PopCo

by Scarlett Thomas

  My grandmother nods. ‘Unless there is no treasure,’ she repeats. ‘Although, if there wasn’t any treasure, it doesn’t really matter. Your grandfather solved the puzzle, and that’s what he would like known.’

  ‘And the answer is really here, in my necklace?’ I say.

  ‘Yes.’

  ‘Do you know what it is?’

  ‘No.’

  We now know, thanks to many hours of boredom and toast, the number of words and letters on every page of the Voynich Manuscript. But now my grandfather wants me to come up with the prime factors of all these numbers. Until he started talking about prime factorisation, I didn’t know how complicated prime numbers were. Every number, it turns out, is either prime or can be expressed as the product of prime numbers, which is why primes are sometimes known as the building blocks of the universe. The number 2 is prime, as are 3, 5, 7, 11, 13, 17 and 19 and so on, all the way to infinity (or aleph-null). If a number is prime, then it cannot be divided by any whole numbers apart from 1 and itself. The number 4 is not prime as it is comprised of 2 × 2. The number 361 is 19 × 19, or 192. The number 105 is made of the primes 3 × 5 × 7. The number 5625 is made up of 32 × 54, or 3 × 3 × 5 × 5 × 5 × 5.

  Apparently, once we know this data for all the pages of the Voynich Manuscript, my grandfather will assess it. He has had all kinds of hypotheses in his head all along. Will the numbers, or the prime factors, once we have them, form a pattern? Will there be square numbers of words on every page (there aren’t), or a Fibonacci number of letters (he doesn’t know yet)? Will all the numbers connected with the book turn out to be prime? These sorts of baffling questions are the reason for him wanting me to do all this work, and, while I am excited about being trusted with such an important task, even I realise that it is going to take ages. Counting the words and the letters on each page took for ever. This is going to take longer than for ever and a day.

  My old calculator is going a bit wrong so on Saturday we go into town and I am allowed to choose a shiny new scientific calculator all of my own, with loads of buttons. I also, of course, want a ZX Spectrum, and games, and all the pens and pencils in the shop but my new calculator is so shiny and big that I soon forget all of this. I expect it will have a button that will enable me to complete these prime factorisations in an instant, but when I ask my grandfather that evening, he just laughs.

  ‘Ah,’ he says when he stops.

  ‘What’s “Ah”?’ I say.

  ‘Well. Yes. That’s the thing about prime factorisation. No one’s ever found a short cut. No one knows very much about how primes behave, that’s the problem. Problems to do with primes have puzzled the greatest mathematicians. Now your grandmother …’

  ‘What about me?’ she says, coming down the stairs.

  ‘I was just about to tell Alice that your work might one day help to predict primes and lead to quicker ways to do prime factorisation.’

  ‘Mmm. Yes,’ she says, uncertainly. ‘Maybe one day.’

  ‘But in the meantime, Alice, I’m afraid it’s going to be a bit of a long old job for you.’

  ‘Have you got that poor girl doing your prime factorisation for you?’ my grandmother asks, as my grandfather gets up to pour her drink. ‘Shame on you.’

  But they both laugh, as if prime factorisation is just another bypass.

  This is a challenge all right. Still, maybe I will learn the secret short cut as I go through these numbers. It’s complicated enough for me to quite enjoy it, although I don’t know how long all this is going to take. You need a list of the primes, to start with, which I have obtained from my grandmother’s study and copied out on to fresh sheets of paper. I have written out the first hundred from 2 to 541, which I hope will be enough, although my grandmother has more than ten thousand primes up there, like they’re pets she collects. The hundredth prime squared, however, is 292,681, which is far bigger than any of my numbers, so I think I will be all right.

  To do prime factorisation, you have to remember the following rule. Every number that exists is either prime or can be expressed as a product of prime numbers (or ‘prime factors’). A number that can broken down to prime factors is called ‘composite’. 7 is prime, because it is only divisible by 1 and itself. But 9 is not prime. 9 is composite because it has a prime factor of 3. The number 21 has two prime factors: 3 and 7. Prime factorisation, then, means taking a number and trying to work out which primes divide into it. This is a trial-and-error process. And it really does take ages.

  There’s something I don’t understand about this, though. I am a child and, although I am quite good at prime factorisation, I wouldn’t trust me to do it, if I was my grandfather. I have a suspicion that he checks all my results as they come in, but if he’s doing all that, why not do the prime factorisation himself? It’s confusing. I suppose it is much easier to check a result than generate it in the first place but I still think it’s a little odd. I don’t think he checked my results of the numbers of words and letters in the manuscript, either. Perhaps all my calculations are wrong.

  Sometimes I see prime factors in my sleep.

  *

  Eventually, Kieran drifts off and I find I am walking on my own. Well, I am not exactly on my own, since I am walking in a group, but no one is walking alongside me, chatting. My throat is full of broken glass. This is all so beautiful; the landscape swelling around me. But I just want to go to sleep. In fact, when we get to what we think is Goshawk and sit on the slightly damp grass to start to meditate, I take the opportunity for a little nap, leaning against a big old tree, and have to be woken up by Ben afterwards. When we set off again my legs are full of molasses and feel too heavy to take even a single step.

  Somehow, using this bizarre method of meditation and compass reading (neither of which I am doing myself but I can independently verify that most other people here are), we do eventually end up on the banks of the River Meavy, just after two o’clock. There is a sign confirming that it is the correct river, and everyone cheers. And, as we follow the river down, preparing to try to ‘see’ the original corn rabbit, we come upon a pub which we all fall into, breathless and hungry. I eat a bowl of soup and drink a Bloody Mary but this cold is too far gone now. I will not be saved. After we have eaten, I can’t take it any more. There is an open fire in the pub, and it makes the whole place feel hot and syrupy. There are horrible things like stuffed stag’s heads and hunting photographs on the walls. These bleed into nothingness as I close my eyes, put my head down on the table, saying goodbye to it all.

  ‘I’ll take her back,’ I hear Ben saying. ‘She isn’t feeling too good.’ Then gentle arms, cool air outside and a car engine. Finally, the sound of gravel confirms that we are back.

  *

  At the same time as I work on these prime factorisations, I read the book my grandmother lent me, the one about Kurt Gödel. Apparently my grandfather was obsessed with Gödel’s work a long time ago. You can see why. With the same kind of dour anarchism to which my grandfather is prone, Gödel set out to show that you can never completely prove a mathematical theorem is true, not exactly because mathematics isn’t consistent but because it will never be completely flawless.

  In 1900, a German mathematician called David Hilbert gave a famous lecture in which he set out the twenty-three mathematical problems he felt would be key challenges for the new century. The first problem was the Continuum Hypothesis; the theory that there is nothing between the infinities aleph-null and aleph-one; no value to be found between Cantor’s concepts of the countable and uncountable (or the ‘continuum’). The Riemann Hypothesis was number 8 on Hilbert’s list. But Hilbert also called for the very principles and foundations of mathematics – its axioms – to be sorted out once and for all. This was problem number 2. People were already starting to worry about whether or not the closed system of mathematics was actually consistent, and whether the axioms were correct. If it wasn’t consistent then all the proofs of all the theorems to date would amount to nothing (if anyone ev
en knew what nothing was). What if, say, the Riemann Hypothesis was true and false at the same time? If 1 + 1 = 2 and 1 + 1 = 3 at the same time? That sort of thing would never do.

  Axioms are the very foundations of mathematics. Axioms are things that you can’t necessarily prove but form the basis for all mathematical proofs. Proofs, in mathematics, are logical evidence that something will always be the same way. Euclid formulated a proof that there are an infinite number of primes, for example, and Cantor narrowed this infinity down to aleph-null, or ℵ0. A proof is never the same as experimental evidence though. A proof of Pythagoras’s theorem (and I know what this is now, because it’s in this book – it says that the square of the hypotenuse of a right-angled triangle is always equal to the sum of the squares of the other two sides) is not based on someone looking at lots of right-angled triangles, measuring the lengths of the sides and saying, ‘Yup, everything seems to be in order here.’ A proof, elegant and simple, will explain why this will be so for eternity, for all right-angled triangles. There are many proofs of Pythagoras’s theorem.

  Axioms, the things on which proofs are based, like 1 + 1 = 2, are sometimes referred to as ‘self-evident’; others have been proved. You can always join two points with a straight line. All right angles are equal to one another. All composite whole numbers are the product of smaller primes. These are axioms. Axioms are a bit like starting points on a journey. You can start at one point and, using a set of directions, walk to another place. However, you need to know where your starting point is before you can obtain or use the directions. If you got a set of directions that were correct, but you had started in the wrong place, you would end up somewhere very unexpected. If you formulate a proof using axioms that are incorrect, you will end up in the wrong place.

  By the time of Hilbert’s lecture, set theory had thrown up a lot of problems in mathematics. You need sets in consistent mathematics. They tell you what things are and what things are not; which ideas share the same properties or rules (as well as what sorts of different infinities you might get). Axioms are based on them. You can say, ‘A set of triangles is a set of all three-sided, two-dimensional shapes with three angles adding up to 180 degrees,’ and, as long as you were talking about triangles on the plane, not triangles on a sphere, you’d be OK. But in 1903, Bertrand Russell came up with various paradoxes to illustrate the problem that a set (or class) cannot contain itself. Imagine the Barber of Seville. He shaves every man who does not shave himself. So does the barber shave himself? If he does, he doesn’t and if he doesn’t, he does. It’s just like the liar paradox! Despite his clear love of paradoxes, Russell went on to try to sort out these sorts of problems by writing the Principia Mathematica with his teacher Alfred North Whitehead, which was published in 1910. In three vast volumes, this work set down the basic axioms and rules for mathematics. Everything was OK in mathematics after this, or as OK as it ever was, with no pesky paradoxes spoiling everything, until Kurt Gödel came along and messed everything up again in 1930, when he proved two theorems which would together become known as Gödel’s Incompleteness Theorem. In these theories, he explained how you could find fundamental paradoxes within the system of mathematics. He did this using code.

  As I understand it (and I am, after all, a child, so this is the simple version), Gödel worked out a clever way of assigning number codes to statements. The way he did this was by assigning numbers to all parts of mathematical (or other) statements, and then using these numbers to create a unique, large number. It turns out that this is just like making a secret code! Gödel’s code was a little more complicated, but say you assigned the following values to mathematical symbols:

  Symbol Code Number

  × 1

  ÷ 2

  + 3

  - 4

  = 5

  1 6

  2 7

  3 8

  All the symbols now have a number that you can work with. The statement 1 + 1 = 2 would, in this system, be represented by the sequence, 6, 3, 6, 5, 7. Now comes the clever bit. To turn this into a unique large number, you have to use primes. You take the series of prime numbers – remember, the series of prime numbers starts 2, 3, 5, 7, 11, 13, 17, 19 … – and then you raise the first prime to the power of your first number, the second prime to the power of your second number and so on. Then you multiply them all together. In this case you would get the result of 26 × 33 × 56 × 75 × 117, which is 8,843,063,840,920,000,000. That’s a huge number! It won’t even fit on my calculator properly.

  Every composite number is a unique product of its particular prime factors. 3 × 7 only makes 21. It never makes any other number. It’s the same with this large number we have created. It can only be the product of that particular arrangement of primes. As it can only be the product of that particular arrangement of primes, then all you have to do to get your original statement would be to prime factorise the number. But really! It takes me over an hour to prime factorise three-digit numbers, sometimes. Who would sit down and crack that one apart, just to find out that 1 + 1 = 2? But it turns out that this system of encoding is not intended for practical use. It is just there to demonstrate what could happen. Gödel’s theorem says that any statement at all could be encoded this way. It doesn’t matter whether or not you can easily do the resulting calculations; it’s the point that counts. Gödel proved that, with his system, it was possible to have a situation whereby the number 128,936 (for example) was the code for the statement: ‘Statement number 128,936 cannot be proved.’ Not altogether likely, perhaps, but possible all the same.

  Before Gödel, people believed that if you did find something wrong with the foundations of mathematics, a break or a gap, you would just patch it up with a new axiom or two, or maybe a new proof of something. What Gödel proved was that it doesn’t matter how much you do this; using his coded statements you can always create (or have the possibility of creating) self-referential, paradoxical statements. It’s not exactly ‘1 + 1 = 3’. It’s more: ‘If 1 + 1 = 2, then 1 + 1 ≠ 2’.

  This is the liar paradox all over again. And the fact that, using just maths, you could create this type of paradox, where something is true and false at the same time, meant that mathematics was inherently, well, not so much inconsistent, but inconclusive. This kind of thing can give you a headache if you think about it too much. Anyway, poor Hilbert was going to have to deal with the fact that mathematics could not be tidied away neatly. Imagine. You set a problem for people to solve, hoping that the answer is going to be reassuring, and it turns out to be anything but. And poor Gödel. Convinced he had a heart condition, he became paranoid and thought that all his food was poisoned. The only person he trusted to feed him was his wife, Adele. When she went into hospital, he literally starved to death.

  My grandfather is keen to know how I am getting on with this book; I don’t know why. All I want to know is what happened next. Did mathematics collapse? And if not, why not? Was Gödel wrong?

  My grandmother smiles when I ask her this one evening in her study.

  ‘If it collapsed, then how could I still be doing it?’

  ‘But …’

  ‘Gödel did not destroy mathematics. He inspired it. Everyone was inspired by Gödel, particularly Turing. Cantor proved that you could always add infinities to infinity. Gödel proved that you can always add new axioms to mathematics – and never be sure that it’s possible to prove something that is true. Turing proved that there are some computer programmes that may simply never terminate. It’s very exciting, when you think about it.’

  ‘Never terminate?’ I say.

  ‘That’s right.’ She smiles. ‘Say you give a computer a really hard problem to solve. It could take a million years to come up with an answer but it will come up with an answer – at least, you think it will. But how would you know? You won’t be around in a million years to check, so how can you know, in advance, if something is computable or not? Turing tried to prove that there would be a way of finding out but, in the end, he had to
conclude that the problem was undecidable. Sometimes, you just can’t know if a problem has a solution or not.’ She turns to the computer and fires up one of her homemade programmes. ‘I think you might be ready for the next part of the story,’ she says.

  Part Three

  A bit beyond perception’s reach I sometimes believe I see that life is two locked boxes, each containing the other’s key.

  Piet Hein

  Chapter Twenty-one

  A dream: I am lost in a forest, with no one there apart from me. I can hear strange whispers which I try to follow but I know there isn’t any point. Soon, I come to a cottage, with wild roses growing outside, and walls green with ivy. I think, I am in a dream and can therefore enter this cottage: this is the kind of thing it’s OK to do in dreams. Inside, I find that the cottage walls are covered with letters and symbols. The aleph-null symbol is there, repeated like wallpaper over the hallway. The numbers from my necklace are there too: 2.14488156Ex48. The rest of the hallway is randomly decorated with images and ideas from the last week: the Green Man, the PopCo code, a diagram from Mark Blackman’s seminar.

  I enter the living room to find it arranged like a library. There are DVDs, videos and books lining each wall. I remember some conversation where I claimed not to have these collections myself, or be interested in anyone else’s. However, I am impressed by this one. All the films are favourites of mine, or my grandparents. There are maths films, war films, code-breaking films, films that make you cry because the world has changed and people don’t help each other any more. I look at the books on one shelf and realise I am looking at my grandfather’s collection. Books about Gödel, books about astrology and flowers and alphabets. There is a biography of someone who put together an ancient language from mere fragments, and my grandfather’s most well-thumbed code book, Secret and Urgent: The Story of Codes and Ciphers by Fletcher Pratt. This book, published in 1939, contains my grandfather’s favourite frequency table of occurrences of letters in English.