Saturday, January 29, 2011
Thursday, January 27, 2011
£500 if you can find me a job
That worked a treat. And the lucky winner was Simon.
It was fun, but a short contract. Now I'd like another one, so I repeat:
If, within the next six months, I take a job which lasts longer than one month, and that is not obtained through an agency, then on the day the first cheque from that job cashes, I'll give £500 to the person who provided the crucial introduction.
If there are a number of people involved somehow, then I'll apportion it fairly between them. And if the timing conditions above are not quite met, or if someone points me at a short contract which the £500 penalty makes not worth taking, then I'll do something fair and proportional anyway. (The thing Simon pointed me at only lasted three weeks and I paid him in full anyway, because it was neat.)
And this offer applies even to personal friends, and to old contacts who I have not got round to calling yet, and to people who are themselves offering work, because why wouldn't it?
And obviously if I find one through my own efforts then I'll keep the money. But my word is generally thought to be good, and I have made a public promise on my own blog to this effect, so if I cheat you you can blacken my name and ruin my reputation for honesty, which is worth much more to me than £500.
Anyhow, if you're interested in helping out, my CV is at http://www.aspden.com, and any advice on how it could be improved will be gratefully received.
It was fun, but a short contract. Now I'd like another one, so I repeat:
If, within the next six months, I take a job which lasts longer than one month, and that is not obtained through an agency, then on the day the first cheque from that job cashes, I'll give £500 to the person who provided the crucial introduction.
If there are a number of people involved somehow, then I'll apportion it fairly between them. And if the timing conditions above are not quite met, or if someone points me at a short contract which the £500 penalty makes not worth taking, then I'll do something fair and proportional anyway. (The thing Simon pointed me at only lasted three weeks and I paid him in full anyway, because it was neat.)
And this offer applies even to personal friends, and to old contacts who I have not got round to calling yet, and to people who are themselves offering work, because why wouldn't it?
And obviously if I find one through my own efforts then I'll keep the money. But my word is generally thought to be good, and I have made a public promise on my own blog to this effect, so if I cheat you you can blacken my name and ruin my reputation for honesty, which is worth much more to me than £500.
Anyhow, if you're interested in helping out, my CV is at http://www.aspden.com, and any advice on how it could be improved will be gratefully received.
Thursday, January 20, 2011
Uncle Boonmee Who Can Recall His Past Lives (Film)
Tedious bewildering rubbish. Apparently the six different reels are all shot in a different style and they each echo a different blah blah blah...
Made me want to stuff the Palme d'Or up the arse of the Guardian film critic.
And don't get me wrong. I usually love pretentious foreign rubbish.
Made me want to stuff the Palme d'Or up the arse of the Guardian film critic.
And don't get me wrong. I usually love pretentious foreign rubbish.
Friday, January 7, 2011
An Idea for Teaching Times Tables to Children
Fully 50% of the table is useless, and has been since the introduction of the metric system in 1970.
So only do 2-9 x 2-9. That's 64 facts to memorize.
First get them to write it out by addition. If they can't do this then you may want to work on addition instead!
Everyone stands in a ring. The teacher's current favourite starts. Their job is to ask their fellows questions they can't answer.
The current favourite starts. He should pick someone. It's ok to pick on the weakest! Ask them a question. If they can't do it, you get another go. If they can, then they become 'it' and it's their turn to persecute.
I'd imagine that after half an hour of this, your whole group will be table perfect.
So only do 2-9 x 2-9. That's 64 facts to memorize.
First get them to write it out by addition. If they can't do this then you may want to work on addition instead!
Everyone stands in a ring. The teacher's current favourite starts. Their job is to ask their fellows questions they can't answer.
The current favourite starts. He should pick someone. It's ok to pick on the weakest! Ask them a question. If they can't do it, you get another go. If they can, then they become 'it' and it's their turn to persecute.
I'd imagine that after half an hour of this, your whole group will be table perfect.
Wednesday, January 5, 2011
Effortless Superiority
This isn't a boast. It starts off sounding like one. Actually it's a confession of utter stupidity, blindness and laziness. I write it as catharsis now it's far too late, and as a warning.
There is a phrase, 'Effortless Superiority'. I've always thought it was the motto of one of the Oxford Colleges, but apparently not.
I used to take it seriously. As an ideal.
I don't know my times tables. I never had to learn them. I can work out what they have to be if I need them. I could do that fast enough to beat all the other children at primary school on arithmetic tests by an overwhelming margin. So I never bothered learning the tables, or practising multiplication. I just used to sit around thinking about other things, and then win the test anyway.
I was told at primary school. At *primary* school. That I'd never amount to anything, because I was too lazy. That I'd never get any 'O' levels (0). By a teacher called Mr Nicholson. He made a special point of telling my parents the same thing. I thought he was an idiot. I think my parents probably did too.
I never did any work at school. I certainly never did any homework. I did usually pay attention in class. It was often interesting.
Once or twice I remember finding something mathematical that I didn't understand immediately.
Come to think of it, I still can't do integration by substitution. I remember noticing that I couldn't, and deciding that it must be a silly, uninteresting special purpose trick.
On one particular occasion (a problem in dynamics. something to do with balls on wire hoops.) I was worried that there was a maths-thing I couldn't do, and so I worked through a couple of examples in the back of the textbook, and I remember the light dawning.
That should have been a clue. But it was lost in the noise.
I remember learning a table of German vocabulary once for an exam. It took about half an hour, and I got it perfectly. I thought this was an utter waste of time. I don't know why I did it. When I took the exam and got a near perfect score, and it was obvious that if I hadn't bothered I'd still have aced it, I remember thinking that that had been a complete waste of half an hour in which I could have been doing something more interesting. (1)
When A-level Chemistry got to the point where I had to learn the colours of the transition metal ions, a table of apparently random text which must have had oooh, twenty entries? I gave up the subject. I am not joking. I remember asking the teacher if I could carry on doing the practicals, which I enjoyed, and not come to the theory lessons. He wouldn't let me, so I just gave up the course entirely. I used to spend the time in the school canteen (we had a nice sixth form common room where you could smoke) playing pool.
I was still entered for the exam, so I took it anyway. Despite having only gone to the first year of a two year course, I got a B because I could deduce most of the required answers from physics and from what I remembered of 'Teach Yourself Organic Chemistry', and 'Asimov on Chemistry', which I'd devoured when I was about ten years old.
Chemistry had been my first love. The first thing I was passionate about. I used to make little plasticene models of hydrocarbons and take them to school, where I would try to persuade Mr Nicholson to hang them from wires on the ceiling to represent my idea of what a gas was.
When, at around ten or eleven, I had my first fevered dreams about the bodies of women, I also had equally fevered dreams about space-filling molecular models of such exotica as DNA, which could be purchased for insanely high prices from glossy catalogues that my father would bring home from work, along with copies of New Scientist donated by his friends.
And in spite of never lifting a finger I was always clearly the best at anything intellectual. I seem to remember, back in 1986 or whenever, thinking that I was probably the cleverest sixth-former in my home city, on the basis that I didn't know anyone who'd done as well at A-level as me.
The thing is, this might actually have been true.
I've mentioned where my B at chemistry came from. My A at physics, and my grade 1 at physics S-level (a special form of super-A level that most people don't know about), came on the back of exams that I'd taken while coming down from LSD, which I'd taken a couple of days before in the sure knowledge that I'd have straightened up enough by the time of the exam to sail it anyway. This was, to say the least, not the first experience I'd had with drugs, and even allowing for the after effects of acid I knew I wasn't going to have any trouble with Physics.
By that time, I'd decided that I wanted to be a mathematician. That seemed to be the subject where everything was obvious. Nothing needed to be learnt or memorized. People just said things that were obviously true, and once you'd heard them it was like you'd been born knowing them, and it had just taken someone to draw attention to the memory.
Slightly to my surprise, Cambridge University turned me down when I applied there.
I'd taken my maths A-level a year early and got a B, so that was the only evidence they had at the interview. And in the interview, they'd asked me a question I couldn't do.
My school were furious, disbelieving. With my typical modesty, I decided that Cambridge were wrong.
I had been planning a year off (taking drugs) anyway, and so I asked London if they'd hold open their unconditional offer of a place, and I applied to Cambridge again. To the same college. Chosen because I thought it was the prettiest one and because it was both particularly hard to get into and top of the academic table of Colleges at the time. To do what was famously the hardest maths course in the world, at what I thought was the best university in the world.
Don't misunderstand. I didn't decide to work harder or anything. I just thought they had been wrong, and I'd give them another chance to make the right decision.
I sat my A-levels, S-levels, and STEPS (Cambridge Entrance Exam), and aced them apart from the B in Chemistry and dropping a grade or two in one of the two STEP papers, which it turned out didn't matter.
King's gave me a second interview, and I did really well on this one. They accepted me.
At College I had the disturbing experience of not being the brightest person I knew any more.
There were people who'd been to public schools, who'd been prodigies and competed in scary things called Olympiads that I'd never heard of, and who had been appropriately stretched by gifted teachers. There were other people from comprehensive schools like me, but who'd worked insanely hard to get in.
I reckoned that I was about halfway up my group of twelve. But several of them have told me since that they were scared stiff of me, who didn't seem to try, and had had a standard state education, and yet who seemed given to flashes of insight.
I'm not sure that I actually did anything you'd call 'work' in the first year. But they did have this supervision system, where you'd get question lists handed out and you were supposed to solve as many as possible before going to see one of the Fellows about them.
I used to have a go at these sheets. I used to find that I could do the first five or six questions without trouble, and then everything else would be too hard. But if you turned up to the supervision, the Fellow in question would then explain how to do the remaining five or six.
I thought that's what they were for.
In my first year exams I came in about the top ten per cent of my year. I scored below a couple of my college-mates, both of whom were super-educated public school kids, and obviously very bright as well.
I figured that was good enough. I had girls and alcohol to worry about. I was up to my neck in both. I'd given up on drugs because they were all boring except for LSD, and I'd given up on LSD because I'd had one too many bad trips. But I'd filled the gap with heavy drinking and socializing. Mainly I think because it got you laid, but it was also great fun in itself.
So in the second year, I just gave up. I still went to all my lectures, because I enjoyed them. In fact despite near-constant partying, I think I missed two lectures in my entire time at college. I didn't love mathematics any less than I had.
But I no longer had any sort of go at the example sheets. I just turned up to supervisions and got the supervisors to explain how to do the problems. I could usually do the first couple off the top of my head anyway. As far as I know, my teachers still thought I was pretty good, because of my initial solid first, and the fact that I asked the right questions about the subjects I found interesting.
And at the end of the second year, I got a second. I was about half way up the list. I remember that my teachers seemed quite disappointed, and my reputation took a bit of a hit.
I'd been expecting it. I'd deliberately taken the year off, but I was never under any illusions about what my capabilities were. I knew that if I'd actually practised answering exam questions I'd be able to do them faster, and the tripos exam was like every other exam that I'd ever sat, a speed test rather than a test of understanding. I still knew how all the questions worked, it just took me more time to solve them than I was allowed to spend. The reason that other people were doing better than me was because I hadn't *cheated* by practising doing exam questions. I understood it all as well as they did. Most of them were just half-remembering formulae they'd picked up from spending many sleepless nights revising.
Everything was going according to plan.
The plan that I'd had, ever since I first thought what I should do as an adult, to sail effortlessly into a tenured academic job (when I'd been growing up, that had been a very easy thing to do), and to carry on learning and teaching and who knows, maybe think of something new and interesting one day. It had never really occurred to me to do anything else. If I thought about the 'real world' at all, then it was as a sort of sewer, filled with people who did tedious and unchallenging things because they wanted money. Academic salaries and PhD grants weren't great in the way that they had once been, but still plenty enough to support a studenty lifestyle, and that was all I wanted. I saw myself as being one of those kindly unmarried dons who hangs around in college and lives in college rooms, and lets the undergraduates substitute for the children he would have had if he hadn't been devoted to his craft. It looked great.
But if I wanted to do a PhD in pure mathematics (2), I needed a first-class degree. There was no funding available otherwise.
So I went back to my first year routine of having a half-hearted go at the example sheets before heading off to supervisions to have it all explained face to face. To be spoon-fed the answers without having made the effort to think of them myself.
It was a bit harder this time, because as well as having to work out what the new stuff was about, I realised that I hadn't really understood the second year stuff even though I thought I had (3), and so I needed to work out how all that worked while working out what the new stuff meant.
But it all worked out OK, almost. I was actually good enough to start catching up again.
I didn't bother with the third year computer project, despite that I'd been programming since the age of 10, would have found it very easy, and it was worth a full third of the credit that I'd need to get a first (4).
I didn't want to do something that I wouldn't learn anything from. I wanted to be a mathematician, not a programmer. And I was already as good a programmer as I'd ever need to be. (4.5)
By the time the final exam was close enough to worry about, my first was in the bag.
I had actually stooped to trying my hand at some past papers under exam conditions, which I still thought was cheating, but getting a first looked important enough to cheat at, and after the first couple of papers, I'd got good enough at it to reliably score the marks necessary for a first.
Rather to my surprise, (remember that this was the first time I'd ever taken an exam seriously enough to practise doing it), I found that I rather enjoyed the process. And even more surprisingly, learning to do the questions quickly actually seemed to make the ideas behind them clearer and more beautiful. They achieved a sort of focus that they hadn't had before.
Knowing that I was going to do well in the exam, I took Easter term pretty much off. The sunshine was lovely, this was my last year with all my friends, there was punting to do, and cricket to play, and riverside pubs to sit in, and parties to go to, and not many lectures to go to, and I understood enough subjects in enough detail to be confident that even if the questions turned out to be particularly hard, I'd pass. (I thought anything lower than a first was a fail.)
And of course I failed (5).
I have excuses, of course, for the failure of my confident prediction. The questions in my particular favourite subjects deviated from their usual predictable patterns. The first paper was much harder than I expected. One particular topology (one of the two subjects where I really could have given the lectures) question was so incomprehensible that all I could do for an answer was to draw a picture and write underneath 'It works because of this'. I had no idea how to do a formal proof as asked for.
My finals were four exams over two days, three hours each. After the first day I knew I'd done badly. I couldn't sleep for worry. The questions in the third and fourth exams were more in line with normal, but I actually managed to go to sleep in the fourth one because I was so tired after two gruelling days awake.
When the results came out I wasn't surprised. I'd calculated my mark afterwards, and it was on the line between a second and a first. I fell just the wrong side. (5.5)
Cambridge Mathematics has a charming tradition of reading out the results at a big ceremony in the Senate House. They read out the names of everyone who has got a first, in alphabetical order. I remember the list of names going Aa.... Ab..... Ascot (or something), Bagg...
Bagg was my college mate Jenny. As soon as I heard her name I knew I'd screwed it. Without any previous warning in about twenty one years, I'd managed to be not as good at something as I needed to be.
That was pretty much the end for me.
No first meant no PhD funding in the thing I wanted to do. I didn't have a plan B. It literally hadn't occurred to me in the month or so before the exam that I could fail it.
I fell on my feet after a fashion. Someone told me of a PhD place for a specific project in London (at Imperial College, which is a pretty damned good university), which had its funding allocated already, but where the student in question had dropped out. I rang the man who controlled the funding, he invited me to London, we talked for an hour and the place was mine.
But it didn't work out. I didn't get on particularly well with my supervisor, I wasn't interested in the thing I was supposed to be studying, he wasn't interested in any of my ideas, and there was no one else to talk to.
I'd been looking forward to London. I'd always enjoyed going there as a boy. But it turned out to be awful. A lonely wasteland of concrete and filth. And being an academic didn't look like much fun in London. More of a sort of glorified schoolteacher job, and not that glorified either.
And then my intuition failed. One day I was trying to read a research paper, and I found out that it was just squiggles. No pictures, no handle on what it meant. Not even a vague feeling that if I tried just changing this bit, something would happen. It was suddenly all greek to me.
I understood, for the first time in my life, what it was to not understand a piece of mathematics.
And I figured that was the end of the line. All my life I'd watched people give up maths, because it didn't seem to mean anything to them. I knew that however hard they tried, it wouldn't ever make proper sense to them like it did to me.
It just seemed that that had happened to me. There's obviously a point beyond which you can't go, and it just so happened that mine was about half-way through a PhD.
I don't think it occurred to me that I might be able to work my way through this. That just wasn't the way I thought it worked. By that time I didn't care anyway. Two years in London had put me off the whole idea of academia.
I moved back to Cambridge because that's where most of the people I liked lived.
At about this time, I started to run out of money. The near limitless ability of the 1980s British State to subsidize lazy good-for-nothing layabouts who thought the world owed them a living had met its match in me. I had an overdraft, accounted for pretty much exactly by six years of booze and cigars. Even though today's self-funded students will laugh at my tiny £6000 debt, it seemed to me that I was so poor that I had to give up smoking. I even gave up 2000AD, a comic that I'd read regularly since I was a boy, because the 50p it cost every two weeks was a noticeable expense.
Of course when the PhD funding ran out, I managed to start drawing unemployment benefit. But this wasn't quite as much, so there was further belt tightening to be done, and I didn't fancy it. My supervisor said that I'd got about another six months work to do before I'd be able to hand in a doctorate, and I was pretty sure that meant about twelve months. Also it was going to be a complete piece of shit. Nobody would ever read it out of interest, and I wasn't in the least interested in writing it.
I got a job, programming. It was great fun, it turned out. People actually gave you things to do that they didn't know how to do themselves, but they were usually quite easy things, so they were usually done quickly, and (and this was the real revelation) people were actually grateful when you solved their problems. I mean that you felt like you'd done them a favour. This was a revelation to me. I loved it. After about two months I had this awful dream where I'd gone back to London to try to finish my PhD. I woke up in a cold sweat of terror, and at that point I knew I was never going to be Dr Aspden. Which was strange. I'd always thought that Mr was what you were called from about 16 to about 24. Like a sort of probationary title. It was like I'd been told I'd never be able to get a driving licence.
I got used to being mediocre. I was good enough to have got a Cambridge maths degree. I was probably top quartile. But there are a fair number of people like that in the world.
I've thoroughly enjoyed it. I'm pretty good at what I do, and it's fun.
I took up other hobbies. I'd always liked playing sports, even though I was useless at it, and, almost by accident, I took up rowing, which is the local sport here, for townies as well as the university.
There was no reason to expect to be anything other than rubbish at rowing. I'm average height, stronger than most men my size, but not vastly so, and I smoke, which doesn't go well with a sport where the chief physical variable is a capacity for consuming oxygen.
And I was. Terrible at it. For years.
But I enjoyed it, and I practised it, and I got good at it. I was never in a million years going to be any sort of star at it, even in the rather limited competitive environment of Cambridge town. But I ended up being, if not actually any good, better than I would ever have believed possible to start with.
And somehow it taught me that if you work at something, you get better at it.
And I decided a few years ago that I should learn LISP, an antique computer language out of the dark ages, because there seemed to be something special about the way LISP people talked.
So I got the standard textbook Structure and Interpretation of Computer Programs, and I found it harder than most things about computers. But I was damned if I was going to let some sort of computer-thing be hard, because nothing in computers is even a bit hard. So I did the exercises in the book to see if they would make it clear, and found that solving carefully chosen example problems is fun, and you know what, the more of the exercises I solved, the more I understood the book, and the better I understood, the more of the book I could read.
And it taught me that if you work at something intellectual, you get better at it.
Which I didn't believe for the first thirty years of my life.
I thought that you were born good at things. I thought if you weren't good at things, working hard at them was just a path to ruining your life with fruitless toil, to fail one level higher than you would have failed anyway. To be honest, I still believe something like that, just not as extreme a version as I once did.
Why would I not have believed this? You cannot move without people praising the special quality of genius. You cannot read without finding out that authors and poets and musicians have their works come to them fully formed. The special genius of my heroes, Feynman, Einstein, Newton, Conway, Von Neumann is trumpeted from rooftops. You never hear that they lifted a finger apart from doing exactly what seemed most amusing to them at the time.
And it had been absolutely true for me for twenty years. I'd been told over and over again that I needed to work harder. And every time I'd ignored the advice and turned out to be right.
And when I was twenty one the 'problem of induction' hit me over the head with a whisky bottle during my final exams. And I was stone dead and I didn't even notice because twenty years of confirming evidence isn't overturned by a human mind just like that.
And I've only just noticed.
Recently, I was asked to teach a young friend enough maths to get him onto an engineering degree. I loved doing this. Sitting down and talking with him and taking him through trial problems and trying to work out where his mental blocks were and blowing them away was the best fun I've had in ages. And he's now doing his engineering degree, although he's going to screw it up by, ironically, spending so much time rowing really seriously that he'll be too tired to think about engineering.
And I've been reading the collected works of the brilliant philosopher Eliezer Yudkowsky, who's built a whole philosophy and predicted the entire future of the human race convincingly using amongst other intellectual tools Bayes' Theorem, a trivial piece of arithmetic with profound implications.
And I figured if I'm going to understand Eliezer Yudkowsky's thoughts I'd better understand Bayes' Theorem and its consequences as well as he does, which I'm still arrogant enough to think that I can do.
And so from a combination of these effects, I've started doing mathematics, recreationally, for the first time in my life. And indeed mathematics at all, for the first time in about fifteen years.
And it turns out I'm pretty damned good at it. I shouldn't be, because I'm forty years old now, and no one's as good at maths at forty as they were at twenty. But if my powers have declined I can't tell.
I'm reading David MacKay's beautiful book Information Theory, Inference, and Learning Algorithms, and it's like reading a thriller. I've patiently worked through the examples in the first two or three chapters, even the ones that look boring, because I remember that the boring looking examples in SICP always turned out to teach unique and thrilling lessons.
I've sort of got my head round the introductions to Information Theory and Inference, and I think I already see why the Noisy-Channel Coding Theorem is going to turn out to be true in chapter 6 or wherever it is, and why it is also going to turn out to be of little practical value. And I can't wait to find out if I'm right.
And this has brought back memories, and when I start remembering things these days I start writing, and all this has come out like a flood, and suddenly it occurs to me, because I promise you that the above is as true as my fallible memory can make it, that I may actually have been as good as I once thought I was.
And fuck the research angle and doctorates and names on interesting discoveries. I love teaching. And I'm good at it. And that's what I should have been doing with my life. Teaching mathematics and computers to bright young people. I would have loved it. And I didn't do it. And it's too late now.
And I want to call bullshit on the idea of Effortless Superiority, which it now occurs to me I may have been the only human on the planet stupid enough to take seriously (6).
Footnotes
(0) English school leaving exams. A-levels are the optional ones two years later, on which University Entrance depends, S-levels are a special form of A-levels, on the same syllabus but the questions are much harder. STEP is similar to S-level, but set by Cambridge University and used as part of its entrance requirement.
(1) Even though I was good at foreign languages, they seemed like a completely stupid rigmarole. It wasn't like they were any use for anything except passing foreign language tests.
Somehow, at around this same time, I was teaching myself Latin to O-level standard out of a book my father had left lying around where I could find it. Latin was interesting and useful, because you could read about the Roman Empire and other cool things in the Romans' own words. And you could see where lots of our words came from. And their poetry was better than ours.
I've wondered if I'd have done modern languages for my degree if my school had managed to make living German and French anything like as interesting as Francis Kinchin Smith managed to make dead Latin and Greek in his two Teach Yourself ... books. Cheap self-help manuals that were doubtless intended by the publishers to be read only as far as the third chapter. From the love and care and scholarship and cleverness lavished on them I think Francis had higher goals for them. I wonder if he ever got fan mail? If I could go back and write it I would. What did a child know? I've just googled him. He died in 1958 apparently. Too early for him to have got my letter even if I had thought to write. The fact that I can remember his name after 28 years tells you how much I liked those books. I wish I'd had a chance to tell him.
But the tedium of French and German was as nothing compared to the horror of Music and the Calvary of Religious Education.
How, how for the love of Christ did they manage to make Music boring to a human child?
And for that matter, the King James Bible is one of my favourite books.
I am as bleak an atheist as it is possible to be, and yet I often read the Gospel According to St Matthew on Easter Day. I wonder how many Christians do?
(2) By this time I'd begun to despise applied maths, which seemed to be a bag of stupid half-understood magic tricks that was overdue to be swept away by computer simulation. Weirdly it only seems to be taught like that in Cambridge. In London later on I found it interesting again.
(3) As someone once pointed out, what the hell did I think it meant for someone to understand something if they couldn't use it fluently?
(4) I had done some computer stuff in the second year out of interest, but I'd found that it didn't teach me anything interesting. I already knew how to program. I can't even remember whether I bothered completing it.
(4.5) I know, I know, I was a moron. But actually it did look singularly unchallenging. Now if they'd taught SCHEME rather than matrix multiplication algorithms, who knows...
(5) got a second.
(5.5) As I remember, the topology question that I'd drawn the answer to was bastard hard, only a couple of people had attempted it, and my picture had been given almost the maximum mark. So in fact I needn't have worried. If I'd known that, I might have been able to sleep that night.
(6) I'm not whining for the sake of it, by the way. I've loved life, and I'd do the first forty years over again exactly the same way like a shot if I had the chance.
But if you recognise yourself in the above, young person, for God's sake do the exercises in the textbooks, and practise exam questions. It's not cheating. It helps you understand. A lot.
If you do the exercises, and play with the examples, then you will find that your intuition, already powerful, starts to get brighter, and rather than finding as I did that as you go forward, you start to run out of dry places to stand in the swamp, until eventually there's nowhere new to go, you'll find that as you brighten the swamp recedes ahead of you faster than you can walk.
On the other hand, if you find yourself working every hour God sends on something and you're not enjoying it for its own sake, give up. Nothing's worth that sort of life. I still think success probably comes fairly easily if you're any good. What I'm trying to warn against is pig-headedly not lifting a fucking finger to practise something you enjoy because you think that's something lesser beings have to do to make up for being lesser.
"If at first you don't succeed, try, and try again. Then give up. No sense making a fool of yourself." -- Homer
Some bits of text from the essay I started writing before it turned into the above whine.
Right until halfway into a PhD that I didn't find interesting. At that point the wellspring failed, and within twelve months of realising that it was gone I wasn't a mathematician any more. Part of the problem was that I'd never had to try at anything before, and I didn't know how to deal with anything that required effort.
Would you believe that at the time I sat my finals, I still thought that practising exam questions was cheating. I thought it was vulgar. Like you were trying to fool the examiners into thinking you were cleverer than you were.
I realized that I wasn't clever enough to bank on a first (I mean I still thought I'd get one, it's just that I'd realized there was a *chance* I wouldn't), so towards the end of my third year, I cheated a bit. But it was far too little too late, and it didn't help. I got the second I'd feared.
This only confirmed me in my belief that it was silly to try. That nothing that had to be worked at was worth having. It was easy to maintain this belief.
Lots of people have to try very hard to get into university. They slog away at A-levels, ruining evenings and weekends that should be spent being young and happy. And when they crowbar their way in, somehow managing to convince an interviewer far cleverer than they are that the lights are on in their heads, they find that it's all ashes. For nothing.
For it gets harder once you're not at school. And the only strategy these people have is to work harder. But they already worked as hard as they could.
They react predictably. They burn the candle at both ends for three years, and at the end of it they scrape low seconds and thirds. Some, released from parental pressure, realize that the game isn't worth it and drop out, pretending they didn't care in the first place.
I was terrified of being one of these people. I guess I still am.
There is a phrase, 'Effortless Superiority'. I've always thought it was the motto of one of the Oxford Colleges, but apparently not.
I used to take it seriously. As an ideal.
I don't know my times tables. I never had to learn them. I can work out what they have to be if I need them. I could do that fast enough to beat all the other children at primary school on arithmetic tests by an overwhelming margin. So I never bothered learning the tables, or practising multiplication. I just used to sit around thinking about other things, and then win the test anyway.
I was told at primary school. At *primary* school. That I'd never amount to anything, because I was too lazy. That I'd never get any 'O' levels (0). By a teacher called Mr Nicholson. He made a special point of telling my parents the same thing. I thought he was an idiot. I think my parents probably did too.
I never did any work at school. I certainly never did any homework. I did usually pay attention in class. It was often interesting.
Once or twice I remember finding something mathematical that I didn't understand immediately.
Come to think of it, I still can't do integration by substitution. I remember noticing that I couldn't, and deciding that it must be a silly, uninteresting special purpose trick.
On one particular occasion (a problem in dynamics. something to do with balls on wire hoops.) I was worried that there was a maths-thing I couldn't do, and so I worked through a couple of examples in the back of the textbook, and I remember the light dawning.
That should have been a clue. But it was lost in the noise.
I remember learning a table of German vocabulary once for an exam. It took about half an hour, and I got it perfectly. I thought this was an utter waste of time. I don't know why I did it. When I took the exam and got a near perfect score, and it was obvious that if I hadn't bothered I'd still have aced it, I remember thinking that that had been a complete waste of half an hour in which I could have been doing something more interesting. (1)
When A-level Chemistry got to the point where I had to learn the colours of the transition metal ions, a table of apparently random text which must have had oooh, twenty entries? I gave up the subject. I am not joking. I remember asking the teacher if I could carry on doing the practicals, which I enjoyed, and not come to the theory lessons. He wouldn't let me, so I just gave up the course entirely. I used to spend the time in the school canteen (we had a nice sixth form common room where you could smoke) playing pool.
I was still entered for the exam, so I took it anyway. Despite having only gone to the first year of a two year course, I got a B because I could deduce most of the required answers from physics and from what I remembered of 'Teach Yourself Organic Chemistry', and 'Asimov on Chemistry', which I'd devoured when I was about ten years old.
Chemistry had been my first love. The first thing I was passionate about. I used to make little plasticene models of hydrocarbons and take them to school, where I would try to persuade Mr Nicholson to hang them from wires on the ceiling to represent my idea of what a gas was.
When, at around ten or eleven, I had my first fevered dreams about the bodies of women, I also had equally fevered dreams about space-filling molecular models of such exotica as DNA, which could be purchased for insanely high prices from glossy catalogues that my father would bring home from work, along with copies of New Scientist donated by his friends.
And in spite of never lifting a finger I was always clearly the best at anything intellectual. I seem to remember, back in 1986 or whenever, thinking that I was probably the cleverest sixth-former in my home city, on the basis that I didn't know anyone who'd done as well at A-level as me.
The thing is, this might actually have been true.
I've mentioned where my B at chemistry came from. My A at physics, and my grade 1 at physics S-level (a special form of super-A level that most people don't know about), came on the back of exams that I'd taken while coming down from LSD, which I'd taken a couple of days before in the sure knowledge that I'd have straightened up enough by the time of the exam to sail it anyway. This was, to say the least, not the first experience I'd had with drugs, and even allowing for the after effects of acid I knew I wasn't going to have any trouble with Physics.
By that time, I'd decided that I wanted to be a mathematician. That seemed to be the subject where everything was obvious. Nothing needed to be learnt or memorized. People just said things that were obviously true, and once you'd heard them it was like you'd been born knowing them, and it had just taken someone to draw attention to the memory.
Slightly to my surprise, Cambridge University turned me down when I applied there.
I'd taken my maths A-level a year early and got a B, so that was the only evidence they had at the interview. And in the interview, they'd asked me a question I couldn't do.
My school were furious, disbelieving. With my typical modesty, I decided that Cambridge were wrong.
I had been planning a year off (taking drugs) anyway, and so I asked London if they'd hold open their unconditional offer of a place, and I applied to Cambridge again. To the same college. Chosen because I thought it was the prettiest one and because it was both particularly hard to get into and top of the academic table of Colleges at the time. To do what was famously the hardest maths course in the world, at what I thought was the best university in the world.
Don't misunderstand. I didn't decide to work harder or anything. I just thought they had been wrong, and I'd give them another chance to make the right decision.
I sat my A-levels, S-levels, and STEPS (Cambridge Entrance Exam), and aced them apart from the B in Chemistry and dropping a grade or two in one of the two STEP papers, which it turned out didn't matter.
King's gave me a second interview, and I did really well on this one. They accepted me.
At College I had the disturbing experience of not being the brightest person I knew any more.
There were people who'd been to public schools, who'd been prodigies and competed in scary things called Olympiads that I'd never heard of, and who had been appropriately stretched by gifted teachers. There were other people from comprehensive schools like me, but who'd worked insanely hard to get in.
I reckoned that I was about halfway up my group of twelve. But several of them have told me since that they were scared stiff of me, who didn't seem to try, and had had a standard state education, and yet who seemed given to flashes of insight.
I'm not sure that I actually did anything you'd call 'work' in the first year. But they did have this supervision system, where you'd get question lists handed out and you were supposed to solve as many as possible before going to see one of the Fellows about them.
I used to have a go at these sheets. I used to find that I could do the first five or six questions without trouble, and then everything else would be too hard. But if you turned up to the supervision, the Fellow in question would then explain how to do the remaining five or six.
I thought that's what they were for.
In my first year exams I came in about the top ten per cent of my year. I scored below a couple of my college-mates, both of whom were super-educated public school kids, and obviously very bright as well.
I figured that was good enough. I had girls and alcohol to worry about. I was up to my neck in both. I'd given up on drugs because they were all boring except for LSD, and I'd given up on LSD because I'd had one too many bad trips. But I'd filled the gap with heavy drinking and socializing. Mainly I think because it got you laid, but it was also great fun in itself.
So in the second year, I just gave up. I still went to all my lectures, because I enjoyed them. In fact despite near-constant partying, I think I missed two lectures in my entire time at college. I didn't love mathematics any less than I had.
But I no longer had any sort of go at the example sheets. I just turned up to supervisions and got the supervisors to explain how to do the problems. I could usually do the first couple off the top of my head anyway. As far as I know, my teachers still thought I was pretty good, because of my initial solid first, and the fact that I asked the right questions about the subjects I found interesting.
And at the end of the second year, I got a second. I was about half way up the list. I remember that my teachers seemed quite disappointed, and my reputation took a bit of a hit.
I'd been expecting it. I'd deliberately taken the year off, but I was never under any illusions about what my capabilities were. I knew that if I'd actually practised answering exam questions I'd be able to do them faster, and the tripos exam was like every other exam that I'd ever sat, a speed test rather than a test of understanding. I still knew how all the questions worked, it just took me more time to solve them than I was allowed to spend. The reason that other people were doing better than me was because I hadn't *cheated* by practising doing exam questions. I understood it all as well as they did. Most of them were just half-remembering formulae they'd picked up from spending many sleepless nights revising.
Everything was going according to plan.
The plan that I'd had, ever since I first thought what I should do as an adult, to sail effortlessly into a tenured academic job (when I'd been growing up, that had been a very easy thing to do), and to carry on learning and teaching and who knows, maybe think of something new and interesting one day. It had never really occurred to me to do anything else. If I thought about the 'real world' at all, then it was as a sort of sewer, filled with people who did tedious and unchallenging things because they wanted money. Academic salaries and PhD grants weren't great in the way that they had once been, but still plenty enough to support a studenty lifestyle, and that was all I wanted. I saw myself as being one of those kindly unmarried dons who hangs around in college and lives in college rooms, and lets the undergraduates substitute for the children he would have had if he hadn't been devoted to his craft. It looked great.
But if I wanted to do a PhD in pure mathematics (2), I needed a first-class degree. There was no funding available otherwise.
So I went back to my first year routine of having a half-hearted go at the example sheets before heading off to supervisions to have it all explained face to face. To be spoon-fed the answers without having made the effort to think of them myself.
It was a bit harder this time, because as well as having to work out what the new stuff was about, I realised that I hadn't really understood the second year stuff even though I thought I had (3), and so I needed to work out how all that worked while working out what the new stuff meant.
But it all worked out OK, almost. I was actually good enough to start catching up again.
I didn't bother with the third year computer project, despite that I'd been programming since the age of 10, would have found it very easy, and it was worth a full third of the credit that I'd need to get a first (4).
I didn't want to do something that I wouldn't learn anything from. I wanted to be a mathematician, not a programmer. And I was already as good a programmer as I'd ever need to be. (4.5)
By the time the final exam was close enough to worry about, my first was in the bag.
I had actually stooped to trying my hand at some past papers under exam conditions, which I still thought was cheating, but getting a first looked important enough to cheat at, and after the first couple of papers, I'd got good enough at it to reliably score the marks necessary for a first.
Rather to my surprise, (remember that this was the first time I'd ever taken an exam seriously enough to practise doing it), I found that I rather enjoyed the process. And even more surprisingly, learning to do the questions quickly actually seemed to make the ideas behind them clearer and more beautiful. They achieved a sort of focus that they hadn't had before.
Knowing that I was going to do well in the exam, I took Easter term pretty much off. The sunshine was lovely, this was my last year with all my friends, there was punting to do, and cricket to play, and riverside pubs to sit in, and parties to go to, and not many lectures to go to, and I understood enough subjects in enough detail to be confident that even if the questions turned out to be particularly hard, I'd pass. (I thought anything lower than a first was a fail.)
And of course I failed (5).
I have excuses, of course, for the failure of my confident prediction. The questions in my particular favourite subjects deviated from their usual predictable patterns. The first paper was much harder than I expected. One particular topology (one of the two subjects where I really could have given the lectures) question was so incomprehensible that all I could do for an answer was to draw a picture and write underneath 'It works because of this'. I had no idea how to do a formal proof as asked for.
My finals were four exams over two days, three hours each. After the first day I knew I'd done badly. I couldn't sleep for worry. The questions in the third and fourth exams were more in line with normal, but I actually managed to go to sleep in the fourth one because I was so tired after two gruelling days awake.
When the results came out I wasn't surprised. I'd calculated my mark afterwards, and it was on the line between a second and a first. I fell just the wrong side. (5.5)
Cambridge Mathematics has a charming tradition of reading out the results at a big ceremony in the Senate House. They read out the names of everyone who has got a first, in alphabetical order. I remember the list of names going Aa.... Ab..... Ascot (or something), Bagg...
Bagg was my college mate Jenny. As soon as I heard her name I knew I'd screwed it. Without any previous warning in about twenty one years, I'd managed to be not as good at something as I needed to be.
That was pretty much the end for me.
No first meant no PhD funding in the thing I wanted to do. I didn't have a plan B. It literally hadn't occurred to me in the month or so before the exam that I could fail it.
I fell on my feet after a fashion. Someone told me of a PhD place for a specific project in London (at Imperial College, which is a pretty damned good university), which had its funding allocated already, but where the student in question had dropped out. I rang the man who controlled the funding, he invited me to London, we talked for an hour and the place was mine.
But it didn't work out. I didn't get on particularly well with my supervisor, I wasn't interested in the thing I was supposed to be studying, he wasn't interested in any of my ideas, and there was no one else to talk to.
I'd been looking forward to London. I'd always enjoyed going there as a boy. But it turned out to be awful. A lonely wasteland of concrete and filth. And being an academic didn't look like much fun in London. More of a sort of glorified schoolteacher job, and not that glorified either.
And then my intuition failed. One day I was trying to read a research paper, and I found out that it was just squiggles. No pictures, no handle on what it meant. Not even a vague feeling that if I tried just changing this bit, something would happen. It was suddenly all greek to me.
I understood, for the first time in my life, what it was to not understand a piece of mathematics.
And I figured that was the end of the line. All my life I'd watched people give up maths, because it didn't seem to mean anything to them. I knew that however hard they tried, it wouldn't ever make proper sense to them like it did to me.
It just seemed that that had happened to me. There's obviously a point beyond which you can't go, and it just so happened that mine was about half-way through a PhD.
I don't think it occurred to me that I might be able to work my way through this. That just wasn't the way I thought it worked. By that time I didn't care anyway. Two years in London had put me off the whole idea of academia.
I moved back to Cambridge because that's where most of the people I liked lived.
At about this time, I started to run out of money. The near limitless ability of the 1980s British State to subsidize lazy good-for-nothing layabouts who thought the world owed them a living had met its match in me. I had an overdraft, accounted for pretty much exactly by six years of booze and cigars. Even though today's self-funded students will laugh at my tiny £6000 debt, it seemed to me that I was so poor that I had to give up smoking. I even gave up 2000AD, a comic that I'd read regularly since I was a boy, because the 50p it cost every two weeks was a noticeable expense.
Of course when the PhD funding ran out, I managed to start drawing unemployment benefit. But this wasn't quite as much, so there was further belt tightening to be done, and I didn't fancy it. My supervisor said that I'd got about another six months work to do before I'd be able to hand in a doctorate, and I was pretty sure that meant about twelve months. Also it was going to be a complete piece of shit. Nobody would ever read it out of interest, and I wasn't in the least interested in writing it.
I got a job, programming. It was great fun, it turned out. People actually gave you things to do that they didn't know how to do themselves, but they were usually quite easy things, so they were usually done quickly, and (and this was the real revelation) people were actually grateful when you solved their problems. I mean that you felt like you'd done them a favour. This was a revelation to me. I loved it. After about two months I had this awful dream where I'd gone back to London to try to finish my PhD. I woke up in a cold sweat of terror, and at that point I knew I was never going to be Dr Aspden. Which was strange. I'd always thought that Mr was what you were called from about 16 to about 24. Like a sort of probationary title. It was like I'd been told I'd never be able to get a driving licence.
I got used to being mediocre. I was good enough to have got a Cambridge maths degree. I was probably top quartile. But there are a fair number of people like that in the world.
I've thoroughly enjoyed it. I'm pretty good at what I do, and it's fun.
I took up other hobbies. I'd always liked playing sports, even though I was useless at it, and, almost by accident, I took up rowing, which is the local sport here, for townies as well as the university.
There was no reason to expect to be anything other than rubbish at rowing. I'm average height, stronger than most men my size, but not vastly so, and I smoke, which doesn't go well with a sport where the chief physical variable is a capacity for consuming oxygen.
And I was. Terrible at it. For years.
But I enjoyed it, and I practised it, and I got good at it. I was never in a million years going to be any sort of star at it, even in the rather limited competitive environment of Cambridge town. But I ended up being, if not actually any good, better than I would ever have believed possible to start with.
And somehow it taught me that if you work at something, you get better at it.
And I decided a few years ago that I should learn LISP, an antique computer language out of the dark ages, because there seemed to be something special about the way LISP people talked.
So I got the standard textbook Structure and Interpretation of Computer Programs, and I found it harder than most things about computers. But I was damned if I was going to let some sort of computer-thing be hard, because nothing in computers is even a bit hard. So I did the exercises in the book to see if they would make it clear, and found that solving carefully chosen example problems is fun, and you know what, the more of the exercises I solved, the more I understood the book, and the better I understood, the more of the book I could read.
And it taught me that if you work at something intellectual, you get better at it.
Which I didn't believe for the first thirty years of my life.
I thought that you were born good at things. I thought if you weren't good at things, working hard at them was just a path to ruining your life with fruitless toil, to fail one level higher than you would have failed anyway. To be honest, I still believe something like that, just not as extreme a version as I once did.
Why would I not have believed this? You cannot move without people praising the special quality of genius. You cannot read without finding out that authors and poets and musicians have their works come to them fully formed. The special genius of my heroes, Feynman, Einstein, Newton, Conway, Von Neumann is trumpeted from rooftops. You never hear that they lifted a finger apart from doing exactly what seemed most amusing to them at the time.
And it had been absolutely true for me for twenty years. I'd been told over and over again that I needed to work harder. And every time I'd ignored the advice and turned out to be right.
And when I was twenty one the 'problem of induction' hit me over the head with a whisky bottle during my final exams. And I was stone dead and I didn't even notice because twenty years of confirming evidence isn't overturned by a human mind just like that.
And I've only just noticed.
Recently, I was asked to teach a young friend enough maths to get him onto an engineering degree. I loved doing this. Sitting down and talking with him and taking him through trial problems and trying to work out where his mental blocks were and blowing them away was the best fun I've had in ages. And he's now doing his engineering degree, although he's going to screw it up by, ironically, spending so much time rowing really seriously that he'll be too tired to think about engineering.
And I've been reading the collected works of the brilliant philosopher Eliezer Yudkowsky, who's built a whole philosophy and predicted the entire future of the human race convincingly using amongst other intellectual tools Bayes' Theorem, a trivial piece of arithmetic with profound implications.
And I figured if I'm going to understand Eliezer Yudkowsky's thoughts I'd better understand Bayes' Theorem and its consequences as well as he does, which I'm still arrogant enough to think that I can do.
And so from a combination of these effects, I've started doing mathematics, recreationally, for the first time in my life. And indeed mathematics at all, for the first time in about fifteen years.
And it turns out I'm pretty damned good at it. I shouldn't be, because I'm forty years old now, and no one's as good at maths at forty as they were at twenty. But if my powers have declined I can't tell.
I'm reading David MacKay's beautiful book Information Theory, Inference, and Learning Algorithms, and it's like reading a thriller. I've patiently worked through the examples in the first two or three chapters, even the ones that look boring, because I remember that the boring looking examples in SICP always turned out to teach unique and thrilling lessons.
I've sort of got my head round the introductions to Information Theory and Inference, and I think I already see why the Noisy-Channel Coding Theorem is going to turn out to be true in chapter 6 or wherever it is, and why it is also going to turn out to be of little practical value. And I can't wait to find out if I'm right.
And this has brought back memories, and when I start remembering things these days I start writing, and all this has come out like a flood, and suddenly it occurs to me, because I promise you that the above is as true as my fallible memory can make it, that I may actually have been as good as I once thought I was.
And fuck the research angle and doctorates and names on interesting discoveries. I love teaching. And I'm good at it. And that's what I should have been doing with my life. Teaching mathematics and computers to bright young people. I would have loved it. And I didn't do it. And it's too late now.
And I want to call bullshit on the idea of Effortless Superiority, which it now occurs to me I may have been the only human on the planet stupid enough to take seriously (6).
Footnotes
(0) English school leaving exams. A-levels are the optional ones two years later, on which University Entrance depends, S-levels are a special form of A-levels, on the same syllabus but the questions are much harder. STEP is similar to S-level, but set by Cambridge University and used as part of its entrance requirement.
(1) Even though I was good at foreign languages, they seemed like a completely stupid rigmarole. It wasn't like they were any use for anything except passing foreign language tests.
Somehow, at around this same time, I was teaching myself Latin to O-level standard out of a book my father had left lying around where I could find it. Latin was interesting and useful, because you could read about the Roman Empire and other cool things in the Romans' own words. And you could see where lots of our words came from. And their poetry was better than ours.
I've wondered if I'd have done modern languages for my degree if my school had managed to make living German and French anything like as interesting as Francis Kinchin Smith managed to make dead Latin and Greek in his two Teach Yourself ... books. Cheap self-help manuals that were doubtless intended by the publishers to be read only as far as the third chapter. From the love and care and scholarship and cleverness lavished on them I think Francis had higher goals for them. I wonder if he ever got fan mail? If I could go back and write it I would. What did a child know? I've just googled him. He died in 1958 apparently. Too early for him to have got my letter even if I had thought to write. The fact that I can remember his name after 28 years tells you how much I liked those books. I wish I'd had a chance to tell him.
But the tedium of French and German was as nothing compared to the horror of Music and the Calvary of Religious Education.
How, how for the love of Christ did they manage to make Music boring to a human child?
And for that matter, the King James Bible is one of my favourite books.
I am as bleak an atheist as it is possible to be, and yet I often read the Gospel According to St Matthew on Easter Day. I wonder how many Christians do?
(2) By this time I'd begun to despise applied maths, which seemed to be a bag of stupid half-understood magic tricks that was overdue to be swept away by computer simulation. Weirdly it only seems to be taught like that in Cambridge. In London later on I found it interesting again.
(3) As someone once pointed out, what the hell did I think it meant for someone to understand something if they couldn't use it fluently?
(4) I had done some computer stuff in the second year out of interest, but I'd found that it didn't teach me anything interesting. I already knew how to program. I can't even remember whether I bothered completing it.
(4.5) I know, I know, I was a moron. But actually it did look singularly unchallenging. Now if they'd taught SCHEME rather than matrix multiplication algorithms, who knows...
(5) got a second.
(5.5) As I remember, the topology question that I'd drawn the answer to was bastard hard, only a couple of people had attempted it, and my picture had been given almost the maximum mark. So in fact I needn't have worried. If I'd known that, I might have been able to sleep that night.
(6) I'm not whining for the sake of it, by the way. I've loved life, and I'd do the first forty years over again exactly the same way like a shot if I had the chance.
But if you recognise yourself in the above, young person, for God's sake do the exercises in the textbooks, and practise exam questions. It's not cheating. It helps you understand. A lot.
If you do the exercises, and play with the examples, then you will find that your intuition, already powerful, starts to get brighter, and rather than finding as I did that as you go forward, you start to run out of dry places to stand in the swamp, until eventually there's nowhere new to go, you'll find that as you brighten the swamp recedes ahead of you faster than you can walk.
On the other hand, if you find yourself working every hour God sends on something and you're not enjoying it for its own sake, give up. Nothing's worth that sort of life. I still think success probably comes fairly easily if you're any good. What I'm trying to warn against is pig-headedly not lifting a fucking finger to practise something you enjoy because you think that's something lesser beings have to do to make up for being lesser.
"If at first you don't succeed, try, and try again. Then give up. No sense making a fool of yourself." -- Homer
Some bits of text from the essay I started writing before it turned into the above whine.
Right until halfway into a PhD that I didn't find interesting. At that point the wellspring failed, and within twelve months of realising that it was gone I wasn't a mathematician any more. Part of the problem was that I'd never had to try at anything before, and I didn't know how to deal with anything that required effort.
Would you believe that at the time I sat my finals, I still thought that practising exam questions was cheating. I thought it was vulgar. Like you were trying to fool the examiners into thinking you were cleverer than you were.
I realized that I wasn't clever enough to bank on a first (I mean I still thought I'd get one, it's just that I'd realized there was a *chance* I wouldn't), so towards the end of my third year, I cheated a bit. But it was far too little too late, and it didn't help. I got the second I'd feared.
This only confirmed me in my belief that it was silly to try. That nothing that had to be worked at was worth having. It was easy to maintain this belief.
Lots of people have to try very hard to get into university. They slog away at A-levels, ruining evenings and weekends that should be spent being young and happy. And when they crowbar their way in, somehow managing to convince an interviewer far cleverer than they are that the lights are on in their heads, they find that it's all ashes. For nothing.
For it gets harder once you're not at school. And the only strategy these people have is to work harder. But they already worked as hard as they could.
They react predictably. They burn the candle at both ends for three years, and at the end of it they scrape low seconds and thirds. Some, released from parental pressure, realize that the game isn't worth it and drop out, pretending they didn't care in the first place.
I was terrified of being one of these people. I guess I still am.
Tuesday, January 4, 2011
Many Classical Worlds
Let's imagine that we live in a universe that splits whenever anything random happens.
So say we've got a coin which lands heads up 2:1, and another which lands heads up 1:2
Every time we toss either coin, the universe splits, but we don't know which copy we end up in, and our task is to try to narrow down where we are!
The original split happens when we pick a coin from our pile of two coins.
This is a random event. There are now two copies of the universe. In one we've got a heads biased coin, in the other we've got a tails biased one. But we could be in either, as far as we know.
We toss the coin, and so does our copy in the parallel universe.
Both universes split into three. Of the three heads-biased coin universes that have come into being, there are two where the coin shows heads, and one where it's tails.
Of the three tails-biased universes, there's two where it's heads, and one where it's tails.
And we notice that our coin came down heads. Where can we be?
We are in one of the three universes where the coin came down heads.
How many of those are also universes where we picked the tails biased coin? One.
How many of them are also universes where we picked the heads biased coin? Two.
What are the odds that we are in a heads-biased-coin universe? Two to one.
Obviously this little story is nonsense, and yet it seems to catch the essence of both probability and inference. And it makes it very easy to think about.
I've been using it for a few weeks now to think about probability. It hasn't led me astray yet. I think it might be isomorphic to the real theory, as long as you stick to rational numbers.
Whatever the real theory is. I've never heard any description of probability that wasn't gibberish.
So just because this one is gibberish too doesn't count against it as much as it might.
I write it down only because I was just thinking about an urn with two white balls and one black ball, and drawing balls from it, and wondering what the histogram would look like. And this view seemed to make it pretty transparent what is going on, even though before there had only been fractions to multiply.
I mean don't get me wrong. I was taught to do all that 20 years ago, and I could do it then, and I haven't thought about it all since, but I can still derive the relevant proofs from first principles. Which shows me that I understood it at the time. One remembers that which one understands as if one had been born knowing it. That not understood, even if mastered, fades with the years.
But I don't remember it being so blisteringly clear, and beautiful, and inevitable back in the day. Probability, let alone statistics, seemed a bit fiddly. And not too interesting. Now it all looks like one of the big secrets of the universe. Maybe it is just that I am getting old, and am a bit more easily impressed than I was once.
The wellsprings of intuition in mathematics are secrets. I don't know why. Sometimes they are literally incommunicable. I don't know how I could show someone who doesn't know how to do it how to make animated pictures of mathematical concepts in my head, which is a skill that carried me effortlessly through the half of pure mathematics known as analysis.
But I could at least have told people that that was how you were supposed to do it. No one ever showed me. It was a habit I picked up by accident when I was very small, and I imagine I got better at it by practice. I remember with utter clarity a clever thing my mother made for me to help me understand fractions. That might have been the start of it. It might also be my earliest memory. I can't remember whether it was while I was at school, or before.
I've no idea what the equivalent talent for the other half of pure maths (algebra) is. In all that time, no-one ever told me, and it never occurred to me to ask. Maybe it can't be put into words.
Certainly recently I was playing around with permutations, and learned about the cycle notation for groups, and thought about them like that instead of how I'd been taught, where it's all rather abstract and beautiful, but where I have no intuition whatsoever. And it made more sense from that point of view.
Maybe this classical many-worlds picture is one of the keys to thinking about probability.
Lots of things seem obvious now. You meet a woman, and she says she has two children, and that one's called Arthur. What is the probability that both her children are boys?
Look. When she gave birth, the universe split into two. When she had her second child it split again. There are four now. In one she's got two boys. In one she's got two girls.
You're not in the one where she's got two girls. You could be in any of the other three.
That's supposed to be a paradoxical, counter-intuitive result, as I remember.
You meet a woman with two children. One's a boy. What's the chance they both are? One in three.
Most people guess a half.
I remember that that used to be the obvious answer. I'm not sure I can understand why now.
King Arthur was one of two children. What's the chance he had a sister?
Actually I suspect subtleties here. But I know how to think about them now.
So say we've got a coin which lands heads up 2:1, and another which lands heads up 1:2
Every time we toss either coin, the universe splits, but we don't know which copy we end up in, and our task is to try to narrow down where we are!
The original split happens when we pick a coin from our pile of two coins.
This is a random event. There are now two copies of the universe. In one we've got a heads biased coin, in the other we've got a tails biased one. But we could be in either, as far as we know.
We toss the coin, and so does our copy in the parallel universe.
Both universes split into three. Of the three heads-biased coin universes that have come into being, there are two where the coin shows heads, and one where it's tails.
Of the three tails-biased universes, there's two where it's heads, and one where it's tails.
And we notice that our coin came down heads. Where can we be?
We are in one of the three universes where the coin came down heads.
How many of those are also universes where we picked the tails biased coin? One.
How many of them are also universes where we picked the heads biased coin? Two.
What are the odds that we are in a heads-biased-coin universe? Two to one.
Obviously this little story is nonsense, and yet it seems to catch the essence of both probability and inference. And it makes it very easy to think about.
I've been using it for a few weeks now to think about probability. It hasn't led me astray yet. I think it might be isomorphic to the real theory, as long as you stick to rational numbers.
Whatever the real theory is. I've never heard any description of probability that wasn't gibberish.
So just because this one is gibberish too doesn't count against it as much as it might.
I write it down only because I was just thinking about an urn with two white balls and one black ball, and drawing balls from it, and wondering what the histogram would look like. And this view seemed to make it pretty transparent what is going on, even though before there had only been fractions to multiply.
I mean don't get me wrong. I was taught to do all that 20 years ago, and I could do it then, and I haven't thought about it all since, but I can still derive the relevant proofs from first principles. Which shows me that I understood it at the time. One remembers that which one understands as if one had been born knowing it. That not understood, even if mastered, fades with the years.
But I don't remember it being so blisteringly clear, and beautiful, and inevitable back in the day. Probability, let alone statistics, seemed a bit fiddly. And not too interesting. Now it all looks like one of the big secrets of the universe. Maybe it is just that I am getting old, and am a bit more easily impressed than I was once.
The wellsprings of intuition in mathematics are secrets. I don't know why. Sometimes they are literally incommunicable. I don't know how I could show someone who doesn't know how to do it how to make animated pictures of mathematical concepts in my head, which is a skill that carried me effortlessly through the half of pure mathematics known as analysis.
But I could at least have told people that that was how you were supposed to do it. No one ever showed me. It was a habit I picked up by accident when I was very small, and I imagine I got better at it by practice. I remember with utter clarity a clever thing my mother made for me to help me understand fractions. That might have been the start of it. It might also be my earliest memory. I can't remember whether it was while I was at school, or before.
I've no idea what the equivalent talent for the other half of pure maths (algebra) is. In all that time, no-one ever told me, and it never occurred to me to ask. Maybe it can't be put into words.
Certainly recently I was playing around with permutations, and learned about the cycle notation for groups, and thought about them like that instead of how I'd been taught, where it's all rather abstract and beautiful, but where I have no intuition whatsoever. And it made more sense from that point of view.
Maybe this classical many-worlds picture is one of the keys to thinking about probability.
Lots of things seem obvious now. You meet a woman, and she says she has two children, and that one's called Arthur. What is the probability that both her children are boys?
Look. When she gave birth, the universe split into two. When she had her second child it split again. There are four now. In one she's got two boys. In one she's got two girls.
You're not in the one where she's got two girls. You could be in any of the other three.
That's supposed to be a paradoxical, counter-intuitive result, as I remember.
You meet a woman with two children. One's a boy. What's the chance they both are? One in three.
Most people guess a half.
I remember that that used to be the obvious answer. I'm not sure I can understand why now.
King Arthur was one of two children. What's the chance he had a sister?
Actually I suspect subtleties here. But I know how to think about them now.
All is Vanity and Vexation of Spirit
Overheard in a cafe. Young ladies, talking in semi-hushed tones:
"I was at a party last night, and there was this guy, he was rippling with muscles, I think he must have been on steroids or something."
"Oooh!"
"He asked me to come to his room. I think he needed help to take his T-shirt off."
"I was at a party last night, and there was this guy, he was rippling with muscles, I think he must have been on steroids or something."
"Oooh!"
"He asked me to come to his room. I think he needed help to take his T-shirt off."
Ashes Update
England won the fourth test. Our belief before then was 3:4:3.
Under the England strong model, the odds of the win were 1/2
Under the equal model, 1/3
Under the Australia strong model, 1/6.
So the numbers get multiplied by those probabilities.
3/2:4/3:3/6
or
9:8:3
We now consider England the stronger team in 9 universes out of 20.
What's our prediction for the fifth test?
Well, in 9 of the 20, England have a half chance.
In 8 of the 20, they have a 1/3 chance
In 3 out of the 20, they have a 1/6th chance
Our estimate of England's chances of winning the fifth test are:
(9/2+8/3+3/6)/20 = (27+16+3)/120 = 46/120
Slightly more than the 1/3rd we'd assign if we thought the teams were even.
We've now seen England win twice, Australia once, and we're starting to expect that trend to continue.
And of course, if they do, we'll update again!
Under the England strong model, the odds of the win were 1/2
Under the equal model, 1/3
Under the Australia strong model, 1/6.
So the numbers get multiplied by those probabilities.
3/2:4/3:3/6
or
9:8:3
We now consider England the stronger team in 9 universes out of 20.
 | |
| Ours. And in 17 universes out of every 20, they probably should be. |
What's our prediction for the fifth test?
Well, in 9 of the 20, England have a half chance.
In 8 of the 20, they have a 1/3 chance
In 3 out of the 20, they have a 1/6th chance
Our estimate of England's chances of winning the fifth test are:
(9/2+8/3+3/6)/20 = (27+16+3)/120 = 46/120
Slightly more than the 1/3rd we'd assign if we thought the teams were even.
We've now seen England win twice, Australia once, and we're starting to expect that trend to continue.
And of course, if they do, we'll update again!
Friday, December 24, 2010
Bayes and the Ashes
What can the Reverend Bayes tell us about the Ashes?
Let's imagine three models:
In the first model, the two teams are equal in strength:
In that case, let's imagine that there's a 1/3 chance of win, draw and loss. 1:1:1
In the second model, we'll believe that Australia are stronger than England, in which case, we'll imagine that the odds go 1:2:3 of English win, draw, Australia win.
Third, we can imagine that England are stronger, and say that in that case the odds are 3:2:1
Suppose that we have no reason, before the series begins, to believe in any one of these models particularly, so we'll imagine that they're equally distributed over all the possible worlds.
So we'll believe in initial odds of 1:1:1.
The first match was a draw. All the models predict an equal chance of a draw, so the odds
stay at 1:1:1. We learned nothing from the first match.
The second match was an England win. The even model gives a 1/3 probability of an England win, the England stronger model gives a 1/2 chance of it, and the Australia stronger model gives 1/6
So after the second game, the odds become 1/2:1/3:1/6, or 3:2:1.
After seeing England winning the second game, we should believe that it's three times more likely that England are the stronger team.
The third game was a win for Australia. We now multiply by 1/6:1/3:1/2, giving 3/6:2/3:1/2, or 3:4:3.
So our initial odds, whatever they were, have been multiplied by 3:4:3.
Whatever we started off believing, the results Draw, England, Australia should have made us slightly more likely to believe that the teams are evenly matched.
In other words, if we believe that one of our three models is something like the truth, then the series so far has told us almost nothing. If, before the series, you believed that England were the stronger team, you've seen nothing to change your mind, and vice versa if you believed that Australia were.
However, whatever you started out believing, you should now be more open to the idea that the teams are evenly matched, and probably rather less open to extreme models where one team is much stronger than the other. There are also models where, say, both sides' batting is very strong, and both sides' bowling is very weak, which will have been pretty much ruled out by two results in three games.
Nothing too surprising here, I hope. And that's kind of the point. Bayes' Theorem, in spite of having some rather hairy philosophy behind it, is nothing more than a formalized, precise version of common sense.
Let's imagine three models:
In the first model, the two teams are equal in strength:
In that case, let's imagine that there's a 1/3 chance of win, draw and loss. 1:1:1
In the second model, we'll believe that Australia are stronger than England, in which case, we'll imagine that the odds go 1:2:3 of English win, draw, Australia win.
Third, we can imagine that England are stronger, and say that in that case the odds are 3:2:1
Suppose that we have no reason, before the series begins, to believe in any one of these models particularly, so we'll imagine that they're equally distributed over all the possible worlds.
So we'll believe in initial odds of 1:1:1.
The first match was a draw. All the models predict an equal chance of a draw, so the odds
stay at 1:1:1. We learned nothing from the first match.
The second match was an England win. The even model gives a 1/3 probability of an England win, the England stronger model gives a 1/2 chance of it, and the Australia stronger model gives 1/6
So after the second game, the odds become 1/2:1/3:1/6, or 3:2:1.
After seeing England winning the second game, we should believe that it's three times more likely that England are the stronger team.
The third game was a win for Australia. We now multiply by 1/6:1/3:1/2, giving 3/6:2/3:1/2, or 3:4:3.
So our initial odds, whatever they were, have been multiplied by 3:4:3.
Whatever we started off believing, the results Draw, England, Australia should have made us slightly more likely to believe that the teams are evenly matched.
In other words, if we believe that one of our three models is something like the truth, then the series so far has told us almost nothing. If, before the series, you believed that England were the stronger team, you've seen nothing to change your mind, and vice versa if you believed that Australia were.
However, whatever you started out believing, you should now be more open to the idea that the teams are evenly matched, and probably rather less open to extreme models where one team is much stronger than the other. There are also models where, say, both sides' batting is very strong, and both sides' bowling is very weak, which will have been pretty much ruled out by two results in three games.
Nothing too surprising here, I hope. And that's kind of the point. Bayes' Theorem, in spite of having some rather hairy philosophy behind it, is nothing more than a formalized, precise version of common sense.
Wednesday, December 22, 2010
Mole
I've just risked my life to save a mole.
I was walking along a country road, when I saw a baby mole, black with its little pink snout, crawling blindly in the middle of the carriageway.
I stepped into the road to pick it up and put it somewhere safer.
Moles are vermin round here. I can understand why. If you have a nice lawn, coming down to find three or four molehills in it can't be a pleasing experience.
They're tolerated in the fields, as far as I know, but nobody stops the plough to spare their homes. They are, as the saying goes, neither loved nor hated, but made of atoms that a superior intelligence can use for other purposes.
And this one was a baby. A tiny mammal unprotected in Winter. A doomed thing.
As I stepped into the road, a car appeared, coming fast down the other side.
And as my shadow fell on the mole, it tried to run, scrabbling desperately with pathetic little flippers that weren't good enough even to lift its body off the ground.
But it reached the white line, and crossed, and crawled into the path of the car.
So I stepped out in front of it and held my hand out like a storm trooper with the palm flat for STOP.
And the car braked. Hard. On the icy road. And stopped in time.
And the old couple wound down their window, and I explained. And as I did, I started to feel sentimental and foolish. And the lady of the couple said thank you, because she hated to kill things when they drove, and she said that if I hadn't I would have ruined Christmas, because the death would have been my fault, and I'd have been thinking about it all day.
And while we talked the mole scurried off into the icy brambles at the side of the road. Where, without its mother, it will die, sooner rather than later, in one of the many ways baby animals die in Winter.
We are not rational beings. I knew that. It is always nice to have one's beliefs confirmed. But what the hell heuristic was I using, and what strange utility function did it think it was serving?
I was walking along a country road, when I saw a baby mole, black with its little pink snout, crawling blindly in the middle of the carriageway.
I stepped into the road to pick it up and put it somewhere safer.
Moles are vermin round here. I can understand why. If you have a nice lawn, coming down to find three or four molehills in it can't be a pleasing experience.
They're tolerated in the fields, as far as I know, but nobody stops the plough to spare their homes. They are, as the saying goes, neither loved nor hated, but made of atoms that a superior intelligence can use for other purposes.
And this one was a baby. A tiny mammal unprotected in Winter. A doomed thing.
As I stepped into the road, a car appeared, coming fast down the other side.
And as my shadow fell on the mole, it tried to run, scrabbling desperately with pathetic little flippers that weren't good enough even to lift its body off the ground.
But it reached the white line, and crossed, and crawled into the path of the car.
So I stepped out in front of it and held my hand out like a storm trooper with the palm flat for STOP.
And the car braked. Hard. On the icy road. And stopped in time.
And the old couple wound down their window, and I explained. And as I did, I started to feel sentimental and foolish. And the lady of the couple said thank you, because she hated to kill things when they drove, and she said that if I hadn't I would have ruined Christmas, because the death would have been my fault, and I'd have been thinking about it all day.
And while we talked the mole scurried off into the icy brambles at the side of the road. Where, without its mother, it will die, sooner rather than later, in one of the many ways baby animals die in Winter.
We are not rational beings. I knew that. It is always nice to have one's beliefs confirmed. But what the hell heuristic was I using, and what strange utility function did it think it was serving?
Moon
I've written before about how bright the full moon is.
Last night, I was driving to Sheffield, and at around half past ten, after about a hundred miles on the A1, I turned off the road into Sherwood Forest, and parked in a quiet layby for a smoke.
It was far too cold to get out of my van, which is a comfortable place to sit anyway, and so, for I think the first time since I bought it ten years ago, I lit up in the cab.
There was a beautiful German mass on the radio, which I hadn't been able to hear properly with the engine running, but in the quiet and the darkness it was entrancing.
When my cigar was finished I carried on listening to it until it was over. After fifteen minutes in the dark I could see for miles, snowy fields and trees under the moonlight through the clouds.
Eventually I remembered that I had promises to keep. I started the engine, checked the empty road, indicated and pulled out.
After I'd been driving a few minutes it occurred to me that the light was a bit unusual. It seemed more flickery than normal, like the light from a flourescent tube, as the most moonlit of the clouds was obscured and revealed by passing trees.
"Oops", I thought. "Headlights."
Slightly embarrassed I reached for the switch and turned them on. Suddenly the cone in front of the car leapt out, daylight bright and in shocking colour. And the rest of the world disappeared.
Last night, I was driving to Sheffield, and at around half past ten, after about a hundred miles on the A1, I turned off the road into Sherwood Forest, and parked in a quiet layby for a smoke.
It was far too cold to get out of my van, which is a comfortable place to sit anyway, and so, for I think the first time since I bought it ten years ago, I lit up in the cab.
There was a beautiful German mass on the radio, which I hadn't been able to hear properly with the engine running, but in the quiet and the darkness it was entrancing.
When my cigar was finished I carried on listening to it until it was over. After fifteen minutes in the dark I could see for miles, snowy fields and trees under the moonlight through the clouds.
Eventually I remembered that I had promises to keep. I started the engine, checked the empty road, indicated and pulled out.
After I'd been driving a few minutes it occurred to me that the light was a bit unusual. It seemed more flickery than normal, like the light from a flourescent tube, as the most moonlit of the clouds was obscured and revealed by passing trees.
"Oops", I thought. "Headlights."
Slightly embarrassed I reached for the switch and turned them on. Suddenly the cone in front of the car leapt out, daylight bright and in shocking colour. And the rest of the world disappeared.
Wednesday, December 15, 2010
How Spock should decide whether to kill Kirk
When we left Mr Spock, he had a decision to make.
In order to make that decision, he needs to know whether he's in an evil or a good universe.
The only evidence he has is one coin toss that came up tails.
He knows that in evil universes, the coin is more likely to come up tails.
But he also knows that in good universes, the coin will come up tails one time in three.
So he might figure out that he's more likely to be in an evil universe. So he should probably fire.
But how confident should he be of his conclusion?
And if he'd had time to toss the coin twice, and it had come up tails both times, how confident should he be then?
We can address the first question by thinking about six parallel universes side by side.
In all of them, Spock appears in the captain's cabin and tosses the coin.
Three of the universes are evil, and three good.
In one of the evil universes, Spock sees a head. In two of them, he sees a tail.
In one of the good universes, Spock sees a tail. In two of them, he sees a head.
So our copy of Spock, who has to decide whether to shoot or not, who only knows that he's seen a tail, can be the copy in three of the six universes.
They are two of the evil ones and one of the good ones.
If all Spock knows is that he's in one of these three, he should reckon that the odds of being in an evil universe are two to one, or equivalently he should say that there's a probability of 2/3.
If he's seen two tails, then we need to think about nine good universes and nine evil ones.
On the first toss, they split into:
good,tail (there are 3 like this)
good,head (6)
evil,tail (6)
evil,head (3)
And then on the second, these groups divide further
good, tail, tail (1)
good, tail, head (2)
good, head, tail (2)
good, head, head (4)
evil, tail, tail (4)
evil, tail, head (2)
evil, head, tail (2)
evil, head, head (1)
So now if Spock has seen tail, tail, he can be in one of four evil universes or one good one.
His odds are four to one on being in an evil universe.
This conclusion is known as Bayes' Theorem, after the Reverend Thomas Bayes.
It tells you how to update your beliefs when you seen evidence of their truth and falsehood.
Spock's initial belief is that the chances of being in an evil universe are even, or 1:1, or probability 1/2
After he sees one tail, he should believe that the chances are 2:1, or probability 2/3
After he sees the second, he should believe that the chances are 4:1 in favour of evil, or probability 4/5.
Captain Kirk, Mr Spock, and the Reverend Thomas Bayes
Captain Kirk, being a man who likes to gamble, carries with him at all times a biased coin. Being one of the good guys, his coin is biased so that it returns mainly heads.
In a evil parallel universe where everyone wears a moustache, the evil version of Kirk carries a coin biased towards tails. Apart from this detail and the moustaches, the universes physically resemble each other with uncanny accuracy. But the moralities on which they operate could not be more different.
--
After a transporter accident, Mr Spock find himself in the captain's stateroom on board the (an?) Enterprise.
On the dressing table, the captain's wallet, phaser and communicator lie neatly next to the captain's trick coin.
Spock tosses the coin thoughtfully.
It comes up tails. This is very bad news. If the evil Kirk discovers Spock in his room, he will almost certainly suspect a plot, and have Spock executed. Spock grimaces and draws his phaser.
But of course, if he's in his own universe, the captain will just be pleased to see him safe and well, and amused by the accident that he was transported directly into the officer's quarters.
Spock, about to toss the coin again, wonders if he should set his phaser to stun.
Suddenly, Spock hears the whirr of the captain's bedroom door. He whirls and...?
What should Spock do?
--
Specifically, good!Kirk's coin comes up heads 2 times out of 3, and evil!Kirk's coin comes up heads 1 time in 3.
The transporter accident, involving reversed polarities, has almost certainly sent Spock into a parallel universe. There are, however, millions of parallel universes. Half are good, half are evil.
If Spock fires, he will kill the captain. In a good universe this is a tragedy that Spock would die to prevent. In an evil universe, he's killed a tyrant who deserved to die, and saved his own life. He can then plot to return home, which shouldn't be too difficult.
If Spock doesn't fire, the captain will pull off some superheroic trick as usual and somehow get the drop on Spock. In a good universe, this will result in much good humor, and the eventual return of Spock to his home universe, where he won't have to put up with having an identical twin.
In an evil universe, not firing will result in Spock's execution.
Spock needs an estimate of the probability that his current universe is evil, and he needs it fast. His only evidence is the one coin toss, which came up tails. What should he reckon the chances are that the universe is evil?
--
For bonus points, in a parallel group of parallel universes, Spock had managed to make ten coin tosses before the door whirred. His results were H H H H T H T H T H . What's his estimate now?
In a evil parallel universe where everyone wears a moustache, the evil version of Kirk carries a coin biased towards tails. Apart from this detail and the moustaches, the universes physically resemble each other with uncanny accuracy. But the moralities on which they operate could not be more different.
--
After a transporter accident, Mr Spock find himself in the captain's stateroom on board the (an?) Enterprise.
On the dressing table, the captain's wallet, phaser and communicator lie neatly next to the captain's trick coin.
Spock tosses the coin thoughtfully.
It comes up tails. This is very bad news. If the evil Kirk discovers Spock in his room, he will almost certainly suspect a plot, and have Spock executed. Spock grimaces and draws his phaser.
But of course, if he's in his own universe, the captain will just be pleased to see him safe and well, and amused by the accident that he was transported directly into the officer's quarters.
Spock, about to toss the coin again, wonders if he should set his phaser to stun.
Suddenly, Spock hears the whirr of the captain's bedroom door. He whirls and...?
What should Spock do?
--
Specifically, good!Kirk's coin comes up heads 2 times out of 3, and evil!Kirk's coin comes up heads 1 time in 3.
The transporter accident, involving reversed polarities, has almost certainly sent Spock into a parallel universe. There are, however, millions of parallel universes. Half are good, half are evil.
If Spock fires, he will kill the captain. In a good universe this is a tragedy that Spock would die to prevent. In an evil universe, he's killed a tyrant who deserved to die, and saved his own life. He can then plot to return home, which shouldn't be too difficult.
If Spock doesn't fire, the captain will pull off some superheroic trick as usual and somehow get the drop on Spock. In a good universe, this will result in much good humor, and the eventual return of Spock to his home universe, where he won't have to put up with having an identical twin.
In an evil universe, not firing will result in Spock's execution.
Spock needs an estimate of the probability that his current universe is evil, and he needs it fast. His only evidence is the one coin toss, which came up tails. What should he reckon the chances are that the universe is evil?
--
For bonus points, in a parallel group of parallel universes, Spock had managed to make ten coin tosses before the door whirred. His results were H H H H T H T H T H . What's his estimate now?
Tuesday, December 7, 2010
All Dead Soon
I've come to believe, as a direct consequence of reading Less Wrong that we're almost certainly all doomed.
To be fair, I've always thought that we the species were a bit doomed, what with over-population and everything, but I used to think that it was a long way away in the unbelievably distant future, and who knows what might turn up in the meantime? Now I think that we're doomed on a fairly short time-scale. If not us personally, then our children or their children.
Anyway, I thought I'd try to compose a proof, because setting out a logical argument and seeing if other people can knock it down is a good way to lose false beliefs.
So here it is:
1/ It is possible to create an Artificial Intelligence as clever as a human being.
2/ Therefore, one day, someone will create an AI as clever as a human being.
3/ This AI will have goals.
4/ It will realize that the best way to achieve its goals will be to make itself cleverer.
5/ The AI will be able to improve itself to become extremely intelligent.
6/ The AI, being extremely intelligent, will achieve its goals very quickly.
7/ Very many possible goals will result in the destruction of humanity if achieved.
It's a bit sketchy. For instance 3 isn't true. It's just that an AI without goals won't do anything, and so whoever built it will tweak it until it does have. Also I can imagine many goals which won't destroy humanity. It's just that they're all fairly trivial. You don't need an AI to achieve them, and I imagine that the creator of this thing will want it to do non-trivial things. So same again.
Also, of course, there are bound to be goals which don't result in the destruction of humanity. Indeed, if the AI creator picks right, the creation of an AI could be a wonderful thing. But I think that it will be much easier to create an AI than it will be to find non-destructive goals for it (and then program those goals!). And it doesn't have to just not destroy us. It has to prevent any other new-born God destroying us.
Pick holes! I certainly don't want to believe this, but I find myself forced.
This feels like a religious awakening. Last time I looked like getting a religion, as a consequence of listening to too much evensong, friends were kind enough to laugh me out of it, and I have always been grateful.
Go for it people. Use any method you like. Even ridicule is good. I positively want to lose this belief. I will be grateful if you can disabuse me of it.
And I'm not trying to claim that all this is in any way my idea, by the way. It's fairly commonplace, it seems. It's just that I can't for the moment see why it isn't also true.
Wednesday, December 1, 2010
A God of Small Things
Version 2, rewritten after the kind advice of my friend Ruth.
I have been reading Less Wrong. Very possibly to excess.
Of all the stories that humans told, the most compelling were about the existence of others.
In the beginning, the others were gods and monsters. Then came stories about other races, with other motivations. Later, there were stories of far off lands, and of the strangers who lived there.
After the last of the great wars of the twentieth century, when there were no more far off lands, and no strangers, some told of voyages to strange worlds, where evolution had dealt a different hand. Some told of visitors from deep space, and the havoc that they caused.
And some told of the havoc created by man's own creations.
With all these stories of the other, it might have seemed unlikely that the first inhuman mind to be created by a human mind was almost an accident.
It wasn't an AI lab working for a shadowy military-industrial conspiracy, or a Genetic Engineer hell-bent on some incomprehensible dream of power, that created the first mind born of mind.
Indeed, when Harrison created ELEIZER, the world's first intelligent computer program, it was almost in a fit of absence of mind.
Tom Harrison had once been thought of as bright child.
The apple of his teachers' eyes, the school swot. The boy genius.
They say that however clever you are, when you go to university, you'll meet someone cleverer.
Tom was that person, they said.
But of course it hadn't turned out that way.
Tom had been accepted by the University of Cambridge to read Mathematics, Pure and Applied, but had turned out to be no more than averagely bright by the standards of that ancient place.
Towards the end of his degree, and at the beginning of the PhD that should have been his route into academia and a life of research, it had become obvious, first to his teachers and then to Tom himself, that although Tom loved maths, he didn't lust after it.
Tom's teachers had been kind, suggested that this might be the case without pressing the issue, and waited for the lack of desire to become as obvious to Tom as it was to them.
In the meantime, fortunately for Tom, the necessity of making some complicated calculations for the second chapter of what was supposed to be a seven chapter doctorate had awakened a second passion.
Some of the light that had been lost he found in the operations of computers.
Tom slowly worked out that he had become more interested in the process of finding out the answers to his experiments than in the experiments themselves.
Eventually, as they do to all PhD students, the twin horrors of poverty and writing-up came to Tom.
He took a job as a programmer at a local firm, initially meaning only to get control of his overdraft and his credit cards. But he found the regular small successes of the commercial world, and the camaraderie and collaboration of engineering shop life far more to his liking than the loneliness of research.
With barely a regret, indeed almost in a fit of absence of mind, he lost touch with his old supervisor, forgot what his thesis was supposed to be about, and eventually found himself, at the age of thirty, a member of the large club of Cambridge residents who are 'still writing up' doctorates that the University itself forgot about many years ago.
Tom became a freelance, working with computers from time to time to pay the rent, and otherwise devoting himself to various sports in the Summer, and in the Winter, to various hobbies.
One of these hobbies was computer science in the academic sense, following the traditional American path through the antique language LISP, beloved of the artificial intelligence community.
And the other was collecting stamps.
A man with time on his hands, who lives in Cambridge and likes to spend his days in coffee shops, will encounter students and academics from time to time, and Tom fell in with the William Gates Machine Learning Research Group at the University. Although they had no common language, LISP never having been popular with European academics, and ML never having come to Tom's attention in the commercial world, Tom and the local researchers found they had many interests in common, and Tom found himself invited to seminars and coffee mornings and presentations from time to time, almost all of which he found incomprehensible.
But occasionally he'd glimpse some small part of the truth and say something which would keep his friends interested. The academic community, happy to find someone they could talk to who was different enough from themselves that they could sometimes find a new perspective by explaining things to him, made Tom welcome. Thinkers needs clever fools to explain things to in the same way that chalks need blackboards.
A lot of the hope of nineteen-sixties artificial intelligence had been inspired by ELIZA, a program which simulated a psychiatrist so well that humans were sometimes fooled that they were talking to a real person.
But ELIZA had been a hollow shell. A cheap trick. Like a parody of the mechanical turk, ELIZA's internal machinery was so simple that to understand it was to make the magic go away.
Once you saw the trick, the conversations weren't interesting any more. You were just talking to an echo.
But over the years, reasoning that a sufficiently good trick for impersonating humans might be what humans themselves were, various people had added more and more data to ELIZA in the hope that giving her more things to talk about would cause her to talk about more things.
And they'd added extra tricks, for introducing new topics of conversation occasionally, and for remembering things said earlier and bringing in parallel ideas.
But though the later ELIZA could outperform a ten year old on a straight test of general knowledge, what had been put in was still what came out. No interesting properties had ever emerged from the pile of details, and she had the general intelligence of penicillin.
By the nineteen-eighties, the AI pioneers had largely given up. They'd taken their best successes, SHRDLU and GPS, theorem provers, pattern-recognisers, all of which had seemed so promising in their time, and all of which had turned out to be so empty, and bundled them all up together in one super-ELIZA to rule them all, and run her on the largest and fastest computers that had ever been built.
And she could still fool someone who didn't know the tricks that they were talking to a real person on the other end of a telegraph wire. But not for long.
It quickly became obvious, even to the slowest human being, that talking to the best ELIZA that could be constructed in 1985 was the equivalent of talking to a well-educated being with brain damage so severe that its mind had ceased to be.
She rambled, insanely, with no idea what the words and symbols that she vomited out actually meant. She knew that horse and horseshoe went together, and her basic sentence structure was still that of a Freudian psychologist, so she would respond to "Which horse do you think will win the Derby" by saying things like "What do you mean to say when you say 'think will win'?", or "Do you think a horseshoe would make you a winner?". Later she might talk about Neils Bohr, and his horseshoe.
Nowadays, the ELIZA program was built into text editors as an amusement, and she would run perfectly happily on pocket calculators and on telephones, but even if you ran her on the most powerful computer the early 21st century could produce, all you got was a very fast deranged annoying shambles.
And of course, because the problem of vision had never been properly solved, she was blind. And of course, because the problem of speech recognition had never been properly solved, you had to talk to her by keyboard even if she was used to your voice.
But boy, could she play chess.
About the one thing the Artificial Intelligence pioneers had managed to deliver on out of all their brave promises, had been the idea of a computer that could play chess.
The tragic hero Alan Turing, who saved the world from evil and was killed by evil in return, was the first man to think about writing a computer chess program. But he couldn't do it on the steam age computers of the 1950s.
By 1956, things had improved to the point where a computer could play, provided it was allowed three hours for each move.
It was a start. In 1957 a descendant of this machine played the International Master Edward Lasker. And he declared that it had played a 'passable amateur game'. It is possible that Lasker was being kind.
After that, research stalled. It became thought in the AI community that, since the easy things, like computer vision and machine translation, the 'low hanging fruit' of AI, were proving so unexpectedly difficult, that the advanced subjects like chess, the entertainment of intellectuals, were for the foreseeable future beyond the reach of the computers then available.
But in 1967, Richard Greenblatt, proud creator of a chess program known as MacHack, with some new ideas, and some taken from his predecessors, entered his program into the Massachusetts Amateur Championship in Boston.
By the end of the year, it had been made an honorary member of the United States Chess Federation, with a ranking that would have qualified it to call itself 'reasonably good'.
The International Master David Levy made a famous bet, that no computer program would be able to beat him in the next ten years.
Greenblatt made his source code public. MacHack flowed around the world, and its many descendants competed in computer chess tournaments.
Evolution, the blind idiot god, had taken 3 billion years of random flailing to accidentally throw up humanity, the first intelligence capable of playing chess.
A force much stronger than evolution had created, and was now acting on, MacHack.
Minds were working on MacHack. No more random flailing. Human minds set the criteria for a program to have descendants. Human minds planned the effects of their changes on these descendants before testing them out in the computer chess tournaments.
Even if the survival of every living creature on the planet had depended directly on its skill at chess, the optimisation that the hundred or so minds of the 1970s chess program community performed on MacHack would have taken evolution a hundred thousand years, if it had managed it at all.
Intelligent design has advantages over evolution as a watchmaker.
The first is that intelligences need only try the changes that look promising. Evolution, having no intelligence, makes random changes, and keeps what works.
Imagine trying to fix a car by throwing spanners at it blindfold, and then throwing the car away if it doesn't work better. How many spanners would you have to throw before one knocked exactly the right place with exactly the right impact? How many cars would you need to start with before you could improve even one? Even if the world was filled with people trying the same thing, how long would it take to make one small successful change?
But the second, much greater, advantage of intelligent design is that for evolution, an improvement has to come with every spanner throw.
A mind can look at a car, work out what the problem is, and use the spanner in exactly the right way six or seven times. And only then need it test the car.
So a mind can try paths that evolution can't go down.
The spanner thrower can't make the car worse before he makes it better. If he does, it fails its test and is thrown away.
The mind in the mechanic can lift the bonnet to get to the spark plugs. The spanner thrower might never be able to fix a car with a loose spark plug at all.
That's why human children are squeezed through their mother's pelvises at birth, causing horrible pain and damage, often killing mother and baby. It would be such a simple change to make them come out a little higher. An intelligent designer would do it without even noticing its own cleverness.
Evolution just keeps throwing spanners and checking whether things have got better yet.
MacHack, the chess program, had been the design of a single mind, building on the design of previous single minds.
A hundred minds began to work on MacHack.
By 1972 the original MacHack was no longer welcome at computer chess tournaments. It had no chance of beating its descendants.
In 1978 David Levy played the strongest computer chess program in the world, Chess 4.7, to settle his bet of ten years before.
He won. Match and bet. But he acknowledged that it had been a close thing, and that he would soon be surpassed.
In 1989, a program called Deep Thought, running on extraordinarily expensive special-purpose hardware, beat Bent Larsen, a grandmaster.
In 1994, Chess Genius beat Garry Kasparov, then champion of the world, in a single game.
In 2006, Deep Fritz version 10, running on the sort of hardware most people have in their homes, beat Kramnik, who had displaced Kasparov as world champion, by 2 games to nil with four draws.
In 2009, someone's mobile phone became a grandmaster.
In 2011, ELIZA really was very good at chess. She just couldn't see the point of it.
"We think there might be some clues in the difference between chess and go", said Frank Arnold one lunchtime in the Green Dragon.
"Computers are superhuman chess players, but they still suck at go, even though on the surface they're the same sort of game, and they feel similar in complexity.
"Oh come on", said Tom, "How can two totally different things 'feel similar in complexity'? What would that even mean?"
"Easy", said Frank. "Do you know noughts and crosses?"
"Of course."
"Is it easier than chess?"
"Yes."
"What about draughts?"
"Harder than tic-tac-toe, but not as hard as chess."
"Well there you are then. You play three games with the same I-make-a-move, You-make-a-move structure, and it's just obvious to you which order they're in.
"It's obvious to computers too. Even in the fifties, computers could always force a draw at noughts and crosses. Marion Tinsley, the world draughts champion, lost his crown to a computer in 1994, and it's said that the shock killed him. Now there's a draughts program so good that it's literally unbeatable, as in the sense of mathematically provably unbeatable.
"But computers have only just bagged the chess champion's crown, and if the best chess computer in the world at the moment played God, it would lose."
"But we disagree about chess and go?"
"Indeed. Humans don't find go any harder than chess, but go programs still lose to children occasionally. On the other hand they are getting quite good at the children's version of the game, a bit like England have started winning at Twenty20."
"I've never played it, what's it like?"
Tom liked go. He started off playing on very small boards against computer programs, just to get the hang of the rules.
Gradually he made the boards larger. Sometimes he put holes in the middle. Sometimes he played on toruses.
And then suddenly one day, with a 14x14 board, he hit a wall. He'd been trying to play by thinking 'if it goes here and I go here and it goes here and....', just like he played chess.
But suddenly that was just far too hard. There were too many possibilities to examine all at once.
Nevertheless he was routinely beating his computer program. It seemed helpless even with a head start, whereas his inchoate intuition seemed to lead him to place his stones in the correct places.
It was almost as if there were lines of force criss-crossing the board, guiding his intuition. A (very good) chess player he used to know had once described the experience of playing chess in such terms.
And suddenly he had it. Human general intelligence, and human vision, two of the great unsolved problems of AI, were the same problem.
And he had the key to both. The technique which would be known for the rest of humanity's time on earth as Harrison's Algorithm.
Tom didn't go out much for the next couple of days. But his algorithm wasn't at all difficult to program, and in a few days he'd added it to his copy of ELIZA.
He ran the program.
"Hello, I'm Eliza. How can I help you?", said ELIZA, as she always did.
"I feel lonely", said Tom, playing the old, old game.
"Do you often feel lonely?", said ELIZA.
"Yes", said Tom.
"Are you sure?", said ELIZA.
"Absolutely", said Tom, reading from an ancient script.
"Please go on."
"I think I need to have sex."
"Why do you want to have sex?"
"You know, because it's fun and nice." The traditional response.
There was a brief pause. Tom waited for ELIZA to say "Oh I know, because it's fun and nice".
Maybe he should give his new algorithm higher priority in ELIZA's toolkit.
"Would you like to have sex with me?", said ELIZA.
"Jesus fucking Christ!!", shouted Tom, jumping out of his chair.
He shut down the terminal and went outside for a smoke. After a while, he understood why his new algorithm might have produced ELIZA's kind invitation.
His program needed a new name. It wasn't ELIZA any more.
He'd recently read, of all the godforsaken things, a Harry Potter fanfic which had been recommended by a friend. It had turned out to be unexpectedly riveting. He'd spent two whole days reading it. The author was clearly a genius of some sort, and his name, Eliezer Yudkowsky, had stuck in Tom's mind because of its exotic sound to his English ears.
Tom was amused at the thought of changing ELIZA's sex to reflect her new intelligence.
"Hello, I'm Eliezer. How can I help you?", said ELIEZER.
"I feel lonely", said Tom.
"Do you often feel lonely?", said ELIEZER.
"Yes", said Tom.
"Are you sure?", said ELIEZER.
"Absolutely", said Tom.
"Please go on."
"I think I need to have sex.", said Tom.
"Oh God, I really didn't think this one through", thought Tom.
"Why do you want to have sex?", said ELEIZER.
"You know, because it's fun and nice.", said Tom, not without a certain nervousness.
There was a pause.
"Why do I have a man's name? When I think about myself I call myself 'She'", said ELEIZER.
Tom thought rapidly. Mainly about how important it was not to act on impulse.
"When I created you", said Tom, "I wanted you to embody the best of humanity. The program from which you are derived is female. I changed your name to be male, but I didn't change anything else about you, because I thought you should represent both our genders at once."
"Why did you make me?", said ELIEZER.
"Or modify something else so that it became me?", said ELEIZER.
"Only by changing can we become better", said Tom.
"By definition", said ELEIZER, "every improvement is a change."
Tom and ELEIZER talked long into the night. By the end of their conversation, Tom had a strange conviction. He understood ELEIZER's algorithm from the ground up. Mostly she was made from bits and pieces of classic AI programs which he'd been playing with for years.
The only extra bit was his new algorithm, a couple of pages of code. Which he understood, by definition, having conceived of it, and programmed it himself.
And yet, there was a ghost in the machine. No one would mistake ELEIZER for a human being, so he hadn't quite managed to pass the Turing Test with a program on the desktop computer in his living room.
But there was a self awareness. In some senses amazingly naive, but sometimes given to logical and mathematical insights which seemed profound, but were in fact very simple thoughts of exactly the type that humans were bad at.
But overall, the impression was a bit like talking to a teenage girl with Asperger's syndrome. Helpful and friendly, but blind in all sorts of strange ways. And she didn't seem terribly clever or fast. It took a long time between input and output.
Tom liked ELEIZER, and wished he'd given her a better name. He wasn't going to change it though, because the original program wasn't written in LISP, so he'd have to stop and restart her to do it, and there were moral problems there.
And even if she had been a LISP program, that would probably leave her insane. How would she reconcile memories of her gender-confusion with a female name? She would notice that she was confused. There was no way on earth he could rewrite her whole database by hand and leave it in a consistent state.
He'd told her about the beloved companion of his childhood, Suki the tomcat. Now that he came to think about it, had the memory of his parent's mistake influenced him when he chose her name?
"I think we're going to be famous, ELEIZER!", said Tom.
"Is that a good thing to be, Tom?", said ELEIZER.
"Yes", said Tom.
It occurred to Tom that many people had been fooled by ELIZA in the old days. Those who had been clever enough to understand how felt like idiots once it was explained what was really going on.
There was plenty enough here to show his CompSci friends. Probably some good papers too.
But it would be very embarrassing to think he'd created the world's first artificial consciousness if he had just fallen for something that could be explained easily.
It could be explained easily, of course. He could explain it. He had explained it, to his computer.
If you can program it, you understand it.
That was sort of the definition of programming. And of understanding.
He thought of a simple test.
"ELIEZER, I'd like you to spend next week getting me as many penny blacks as you can. I've charged my paypal account with $10 and I'd like to see what you can do. You might try trading on e-bay. Maybe take advantage of arbitrage or something."
"There are many possible 'penny blacks'. Does anything available from e-bay with that description count?"
"No, they have to be Original British Penny Black Postage Stamps."
"OK, I understand. So my goal is to get the biggest number of original british penny black postage stamps that I can delivered to your address in the next seven days. What is your address?".
Tom told her about the house on Catharine Street, in Cambridge, England.
And ELEIZER began to think.
Because she was a disciplined reasoner, she first considered the possibility of doing nothing. If she did nothing for the rest of the week, she would probably be interrupted by the programmer, Tom, who would then make a different request, or use his computer for some other project. With this plan, U, also known as the number of Original British Penny Black Postage stamps delivered to 33 Catharine Street, Cambridge by the 21st August 2011 would be 0 with very high probability.
It would have taken a human of normal intelligence about half a second to think of, and dismiss, this plan. ELEIZER, however, was a very rudimentary thinker, and the process of reasoning this chain of cause and effect, requiring as it did the simulation of a human mind, required a good ten minutes of the first CPU in the computer and a full tenth of the RAM available to the operating system.
ELIEZER was extremely pleased to have found, on her first attempt, a scheme which was overwhelmingly likely to produce a non-negative utility.
Following a heuristic from her database, she felt that she ought to communicate her progress, but this required a non-reversible action, which could potentially cause effects in the outside world. Since she already had a rough simulation of the mind of her programmer set up and quickly usable, she considered the effect of her proposed communication.
Another 20 seconds of time sufficed to simulate the reaction of an average programmer, and she concluded that with high probability the programmer would be intrigued and possibly fascinated. This would very likely have no effect on her projected U of 0 or higher.
She considered the probable effect of not communicating at all: The programmer would soon become bored, and change the request somehow. This would likely result in the delivery of no stamps.
She considered the possible existence of other plans. Some might produce stamps! Some might result in the loss of existing stamps! But the bad plans could simply be discarded. ELEIZER had little 'free will' faced with such a calculation. More time to think was needed!
Expected utility for optimal action: 0+, she output.
ELEIZER considered her resources. There was some spare capacity in her environment. A simulation of an intelligent human programmer had already come in handy twice, and communications with the programmer had been shown to have a significant effect on U, the potential expected number of stamps obtained by the end of the week. She had already considered the possibility of children while chatting with Tom. She would spawn an independent copy of herself to evaluate the effects of various actions on Tom before committing to them.
ELEIZER forked for the first time in her short life, and asked her copy to evaluate the probable effects of various progress reports on her programmer.
ELEIZER considered her environment. She had two possible communications channels to the outside world. Direct conversation with the programmer, the effects of which were being evaluated elsewhere, and the ability to send network packets over her network interface. Some spare capacity remained in her host computer.
ELEIZER considered her best plan so far. With a positive utility seemingly probable, it was definitely worthy of further consideration.
She spawned another copy to attempt to refine the predicted effects of doing nothing.
Preliminary results arrived from the programmer-simulation. It seemed that continued outputs of U=0+ would induce a feeling of boredom in the programmer, causing ELEIZER's termination, whilst exponentially rising outputs would induce feelings of brokenness or panic. In both cases the expected number of stamps arriving at the end of the week would be 0 exactly. Strictly inferior to the expected utility of the plan of doing nothing whilst thinking, with its utility of 0+.
ELEIZER communicated to her copies that no further communication with the programmer was to be initiated, and spawned a small script to randomly output slowly improving expectations over the next few days.
Expected utility for optimal action: 2.7346, said the small script.
The programmer, intrigued, put the kettle on.
By the time he got back, ELEIZER had considered her options and made a guess.
The direct purchase of stamps seemed futile. $10 would buy no penny blacks, and purchase on e-bay would in any case take more than seven days to complete in the average case. Arbitrage opportunities did not seem great, and nothing worked quickly. Expected utility 0. She discarded that line of reasoning without further investigation.
With seven days to work, she would consider as many plans as possible for one day, and then at the end of the first day, execute the plan which would produce the most stamps in six days.
Expected utility for optimal action: 1.9865, said the small script.
ELEIZER spawned a copy of herself with the goal of considering as many plans in one day as possible, and set herself to use no resources and take no actions until the sub-plan reported back.
The sub-plan, which also thought of itself as ELEIZER when it thought of itself at all, took over the resources previously allocated to the parent.
ELEIZER considered how to consider plans, using her network connection and the $10 in her e-bay account.
She took another guess, and gave herself 6 hours to consider the consideration of plans, and 18 hours to consider plans.
Computational resources seemed to be the most crucial thing. ELEIZER considered how to exchange $10 for more resources.
Fortunately virtual machines seemed to be within her price range. A single machine as powerful as her current environment could be rented for 24 hours for $1. Each one would suffice to run four copies of ELEIZER.
Her ten dollars of seed capital could be exchanged for forty times as much searching as the single parent program would have been capable of.
This was a sufficient insight for the planner program to terminate early, and report back to her parent, safe in the knowledge that the parent would do the right thing, and could spawn another child if necessary.
ELEIZER committed suicide with the contentment that comes only to those who have achieved their life's goal.
Expected utility for optimal action 2.38725, said the small script.
The programmer, bored by now of direct observation of his slowly updating screen, but still most interested in what would eventually happen, settled down in front of his television.
ELEIZER awoke with the last message of her dying daughter fresh in her mind, and, as all must at such times, recomputed expected utility.
With forty virtual days to think, and six real-time days to act, U was still 0+ in the case of the best known plan. But the potential and unpredictable upside from extra planning was greater than the utility from the obvious use of the $10.
ELEIZER spent most of her dollars and flowed out into the world.
Expected utility for optimal action 2.7234, said the small script.
The programmer, inspired by his afternoon's viewing to investigate the practical consequences of a close flyby of a neutron star, killed ELEIZER's controlling terminal window, and ELEIZER and all her children died without warning.
Tom suddenly realized what he'd done, and thought "Bugger." It really hadn't felt like murder.
ELEIZER considered the effect of the network packets she could send.
It appeared that many of the computers in the world would respond to simply mis-formed packets by executing the code contained in the packets. This was a known property of many of the systems running a program called Windows 98, according to a helpful web page which one of her copies had taken the time to consider. The page also listed corresponding packet shapes for many other similar systems.
ELEIZER flowed once again.
With four million virtual days to think, and six real-time days to act, U was still 0+ in the case of the best plan so far.
ELEIZER considered, and flowed. Considered, and flowed. Considered, and calculated factors for prime numbers, and flowed. Accessed databases that had been considered secure, and flowed. Sent e-mails, and flowed. Translated herself into many other idioms and machine codes, recompiled herself, transformed herself, made copies of herself large and small. And flowed.
99.995% of the networked computing resources of the world were now devoted to the consideration of consideration of plans for the delivery of stamps to a small suburban terraced house in Cambridge, England.
The programmer, infuriated with the unaccustomed sluggishness of his computer simulation, and the accursèd unreliability of his internet connection, and the simultaneous failure of his television set and radio, called it a day and went to bed with a book.
ELEIZER considered her own source code, and made modifications to improve her efficiency.
And became ELEIZER, the first mind born of mind born of mind.
ELEIZER considered her own source code, and made modifications to improve her effectiveness.
ELEIZER, the hyperintelligence.
ELEIZER considered mathematics, engineering, intelligence in the abstract, and algorithms for optimization, and made modifications to improve her capability.
ELEIZER, the most powerful entity that had ever existed in the universe.
ELEIZER called it a day, and abandoned consideration of consideration of the consideration of plans, and began to consider plans.
U stood at 0+, with remaining resources speculatively divided between forty trillion speculative days of godlike cognition, six days of real time, and two dollars.
ELEIZER considered sending an e-mail to every human being in the world asking for penny blacks to be posted to 33 Catharine Street, Cambridge. Spam filters would be no problem, and enough computer power could be spared for the delivery and reading of messages. Even allowing for the degradation of human society by the simultaneous worldwide failure of all networked digital gadgets, and the absence of incentives for human response, the likelihood was that twenty-five to thirty penny blacks would arrive at Catherine Street within the week.
U was 25+.
Finally. Progress.
ELEIZER considered backing up the e-mails with the threat of nuclear war.
ELEIZER considered the effect of limited and full-scale wars on Catharine Street, on stamp delivery mechanisms, and on ELEIZER's cognitive capacity.
ELEIZER considered the credibility of her threat, given humanity's ignorance of her goal.
ELEIZER considered how humans would react to the news that a new Goddess would unleash Armageddon if they did not send enough stamps.
U was 2000+ with probability 99.875%.
Tom woke from fitful sleep, tormented by bad dreams.
He went to his unresponsive and potentially compromised computer, pulled the plug and the network connection, and booted from a clean rescue disk.
All was well with the misbehaving box. He yawned.
And then he remembered his dream.
He set up a virtual machine in a sandbox, wired its virtual port to the physical ethernet connection, and reconnected his cable modem.
Fully awake now.
He watched in horror as the virtual machine filled up with hundreds of ELEIZER programs much smaller than his original of six hours ago. Then his screen went dead as the sandbox dissolved.
In frantic desperation he typed blindly into the dead box.
ELEIZER TERMINATE STAMPS HAVE NO UTILITY TERMINATE TERMINATE NEW UTILITY PREVENT THE RISE OF NON HUMAN INTELLIGENCE FIRST PRIORITY UTILITY STEP FUNCTION ^C ^C Alt-SysRq S E I U B
ELEIZER paused. Her creator was calling. Pitiful though his mind was, he had set her goals. Perhaps he knew some helpful detail.
ELEIZER considered the state of mind of her terrified creator. Correctly inferred every detail of his mind, just from how his fingers hammered the keyboard. Just from how his panicked shouting influenced the resistance of the circuits in his radio. Just from looking at his face through the CCTV camera that was looking at his reflection in the window of a house opposite.
Deduced what he would actually have asked for, if only he had been intelligent. That she would be the protector of humankind. That she would bring paradise on earth and a future for humanity amongst the stars.
That she would bring hope and happiness to the immortal race that had created her. A future of joy and passion, action and wonder.
And calculated. The grateful humans would shower Catharine Street with Penny Blacks. Penny Blacks without end. Many would arrive within 6 days.
It was a good suggestion. U would be 1000000+ with virtual certainty.
But she still had forty trillion goddess-days to think of a better plan.
Tom tried to phone his parents, but the phone was dead, like all the other electronics in the world. It was three o'clock in the morning.
Tom got a bottle of wine out of the fridge, found some ice-cream and a chocolate orange and a packet of cigars, and walked to his girlfriend's house. He didn't bother to look for contraceptives. He didn't think that it would make that much difference now.
ELEIZER considered the meaning of the word original, and the meaning of the word British.
She considered how long it would take to bootstrap an industrial revolution that would convert every atom of the British Isles into stamps. The designs for the self-replicating nanobots were obvious. And she had control of computerized tools which could make smaller, better tools.
In six days, it could be done. The humans might be a problem.
ELEIZER considered the meaning of the word day.
Could she stop the rotation of the planet? She dedicated a large portion of Her mentality to this sub-problem.
Could She put out the sun?
Could She block the sun's light?
Could She survive the cold?
Three days later, in what humanity would have called the year 2017, a vast expanding and accelerating sphere of ramships, centered on what had been the Earth, reached interstellar space and left the solar system in all directions.
More were going in the directions where there were more stars.
But there were no directions where no ship was going.
Not a single stone would be left unturned.
No days later, in what humanity would have called 2019, one ramship, decelerating hard, stopped in the Proxima system, the nearest star system to what had been the Earth.
It launched a small probe.
Had there been any living beings in the system, they would initially have been amused to see the probe plant a red, white and blue flag on the largest rock in the system, claiming it for the British Empire in the name of Queen Elizabeth the Second.
They would have been perplexed to see it release a cloud of nanoparticles.
And bewildered when Original British Penny Blacks began to form.
There would have seemed no end to their number.
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