An excellent way to kill a conversation is to tell the person you are talking to that you are a mathematician. The conversation may limp on for a minute or two, but nearly always it is doomed. However, there is a miracle cure: just tell your interlocutor that you are a musician as well as a mathematician. Even people who know nothing about mathematics have heard that mathematical ability is connected, in some fascinating and counter-intuitive way, to musical ability.
As a mathematician with strong musical interests who grew up in a family of musicians, I have been asked about this connection many times. And I have bad news: although there are some obvious similarities between mathematical and musical activity – and although many musical patterns can be fruitfully analysed mathematically – there is (as yet) no compelling evidence for the kind of mysterious, almost magical connection that many people seem to believe in. I'm partly referring here to the "Mozart effect", where children who have been played music by Mozart are supposedly more intelligent, including at mathematics, than children from a control group. It is not hard to see why such a theory would be popular: we would all like to become better at mathematics without putting in any effort. But the conclusions of the experiment that originally prompted the widespread belief in the Mozart effect were much more modest and have been grossly exaggerated. If you want your brain to work better, then not surprisingly, you have to put in some hard graft; there is no such thing as an intellectual perpetual-motion machine. Baby Mozart CDs and toys that combine maths and music might help, but not much, and the effects are temporary.
Of course, this does not show that there is no interesting connection between mathematics and music. It was always a little implausible that lazily listening to Eine kleine Nachtmusik would earn you extra marks on that maths test tomorrow, but what about learning to read music or spending hours practising the piano? That takes genuine effort. Could it be that the rewards for that effort spill over into other areas of intellectual life, and in particular into mathematics? Is there any evidence that people who have worked hard to become good at music are better at mathematics than people who are completely unmusical? And in the other direction, are mathematicians better than average at music?
Demonstrating a connection of this kind is not as easy as one might think. To begin with, there are plenty of innumerate musicians and tone-deaf mathematicians, so the best one could hope to demonstrate would be a significant positive correlation between aptitudes at the two disciplines. And then one would face all the usual challenges of establishing a statistical connection. For example, if you want to show that professional mathematicians are on average better at music than other people, then you have to decide quite carefully who those "other people" are. You might expect that the kind of person who becomes a professional mathematician is much more likely than average to come from the kind of family that would consider music to be an important part of a child's education, so for that reason alone one would expect at least some "background correlation" between the two. Therefore, not much will be proved if you compare professional mathematicians with the population at large. Identifying and controlling for these kinds of effects is difficult, and as far as I know (though I would be delighted to be corrected), there has been no truly convincing study that has shown that musical ability enhances mathematical ability or vice versa.
And yet, the belief that the two are interestingly related won't go away without a fight. I cannot help observing that among the mathematicians I know, there do seem to be a surprising number who are very good indeed at the piano. (Incidentally, that is a study waiting to be done: are mathematicians more drawn to the piano than to other instruments? Of the mathematicians I can think of who are superb instrumentalists, all but one are pianists.) While we wait for scientific evidence to back up the anecdotal evidence, can we at least argue that it is plausible that there should be a connection?
Indeed we can. For a start, both mathematics and music deal with abstract structures, so if you become good at one, then it is plausible that you become good at something more general – handling abstract structures – that helps you with the other. If this is correct, then it would show a connection between mathematical and musical ability, but not the kind of mysterious connection that people hope for. It would be more like the connection, such as it is, between ability at football and ability at cricket. To become better at one of those then you need to improve your fitness and co-ordination. That makes you better at sport in general and therefore probably helps with the other.
Of course, abstract structures are not confined to mathematics and music. If you are learning a foreign language then you need to understand its grammar and syntax, which are prime examples of abstract structures. And yet we don't hear people asking about a mysterious connection between mathematical ability and linguistic ability. My guess is that that is because the connection exists but not the mystery: grammar feels mathematical, so it would hardly be a surprise to learn that mathematicians are better than average at learning grammar. Music, by contrast, is strongly tied up with one's emotions and can be enjoyed even by people who know very little about it. As such, it seems very different from mathematics, so any connection between the two is appealingly paradoxical.
In an effort to dispel this air of paradox, let me give one example of a general aptitude that is useful in both mathematics and music: the ability to solve problems of the "A is to B as C is to D" kind. These appear in intelligence tests (car is to garage as aircraft is to what?) but they are also absolutely central to both music and mathematics. Consider, for instance, the opening two phrases of Eine kleine Nachtmusik. (In the unlikely event that you don't know this piece, it may help if I tell you that it is the piece that is played in a maddening endless loop on Ryanair flights.) The second phrase is a clear answer to the first. But one can be more precise about what this means. If you try to imagine any other second phrase, nothing seems "right" in the way that Mozart's chosen phrase does. So what is the question to which that phrase is the right answer? It is something like, "The first phrase goes broadly upwards and uses the notes of a G major chord; what would be the corresponding phrase that goes broadly downwards and uses the notes of a D7?" Music is full of little puzzles like this. If you are good at them, then when you listen to a piece, expectations will constantly be set up in your mind. Of course, some of the best moments in music come when one's expectations are confounded, but if you don't have the expectations in the first place then you will miss out on the pleasure.
Here is a fairly simple example from mathematics, which I recommend trying to answer for yourself before reading on: what is to multiplication as zero is to addition? If you are not mathematically inclined, then you will probably be aware that zero is a special number without ever having thought hard about why. The question requires you to relate zero to the general operation of addition. In other words, it requires you to identify what it is about the role that zero plays in the game of adding numbers together that distinguishes zero from all other numbers. That role is the following: adding zero to a number makes no difference. If we now want to solve the puzzle, we need a corresponding statement concerning multiplication. And there is one: multiplying a number by one makes no difference. So the answer is one.
That is by no means the end of the story. If you pursue the analogy far enough, you will find yourself inventing the theory of logarithms and exponentials. For many people logarithms mark the point where they part company with mathematics. Those who are good at "A is to B as C is to D" puzzles are less likely to fall at that hurdle, and the same goes for many subsequent hurdles.
In my view, the general question of whether mathematical ability and musical ability are related is much less interesting than some similar but more specific questions. I have already mentioned the possibility that mathematicians are more drawn to the piano than to other instruments. Are they more drawn to certain composers (Bach, for instance)? Are musical mathematicians more drawn to certain areas of mathematics? Do mathematicians tend to listen to music in a more analytical "A is to B as C is to D" way rather than simply allowing themselves to be caught up in the emotion? One can imagine many interesting surveys and experiments that could be done, but for now this is uncharted territory and all we can do is speculate.
Cambridge University maths professor Tim Gowers will speak at Cheltenham Music Festival on Sunday 10 July: see cheltenhamfestivals.com/music for details