PopCo (31 page)

I am occasionally allowed to sit with my grandmother and feed in the numbers which set the programme off, and we watch while the black blobs expand, expand and die, turn into little flashing lines or just die after a single generation. With this odd background of pixels on a black and white screen, we sometimes talk about other things. I have learnt some more about the Riemann Hypothesis at last, and I know how to use four-dimensional co-ordinates (although I don’t know what the resulting graph would look like). My grandmother has explained that after all these years of working
with four dimensions, she can actually see them in her head, even if it is supposedly impossible. I have read a science-fiction book that says the fourth dimension is time. My grandmother corrects this.

‘Time has only one dimension,’ she says. ‘In the physical world we can experience three dimensions of space, and one of time.’ A fourth dimension of space would be very different, she explains, with outside and insides of things becoming infinitely complicated, and cubes becoming simply the edge of something else.

As the game of Life trickles away in the background, we talk about mathematical proof, and my mother and father, and even what my grandmother was like as a young woman. One day I ask about my necklace again.

‘What did you mean,’ I ask, ‘when you said I’d been turned into a living proof? Is this actually the answer, written on here?’ I pull out my necklace and open the small silver locket. I point at the numbers inside. ‘I know now that the aleph-null symbol is a red-herring, but this – this other bit – is that his proof?’

‘You should ask your grandfather,’ she says, like she used to when I asked her about the weather.

‘I can’t ask him,’ I say, sadly. ‘He won’t talk about it any more.’

The screen on the desk has been full of activity – large geometric patterns in every part of the screen – for the last five minutes. Now, inexplicably, the patterns start to die off, eventually ending as three small lines of three blobs each which flash horizontal and then vertical
ad infinitum
. I think it’s OK to say
ad infinitum
about these patterns as everyone knows they are final positions in this game; they won’t change any more, now.

‘It’s a code,’ she says, after a long pause. ‘A code he thinks you will be able to work out.’

‘But I can’t …’

‘Not now. When you grow up. When he is dead, you will inherit all his papers, you know that, don’t you?’

I didn’t but I nod dumbly anyway.

‘He thinks that, if you want to, you can work out the code then and make your own choice about what to do about it. It’s hard for him. He doesn’t want anyone to recover the treasure, but at the same time he very much wants to be known as the person who solved the Stevenson/Heath puzzle. Perhaps one day there won’t be
a bird sanctuary on the site any more, or perhaps, rather than digging up the treasure yourself, you would want to give the solution to an archaeology department of a university and let them organise a proper dig …’

‘Why doesn’t he do that?’

‘He hates universities.’

‘Oh, yes.’ We both smile.

‘Perhaps war will break out again, or perhaps one day you won’t have all you need like we do now. It’s there and you do have the key. What you do with it will be your choice.’

‘Unless there is no treasure,’ I say, experimentally. I have been considering what my grandfather said about the Stevenson/Heath manuscript being a hoax. Even if Stevenson did exist (which, according to my grandfather’s evidence, he certainly did), it could still be a hoax.

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 19
²
. The number 105 is made of the primes 3 × 5 × 7. The number 5625 is made up of 3
²
× 5
⁴
, 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 even 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 ℵ
₀
. 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.