Saturday, January 29, 2011

It's a fair cop


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.

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.

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.

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 backwards 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.

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."

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.

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!

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