sforzi costruttivi.