The problem with one right answer
Jan. 27th, 2007 10:50 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
imomusOne of the things that frustrated and perplexed me most at school -- and I think this is the reason I quickly abandoned science and maths subjects -- was the way teachers would set us problems to which they assumed there was only one right answer. Students who solved these problems "correctly" were the ones who framed the question in the same way as the teacher, subscribed to the same assumptions, and shared the teacher's utter lack of imagination.
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The kind of absurd, scary arbitrariness of the supposed "solutions" to maths problems is mocked in satirical show Look Around You: Maths. Here's an example of their logic.
Problem 1: Jean is shorter than Brutus but taller than Imhotep. Imhotep is taller than Jean but shorter than Lord Scotland. Lord Scotland is twice the height of Jean and Brutus combined, but only a tenth of the height of Millsy. Millsy is at a constant height of x minus y. If Jean stands exactly one nautical mile away from Lord Scotland, how tall is Imhotep?
Solution: Imhotep doesn't exist.
The answer is a nice rebuke to our idiotically conformist attempts to solve the problem on the basis of maths, when in fact we should have been looking at the existential and religious realm (Imhotep is an idol from Easter Island).
I spent a while yesterday on the Analytical Problems and Puzzles site, utterly frustrated by its compilers' lack of imagination as, time after time, they proposed just one right answer to their conundrums. Even when some of their "lateral thinking" puzzle solutions showed a freshness of thought almost as daring as Look Around You's, they never acknowledged the plethora of other possibilities thrown up by each scenario. So today I thought I'd add a few solutions of my own. I've decided to act as if this website were one of those science teachers I used to sit in silence and fume at.
Teacher's Problem: There was once a recluse who never left his home. The only time anyone ever visited him was when his food and supplies were delivered, but they never came inside. Then, one stormy winter night when an icy gale was blowing, he had a nervous breakdown. He went upstairs, turned off all the lights and went to bed. Next morning, he had caused the deaths of several hundred people. How?
Teacher's Solution: He was a lighthouse keeper who turned off the light.
I Say: BAH! Come on, do lighthouses really save hundreds in one night, these days, when there's sonar and satellite tracking? Get real, teacher! Here are some more likely scenarios. After turning off all the lights, the recluse went out sleepwalking and killed several hundred people with an axe, chainsaw or machinegun. Alternatively, he just dreamed he killed the people. You just said "caused the deaths of people", you didn't say they had to be real people. Alternatively, the recluse's nervous breakdown reversed his attachment to his own home and attached him, instead, to other people's. He went upstairs in other people's houses (you didn't say the upstairs was his own, after all), having first locked the normal occupants out in the storm, where they perished. Isn't it obvious, with hindsight? Aren't you kicking yourself now I've told you?
Teacher's Problem: Not far from Madrid, there is a large wooden barn. The barn is completely empty except for a dead man hanging from the middle of the central rafter. The rope around his neck is ten feet long and his feet are three feet off the ground. The nearest wall is 20 feet away from the man. There is a puddle of water nearby. It is not possible to climb up the walls or along the rafters. The man hanged himself. How did he do it?
Teacher's Solution: He climbed on a block of ice which has since melted.
I Say: For fuck's sake, you smug git! Do you really think that block of ice is the only possible correct answer? If you can conjure that out of thin air, I can conjure a barn built entirely underground, with the rafters at ground level. The man simply slings his rope over the "floor" and drops down into the basement. The puddle is formed when he weeps a bit before doing this. Don't forget emotion! Alternatively, it's a normal barn but the man has a leaky jetpack.
Teacher's Problem: It was a dark stormy night and a couple were in a car racing madly through a foreign city. The car broke down and the husband had to go get help from someone who spoke his language. He was afraid to leave his wife alone in the car so he pulled up the windows and locked the car before leaving. When he came back, the car was in the same state as he had left it but his wife was dead, there was blood on the floor and there was a stranger in the car. What happened?
Teacher's Solution: The wife was about to have a baby. They were driving to the hospital. The baby was born, and the wife didn't survive the birth.
I Say: You unimaginative moron! Do you really think that's the only possible answer? The car broke down because the husband crashed it, killing his wife. The stranger was a policeman, investigating the crash. The man had been afraid to leave his dead wife alone because the area was a notorious necrophilia black spot. In fact, even the policeman wasn't above suspicion -- he was fumbling with the dead woman's clothes. Isn't it fucking obvious, you dunderhead?
Teacher's Problem: A man was driving alone in his car when he spun off the road at high speed. He crashed through a fence and bounced down a steep ravine before the car plunged into a fast-flowing river. As the car slowly settled in the river, the man realised that his arm was broken and that he could not release his seat belt and get out of the car. The car sank to the bottom of the river. He was trapped in the car. Rescuers arrived two hours later, yet they found him still in the river, but alive. How come?
Teacher's Solution: The water in the river only came up to the man's chest.
I Say: Oh for Christ's sake! Is that the best you can do? The car might have had a watertight passenger compartment, and had air trapped inside it, enough to breathe for two hours. Or -- much more likely -- there was a snorkel and oxygen tanks on the back seat, because the man was going diving. Or, who knows, there was a drought and the river chose that moment to dry up. It happens.
Teacher's Problem: A man lives in the penthouse of an apartment building. Every morning he takes the elevator down to the lobby and leaves the building. Upon his return, however, he can only travel halfway up in the lift and has to walk the rest of the way - unless it's raining. What is the explanation for this?
Teacher's Solution: The man is a dwarf. He can't reach the upper elevator buttons, but he can ask people to push them for him. He can also push them with his umbrella.
I Say: You apart-height-endorsing, politically-incorrect fascistic mongol! If the dwarf had this problem reaching the top buttons, why couldn't he carry a stick with him every day, or keep one stowed in the lift somewhere? Why didn't you say "unless there are other people in the lift", in which case the dwarf could just have said "Could you press penthouse please?" And anyway, how come this dwarf owns the best apartment in the building, you idiots? Dwarves are inadequate losers who never have any money and have to join the circus. Haven't you thought this scenario through at all?
But, given the parameters of the problem as you've set it, your pathetic answer is still not the only solution. He suffers from vertigo. The lift is glass walled. He doesn't mind approaching the earth, but does mind leaving it behind. When it's raining and misty, he can't see the earth, so he can go the whole hog. Or... or... it's an open-topped mechanical elevator which you operate under your own power, with a pulley system. Going down is easy, but a small man or a child can only pull the thing up half way. But a bucket of water hangs as a counterbalance. If it's raining the bucket fills up, making the dwarf's task easier.
Isn't it obvious, you cretin? I knew this stuff in first grade.
[Error: unknown template video]
The kind of absurd, scary arbitrariness of the supposed "solutions" to maths problems is mocked in satirical show Look Around You: Maths. Here's an example of their logic.
Problem 1: Jean is shorter than Brutus but taller than Imhotep. Imhotep is taller than Jean but shorter than Lord Scotland. Lord Scotland is twice the height of Jean and Brutus combined, but only a tenth of the height of Millsy. Millsy is at a constant height of x minus y. If Jean stands exactly one nautical mile away from Lord Scotland, how tall is Imhotep?
Solution: Imhotep doesn't exist.
The answer is a nice rebuke to our idiotically conformist attempts to solve the problem on the basis of maths, when in fact we should have been looking at the existential and religious realm (Imhotep is an idol from Easter Island).
I spent a while yesterday on the Analytical Problems and Puzzles site, utterly frustrated by its compilers' lack of imagination as, time after time, they proposed just one right answer to their conundrums. Even when some of their "lateral thinking" puzzle solutions showed a freshness of thought almost as daring as Look Around You's, they never acknowledged the plethora of other possibilities thrown up by each scenario. So today I thought I'd add a few solutions of my own. I've decided to act as if this website were one of those science teachers I used to sit in silence and fume at.
Teacher's Problem: There was once a recluse who never left his home. The only time anyone ever visited him was when his food and supplies were delivered, but they never came inside. Then, one stormy winter night when an icy gale was blowing, he had a nervous breakdown. He went upstairs, turned off all the lights and went to bed. Next morning, he had caused the deaths of several hundred people. How?
Teacher's Solution: He was a lighthouse keeper who turned off the light.
I Say: BAH! Come on, do lighthouses really save hundreds in one night, these days, when there's sonar and satellite tracking? Get real, teacher! Here are some more likely scenarios. After turning off all the lights, the recluse went out sleepwalking and killed several hundred people with an axe, chainsaw or machinegun. Alternatively, he just dreamed he killed the people. You just said "caused the deaths of people", you didn't say they had to be real people. Alternatively, the recluse's nervous breakdown reversed his attachment to his own home and attached him, instead, to other people's. He went upstairs in other people's houses (you didn't say the upstairs was his own, after all), having first locked the normal occupants out in the storm, where they perished. Isn't it obvious, with hindsight? Aren't you kicking yourself now I've told you?
Teacher's Problem: Not far from Madrid, there is a large wooden barn. The barn is completely empty except for a dead man hanging from the middle of the central rafter. The rope around his neck is ten feet long and his feet are three feet off the ground. The nearest wall is 20 feet away from the man. There is a puddle of water nearby. It is not possible to climb up the walls or along the rafters. The man hanged himself. How did he do it?
Teacher's Solution: He climbed on a block of ice which has since melted.
I Say: For fuck's sake, you smug git! Do you really think that block of ice is the only possible correct answer? If you can conjure that out of thin air, I can conjure a barn built entirely underground, with the rafters at ground level. The man simply slings his rope over the "floor" and drops down into the basement. The puddle is formed when he weeps a bit before doing this. Don't forget emotion! Alternatively, it's a normal barn but the man has a leaky jetpack.
Teacher's Problem: It was a dark stormy night and a couple were in a car racing madly through a foreign city. The car broke down and the husband had to go get help from someone who spoke his language. He was afraid to leave his wife alone in the car so he pulled up the windows and locked the car before leaving. When he came back, the car was in the same state as he had left it but his wife was dead, there was blood on the floor and there was a stranger in the car. What happened?
Teacher's Solution: The wife was about to have a baby. They were driving to the hospital. The baby was born, and the wife didn't survive the birth.
I Say: You unimaginative moron! Do you really think that's the only possible answer? The car broke down because the husband crashed it, killing his wife. The stranger was a policeman, investigating the crash. The man had been afraid to leave his dead wife alone because the area was a notorious necrophilia black spot. In fact, even the policeman wasn't above suspicion -- he was fumbling with the dead woman's clothes. Isn't it fucking obvious, you dunderhead?
Teacher's Problem: A man was driving alone in his car when he spun off the road at high speed. He crashed through a fence and bounced down a steep ravine before the car plunged into a fast-flowing river. As the car slowly settled in the river, the man realised that his arm was broken and that he could not release his seat belt and get out of the car. The car sank to the bottom of the river. He was trapped in the car. Rescuers arrived two hours later, yet they found him still in the river, but alive. How come?
Teacher's Solution: The water in the river only came up to the man's chest.
I Say: Oh for Christ's sake! Is that the best you can do? The car might have had a watertight passenger compartment, and had air trapped inside it, enough to breathe for two hours. Or -- much more likely -- there was a snorkel and oxygen tanks on the back seat, because the man was going diving. Or, who knows, there was a drought and the river chose that moment to dry up. It happens.
Teacher's Problem: A man lives in the penthouse of an apartment building. Every morning he takes the elevator down to the lobby and leaves the building. Upon his return, however, he can only travel halfway up in the lift and has to walk the rest of the way - unless it's raining. What is the explanation for this?
Teacher's Solution: The man is a dwarf. He can't reach the upper elevator buttons, but he can ask people to push them for him. He can also push them with his umbrella.
I Say: You apart-height-endorsing, politically-incorrect fascistic mongol! If the dwarf had this problem reaching the top buttons, why couldn't he carry a stick with him every day, or keep one stowed in the lift somewhere? Why didn't you say "unless there are other people in the lift", in which case the dwarf could just have said "Could you press penthouse please?" And anyway, how come this dwarf owns the best apartment in the building, you idiots? Dwarves are inadequate losers who never have any money and have to join the circus. Haven't you thought this scenario through at all?
But, given the parameters of the problem as you've set it, your pathetic answer is still not the only solution. He suffers from vertigo. The lift is glass walled. He doesn't mind approaching the earth, but does mind leaving it behind. When it's raining and misty, he can't see the earth, so he can go the whole hog. Or... or... it's an open-topped mechanical elevator which you operate under your own power, with a pulley system. Going down is easy, but a small man or a child can only pull the thing up half way. But a bucket of water hangs as a counterbalance. If it's raining the bucket fills up, making the dwarf's task easier.
Isn't it obvious, you cretin? I knew this stuff in first grade.
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(no subject)
Date: 2007-01-27 12:13 pm (UTC)(no subject)
Date: 2007-01-27 12:23 pm (UTC)To lay down the "wit" for a while, I do actually agree with you. The sorts of "analytical" problems you present up there are completely worthless and I pity any educational system that actually still uses them for anything except humour. To have any sort of meaning as educational tools they'd have to be closed systems - they'd have to contain the seeds of the "solution" in themselves in such a way that they could be reasonably inferred without adding anything to the equation. As it stands, they are largely exercises not for testing deduction ability but, as you rightly point out with excellent snark, for testing a person's ability to select from an infinity of options the one hackneyed explanation the promulgator happened to have in mind.
However, the idea that any of this has anything to do with how we employ the language of mathematics - quantum or otherwise - is simply too hilarious not to extract the mickey from.
(no subject)
Date: 2007-01-27 12:40 pm (UTC)(no subject)
Date: 2007-01-27 12:48 pm (UTC)(no subject)
Date: 2007-01-27 12:57 pm (UTC)(no subject)
Date: 2007-01-27 01:14 pm (UTC)Of course mathematics is fictional anyway. "It's only a model". So's everything else, I'm pretty sure so-and-so states that somewhere in the first paragraph of the Big Book of Modern Philosophy - "the truth is a mobile army of metaphors". I might not have a higher degree in reading theory but this shit is far from mindfuck level.
Now, Heidegger's beautiful idea that "working from first principles" is as falsely cold and flawed a way of forming a philosophy of nature as one can have (since the true first principle is not the most basic thing, but in fact the most complex thing - the self in the act of experience) - that's interesting.
(no subject)
Date: 2007-01-27 01:19 pm (UTC)I think we have to distinguish here between fictions that don't stand up to scrutiny because they rely on real world parallels (and can therefore fail the plausibility test) and fictions we have to take at face value because they are entirely self-referential, self-validating, air-tight systems. Like maths.
(no subject)
Date: 2007-01-27 02:05 pm (UTC)Likewise these "real world examples", filled as they are with flaws as far as their relationship to "reality" is concerned (as you so ably point out in the OP), evidently also have their own closed system of stupid-puzzle-logic - inadequately justified to pupil by teacher, inadequately communicated by teacher to pupil - in which the proper forms are followed and a single correct answer exists.
The mistake I guess is regarding mathematics as something small and single-faceted, when it's a large many-splendoured thing that's more than capable of containing the beautiful, the humdrum and the downright aesthetically offensive in its disparate parts. In fact mathematics is a field of thought that likes to swallow other fields of thought whole whenever it can. You think you've got your own patch of the epistemological paddock and then some mathematician comes in and models your ass.
(no subject)
Date: 2007-01-27 12:38 pm (UTC)It's true there's a nice correspondence between the breakdown of logical positivism / Newtonian or Marxist scientific determinism in the first half of the 20th C. and the reassessment of absolute truth in relation to literary meaning that occurred concurrently with Empson and so on. But the correspondence only goes so far.
I still claim this whole argument's as it applies to the actual principles of maths and science is rather misconceived - if a question has more than one "correct answer", then from a mathematician's perspective the real correct answer is actually the set of all correct answers.
Mathematicians are pluralists too - heck, mathematicians conceived the imaginary numbers so that quadratic equations "without an answer" wouldn't feel lonely.
shor algorithm
Date: 2007-01-27 03:17 pm (UTC)I certainly wouldn't mention the parallel worlds theory because that isn't really a mainstream concept. Most quantum physicists opt for the Copenhagen interpretation in order to make the fewest assumptions about unmeasurable behaviours. Essentially, the idea is that we can say nothing about a system except perhaps the statistics of clicks on a photo-detector.
Anyway, it doesn't sound quite so literary now. I thought Alan Sokal dealt with this kind of stuff in his excellent book a few years back.
(no subject)
Date: 2007-01-28 02:26 am (UTC)