Of Proof and other things.
Over the last couple of days, I’ve seen a couple of days that made me think hard about all sorts of things. The first, at least, is tangentially related to the subject of this blog, and so I feel justified in writing about it. I’d been waiting to see Proof for a long time; the play comes with a heavy reputation having won the Pulizer prize and a Tony Award for its author, David Auburn. It’s even had the ultimate accolade of being made into a movie, starring Anthony Hopkins of all people.
It deals with the story of young mathematician, struggling with the death of her father – another mathematician, who had done brilliant work in his early 20s and then gone steadily, slowly crazy, looking for messages encoded into the Dewy Decimal system of cataloguing library books. The eponymous proof – of a theorem mathematicians have been wrestling with ‘for as long as there have been mathematicians’, although we never learn its identity – lies in one of the many notebooks that lie scattered around the house.
The production was good, gripping at times, and we, the audience, were drawn into the world of the characters. Yet I came away disappointed and even a little angry. I’d expected a play that was to mathematics what Michael Frayn’s magisterial Copernhagen was to physics; a serious attempt to engage with the ideas and mental landscape of a subject. The language of Copernhagen is filled with physics; we see Bohr and Heisenberg on stage discussing their work and – miracle of miracles – can follow along. When friends and I produced the show in Cambridge (cast : entirely arts students, crew : entirely scientists…) I would stand by the doors during the interval, and listen to our audience talk about the ideas in the play.
Proof, on the other hand, might as well have been about flower arranging. Or molecular genetics. Or stock-car racing, or anything at all for all the presence maths had. A few technical terms are dropped in – a reference to Hilbert spaces and elliptical forms suggests that the playwright had been reading press coverage of the solution to Fermat’s last theorem – but no explanation offered. Characters talked often of prime numbers – but never allowed the audience to glimpse why anyone might care.
For the play to be as good as I’d wanted it to be, I needed to believe in the character’s motivations and thoughts, yet not one of the mathematicians on stage ever gave us a glimpse of that. It was as if the show was about struggling painters, who had forgone any suggestion of belief in their art. Without that, the play felt unfinished.
Actually, it’s funny I mention painters as an analogy. The one theme that did get underlined was that mathematics is an art form, not a science. It’s something that’s been uppermost in my mind since I read a document known on the net as Lockhart’s Lament (warning – 25 page PDF).
It was written a few years ago by Paul Lockhart, a mathematician who left academic life to become a teacher. The first paragraphs capture the gist :
A musician wakes from a terrible nightmare. In his dream he finds himself in a society where
music education has been made mandatory. “We are helping our students become more
competitive in an increasingly sound-filled world.” Educators, school systems, and the state are
put in charge of this vital project. Studies are commissioned, committees are formed, and
decisions are made— all without the advice or participation of a single working musician or
composer.
Since musicians are known to set down their ideas in the form of sheet music, these curious
black dots and lines must constitute the “language of music.” It is imperative that students
become fluent in this language if they are to attain any degree of musical competence; indeed, it
would be ludicrous to expect a child to sing a song or play an instrument without having a
thorough grounding in music notation and theory. Playing and listening to music, let alone
composing an original piece, are considered college, and more often graduate school. .
Is maths in the same situation? Read the rest, and let me know if you agree or disagree.
on October 29th, 2008 at 3:58 pm
I almost started looking online for the play, because I’d be so interested to see that . . .
I loved the Paul Lockhart essay and started a thread about it in the Galaxy Zoo Forum. For me, it summed up entirely why I’d never done well at maths or been interested. It also summed up what is wrong with education – that lessons start at the wrong end. Rather than being encouraged to deduce or invent anything for themselves (apart from the recent fad of letting 15-year-olds draw pictures or substitute play-dough for writing, the nonsense they call “personalised learning”), the children are expected to start by memorising and applying definitions. It’s like starting with the final chapter of a book and memorising somebody else’s summary of events or something, rather than reading it and thinking about it.
I never had the least idea what maths actually *is* until I’d read that essay. Deep down, I knew it was the result of a series of inventions; but I’d never questioned the assumption they’d fed us that you have to get to post-PhD level to think anything out for yourself – and I couldn’t do GCSE Maths because it was so tedious that I couldn’t care less. As soon as I’d seen a few shapes and questions by Paul Lockhart, I spent two weeks obsessively drawing circles and triangles, and speculating wildly about how much of a cube a three-dimensional pyramid takes up whilst refusing to Google the formula.
A quantum physicst at Brighton University blames the culture of “quick results”, fuelled by television. Lessons must start with two “learning objectives” which all children must achieve by the end of the lesson. Whereas the deepest learning takes months or years to sink in. You can’t test an honest intellectual relationship, it doesn’t come with box-ticking, and therefore it fails inspections.
I hope that we are doing something like Paul Lockhart’s teaching at the Galaxy Zoo Forum – sharing, speculating, inventing things (like Waveney’s merger and irregular websites, and the peas projects), having the opportunity to be wrong, to find our own interests, and occasionally to set each other quizzes and problems.
I don’t know enough about maths to say for sure that it is in that situation, though I believe it is. And I’m terribly sad that I don’t know, and I wish I could find out, but current textbooks only confuse me and I know I can’t learn from them.
(Apologies for long post.)
on October 29th, 2008 at 8:47 pm
Hi Chris – Whilst I agree that a play about mathematicians should contain some maths to make it intellectually satisfying, you might like to consider it from the viewpoint of the producer. At Cambridge, I guess that you had, potentially, quite a select audience of pretty high intellect.
On the basis that the average man-in-the-street (most likely including the critics) can’t multiply two fractions together, to most people mathematical concepts are yawn making ! (Even on the GZ forum !)
So I would guess there wouldn’t be many bums-on-seats !
I thought the BBC production of “Copenhagen” was brilliant ! But I suspect that not many people grasped the concepts of Heisenberg/Bohr discussion and it’s implications during the war (WW2).
Fermats Brother
on November 13th, 2008 at 7:12 pm
Hi Chris,
I’m with you on the disappointment front – whilst I’ve not seen Proof, many television documentaries that supposedly address science (or even rare excursions that include mathematics) are often content-free when it comes to the real subject matter!
I read Paul Lockhart’s article (thank you for the length warning!) with a degree of sympathy – I recognised many of the points that he raised from my own, and my children’s school experiences. I did like the Gauss quote about it being better to have notions rather than notations.
I seem to recall Lancelot Hogben in ‘Mathematics for the million’ providing some alternatives to the ‘formal logic proofs’ – maybe we should ask Alan Bennett to write a mathematical version of ‘The history boys’ based on some of Lockhart and Hogben’s material. It’s more likely to rescue mathematics in schools then generations of politicians and ‘educators’.
Cheers, and thank you for this thought provoking item.