You may have noticed that I am not exactly diligent about referencing the pictures and diagrams I use on this blog. In fact the majority of them come from google images, and I usually have no idea where they were originally used (one of the good things about writing a blog is the lack of academic rigour required). However, last night I saw a Coen brothers film called “A Serious Man”, and am pleased to be able to reveal that the header above is actually a screenshot from that film.
As you might guess from the picture, the film itself is not really all that serious, and how Larry Gopnik – the physics professor depicted – managed to write on a board that size is left to the viewer to ponder. Larry’s brother Arthur is also a mathematician, although a rather less functional one. He is quite a strange man , and spends most of the film lying on the couch writing what he calls his “Mentaculus”, which is supposed to be a “probability map of the universe”. This may or may not be a serious endeavour; we are only given a brief glimpse at its contents, and it appears to be no more than a collection of very intricate doodles.
Writing a probability map of the universe is a noble but rather misguided aim; the notion itself doesn’t really make sense, and it would clearly be impossible to pull off. But the urge to explain the workings of the world around us is a major drive for mathematicians. Unfortunately, the real world is messy and chaotic, and there is very little that we can do to accurately describe how things work in practice outside of rough models.
Arthur’s quixotic mission reminds me of the depiction of John Nash in “A Beautiful Mind”. In certain scenes he is shown trying to discover the mathematics behind various phenomena, such as the geometric patterns described by the movements of pigeons and football players. And this is not necessarily quite as wacky an idea. There has actually been quite a lot of research done into flocking behaviour, for example. It has been found that the collective movement of large groups of animals (or humans!) is determined by only three simple rules, and can hence be accurately modelled. Each member of the group follows the direction of its neighbours, stays close enough to them so as not to get separated, and remains far enough away so as not to become crowded. So by letting each individual member of a large group act according to a simple algorithm (head in average direction of neighbours, steer towards average position of neighbours, and keep a certain minimum distance from neighbours) flocking behaviour can be realistically simulated.
This technique is often used in films to generate very realistic crowds. But knowing what algorithm governs the movement of each individual in a crowd is a very different matter from predicting the movements of the crowd itself! That would be a much more difficult proposition, and would involve modelling a chaotic system (something I mentioned in my first post). I suppose that in a way this chaos goes some way to explaining the notion that crowds can be “wise” in a way that its individual constituents are not: a group of people acts in a complex and unpredictable way, even if individually they are behaving according to simple rules.
John Nash would have been unlikely to even attempt to describe pigeons’ movements geometrically. He did however make great advances in a branch of mathematics known as game theory, which deals with behaviour in strategic situations, that is those where there is any kind of incentive to be gained from acting in a certain way. This does not necessarily have to be a game as such, of course. In fact it could be argued that, as our behaviour is generally motivated by incentive, in the sense of game theory – life is a game!
Nash is most famous for his Nash Equilibrium, which is basically a set of optimal strategies (and corresponding payoffs) for a game, such that any one player has nothing to gain from changing their strategy. In the film, Nash has an epiphany whilst in a bar with his friends trying to pick up women. There are a group of women, one of whom is blonde, the other brunettes. All the men want the blonde woman, and one of Nash’s friends remarks that, following the words of Adam Smith (“in competition, individual ambition serves the common good”), their best strategy would be to all go and speak to her. He realises that this would result in none of them getting a girl (she would be intimidated and choose no-one, and then the others would be offended that they were the second choice), and that actually they would all benefit from none of them speaking to the blonde. Disregarding some very dubious assumptions, such as that every girl will fall at the feet of the first man to speak to her, the scene is supposed to demonstrate that cooperative behaviour often yields better results for everyone than competition.*
Nash’s theory concerns situations in which selfish actions can be to the detriment of everyone. It is certainly mathematics, but is unusual in its great potential for application to the real world. For example, a tennis player who continually serves to the opponent’s weaker hand will eventually be at a disadvantage, as the opponent will be able to predict their actions and act accordingly. Shops who continually try to undercut each others’ prices will fare worse than they would have if they had agreed to keep them high. There are various situations in biology in which an organism appears to act altruistically – even in a way that is detrimental to itself – in order to increase overall evolutionary fitness. In all these “games” (tennis, economics, evolution), among many others, Nash’s theory says that the players will tend towards an equilibrium whereby none have any incentive to deviate from their chosen strategy.
This post seems to be getting quite long already, so I’ll just mention one other film before I sign off. Not because it is a particularly good film (in fact the films I mention seem to be decreasing in quality), but because I have some kind of tenuous link to it. It is called “It’s My Turn”, and is about the love-life of a mathematics professor – played by Jill Clayburgh – in New York. On the whole, it is rather formulaic, but it is notable for being possibly the only film I’ve ever seen which contains a scene in which a mathematical theorem is proved (the consensus seems to be that this does not make for riveting cinema). The theorem in particular is the Snake Lemma**, and it occurs right at the beginning of the film. I’m afraid I’m not even going to attempt to explain the Snake Lemma: whilst fundamental to homological algebra – and especially to algebraic topology – it is not a very interesting theorem. The proof is repetitive and involves “diagram-chasing”, which basically means drawing a diagram and following the arrows.
The movie was filmed in New York; in fact it was filmed at the CUNY Graduate Center, which I used to attend. I know this because the professor who taught my algebraic topology class there told me that he had actually been hired to teach Jill Clayburgh the proof of the theorem. Obviously she wouldn’t have been able to understand it, but she was apparently a very diligent pupil, and memorized the whole thing perfectly. Below is the scene – I think you’ll agree that she does quite a good job. If only I could say the same thing about the director!
*Actually the scene doesn’t even really demonstrate the concept of the Nash equilibrium, as if no-one goes for the blonde girl, then no-one gets her, which is not the most desirable outcome. The optimal solution would be for one man (preferably Russell Crowe) to speak to her.
**Interestingly, this is the same theorem which can be seen written on the blackboard behind Dustin Hoffman at the very beginning of The Graduate.