PopCo (32 page)

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:

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 2
⁶
× 3
³
× 5
⁶
× 7
⁵
× 11
⁷
, 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.

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

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.

Another set of shelves also contains books; and I realise that these are all mine. Mary Shelley, Edgar Allan Poe, H. G. Wells, Jules Verne, Samuel Beckett, Raymond Chandler, William Gibson, Umberto
Eco, Marge Piercy, Margaret Atwood. The kits I have created for PopCo are here too, but changed. Instead of colourful boxes carrying the PopCo logo, the material now appears in grown-up books, with my name, Alice Butler, on the spine. For a second I feel like an author, how I have always wanted to feel. Then everything feels odd. I realise that this cottage is the inside of my brain, and I think that it’s probably dangerous to go wandering around inside your own brain, even if it is disguised as a cottage, and you are, objectively, inside a dream.

On the floor, there is a single wooden block, with the letter A on two of its faces. The other four faces are blank. It’s not that messy in here, apart from the block on the floor, but I am thinking, I have to tidy this up! I have to make it make sense! I have to get out of here. I concentrate as hard as I can, almost meditating, and everything in this room starts to melt into code. I wonder where it’s going – the cottage and its contents – and then I realise that it’s filling the blank faces of the wooden cube. I don’t know how this is happening but, some time later, I blink and find that the process is over, the cottage has gone and I am in the forest alone again. When I touch my necklace, as I always do after some dramatic or dangerous event has occurred, I find it has changed shape. I frantically pull the chain out and see that instead of carrying a locket with a number inside it, it now carries a little wooden block, with all the information from the cottage inscribed on its six faces. Even though I know the necklace number by heart and haven’t really lost anything, I shout
No!
so loudly that my voice echoes in the whole forest, and birds fly out of the trees like dust beaten from an old sofa.

When I wake up, it is the middle of the night. I have slept for hours. With a thick head and a gravelly throat, I think about getting up but somehow can’t, so I just lie there on my bed in Room 23 feeling sorry for myself instead. I wonder whether anyone will come in the morning or whether, as usual nowadays, I will have to be ill on my own without anyone’s help or comfort. Even Atari doesn’t usually care that much when I am ill. In the language of Paul Erdös, cats are bosses, not slaves.
Don’t be pathetic, Alice
. There’s no point, anyway. Being pathetic only has a point if there’s somebody watching. Aching all over, I get up and run myself a glass of tap water. I take
a book from the shelves and get back into bed. It’s cold in here, or maybe that’s just the fever. I probably should take something for this.

What have I got with me? Some Arsenicum, some Lycopodium, some Nux Vomica and some Gelsemium. Hmm. Not a very big choice considering that there are tens of thousands of homeopathic remedies. But I feel cold, and I still feel like tidying up all the time. Do I feel nervous about the future? No. Do I think I might die from this? Yes, irrationally, I do. I think every cold I get will kill me. Fastidiousness implies both Arsenicum and Nux, as does coldness. However, I don’t feel particularly angry or snappy (Nux), and the fear of death is certainly Arsenicum. I don’t feel as restless as Arsenicum usually does but, still, I take an Arsenicum 200 and, without even looking at the book I have selected, I go back to sleep.

*

By the time the summer holidays start, I have finished as much of the prime factorisation as I can do without going mad. I ask for a two-week break. My grandfather says I have done enough, anyway, and takes me off the whole project for the time being. My grandmother approves of his releasing me from this huge task to enjoy a ‘normal summer’ like ‘other children’. I feel exhilarated at being able to abandon the rest of this task but, as the holidays get under way, I get an aching sense of having left something unfinished. I hate this feeling. It’s like being away from school on the day of a test: you miss the test but also the nice feeling of the test being over.

On my birthday, my grandparents present me with several gift-wrapped boxes: a new fountain pen to use when I go back to school (I am changing up to the comprehensive this year, which I am excited/nervous about), an RSPB bird-watching book, a set of my own cricket stumps, a David Gower mug, a radio cassette player and a small silver digital watch. This is the best present haul I have ever had! Perhaps it’s because I am a prime number of years old. My grandfather explained to me last year about my prime number birthday but of course I was ten then: 2 x 5. Now everything about me is prime! Once I have finished opening my presents, my grandmother starts to explain about the large tea-chest my grandfather is dragging into the sitting room.

‘It’s from your mother,’ she explains. ‘Books. Diaries. I don’t know if you are too young … But she did say that she wanted you to have them when you turned eleven. So here you are.’

My grandfather finishes dragging it into the middle of the room and then stands there looking at it, puffing slightly. I don’t know what must be happening to my face. My mother? Bloody hell.

‘Well…’ He says. And I think this is going to be a ‘Well…’ that means I am going to have to examine this box now, with them both watching me. Instead, my grandparents exchange a slightly concerned look, and leave the room.

Oh God. I’m sitting there staring at the box, not sure what to do. I didn’t know my mother had left me anything of hers. Why did no one tell me sooner? When I still lived with my father I would have given anything to touch even a hair of hers. If I could have found anything at all – a single earring, a thin bangle, a small torn-off corner of an old shopping list – I would have guarded it more carefully than I guard anything in my whole life, even my necklace. But there was never anything. As if my mother was a habit my father had given up, every tiny trace of her was removed less than a week after her death. When I returned to the flat (I had been staying in my grandparents’ old house in town) everything to do with her was gone.
Everything
. There were no bras, no books, no chewed old biros, no brown bread, no postcards of ballerinas, no tins of mints, no pot plants, no cassette player, no tapes, no notebooks, no apple cores, no lipstick, no violin, no nothing. And he thought I wouldn’t notice.

Now this. A whole box of things that she has touched; books she has actually read. Diaries? Imagine that. With my eyes wide, I approach the box. It is open at the top. Bloody hell! There aren’t just books in here. There is a teddy bear, old and battered, his head poking out from a pile of novels. There is also a cassette which I later discover contains a series of violin recitals taped from the radio. I spend the rest of my birthday morning slowly and carefully taking my treasure upstairs in small, easy-to-carry piles. Then I take the empty box up and re-fill it with the piles of books. The teddy is now on my pillow. I play the tape in my new stereo. I still haven’t opened any of the books.

I don’t know anything about my mother at all. When she died, there was no one to ask. Everyone was just too upset to deal with
questions from me so I just learned not to ask them. Perhaps everyone thought I was too young to understand. This is why I am surprised when, suddenly, today, my grandparents offer to tell me anything I want to know about her.

‘Really, Alice,’ my grandmother says gently over lunch. ‘Anything at all.’

‘Did she like maths?’ is all I can think of to ask.

They both laugh. ‘Oh, no,’ says my grandmother. ‘Not even through her music. She didn’t have any interest in it whatsoever.’

My grandfather looks a bit sad, so I don’t ask anything else. After lunch, he settles down with the Voynich Manuscript calculations, and my grandmother puts on a silk scarf and goes over to Tracey’s nan’s house to make the birthday tea for my party.

I spend the afternoon on my bed, flicking through novels and music books. There is a diary that I haven’t plucked up the courage to examine yet. When I eventually do, at about four o’clock, I don’t actually read it. I simply sit there touching the writing, wishing things had been different, not allowing myself to cry.

There is hardly any time to get ready for my party. I’m not sure I even want a party now but the village hall is booked and invitations have gone out and the tea is probably already made and on its way to the hall; flowery china plates covered in cling-film, and jugs of orange squash. My tummy hurts just thinking about it. I begged and begged to have this sort of party because all my friends have them but I don’t really like any of the people I have invited. Will anyone from school actually come? I have seventeen returned slips from the invitations – all with little ladybird patterns on them which I liked when I chose them from Woolworth’s but on reflection seem a bit babyish – but I sent out twenty-five. And this was before we broke up for the holidays, so maybe everyone has forgotten. My grandparents agreed to the hall but not to a professional disco, so my grandfather will be in charge of the music, using my cassette player and some borrowed tapes (mainly Rachel’s). Will this be a disaster? Boys are coming, too. Will
this
be a disaster? How many disasters can one party have? I wish this was someone else’s party, not mine.

I end up wearing my blue dress and red sneakers. I am not sure they go together but I don’t really care. I brush my hair and braid it carefully into two plaits, the way I have been doing since I was
about six. Then I rub a flannel over my face and go downstairs. My grandfather is still there with the Voynich Manuscript.

‘Time to go,’ I say.

The village hall is like a witch’s house; small, grey and pointy with climbing plants all over it. You enter it through a vast, wooden, creaking door that can be held open with a huge brass hook (which is good, because this door is so heavy, and on such a tight spring, that children could easily be crushed to death by it trying to close). This door leads to a tiny foyer, cobwebby and dark, with five rusty coat-hooks and a broom. Then another door, thin wooden slats, leads to the hall itself. This hall is a repository for memories of bingo, bridge tournaments, chess club (which my grandfather created but which is no longer going), Brownies, Scouts, Guides, Boys’ Brigade, Girls’ Brigade, the Woodcraft Folk (also defunct), the Wallflower Theatre Society, the Women’s Institute and the Amateur Operatic Club. A local rock band also practise here on a Thursday evening. Even our house, which is half a mile from the hall, vibrates on a Thursday evening. The band has a girl singer who always dresses in black. She smokes long, thin cigarettes, too. When I first saw her, I wanted to be her so much that my stomach ached for two weeks. She was carrying an electric guitar.

In the afternoon, the sun coming through the windows of the village hall is like something coming from heaven, and you imagine angels playing in the dust-storms inside. Directly in front of the doors, about twenty paces beyond them, is the stage, made of dark, shiny wood. You can’t climb on to the stage from the hall if you are a child; it’s too high. Instead, if you want to go on the stage, you have to negotiate the maze of little back rooms – kitchen, backstage, dressing rooms, broom cupboard and store – to get to the actual stage door, beyond which there are seven shiny wooden steps which lead up to it. I used to come to Brownies here, which is how I know all of this.

Everyone at my party wants to get on to the stage. I am the queen of all hostesses in my blue dress, because only I know the ancient secrets of the stage. While the grown-ups stand around the fold-out tables talking, and my grandmother regards the geometry of the birthday tea with evident satisfaction (right-angled jam sandwiches, cheese footballs as neatly spherical as Riemann’s zeros, jellies in the
shape of ellipses) we all take off our shoes and creep through the haunted passageways until we get to the seven magical steps and then we spend about half an hour skidding up and down the shiny stage in our stockinged feet. My grandfather fires up my new cassette player and selects a tape from Rachel’s collection. The first five or six songs are recent pop hits that everyone knows from the weekly chart countdown on the radio. My guests and I create a game where we stand on the very edge of the stage, waiting for the climax of the song, at which we all jump off simultaneously. This seems to have been inspired by something called
The Kids from Fame
, although I don’t know what this is. After we have jumped off the stage we immediately race around the back again, almost forgetting about the ghosts, ready to jump off at the next climactic bit of the song. Even the boys do this! Could I be having the best party ever?