I wrote this article years ago, after the fall of the Northern Rock bank in England, which was the first run on an English bank in a hundred years. A while later, when it looked as though Greece was going to default on its debts, I put it back to the top since it had started to look like a prophecy.
It's now 17th June 2011. Last year Greece was bailed out on promise of economic reforms, but those reforms were exceedingly unpopular and have just brought down its government and it is again on the verge of default. The IMF has refused to supply money, effectively blackmailing Germany into paying Greece's debts.
This isn't a long term solution. The problem will recur. Similar problems have occurred in Ireland and Portugal, and been 'solved' by similar methods. The longer this sort of solution is attempted, the more catastrophic the eventual reckoning will be.
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Note of 10th May 2010
I've put this back to the top because of the financial crisis in Greece. It was originally written after the UK banking crisis that turned into the (first half of the most recent) depression. Sovereign debt default is the next stage of an inevitable process. We have to ask ourselves whether it is better to take the hit now, or put the crisis off again. Next time there may not be an entity large enough to organise a bailout.
I'm going to stick my neck out and make a prediction. Today is 10th May 2010. I predict that either Greece will default on its debts, or the EU will bail it out, and then within the next five years other EU nations will default. The resulting financial effects will domino, and there will be widespread bank defaults. No government will be able or willing to guarantee the banks this time. The turmoil will cause Japan to default on its debt, and the result of that will be a terrifying financial apocalypse the like of which has never been seen. There will be revolutions and wars.
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Original Article:
There are few good arguments from analogy. Anyway, here's mine:
Once upon a time, the forestry service put out wild fires.
Nowadays they start them.
Once upon a time, banks built solid marble buildings with Doric columns.
Nowadays they rent cheap offices with plastic chairs and nasty carpets.
And I think these facts are related.
Let's look at the war going on in the woods.
The natural enemies of plants are trees. If you have an expanse of open earth, the first things that grow there will be grasses and nettles. Grasses grow fastest of all, finding new earth with their wind-blown seeds and rapidly colonizing everywhere they can fix roots.
Nettles are next. Nettles have the trick of making poisons. Grazing animals don't eat them. Because animals eat the grass, the nettles are taller than the grass. So they can see the sun. Whilst the grasses, being short, die in the shade of the nettles.
A patch of open earth will soon be covered in grass. Soon after that it will be covered in nettles. The thick covering of nettles means that the grazing animals will not go there.
This opens up a place where trees can grow. Trees have the knack of making wood. Wood allows them to grow taller than the nettles. The trees get so tall that they can rise out of the nettle patch entirely, and then begin to spread their branches wide, spreading a vast carpet of leaves. Blotting out the sun. The nettles, which protected the trees from the grazing animals, die. But by this time the trees have grown so tall, and with such thick bark, that the grazing animals can't reach any edible bits of the tree. So the patch is covered in trees. And this is the final state of the world.
So we might wonder why grasses and nettles still exist, if they always lose.
The answer is fire.
A forest fire is an awful thing. Dry trees burn very well, and there is sufficient wood to create an inferno which is hot enough to set light to a tree.
Every so often, even in the oldest and most permanent forest, there will be a period of drought. The trees will become so dry that a lightning strike can set one burning. And the flames will set light to a nearby tree. And the fire will begin to rage through the dry woods, incinerating everything in its path. And nothing will stop it. Not even the arrival of the rains. Until the whole of the forest is a smouldering wreck, and everything is dead except the fire-tolerant seeds hidden in the ground.
Which will begin to grow. Grasses grow fastest, because they don't need to make wood or poison. But grazing animals eat them........
A forest fire is a dreadful thing. It moves faster than you can run. It destroys everything and everyone. You never want to be anywhere near one. Nothing escapes but birds.
So the forestry service used to put out forest fires. And this was a very good thing.
But the trees grew and kept growing until the density of wood in the forests was so high that one day there was a forest fire and it had so much fuel to feed on that the foresters could not put it out. And because there had not been a fire for so many years, a lot of people had got a little careless, and built their homes in the woods.
And the fire that burned was bigger and fiercer than a normal forest fire, because there was so much wood. And you do not ever want to be near even a normal forest fire.
So these days the forestry service starts forest fires. They keep the density of the trees low, and if you are skilled in the ways of the woods you can predict where the fires will go, and to an extent control them and protect people.
You never get the embarrassing situation that happens when there hasn't been a dry patch for a while, and lightning hasn't felt like striking much, and the woods are overgrown and dry and waiting like a tinder box for the spark that will unleash hell.
And who knows if this latest innovation of the foresters will have an unintended side-effect? So far it seems to work well.
What about banks?
Banking is a very risky business. You borrow money from people. And you lend money to people. But there is no symmetry.
People lend you money so that it will be safe. Once upon a time, you would build a nice strong building and charge people to keep their money in it, where it would be safe. This model has fallen out of fashion. But at any rate, people's primary motivation is that their money should not fall into the hands of the sort of person who is forever talking about giving it to the poor but whose favourite Maid is suspiciously well dressed for a girl who lives in the woods.
So you are sitting in a nice strong building full of money. And you can practically hear the outlaws plotting in the woods about tunnels and bulldozers and other ways of making what is inside the building cease to be inside it.
So you need to find some way to get rid of the stuff. And it turns out that there are plenty of people who are willing to take it off your hands and keep it safe for a while themselves. They do this by buying houses and setting up businesses and other wholesome things, and they have promised to pay the money back slowly, with some interest, and most importantly both you and the Sheriff know where they live.
And everything is going so swimmingly that you can barely believe your luck, until Crazy Jake walks into a bar and tells everyone that Mr Shylock put all of the town's money into a scheme to mine gold in the hills and it turns out that there ain't no more gold in them thar hills and all the money is gone.
Now Crazy Jake is a big fat liar. At the most you only put a very small amount of spare money of your own into that scheme and yes, it is gone, but no one need care particularly because there is plenty of money in the bank to go round.
But people are a suspicious lot, and a few people pop round, not at all concerned, really, to ask for their money back so that they can keep it under the bed instead.
And a few people come round as normal to ask if they can have their money back because they have bills to pay.
And towards the end of the day you realise that the bank is out of cash. But everything will be alright tomorrow, you promise, because there are lots of interest payments on loans due in.
Tomorrow there is a line round the bank half a mile long. Because a couple of non-crazy people have been talking about how they asked the bank for some of their money back and the bank didn't have any money.
And you are a dead man. If you don't actually get hung from a lamp-post on the spot, you will be carted off to a dungeon somewhere and kept there forever. Crazy Jake has killed you as randomly and with as little malice as a lightning bolt hits a tree.
And the only people who are happy are the people that you lent all the money to, because their creditor is dead. And maybe the mess will all get sorted out eventually and they will be paying their interest to the people who've lost all their money by trusting it to you. And maybe it won't.
This is called a bank run, and it is why banking is a risky business.
Bank runs used to happen all the time. You can see black and white footage of huge queues round banks. Bank runs feature frequently in Westerns.
Banking is a very risky business, entered into by gamblers of the bravest and most reckless kind.
There is no way round it. The problem is that you are making money by lending long term and borrowing short term. That is what a bank does. If you aren't doing that, you aren't a bank.
Imagine that you are a gambler of the most reckless kind, who has a way to make large amounts of money at the price of occasionally and unpredictably being hung from a lamp-post.
To make it work, you need to get people to trust you with their money. These people may or may not be fools, but they certainly know about bank runs. They know that if a bank is behaving recklessly, it may suddenly cease to exist.
What should you build your building out of?
You should build it out of stone. And you should face the stone with marble. It should look as though it has been built out of pure money, by people who have lots of money, and intend to be around for a very long time.
Sober, serious, professional people. People who wear suits.
The minute you start to express the slightest bit of creativity, imagination or flair, the minute you're caught renting an office, as though you hadn't got the slightest intention of lasting for ten thousand years, then people will begin to take their money out and put it with more serious people. And we know what happens then.
Now once upon a time, this was what banks looked like. English towns are full of great big solid old banks built along classical lines, with lovely solid sounding names like 'The Trustees Savings Bank", or "The National Westminster Bank". Boring as hell, but lovely safe places to trust with money. Of course, they weren't actually. That's why they had to try so hard to look as though they were.
When I was a little boy, there was a new bank startup in England called "William's and Glyn's". I thought that that was a really nice name, and sounded very human and decent. They offered high interest rates and good service and they advertised heavily. And my father wouldn't put his money anywhere near them, because they sounded like they were just a couple of guys who'd started a bank.
But nowadays, banks don't look all solid like they used to. All the lovely solid marble buildings that look as though they will last for ten thousand years have been converted into public houses.
The last time I went into a physical bank, I found myself sneering at the crappy quality of the furniture. And anyone who knows me will tell you that I am not one to care too much about the quality of furniture. But this stuff was made out of formica, and it wobbled.
They don't even have sensible names any more. They're called "Natwest" or "Floyd's TSB" or "Egg". They're behaving like people who've been christened Thucydides by unworldly parents, and who have decided that they prefer to be called Diddy.
What has changed? Nothing recently. There hasn't been a serious bank run for years and years.
What has happened is that recently, the folk memory of bank runs has vanished. No one can remember what they were any more. So trusting money to a bank has stopped being scary. It will be safe. They will look after it carefully.
In fact trusting money to a bunch of gamblers to gamble with has started to seem like the boring option. Real men stick their life's savings in stocks and shares, or in exotic derivative instruments like property prices in areas where no more property can be built. Safe as houses.
What changed?
Government got involved in banking. A government can prevent a bank run. All it has to do is say: "Don't worry about your money. We guarantee that we will pay it all back if the bank fails."
The government can make this guarantee, because it has guns and tanks. So if it needs money it can steal it.
So if the government makes the guarantee, it is believed. Panic over. No-one needs to worry about whether their bank is solvent. No bank runs.
That's why we haven't seen a bank run in a while.
I believe that bank runs work the same as wildfires:
Lightning struck a while back. Through heroic efforts by all the governments in the world, a vast amount of money was stolen, and the fire was put out by pouring all this money over it. No one got hurt.
But the woods are still full of dry tinder. Be very afraid.
Friday, June 17, 2011
Thursday, June 9, 2011
Overconfidence Bias
The other day, a friend and I are out throwing cricket balls to one another.
I believe that when practising a skill, you should set the difficulty of the thing you're practising to the point where you fail one time in ten. Then you're getting positive emotional reinforcement from your successes, while constantly practising things that are still challenging, and pushing yourself periodically. If you're practising a skill that is in fact dangerous, then this also gives you a reasonable chance of avoiding injury, if the usual failure case is to not get injured.
I felt that the throws Joe was giving me were getting below the one in ten level, i.e. I seemed to be catching most of them routinely without having to try too hard, and I was about to ask him to make them harder.
Instead, we started counting successes and failures.
Over the next twenty or so throws, the total number of catches was 14, and I dropped the ball seven times. i.e. the true rate was about one in three.
Both Joe and I thought that this was due to getting freaked out by suddenly making success/failure a thing that was noticed, or possibly getting distracted by the effort of counting. So we carried on. Neither of us thought that the throws had got more difficult because of the counting.
Over the next twenty throws, the proportion of catches rose, until after about seventy-five, the proportion was 60 to 15. I'd completely stopped worrying about counting, and I'm pretty sure that the effort of remembering the two numbers wasn't interfering.
I think we were observing learning actually happening, as the rate changed from one in three to one in four.
We ended the session there, since we were both bored, but I really can't now believe that I'm capable of catching nine out of ten similar throws.
And so I wondered for a bit what it is that makes a person (I'm assuming I'm typical!) who can do something one time in three, and has done it lots recently, and presumably failed one third of the time, believe sincerely that they are likely to succeed nine times out of ten.
A little research brings up the term Overconfidence Bias, which seems somewhat related, where people back their own judgements to be right 'with 90% confidence', and then get it right about 40% of the time.
Afterwards the thought occurred to me that I must have been miscalibrating my 9/10 rule all my life.
I used it to learn to ski and to learn to row, and at the moment I'm trying to use it to learn to catch cricket balls, and I imagine in various other contexts where I was learning to do some physical skill.
And then it occurred to me that I had no idea why I believed that that was the right ratio anyway.
I just seem to have come up with it once, tried it, found that it worked, and then adopted it as an article of belief without ever trying any other way of learning physical skills.
That also sounds like overconfidence, but actually it's called the Congruence Bias.
If you have a theory, you test it by doing what it says to do and seeing if it works.
And in fact if that's all you do, that's a terrible way of testing a theory. You should be looking into the dark and asking if, when you try something else, do you still get good results.
But it's hard to do this.
I'm now thinking 'I should try catching really hard throws'.
But I don't want to. I know what will happen. I'll get my fingers broken and my hands hurt and I'll teach myself to fear the ball even more than I already do.
I'm thinking 'I should try to catch really easy throws'.
But I don't want to. It will be very boring and I'll learn nothing from it.
It is difficult, this 'rationality'. Once you start looking at what you believe and why, and what you do and why, you find all sorts of odd things going on.
I believe that when practising a skill, you should set the difficulty of the thing you're practising to the point where you fail one time in ten. Then you're getting positive emotional reinforcement from your successes, while constantly practising things that are still challenging, and pushing yourself periodically. If you're practising a skill that is in fact dangerous, then this also gives you a reasonable chance of avoiding injury, if the usual failure case is to not get injured.
I felt that the throws Joe was giving me were getting below the one in ten level, i.e. I seemed to be catching most of them routinely without having to try too hard, and I was about to ask him to make them harder.
Instead, we started counting successes and failures.
Over the next twenty or so throws, the total number of catches was 14, and I dropped the ball seven times. i.e. the true rate was about one in three.
Both Joe and I thought that this was due to getting freaked out by suddenly making success/failure a thing that was noticed, or possibly getting distracted by the effort of counting. So we carried on. Neither of us thought that the throws had got more difficult because of the counting.
Over the next twenty throws, the proportion of catches rose, until after about seventy-five, the proportion was 60 to 15. I'd completely stopped worrying about counting, and I'm pretty sure that the effort of remembering the two numbers wasn't interfering.
I think we were observing learning actually happening, as the rate changed from one in three to one in four.
We ended the session there, since we were both bored, but I really can't now believe that I'm capable of catching nine out of ten similar throws.
And so I wondered for a bit what it is that makes a person (I'm assuming I'm typical!) who can do something one time in three, and has done it lots recently, and presumably failed one third of the time, believe sincerely that they are likely to succeed nine times out of ten.
A little research brings up the term Overconfidence Bias, which seems somewhat related, where people back their own judgements to be right 'with 90% confidence', and then get it right about 40% of the time.
Afterwards the thought occurred to me that I must have been miscalibrating my 9/10 rule all my life.
I used it to learn to ski and to learn to row, and at the moment I'm trying to use it to learn to catch cricket balls, and I imagine in various other contexts where I was learning to do some physical skill.
And then it occurred to me that I had no idea why I believed that that was the right ratio anyway.
I just seem to have come up with it once, tried it, found that it worked, and then adopted it as an article of belief without ever trying any other way of learning physical skills.
That also sounds like overconfidence, but actually it's called the Congruence Bias.
If you have a theory, you test it by doing what it says to do and seeing if it works.
And in fact if that's all you do, that's a terrible way of testing a theory. You should be looking into the dark and asking if, when you try something else, do you still get good results.
But it's hard to do this.
I'm now thinking 'I should try catching really hard throws'.
But I don't want to. I know what will happen. I'll get my fingers broken and my hands hurt and I'll teach myself to fear the ball even more than I already do.
I'm thinking 'I should try to catch really easy throws'.
But I don't want to. It will be very boring and I'll learn nothing from it.
It is difficult, this 'rationality'. Once you start looking at what you believe and why, and what you do and why, you find all sorts of odd things going on.
Monday, May 9, 2011
Super Injunction
How is a super injunction supposed to work? Am I allowed to say that Jeremy Clarkson is having an affair with Jemima Khan? Or not?
Sunday, April 17, 2011
Mother's Day Flowers
Credit where credit is due. This year I remembered Mother's Day the day before, and it's a bit too far to drive for a day (I think. I know some people do 300 mile round trips for fun!).
I thought about asking my sister to take something to Mum for me, but that seemed a bit lame, so I found a Sheffield florist on-line and asked them if they could deliver on the day, which was a Sunday.
I asked for their advice on what to send, and then managed to give them an old credit card number which had expired. They rang back later and told me very apologetically that it hadn't got through, but luckily I realized what I'd done and gave them the new one.
They produced a really nice bunch of tulips for about £35 and delivered them at 11 o'clock on Sunday morning. I think they did the delivery themselves rather than using a parcel service. My parents live a fairly long way out of Sheffield itself so it must have been a long trip on what I assume is their busiest day of the year.
I was at a cricket net at the time, and when I got home I had 6 missed calls, all from Mum. (She doesn't normally do that.) Of course I rang back. A couple of days later she sent me a hand-made thank you card and the picture above.
So I'm pretty happy with Katie Peckett ( http://www.katiepeckett.com/ ), Sheffield Florist, 0114 2664985, and I thought I'd give them a plug on my blog.
I've no idea whether they're a big or a small operation, but although their website makes them look like a large and expanding firm, their service was as personal and obliging as if I'd rung a friendly corner shop.
I asked for their advice on what to send, and then managed to give them an old credit card number which had expired. They rang back later and told me very apologetically that it hadn't got through, but luckily I realized what I'd done and gave them the new one.
They produced a really nice bunch of tulips for about £35 and delivered them at 11 o'clock on Sunday morning. I think they did the delivery themselves rather than using a parcel service. My parents live a fairly long way out of Sheffield itself so it must have been a long trip on what I assume is their busiest day of the year.
I was at a cricket net at the time, and when I got home I had 6 missed calls, all from Mum. (She doesn't normally do that.) Of course I rang back. A couple of days later she sent me a hand-made thank you card and the picture above.
So I'm pretty happy with Katie Peckett ( http://www.katiepeckett.com/ ), Sheffield Florist, 0114 2664985, and I thought I'd give them a plug on my blog.
I've no idea whether they're a big or a small operation, but although their website makes them look like a large and expanding firm, their service was as personal and obliging as if I'd rung a friendly corner shop.
Tuesday, April 5, 2011
The Rise of the Machines
I made the last post from Chrome rather than Firefox. I haven't installed adblock in Chrome. My screen is now covered in adverts for pest control firms. I think the singularity may be some years off yet.
The Demon out of Nightmare
Imagine what it's like to be hunted by the basilisk.
Just a look will turn you to stone. You are naked, weaponless, powerless to hurt it unless you can get close enough to kill it with your bare hands.
The basilisk can disguise itself well. It can lie motionless for hours, and when it does you can't see it until you're very close, but if it sees you first, you are dead.
It can move faster than you. It can fly across the ground faster than any living creature can run. It can rise into the air and see a huge area at once, and it can kill over all that radius. It has sharper eyes than any natural being.
It has the power to control the world itself. Sometimes, even when it is nowhere near, plants or rocks will suddenly coil and deform, wrapping round you. Very often, when you are running from it, you will find that it has created a ravine or a gully or a trap in front of you, cornering you. Sometimes the ground itself will grow teeth, and bite into your legs, holding you in place. Once you are trapped, all you can do is wait in agony for the basilisk to come.
Often it will not even bother to come and kill you. It will just leave you to starve in agony, tearing at your wounds, trying to get free.
It knows your habits and your favourite places. Sometimes it can sense you over the horizon.
It can communicate silently with other basilisks, summon them to join its hunt, coordinate so that there is always one basilisk in your path, waiting for you.
Sometimes a basilisk will not kill you when it can.
In those cases your fate will be worse than death. You will be enslaved and worked to death, or you will be imprisoned forever in a tiny cage, to go mad with boredom and misery and lack of companionship. Either you will be castrated, or the children of your brief, brutal, forced matings will share your fate.
Often after years of imprisonment and every conceivable form of violation you will be eaten anyway.
Until recently, basilisks were only mythical demons. They were said to live in forests and savannahs where the wise did not go. They were few, and they did not stray from their territories.
But recently, they have discovered a new source of magical power. Always the deadliest things in the world, they have grown in deadliness to surreal levels. Nothing can stand against them. All creatures fear them, except a few inexplicable terrible allies, slaves swollen on the profits of their collaboration with evil.
And the basilisk breed. Where once they were few, a deadly but rare predator, now they grow without limit. Suddenly they are millions, tens of millions, hundreds, billions and they will not stop multiplying until there is nothing left of the world but basilisks. And there is nothing in the world that can stop them, or that will even dare to try.
Perhaps I am laying it on a bit thick here? My monsters are too terrifying to be believed, you say?
I killed an ant yesterday. It was walking across our stainless steel sink. I turned the tap on and it swirled down the plughole. I don't have a big problem with ants. In fact I like them. It's just that they're not the sort of thing one likes to have in the house. If I hadn't been feeling a bit ill, and only dressed in my dressing gown, I'd have caught it in a glass and put it outside, like I did with yesterday's enormous hornet.
I think most people would have used fly spray on the (huge and terrifying) hornet. I was feeling brave.
Oh yes. The basilisks can magic the very air so that it kills you when you breathe it by filling your lungs with knives. There's no way to tell the magic air. The first warning you get is when your lungs start to fill with knives.
Sometimes they use different magic that makes you go to sleep so they can put you in their cages. Sometimes they use a type which burns your eyes and skin away. Sometimes they just set the entire world around you on fire. There's no way to tell what sort of mood they'll be in.
Imagine what it's like to be a creature 'sharing' a world with a higher intelligence.
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!
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