I have just read Benjamin Wardhaugh's fascinating book Poor Robin's Prophecies: A curious Almanac, and the everyday mathematics of Georgian Britain (Oxford). It is a fascinating account of popular mathematics during, essentially, the long eighteenth century, based on a comic almanac "written by POOR ROBIN Knight of the Burnt-Island, a well-willer to the Mathematicks", first published in 1663 and which lasted - latterly as Old Poor Robin - till 1828. Poor Robin's almanac contained parodies, as well as standard almanac information, and Wardhaugh uses the almanac and other popular publications to explore the uses of mathematics over a period when it was, for some, a means to social advancement or a route into a career, and when sophisticated mathematical problems appeared in publications like The Ladies' Diary.
Does spirit of Poor Robin of Saffron Walden (for the meridian of which the original almanac was calculated, according to its title-page) survive today? I think Poor Robin's attitude to mathematics can be found in TV shows like Dara O Briain's School of Hard Sums and in the mathematical jokes and puzzles which seem to achieve a wide circulation on Facebook (and, if my Facebook stream is typical, are appreciated and reposted by many who have no particular mathematical background). Wardhaugh's book reminds us that the value of mathematics has always been questioned, but that even so there has always been popular interest in all kinds of mathematics: both traits are evident today!
A very occasional blog about maths things which interest me.
Wednesday, 16 January 2013
Sunday, 6 January 2013
Computer chess and human error
When I was a teenager I was a keen chess player, and when I discovered the joys of programming a computer I was naturally interested in how computers could be programmed to play chess. This was in the days before PCs, the mid 1970s, when computer time was a rare resource, and when the possibility that a computer could beat a good human chess player seemed at best a long way away.
It was also a time when the less perspicacious of us, like me, believed that playing chess was a serious test of computer power; a demanding human activity which was a pinnacle of intelligence. Now, of course, we realise how much more difficult it is to programme a machine to recognise faces or interpret speech or read handwriting, and other activities which are very difficult for a computer but which human beings do automatically. (Well, I am embarrassingly bad at recognising faces, as it happens!) Of course, in his seminal paper 'Computing Machinery and Intelligence' Alan Turing had actually included solving a chess problem as a suitable task for a potentially intelligent computer, but he had, with is usual perceptiveness, also included writing poetry.
Anyway, in the 1970s we seemed a long way away from computers being able to play chess at the top human level. Indeed, the Scottish chess player David Levy made, and won, a series of bets against proponents of computer chess at the time that no computer could beat him.
I was delighted when I found in the university library Alex Bell's book The machine plays chess? (1978), which was a very entertaining account of our attempts to get computers to play chess.
I remember his hilarious account of an early programme, which (playing against a weak human opponent) got into a wining position. The opponent wanted to resign when the computer got to an ending with king and two queens against king but the programmers insisted the game be played out to its conclusion. Unfortunately, the simple way to force mate was slightly too long for the computer to calculate, so, knowing that pieces are most powerful in the centre, it moved its queens to the centre of the board. Its strategy was essentially to keep its queens in the centre, but it knew about draws by repetition of position, so as the game continued the computer's queens gradually spiralled away from the centre, but failed to progress towards a mate.
Bell's book was very funny, and full of lessons and entertainment for any aspiring programmer, but growing computer power meant that humans were no longer unbeatable. First, a computer beat the backgammon world champion, albeit with a huge amount of luck, and eventually in 1997 the computer Deep Blue beat the great Garry Kasparov in a best-of-six-games chess match.
Having enjoyed Bell's book so much, I was fascinated to read in Nate Silver's recent book about mathematical predictions, The signal and the noise, an account of Kasparov's defeat which presents Deep Blue's triumph as the result of a programming error! Kasparov won the first game, but was puzzled when, in a losing position, Deep Blue chose what appeared to be an inferior move, rather than one which would seem to have held off defeat for longer. According to Silver, Kasparov tried to work out Deep Blue's logic, and concluded that the computer was looking so far ahead that it could see that the "better" move lost just as badly as the "inferior" one.
In the second game, Deep Blue had a slight advantage. At a key point, it could either play a move which would lead to a complex tactical situation - which one would expect to favour the computer - or one which led to a simpler game in which Deep Blue had an edge which might not be sufficient to force a win. To everyone's surprise, it chose the latter. In fact Kasparov had the opportunity to force a draw, but missed it. It seems that, knowing from the first game how far ahead Deep Blue was calculating, Kasparov assumed that there could not possibly be a way for him to draw, or Deep Blue would have played the other move. Since he assumed the game was lost, inevitably he missed the draw.
Demoralised by this defeat in the second game, Kasparov then blundered in the final game and lost the match by 3.5 to 2.5.
But according to Silver, Deep Blue's choice of losing move in the first game wasn't due to its seeing a long way ahead. It was due to a programming error! So Kasparov's attributed the move to deep analysis when in fact it was a simple bug in Deep Blue's programme, and this mistaken analysis of Deep Blue's abilities led Kasparov to defeat in the match. Rather than being the triumph of the infallible calculating machine, Deep Blue's victory was due to Kasparov's very human response to a computer error!
As always, there is more to both human and artificial intelligence than a strict logical analysis appears to suggest!
It was also a time when the less perspicacious of us, like me, believed that playing chess was a serious test of computer power; a demanding human activity which was a pinnacle of intelligence. Now, of course, we realise how much more difficult it is to programme a machine to recognise faces or interpret speech or read handwriting, and other activities which are very difficult for a computer but which human beings do automatically. (Well, I am embarrassingly bad at recognising faces, as it happens!) Of course, in his seminal paper 'Computing Machinery and Intelligence' Alan Turing had actually included solving a chess problem as a suitable task for a potentially intelligent computer, but he had, with is usual perceptiveness, also included writing poetry.
Anyway, in the 1970s we seemed a long way away from computers being able to play chess at the top human level. Indeed, the Scottish chess player David Levy made, and won, a series of bets against proponents of computer chess at the time that no computer could beat him.
I was delighted when I found in the university library Alex Bell's book The machine plays chess? (1978), which was a very entertaining account of our attempts to get computers to play chess.
I remember his hilarious account of an early programme, which (playing against a weak human opponent) got into a wining position. The opponent wanted to resign when the computer got to an ending with king and two queens against king but the programmers insisted the game be played out to its conclusion. Unfortunately, the simple way to force mate was slightly too long for the computer to calculate, so, knowing that pieces are most powerful in the centre, it moved its queens to the centre of the board. Its strategy was essentially to keep its queens in the centre, but it knew about draws by repetition of position, so as the game continued the computer's queens gradually spiralled away from the centre, but failed to progress towards a mate.
Bell's book was very funny, and full of lessons and entertainment for any aspiring programmer, but growing computer power meant that humans were no longer unbeatable. First, a computer beat the backgammon world champion, albeit with a huge amount of luck, and eventually in 1997 the computer Deep Blue beat the great Garry Kasparov in a best-of-six-games chess match.
Having enjoyed Bell's book so much, I was fascinated to read in Nate Silver's recent book about mathematical predictions, The signal and the noise, an account of Kasparov's defeat which presents Deep Blue's triumph as the result of a programming error! Kasparov won the first game, but was puzzled when, in a losing position, Deep Blue chose what appeared to be an inferior move, rather than one which would seem to have held off defeat for longer. According to Silver, Kasparov tried to work out Deep Blue's logic, and concluded that the computer was looking so far ahead that it could see that the "better" move lost just as badly as the "inferior" one.
In the second game, Deep Blue had a slight advantage. At a key point, it could either play a move which would lead to a complex tactical situation - which one would expect to favour the computer - or one which led to a simpler game in which Deep Blue had an edge which might not be sufficient to force a win. To everyone's surprise, it chose the latter. In fact Kasparov had the opportunity to force a draw, but missed it. It seems that, knowing from the first game how far ahead Deep Blue was calculating, Kasparov assumed that there could not possibly be a way for him to draw, or Deep Blue would have played the other move. Since he assumed the game was lost, inevitably he missed the draw.
Demoralised by this defeat in the second game, Kasparov then blundered in the final game and lost the match by 3.5 to 2.5.
But according to Silver, Deep Blue's choice of losing move in the first game wasn't due to its seeing a long way ahead. It was due to a programming error! So Kasparov's attributed the move to deep analysis when in fact it was a simple bug in Deep Blue's programme, and this mistaken analysis of Deep Blue's abilities led Kasparov to defeat in the match. Rather than being the triumph of the infallible calculating machine, Deep Blue's victory was due to Kasparov's very human response to a computer error!
As always, there is more to both human and artificial intelligence than a strict logical analysis appears to suggest!
Wednesday, 19 December 2012
Jaguar Paw's powerful use of number
A reassuring article by Tom Holland in Saturday's Guardian discusses the supposed imminent end of the world predicted by Mayan calendar. Holland refers to a recently-discovered Mayan reference to the date (21 December 2012 in our calendar). The king Jaguar Paw was defeated in battle in 695 CE. To restore the confidence of his allies, he associated his time with the distant future by talking about 2012. Holland says this was "designed to place his defeat in a reassuringly cosmological context. Bad news was being veiled behind a recitation of numbers. George Osborne would surely have approved."
So we have an interesting example of the use of number as propaganda to assert one's place in the cosmos.
Incidentally, if you are worried about the end of the world this week (and Holland says that a Mayan inscription refers to the world's still existing in 4772, so there wasn't unanimity in the alleged prediction), the article suggests actions that the Mayans might have taken to reduce the risk. These include "piercing their tongues with thorns, and stabbing their penises with stingray spikes". So those who take this seriously know what they should do.
So we have an interesting example of the use of number as propaganda to assert one's place in the cosmos.
Incidentally, if you are worried about the end of the world this week (and Holland says that a Mayan inscription refers to the world's still existing in 4772, so there wasn't unanimity in the alleged prediction), the article suggests actions that the Mayans might have taken to reduce the risk. These include "piercing their tongues with thorns, and stabbing their penises with stingray spikes". So those who take this seriously know what they should do.
Friday, 7 December 2012
My (current) favourite infinity paradox
I remember one of my undergraduate tutors telling me about this paradox, of which I have just been reminded by Littlewood's Miscellany.
At one minute to noon the numbers 1 to 10 are put into a box, and the number 1 is removed.
At 1/2 minute to noon the numbers 11 to 20 are added to the box, and 2 is removed.
At 1/3 minute to noon the numbers 21 to 30 are added to the box, and 3 is removed.
And so on.
How many numbers are in the box at noon? The answer is obviously 9+9+9+9+... which looks as if it should be infinite. But in fact there are no numbers in the box, because if you suggest that number n might be there, I point out that it was taken out at 1/n of a minute before noon. The box is empty: nine times infinity is zero.
One has to be careful in dealing with infinity!
At one minute to noon the numbers 1 to 10 are put into a box, and the number 1 is removed.
At 1/2 minute to noon the numbers 11 to 20 are added to the box, and 2 is removed.
At 1/3 minute to noon the numbers 21 to 30 are added to the box, and 3 is removed.
And so on.
How many numbers are in the box at noon? The answer is obviously 9+9+9+9+... which looks as if it should be infinite. But in fact there are no numbers in the box, because if you suggest that number n might be there, I point out that it was taken out at 1/n of a minute before noon. The box is empty: nine times infinity is zero.
One has to be careful in dealing with infinity!
Sunday, 2 December 2012
Box paradoxes
It's longer than I would have liked since I last posted - which is because I've spent the last two weekend at maths conferences. First there was MathsJam - which was every bit as good as I expected when I wrote about it recently. Then there was the IMA's Early Careers Mathematicians Conference at Greenwich - where we had the chance to play with lost of unusual puzzles and games from the collections of David Singmaster and Laurie Brokenshire CBE and to try out wonderful linkages with Danny Brown.
There is a lot from MathsJam I could write about - not least a wonderful piece of graph theory from Colin Wright, who showed that factorising large numbers can be reduced to a graph colouring problem! My own short talk was about the two-box paradox(attributed to Schrodinger, in a slightly different form) that I wrote a blog post about in October.
This set me thinking about paradoxes involving boxes. There are several: the Monty Hall problem, Newcomb's Paradox, and Schrodinger's Cat come immediately to mind. What is nice is that they all tell us different things.
Schrodinger's Cat expresses concisely the quantum concept of the superposition of states.
The two-box problem I previously discussed tells us about the difficulty of selecting randomly from an infinite set wihtout being very specific about how you do it.
Newcomb's paradox gives the player the contents of two boxes. They can choose to take box B only, or both boxes A and B. Box A definitely contains one thousand pounds. An infallible predictor has chosen the contents of Box B in advance. If the predictor predicted that the player would take both boxes, they put nothing in Box B. If they predicted that the player would take Box B only, they placed one million pounds in Box B. What should the player do? At the point at which they make their choice, the contents of both boxes are fixed, so logically they get more if they take both boxes. But should they ignore the predictor's infallibility. This paradox, I think, shows us that no such predictor can exist.
The Monty Hall paradox can be viewed as a disguised version of Martin Gardner's prisoners paradox. Gardner wrote of three prisoners. held in solitary confinement, who know that two are to be executed the next day. Each has a 2/3 chance of dying. A thinks he can improve his odds - he points out to the guard that at least one of the other two must die, so that being told which of B or C will die isn't going to give him new information. But when the guard says C will die, A now reckons that his chance of survival is now 1/2 since it is either him or B. Of course, A is wrong (B's chance has improved) and I have a theory that it is memories of Gardner's article, where A's odds don't change, that led many mathematicians (myself included) to quickly jump to the wrong answer.
There is an excellent novel about mathematicians by Sue Woolfe, Leaning towards infinity, which contains a devastating account of the appalling treatment of women by male mathematicians at a fictional conference. I have never seen such behaviour at a maths conference and I found the account implausible until I read in Jason Rosenhouse's account in The Monty Hall Problem of the treatment the journalist Marilyn Vos Savant received from (some) mathematicians when she wrote about this problem. The responses would have been appalling even if Vos Savant had been wrong, but in fact she got it right and her critics didn't. So one useful lesson from the Monty Hall paradox is that on occasion even mathematicians can be jerks.
There is a lot from MathsJam I could write about - not least a wonderful piece of graph theory from Colin Wright, who showed that factorising large numbers can be reduced to a graph colouring problem! My own short talk was about the two-box paradox(attributed to Schrodinger, in a slightly different form) that I wrote a blog post about in October.
This set me thinking about paradoxes involving boxes. There are several: the Monty Hall problem, Newcomb's Paradox, and Schrodinger's Cat come immediately to mind. What is nice is that they all tell us different things.
Schrodinger's Cat expresses concisely the quantum concept of the superposition of states.
The two-box problem I previously discussed tells us about the difficulty of selecting randomly from an infinite set wihtout being very specific about how you do it.
Newcomb's paradox gives the player the contents of two boxes. They can choose to take box B only, or both boxes A and B. Box A definitely contains one thousand pounds. An infallible predictor has chosen the contents of Box B in advance. If the predictor predicted that the player would take both boxes, they put nothing in Box B. If they predicted that the player would take Box B only, they placed one million pounds in Box B. What should the player do? At the point at which they make their choice, the contents of both boxes are fixed, so logically they get more if they take both boxes. But should they ignore the predictor's infallibility. This paradox, I think, shows us that no such predictor can exist.
The Monty Hall paradox can be viewed as a disguised version of Martin Gardner's prisoners paradox. Gardner wrote of three prisoners. held in solitary confinement, who know that two are to be executed the next day. Each has a 2/3 chance of dying. A thinks he can improve his odds - he points out to the guard that at least one of the other two must die, so that being told which of B or C will die isn't going to give him new information. But when the guard says C will die, A now reckons that his chance of survival is now 1/2 since it is either him or B. Of course, A is wrong (B's chance has improved) and I have a theory that it is memories of Gardner's article, where A's odds don't change, that led many mathematicians (myself included) to quickly jump to the wrong answer.
There is an excellent novel about mathematicians by Sue Woolfe, Leaning towards infinity, which contains a devastating account of the appalling treatment of women by male mathematicians at a fictional conference. I have never seen such behaviour at a maths conference and I found the account implausible until I read in Jason Rosenhouse's account in The Monty Hall Problem of the treatment the journalist Marilyn Vos Savant received from (some) mathematicians when she wrote about this problem. The responses would have been appalling even if Vos Savant had been wrong, but in fact she got it right and her critics didn't. So one useful lesson from the Monty Hall paradox is that on occasion even mathematicians can be jerks.
Sunday, 11 November 2012
Why a flawed proof is worrying
In preparation for a class I have been looking at Alfred Bray Kempe's 1879 "proof" of the Four Colour Theorem. Until Percy Heawood pointed out the flaw in Kempe's proof eleven years later it was accepted. We had a clear, relatively straightforward proof of the result: anyone could check it for themselves (very different from the position now, when we all accept Appel and Haken's computer proof!) The error in Kempe's proof is subtle: in preparing the false proof to present to my class, I found it very persuasive! (The story is told in Robin Wilson's excellent book Four Colours Suffice, published as Four Colors Suffice in parts of the world where modern spelling hasn't yet arrived.)
False "proofs" are worrying. As a teenager I was turned off geometry by a well-known fake proof that all triangles are isosceles. The proof relies on an incorrect diagram. After seeing that I found it hard to accept any geometric proof: I became suspicious of geometric arguments (which I think was good) and turned my back on geometry (which was not so good for an aspiring mathematician!) I wasn't worried so much by trick algebraic proofs that 1=2, which relied on division by 0 or on an ill-based induction argument: I appreciated these jokes, but geometric "proofs" like E.A. Maxwell's "Fallacy of the empty circle" destroyed my enjoyment of geometry.
So Kempe's proof worries me because it is so plausible. If a simple argument can be accepted by the leading mathematicians of the day, where do we stand with today's proofs which are accessible only to specialists and which the rest of us cannot take on trust?
Many proofs contain errors but these errors are usually fixable. Kempe's wasn't (although a proof of the weaker Five Colour Theorem was salvaged) and that seems a warning against complacency.
Wilson's book records that Haken's son presented a seminar on the Appel-Haken proof at Berkeley which led to fierce discussion. Those under 40 were reluctant to accept the computer's role in the proof: those under 40 were more suspicious of a proof relying on 700 pages of human calculation. As I have grown older I have gone in the reverse direction. Whereas I once had concerns about computer proofs, I am much less sure now than I was thirty years ago that there is a gold standard for proofs. If we cannot be sure of a proof we have checked for ourselves (and that is what the story of Kempe's proof suggests) then where is mathematical certainty?
False "proofs" are worrying. As a teenager I was turned off geometry by a well-known fake proof that all triangles are isosceles. The proof relies on an incorrect diagram. After seeing that I found it hard to accept any geometric proof: I became suspicious of geometric arguments (which I think was good) and turned my back on geometry (which was not so good for an aspiring mathematician!) I wasn't worried so much by trick algebraic proofs that 1=2, which relied on division by 0 or on an ill-based induction argument: I appreciated these jokes, but geometric "proofs" like E.A. Maxwell's "Fallacy of the empty circle" destroyed my enjoyment of geometry.
So Kempe's proof worries me because it is so plausible. If a simple argument can be accepted by the leading mathematicians of the day, where do we stand with today's proofs which are accessible only to specialists and which the rest of us cannot take on trust?
Many proofs contain errors but these errors are usually fixable. Kempe's wasn't (although a proof of the weaker Five Colour Theorem was salvaged) and that seems a warning against complacency.
Wilson's book records that Haken's son presented a seminar on the Appel-Haken proof at Berkeley which led to fierce discussion. Those under 40 were reluctant to accept the computer's role in the proof: those under 40 were more suspicious of a proof relying on 700 pages of human calculation. As I have grown older I have gone in the reverse direction. Whereas I once had concerns about computer proofs, I am much less sure now than I was thirty years ago that there is a gold standard for proofs. If we cannot be sure of a proof we have checked for ourselves (and that is what the story of Kempe's proof suggests) then where is mathematical certainty?
Sunday, 4 November 2012
The MathsJam conference is coming - what can I talk about?
The MathsJam weekend conference is only two weeks away! It's only been going for two years but it's established as a highlight of the UK mathematical year. Over 100 of us get together to spend a weekend talking mathematics, showing off tricks and toys, and generally sharing our delight in the subject. The diverse attendance - all ages, amateurs and professionals, pure mathematicians and applied, ... - contributes to the joy of the MathsJam weekend.
The maths community is greatly indebted to those who set up and run this wonderful event - the founder Colin Wright, whose efforts ensure the smooth organisation of the event, and his supporting team of David Bedford, James Grime, Matt Parker and Rob Eastaway. Colin tirelessly puts a huge amount of work into the event: mathematicians are wonderful people, but not necessarily easy to organise an event for! The proceedings owe much in spirit to Martin Gardner: the whole mathematical world would have been very different without him.
At the MathsJam weekend all the talks are five minutes - time to present an inspiring idea but not time to teach, as Colin puts it. And I can't think what to talk about! Partly because I used up my best ideas at previous MathsJams. Partly because I am so heavily engaged in preparing new teaching material for a new final year course that my mathematical creativity is all going into teaching just now. I have a few ideas but am hoping for major inspiration in the next few days.
Hope to see you near Crewe later this month!
The maths community is greatly indebted to those who set up and run this wonderful event - the founder Colin Wright, whose efforts ensure the smooth organisation of the event, and his supporting team of David Bedford, James Grime, Matt Parker and Rob Eastaway. Colin tirelessly puts a huge amount of work into the event: mathematicians are wonderful people, but not necessarily easy to organise an event for! The proceedings owe much in spirit to Martin Gardner: the whole mathematical world would have been very different without him.
At the MathsJam weekend all the talks are five minutes - time to present an inspiring idea but not time to teach, as Colin puts it. And I can't think what to talk about! Partly because I used up my best ideas at previous MathsJams. Partly because I am so heavily engaged in preparing new teaching material for a new final year course that my mathematical creativity is all going into teaching just now. I have a few ideas but am hoping for major inspiration in the next few days.
Hope to see you near Crewe later this month!
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