Math in Alice in wonderland
"Who in the world am I?' Ah, that's the great puzzle!"
In chapter 2: Pool of tears
"Let me see: four times five is twelve, and four times six is thirteen, and four times seven is—oh dear! I shall never get to twenty at that rate! However, the Multiplication Table doesn’t signify.."
In the pool of tears, Alice’s attempts at simple multiplication leave her confounded. In regular math, four times six would never be thirteen. 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. However, if you play around with the base systems, things can change. While Alice was calculating in base ten, in this new, crazy wonderland, her answers slipped into higher base systems. People are at risk of getting lost like Alice when they stay anchored to original standards or beliefs in the face of changing systems.
In chapter 5: Advice from a caterpillar
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.
The first clue may be in the pipe itself: the word “hookah” is, after all, of Arabic origins, 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. The restoration was what brought Alice to the mushroom: she was looking for something to eat or drink to “grow to my right size again”, and reduction was what actually happened when she ate some: she shrank so rapidly that her chin hit her foot. De Morgan’s work explained the departure from universal arithmetic – where algebraic symbols stand for specific numbers rooted in a physical quantity – to that of symbolic algebra, where any “absurd” operations involving negative and impossible solutions are allowed, provided they follow an internal logic. Symbolic algebra is essentially what we use today as a finely honed language for communicating the relations between mathematical objects, but Victorians viewed algebra very differently. Even the early attempts at symbolic algebra retained an indirect relation to physical quantities. De Morgan wanted to lose even this loose association with measurement and proposed instead that symbolic algebra should be considered as a system of grammar. “Reduce” algebra from universal arithmetic to a series of logical but purely symbolic operations, he said, and you will eventually be able to “restore” a more profound meaning to the system – though at this point he was unable to say exactly how.
In chapter 6: Pig and Pepper
"The baby grunted again, and Alice looked very anxiously into its face to see what was the matter with it. There could be no doubt that it had a very turn-up nose, much more like a snout than a real nose; also its eyes were getting extremely small, for a baby: altogether Alice did not like the look of the thing at all, “—but perhaps it was only sobbing,” she thought, and looked into its eyes again, to see if there were any tears. No, there were no tears. “If you’re going to turn into a pig, my dear,” said Alice, seriously, “I’ll have nothing more to do with you. Mind now!” The poor little thing sobbed again, (or grunted, it was impossible to say which,) and they went on for some while in silence. Alice was just beginning to think to herself, “Now, what am I to do with this creature when I get it home?” when it grunted again, so violently, that she looked down into its face in some alarm. This time there could be no mistake about it: it was neither more nor less than a pig...."
Carroll craftily paly down the work of mathematician Jean-Victor Poncelet in this section. Poncelet talked about the transformation of geometric figures and perpetuated the belief that geometric figures undergoing a continuous transformation, without any sudden changes or subtractions, are likely to retain some of their original features. However, this might not necessarily be physically tangible and would be possible only through the use of things like imaginary numbers. The baby transforming into a pig is Carroll’s way of showing how absurd and grotesque he found this idea. You can either be a baby or a pig, but no amount of tiny changes can make you both.
In chapter 7: At the mad tea party
"Alice sighed wearily. “I think you might do something better with the time,” she said, “than waste it in asking riddles that have no answers.” “If you knew Time as well as I do,” said the Hatter, “you wouldn’t talk about wasting it. It’s him.” “I don’t know what you mean,” said Alice. “Of course, you don’t!” the Hatter said, tossing his head contemptuously. “I dare say you never even spoke to Time!” “Perhaps not,” Alice cautiously replied: “but I know I have to beat time when I learn music.” “Ah! that accounts for it,” said the Hatter. “He won’t stand beating. Now, if you only kept on good terms with him, he’d do almost anything you liked with the clock. “Is that the way you manage?” Alice asked. The Hatter shook his head mournfully. “Not I!” he replied. “We quarreled last March—— just before he went mad, you know——” (pointing with his teaspoon at the March Hare,). “And ever since that,” the Hatter went on in a mournful tone, “he won’t do a thing I ask! It’s always six o’clock now.”
“Take some more tea,” the March Hare said to Alice, very earnestly. “I’ve had nothing yet,” Alice replied in an offended tone, “so I can’t take more.” “You mean, you can’t take less,” said the Hatter: “it’s very easy to take more than nothing.”"
The tea party floats ambiguously, seemingly around a tree, and is interspersed with butterflies and oversized insects. In Carroll’s tea party, the Doormouse, the Mad Hatter, and the March Hare are all going in a circle around a table in a perpetual tea time as Carroll took away the fourth member of their party, Time. In the mid-1800s, mathematician William Rowan Hamilton had come up with a new number system called quaternions. This was a sort of coordinate system based on four terms, three that designate a place or spatial dimensions, 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 “time”, they would keep rotating round and round in a plane, like the hands of a clock. Carroll was miffed that someone could simply appropriate time as a fourth dimension. By taking away time, he left the other three to keep going around in circles forever, like an incomplete quaternion.
The logic was another weapon of choice for Carroll, and the text of Alice in Wonderland is widely sprayed with riddles and logical statements. The Mad Hatter is trying to tell Alice that she can have more tea, given that she has not yet had anything to drink, but what she cannot do is “take less.” It seems like a typical maths word problem right.
If you the book now, with this mathematical viewpoint you will be able to appreciate Lewis Carroll for his hidden math work in a fantasy fiction story. This is how the mind of a person who understands maths looks.
Note on the author Lewis Carrol
Charles Lutwidge Dodgson, better known by his pen name Lewis Carroll, was an English writer of children's fiction, notably Alice's Adventures in Wonderland and its sequel Through the Looking-Glass. He was noted for his facility with wordplay, logic, and fantasy. The poems Jabberwocky and The Hunting of the Snark are classified in the genre of literary nonsense. He was also a mathematician, photographer, inventor, and Anglican deacon.
Carroll developed the earliest modern use of today's 'logic trees', a graphical technique for determining the validity of complex arguments that he called the 'method of trees'. This was a step towards automated approaches to solving multiple connected problems of logic.
I referred few online resources to write this blog.
