Around the middle of the 19th century, mathematics was undergoing a revolution. Certain Classical notions were being overturned and in their place, something altogether stranger was beginning to appear. Where previously it had been based around observations in reality, maths was now becoming its own language which it was hoped could describe and even anticipate new phenomena.
Conservative mathematicians of the time, raised on the common sense mathematics of Euclid, opposed the change. One such was Reverend Charles Dodgson, pen name Lewis Carroll. As it happens, Alice in Wonderland was written as a satire of this revolution and in the style of Euclid, sought to disprove these ideas through reductio ad absurdum, taking them to their logical extremes to reveal their absurdity. Here I’ll be discussing a few examples and how the mathematics he was poking fun at has actually helped us to answer, or begin to frame, questions about our universe. Enjoy!
Early on in her adventure, Alice finds herself in a place with only one exit. The only problem being, it’s fifteen inches high! She takes a potion to allow her to fit through, wondering as she shrinks down in size whether she’ll eventually reach the point of nothingness, going out altogether like a candle. What exactly is the difference between a very small something and nothing at all?
It’s a very good question, and one that examines the concept of infinity. Is there a smallest unit of something, or can something of finite size really contain infinitely small somethings? Science suggests the former – quantum mechanics tell us that everything comes in discrete packets, or ‘quanta’. For example, photons are the quanta of light.
This idea has been applied in the search for a quantum theory of gravity, where there is a proposed smallest unit of space – the planck length, below which space itself would disappear into a black hole.
The idea of becoming smaller in order to go further is also present in science. On the quantum scale, individual particles are able to tunnel through barriers in a way that more massive things like people can’t. Without this crucial piece of strangeness, stars couldn’t burn and you could never enjoy yogurt, or bees.
‘Oh Dear, How Puzzling It All Is!’
Later, Alice is very big, and having trouble with her maths.
‘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!’
This actually makes a little more sense if you take into account different base systems. Ordinarily we work in base ten, meaning we have zero-through-nine digits, then when we get to ten we move over and put a one in the next column. What’s happening to Alice is that she’s slipping into higher bases of 18, 21 and 24, and suddenly everything she thought she knew has gone awry!
So why do we use different bases? Doesn’t that just confuse matters? Well, it turns out that they come in very handy. In Information Theory, we have the equation S = log(2) N, where S is a measurable quantity of information, N is the number of possible alternatives in a system and the number in brackets, 2, is the base.
For example, let’s say you have two balls – one blue, one red. This corresponds to N=2. When we pick out the red ball from a bag, we immediately have information about what colour the other ball is, and so these possibilities are linked. The advantage of having a base two system here is that log(2) 2 = 1, which is a good number to have for the minimum amount of information possible. We call this ‘one bit’.
This idea lies at the heart of statistical thermodynamics, which involve increasing chaos and the forward motion of time, given by S = k log(e) W, arising from the idea that in a physical system, like a hot cup of tea in a cold room, there are far more possibilities for the tea to cool down and the room to be slightly heated up than the reverse. So, as the heat is dispersed throughout a greater number of individual particles, the specific temperatures of which we can’t easily ascertain, our information about the system always decreases. Einstein had this to say about the theory,
‘It is the only physical theory of universal content, which I am convinced, that within the framework of applicability of its basic concepts will never be overthrown.’
‘Keep Your Temper!’
Following on from all this size-changing, Alice meets a large, blue Caterpillar. ‘Being so many different sizes in a day is very confusing,’ she complains. ‘It isn’t,’ replies the Caterpillar, who’s used to this strange world. He recommends that she ‘keep her temper’, where in Carroll’s time ‘temper’ had another meaning, relating to the proportion in which things are linked.
A huge fan of Euclidean geometry, where ratios between things matter, Carroll’s telling us how he feels about the Non-Euclidean variety beloved by Cthulhu and his chums. For example, we know that on a flat plane the angles of a triangle add up to 180 degrees, with no exceptions. However, on a curved/ spherical plane this doesn’t hold. Suddenly the proportions that have held for thousands of years are all wrong!
It’s fairly straightforward to see how we use this form of geometry in terms of measuring the curvature of objects in space. It can also be used to measure the curvature of space itself, which in turn can tell us about how the universe will end.
‘Why Is A Raven like a Writing Desk?’
Alice ends up at a tea party with some very strange people. One, The Mad Hatter, asks her the now-famous question, ‘Why is a raven like a writing desk?’
Alice asks him why, and he admits he doesn’t know. He was just asking. Alice retorts, ‘I think you might do something better with the time than wasting it in asking riddles that have no answers.’ Ha! Good one, Charles! That’ll show ’em!
Over the years, many have attempted their own answers at this riddle, even eventually Carroll himself, who got sick of people writing in to ask. My personal favourite is Aldous Huxley’s: ‘because there is a ‘b’ in both, and an ‘n’ in neither’.
Does Physics have an answer? Well, yes! Every element is made of atoms, which in turn are made of three central ingredients – protons, neutrons and electrons. While the electron is a lepton, already in its smallest form, protons and neutrons in turn are made up of quarks. So, why is a raven like a writing desk? Well, they’re both made of the same stuff.
Around Dodgson’s time, the physicist William Rowan Hamilton had spent years slowly developing a complicated piece of mathematics called Quaternions, which deal with the physics of rotation in 3D space. For the mathematics to work, Hamilton realised, it would have to involve four dimensions – three of space (length, height, breadth) and one of time. At the tea party, we’re told that The Mad Hatter and Time have had a bit of a falling out, and now it’s always tea time, with no time for cleaning up. Hence, they’re doomed to forever be rotating around the table in search of clean china.
This was Carroll’s way of pointing out the absurdity of appropriating all of time just to rotate things properly. Still, it works, and though they’ll seem obscure even to many mathematicians, they’re frequently used wherever things rotate in 3D space, such as in hinges/ joints/ arms in robotics, pointing satellites and simulating the movements of body parts/ other objects in real life or in video games.
Pig and Pepper
In one scene missing from the film but present in the book, a baby miraculously (for lack of a better way to put it), transforms into a pig. This is Carroll’s attempt to make fun of Topology, a branch of geometry where shapes can transform, treated as though they’re made of rubber. They can be stretched and warped but still retain certain qualities, like the number of holes they have. To a topologist, a coffee cup and a doughnut are the same!
Topology is one of the stranger and more enigmatic things on this list. It’s certainly not obvious what it could be used for. Would it surprise you to know that the Nobel Prize in Physics was awarded for topological work this year? Topology is a key component in organising geometries more complicated than the ones we encounter on a daily basis, such as the overall shape of the universe.
When Alice informs the Cheshire Cat that the baby turned into a pig, he casually replies, ‘I thought it would’.
‘We’re All Mad Here’
‘I don’t want to go among mad people’, Alice remarked.
‘Oh, you can’t help that’, said the Cat. ‘We’re all mad here. I’m mad. You’re mad.’
‘How do you know I’m mad?’, said Alice.
‘You must be’, said the Cat, ‘or you wouldn’t have come here.’
This sums up Carroll’s attitude to Wonderland – a world made from these strange, new types of mathematics. Ultimately though, every single one has been used to move our understanding of reality forward in subtle ways.
Increasingly counter-intuitive as we delve further into the unknown away from our ordinary experiences, science invites us to examine our surroundings in a way that we haven’t necessarily evolved to find easy, or comforting, or sensible. Lewis Carroll may have been wrong about so much of mathematics, but his writing lives on for all of its literary beauty, as well as a reminder that taking a stroll through Wonderland isn’t necessarily such a bad thing.
Though in the end he received little benefit from it, by taking apart these ideas in such a playful way, he hinted at the key to unlocking new discoveries. Many theories are ‘mostly true’ in the sense that they make accurate predictions on a wide variety of things. General Relativity tells us so much that is true about big things, while Quantum Mechanics tells us so much about what is small. They don’t, however, explain each other, and so while we can say that they’re correct, they are at least incomplete. An important part of science is to pull back the deeper mysteries in these ideas to find a common thread between them. In the process we may find that much of what we recognize of them will disappear, but perhaps, with some luck, there will remain a grin. The Universe will smile back at us, for all its strangeness and wonder, as we discover something new.
Merry Newtonmas, folks!
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