Relatively Prime is a new maths podcast from Samuel Hansen. I'm delighted that an early episode, The Toolbox, begins by discussing my favourite area of mathematics, game theory (the subject of my very first post in this blog). Samuel interviews the eminent writer on game theory, Steven J. Brams, whose book The Presidential Election Game I devoured when it first came out many years ago. Brams argues that Shakespeare had an intuitive understanding of game theory and discusses applications of the mathematics in Hamlet and Macbeth.
It is particularly fitting that game theory features in Relatively Prime because game theory contributed to the podcast's very existence. Samuel raised the funding for Relatively Prime through Kickstarter, a crowd-sourcing scheme. Various individuals pledged funds to make the project happen. If Samuel reached his target funding before the deadline, these pledges would be called in; otherwise they would lapse.
I was keen that Relatively Prime should be funded. I admired Samuel's previous work. With Peter Rowlett, he is responsible for the excellent weekly Math/Maths Podcast which has, amazingly, now been going for over two years. (In the interests of transparency I should note that I have contributed to this podcast on a couple of occasions.) Even though Samuel is responsible for the incorrect abbreviation for "Mathematics" in the title of that podcast, I was prepared to consider making a small financial contribution to make Relatively Prime happen.
So consider my situation last summer. I was willing to contribute (a little) to the project, but wasn't sure my contribution would be necessary. From my perspective, the ideal outcome would be that Samuel raised enough from other well-wishers, and I could enjoy the result without paying a penny. But I would rather make a contribution than see the project fail.
No doubt many others are in the same position. We can see how much has been pledged, and if five minutes before the deadline the project had been 5 dollars short, I would certainly have jumped in. But if the project (which sought $8000) were $5000 short just before the deadline, then it's not worth my while wasting time pledging to a lost cause. How do potential donors play it?
Well, I did what no doubt many others did - wait for some time to see what was happening. With a few days to go , there were enough backers to show that there was some interest, but as the deadline day dawned the project was a long way short, but word of mouth was building up. The last few hours were remarkably tense as the pledges started coming in, and suddenly the momentum built up and Samuel reached his target. I was one of the many who contributed on the last day. About 140 people, I think, contributed to Relatively Prime.
As a potential backer one uses game theory here to help achieve the desired outcome, which is that the project gets funded but one doesn't pay more than one's fair share when others were also willing to contribute. How does it work? Nothing significant is going to happen until fairly close to the deadline. I think the intuitive game theorist (like Shakespeare, apparently) would fix a maximum contribution they were prepared to make. They would make a very small contribution at a relatively early stage, to encourage other potential backers by showing that there is some interest, and would then wait until a few hours before the deadline, perhaps committing a little more early on the final day but then hoping that others' contributions will come in. At the last minute, if necessary, they would be prepared to go up to their limit if the target hasn't been reached, but they wouldn't throw all their money in early because that way other intending funders might be let off the hook.
Of course this game theory meant that if Samuel were to reach his target, it would happen at the very last minute (as indeed happened on 3 August last year). The cost of all this game theory was a very nervous few days for Samuel when he was convinced his project would fail. So it is entirely appropriate that Relatively Prime has opened with an illuminating discussion of this fascinating branch of mathematics.
A very occasional blog about maths things which interest me.
Sunday, 30 September 2012
Thursday, 20 September 2012
A mathematical device
Thanks to an ingenious mathematical invention we can now tell the time in the garden.
(Added 22/9/12) It seems fairly accurate though I haven't found the switch for the British Summer Time setting yet. Must be on the back where it's inaccessible!
Sunday, 16 September 2012
Amazing mathematics - the abc conjecture
I heard earlier this week that the solution to an outstanding mathematical problem, the proof of the abc conjecture, has been announced. (The proof is claimed by Shinichi Mochizuki of Kyoto University, and the ideas are so new and deep that it will take a long time for mathematicians to digest the work and be convinced of the validity of the proof, but the claim is being taken seriously.) Why did this news on Twitter make me happy?
You might think this would have been because it was a problem I have been thinking about for years, that I have attempted to solve myself, that I have lost sleep on. Or perhaps because I am aware of the ramifications for mathematics, the new vistas that will be opened up, the new opportunities that will arise.
Well, I had heard of the abc conjecture. Perhaps I once even knew what it was, but I had forgotten. I have no knowledge of the subject area, the applications (if any) or the methods used.
Yet I am excited by the news of the possible proof! How is it that a development about a problem which I know nothing about, which I couldn't even describe in the most general terms, can matter to me?
Well, here (from the excellent Wikipedia article on the conjecture) is a statement of the conjecture: it asserts that the answer to the following question is "yes": "For every ε > 0, are there only finitely many triples of coprime positive integers a + b = c such that c > d (1+ε), where d denotes the product of the distinct prime factors of abc?" What does this mean? Basically if the conjecture is true, then (as I understand it) it means that only exceptionally is the product of the prime factors of abc significantly less than c.
Now this is, frankly, quite an obscure statement about numbers, and I cannot envisage any life-changing applications. What I find wonderful is that human minds like mine can prove this statement. I cannot even begin to imagine how one would set about proving such a conjecture. I still have an instinctive belief (which, rationally, I know is rather naive) that mathematical facts are true regardless of the nature of the human brain, the laws of nature, and so on: thirteen would still be a prime if the human race had never existed, if the laws of physics were totally different, if no sentient creature had ever come into being. That a mind like mine can establish such necessary, deep facts is amazing, a glimpse of something much more true than anything else in our existence.
You might think this would have been because it was a problem I have been thinking about for years, that I have attempted to solve myself, that I have lost sleep on. Or perhaps because I am aware of the ramifications for mathematics, the new vistas that will be opened up, the new opportunities that will arise.
Well, I had heard of the abc conjecture. Perhaps I once even knew what it was, but I had forgotten. I have no knowledge of the subject area, the applications (if any) or the methods used.
Yet I am excited by the news of the possible proof! How is it that a development about a problem which I know nothing about, which I couldn't even describe in the most general terms, can matter to me?
Well, here (from the excellent Wikipedia article on the conjecture) is a statement of the conjecture: it asserts that the answer to the following question is "yes": "For every ε > 0, are there only finitely many triples of coprime positive integers a + b = c such that c > d (1+ε), where d denotes the product of the distinct prime factors of abc?" What does this mean? Basically if the conjecture is true, then (as I understand it) it means that only exceptionally is the product of the prime factors of abc significantly less than c.
Now this is, frankly, quite an obscure statement about numbers, and I cannot envisage any life-changing applications. What I find wonderful is that human minds like mine can prove this statement. I cannot even begin to imagine how one would set about proving such a conjecture. I still have an instinctive belief (which, rationally, I know is rather naive) that mathematical facts are true regardless of the nature of the human brain, the laws of nature, and so on: thirteen would still be a prime if the human race had never existed, if the laws of physics were totally different, if no sentient creature had ever come into being. That a mind like mine can establish such necessary, deep facts is amazing, a glimpse of something much more true than anything else in our existence.
Sunday, 2 September 2012
A lottery oddity
There is a minor news item today about a curious result in the UK National Lottery. Five people who each chose the six correct numbers shared £4.8 million, getting £968,000 each, but the single "runner-up", who matched five of the six main balls plus the bonus ball, won nearly £1.5 million - half as much again as the winners!
This is very unusual: it required several people to pick the same six winning numbers while only a single person chose any of the six possible "runner-up" combinations. Apparently the winning numbers were 15, 30, 36, 39, 41 and 49 and the bonus number was 2.
I am curious about the distribution of the prizes. There are six "runner-up" combinations - any five of the six main balls plus the bonus ball - while only one winning selection, so I would have expected that the pot for the winner would be six times that for the runner-up. Instead it's just over three times. That makes this unexpected result rather more likely than if the relative size of the pots had been proportional to the odds.
Of course, this is good publicity for the lottery (and, as the spokesman says, good news for six people who've won sizeable sums of money).
This is very unusual: it required several people to pick the same six winning numbers while only a single person chose any of the six possible "runner-up" combinations. Apparently the winning numbers were 15, 30, 36, 39, 41 and 49 and the bonus number was 2.
I am curious about the distribution of the prizes. There are six "runner-up" combinations - any five of the six main balls plus the bonus ball - while only one winning selection, so I would have expected that the pot for the winner would be six times that for the runner-up. Instead it's just over three times. That makes this unexpected result rather more likely than if the relative size of the pots had been proportional to the odds.
Of course, this is good publicity for the lottery (and, as the spokesman says, good news for six people who've won sizeable sums of money).
Monday, 27 August 2012
Idiosyncracies?
Saturday's Guardian newspaper feature about Julian Assange contained an amusing quote from a friend from his university days (Assange studied mathematics amongst other subjects): "I've often heard it remarked in the press that Julian has some idiosyncrasies. The people who make such remarks tend not to have hung around mathematics departments very much."
It's perhaps not surprising that mathematicians are associated with eccentricity. We love to tell stories about the idiosyncrasies of Erdos and Godel. Alexander Masters' recent book about the mathematician Simon Norton, The Genius in my Basement is an outstanding reflection on the biographer's art which doesn't do much for the public image of mathematicians. On the other hand, fictional mathematicians who are leading characters in novels such as Iain Banks's The Steep Approach to Garbadale and Ann Lingard's The Embalmer's Book of Recipes are reasonably normal people.
Are mathematicians more eccentric than other creative people? I suspect not: I am sure one can find just as many writers, painters, composers, actors, ... Is it perhaps just the abstract nature of our subject, and the difficulty of talking about the technicalities to non-mathematicians (or even mathematicians who specialise in different areas) which results in a focus on eccentricity? We can't tell our non-mathematical friends about a mathematician's brilliant ideas so we end up talking about their amusing eccentricities.
I'm not sure how far I have convinced myself!
It's perhaps not surprising that mathematicians are associated with eccentricity. We love to tell stories about the idiosyncrasies of Erdos and Godel. Alexander Masters' recent book about the mathematician Simon Norton, The Genius in my Basement is an outstanding reflection on the biographer's art which doesn't do much for the public image of mathematicians. On the other hand, fictional mathematicians who are leading characters in novels such as Iain Banks's The Steep Approach to Garbadale and Ann Lingard's The Embalmer's Book of Recipes are reasonably normal people.
Are mathematicians more eccentric than other creative people? I suspect not: I am sure one can find just as many writers, painters, composers, actors, ... Is it perhaps just the abstract nature of our subject, and the difficulty of talking about the technicalities to non-mathematicians (or even mathematicians who specialise in different areas) which results in a focus on eccentricity? We can't tell our non-mathematical friends about a mathematician's brilliant ideas so we end up talking about their amusing eccentricities.
I'm not sure how far I have convinced myself!
Wednesday, 8 August 2012
Maths history and anecdotes
I have been meaning to write something in response to two recent blog posts on history of mathematics and anecdote. Dennis Des Chene (aka "Scaliger") wrote "On bad anecdotes and good fun" and Peter Rowlett responded with "Mathematics: a culture of historical inaccuracy". I'm also grateful to Thony Christie of the always excellent The Renaissance Mathematicus blog, whose Twitter comment brought Scaliger's blog to my attention.
Scaliger writes about the anecdote that Euler spouted a piece of mathematical nonsense claiming to prove the existence of God, embarrassing Diderot in front of the Empress. The anecdote, as commonly told, is far from true, so why do mathematicians still tell without qualification? Rowlett wonders about the role of such anecdotes, which are arguably part of the culture of mathematics, in training mathematicians.
These are deep issues. I love anecdotes. I like serious history, too, but I am useless as a historian because I don't know enough. Whenever I have worked on historical topics, I have found that, the more I research, the less I feel able to say anything because I am increasingly aware that I do not know enough of the story. I admire both historians, who as a result of years of study are able to enlarge our understanding, and popular writers, who can so often find interesting angles on history without being intimidated as I am by the knowledge of their limitations.
Anecdotes appeal to me when they seem to tell me something, and I don;t think this is always entirely illusory. It doesn't take much thought to see that most anecdotes are constructions. As a teenager I bought a book of literary anecdotes which began with the rather rude "Switter Swatter" story about Sir Walter Ralegh (which is the subject of a contemporary round you can watch being performed here.). Is this anecdote true? It's documented at the time, but how did it get into the public domain? Only, presumably, from Sir Walter himself, and I don't take as gospel any stories by young men about their romantic successes. So despite its impeccable historical credentials I have some scepticism about this story. Most anecdotes are spread because they show someone in a good (perhaps self-depreciating) light (or because they present a negative image of someone's enemy.
Constructions may not be accurate. In his cricket report in last Saturday's Guardian, Vic Marks tells the story of the Durham wicket-keeper Chris Scott, who in 1994 dropped Brian Lara, the best batsman in the world, when he had made 18. Scott (Marks, being a good journalist, adds "it is said"!) moaned "I bet he goes on to get a hundred". Lara in fact went on to 501 not out, the highest score ever made. It's quite likely that Scott made exactly this comment at the time, and I'm not suggesting that this story is not exactly true as it stands. But even if he didn't say it at the time, he might afterwards have said something like "When I dropped him I thought that it would be just my luck if he made another hundred", and in the telling the story at some stage changed to "Scott said at the time that ...". The way memory works seems to be that we reconstruct our memories rather than retrieving them, and anecdotal memories get rewritten so that we come to "remember" what happened in the revised form. Consequently one should interpret even the most "authentic" anecdotes as a slightly idealised form of history.
What I would argue, I guess, is that anecdotes are valuable in passing on the culture of mathematics. We don;t need to express uncertainty about their "truth" because we should all know that anecdotes are not exactly true. When I first read the Euler story as a teenager I may have believed it to be literally true,.but as I became more experienced in such matters I realised that a more nuanced understanding was required. It's good that the background is now readily accessible, but we want to train young mathematicians who can read a story like this and enjoy it without necessarily believing it!
Academics apply rigour when they need to. Having been at an Oxbridge high table when World Cup football was being discussed, I can say with certainty that rigorous thinkers in one field don't necessarily bring the same quality of analysis of other matters. We all apply different standards of rigour in different parts of our lives: that's part of being human.
A final comment: I don't believe it is only mathematicians who take a cavalier approach to the history of their subject. All disciplines and professions have their cultures, built on anecdote and dubious history. Mathematics is no different in that respect.
Scaliger writes about the anecdote that Euler spouted a piece of mathematical nonsense claiming to prove the existence of God, embarrassing Diderot in front of the Empress. The anecdote, as commonly told, is far from true, so why do mathematicians still tell without qualification? Rowlett wonders about the role of such anecdotes, which are arguably part of the culture of mathematics, in training mathematicians.
These are deep issues. I love anecdotes. I like serious history, too, but I am useless as a historian because I don't know enough. Whenever I have worked on historical topics, I have found that, the more I research, the less I feel able to say anything because I am increasingly aware that I do not know enough of the story. I admire both historians, who as a result of years of study are able to enlarge our understanding, and popular writers, who can so often find interesting angles on history without being intimidated as I am by the knowledge of their limitations.
Anecdotes appeal to me when they seem to tell me something, and I don;t think this is always entirely illusory. It doesn't take much thought to see that most anecdotes are constructions. As a teenager I bought a book of literary anecdotes which began with the rather rude "Switter Swatter" story about Sir Walter Ralegh (which is the subject of a contemporary round you can watch being performed here.). Is this anecdote true? It's documented at the time, but how did it get into the public domain? Only, presumably, from Sir Walter himself, and I don't take as gospel any stories by young men about their romantic successes. So despite its impeccable historical credentials I have some scepticism about this story. Most anecdotes are spread because they show someone in a good (perhaps self-depreciating) light (or because they present a negative image of someone's enemy.
Constructions may not be accurate. In his cricket report in last Saturday's Guardian, Vic Marks tells the story of the Durham wicket-keeper Chris Scott, who in 1994 dropped Brian Lara, the best batsman in the world, when he had made 18. Scott (Marks, being a good journalist, adds "it is said"!) moaned "I bet he goes on to get a hundred". Lara in fact went on to 501 not out, the highest score ever made. It's quite likely that Scott made exactly this comment at the time, and I'm not suggesting that this story is not exactly true as it stands. But even if he didn't say it at the time, he might afterwards have said something like "When I dropped him I thought that it would be just my luck if he made another hundred", and in the telling the story at some stage changed to "Scott said at the time that ...". The way memory works seems to be that we reconstruct our memories rather than retrieving them, and anecdotal memories get rewritten so that we come to "remember" what happened in the revised form. Consequently one should interpret even the most "authentic" anecdotes as a slightly idealised form of history.
What I would argue, I guess, is that anecdotes are valuable in passing on the culture of mathematics. We don;t need to express uncertainty about their "truth" because we should all know that anecdotes are not exactly true. When I first read the Euler story as a teenager I may have believed it to be literally true,.but as I became more experienced in such matters I realised that a more nuanced understanding was required. It's good that the background is now readily accessible, but we want to train young mathematicians who can read a story like this and enjoy it without necessarily believing it!
Academics apply rigour when they need to. Having been at an Oxbridge high table when World Cup football was being discussed, I can say with certainty that rigorous thinkers in one field don't necessarily bring the same quality of analysis of other matters. We all apply different standards of rigour in different parts of our lives: that's part of being human.
A final comment: I don't believe it is only mathematicians who take a cavalier approach to the history of their subject. All disciplines and professions have their cultures, built on anecdote and dubious history. Mathematics is no different in that respect.
Sunday, 29 July 2012
Proof that I am a *pure* mathematician
I have always been drawn towards pure, rather than applied, mathematics. As a student I had no feel for mechanics: I could solve the equations but I had no physical intuition. I always felt out of my depth in any kind of applied mathematics, whereas I felt at home with abstract pure mathematics.
Now when I teach mathematics I have sometimes regretted my preference. I see the value of physical applications; I enjoy reading about relativity and quantum theory, and feel that I have gained some understandings of these topics to which, thirty years ago, I couldn't relate at all.
Now I have been looking at Mark Levi's two marvellous books, The Mathematical Mechanic and Why Cats Land on Their Feet. These are books about mechanics. The former uses physics to "prove" pure mathematical propositions, for example using an argument about a rotating fishtank to "prove" Pythagoras's Theorem. (The inverted commas reflect my pure mathematical reluctance to accept that physics can be used in this way!) The latter presents physical paradoxes and their resolution - for example, if, sitting in a seat attached rigidly to the frame of a spacecraft which is stationary in outer space, I push a balloon away from me, what happens to the spaceship?
These books are wonderful. Despite my comment about physics and proof above, I find the contents beautiful and astonishing. But they are also over my head. I have to read carefully and think deeply to get the point, and I need to be told why the paradoxes in the second volume are paradoxical because my physical understanding is so limited I don't "get" it easily.
So what these books confirm is that I have very little intuition about the physics of the world around us. Had I had a different upbringing that might not have been the case, but I have to accept that my mathematical talents do not extend at all to mechanics. I don't particularly regret that - the pleasure of pure mathematics compensates - except when I realise that my appreciation of Levi's books is so limited. I feel as I imagine someone would respond to Matisse who knew the paintings only from monochrome reproduction - there is much to enjoy but I will never be able fully to appreciate them.
This helps me understand why I enjoy quantum theory. I can't do mechanics because I have no physical intuition - but quantum theory makes a nonsense of our intuitions so my weakness isn't a problem!
Can I finish by encouraging any reader to seek out Levi's books, if you don't already know them. If you're an applied mathematician you'll love them; pure mathematicians will also find things to wonder at. Even if I can't fully appreciate them, I can still admire and enjoy.
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