A wonderfully stimulating afternoon at the Tate Modern gallery this afternoon, for a mathematical event in a series entitled "Topology". The afternoon had two parts - a showing of the film "Au Bonheur des Maths" by Raymond Depardon and Claudine Nougaret, followed by discussion between Michael Atiyah and Cédric Villani, two mathematical superstars, chaired by Ian Stewart.
The film comprised a series of short statements by mathematicians, essentially presented in black and white as talking heads, though Villani alone used the blackboard to communicate (more on this later). They all had interesting things to say. I was slightly puzzled by Atiyah's comment afterwards on the diversity of the mathematicians: most seemed to be white, male and middle-aged, and presenting similar views of the subject.
The discussion, ably chaired by Ian Stewart, was absorbing. It helped that the questions from the audience were all interesting (or were made interesting by Villani and Atiyah). It was being recorded, and I hope the result will be made widely available.
There were so many interesting ideas that I can only pick out a few that resonated with me. One was the extent to which mathematics is a frustrating activity: 'You spend most of your time thinking "I can't understand this"' (and if that's the experience of Stewart, Atiyah and Villani, what is it for the rest of us?) This is something that must influence the characters of mathematicians: how can you be arrogant when you are always wrestling with problems you can't solve? It must be off-putting for undergraduates who found mathematics easy at school to find at university that there are hard problems on which they get stuck: appreciating that being stuck is the natural state of even the greatest mathematicians, and isn't a sign of one's inadequacy, might help keep them in the profession. So I'll be quoting this to my students.
I was really impressed by the evident breadth of knowledge of both Atiyah and Villani. They had read Kepler; Villani told us about Voltaire's comment that Kepler's achievement was a humiliation for philosophy in that such a deluded thinker could make such great discoveries. Both seemed keen on reading the writings of the great mathematicians of the past. Villani made fascinating comments about the different ways in which the composers Xenakis and Ligeti made use of maths, and showed how a work of Ligeti refutes a remark in Euler's writing on music. When there is such a focus on research outputs that reading outside one's immediate subject can seem like a luxury one can't afford, I found Atiyah and Villani inspirational in their range of serious interests.
There were insights into similarities between mathematicians and artists - both proceed by breaking rules (for example, non-Euclidean geometry arising by discarding a previously unchallenged axiom) and both work with a mix of individual and collaborative activity. There was wisdom on teaching - the personality of the teacher is the most important factor, and Villani commented that saying that we need more people to study science is not the best way to encourage young scientists.
Villani was asked about the "enemies" a mathematician faces, and identified three - lack of imagination; a priori ideas, which you have to eliminate in order to make progress; and too much information, hiding the things that really matter in fog. Stewart told a fascinating anecdote about a friend discussing three-dimensional topology over the phone with a mathematician who had been blind since birth (a remarkable idea in itself) and noting that the blind mathematician was obviously visualising the problem very clearly but without taking any single point of view - a very interesting concept. There was discussion of ugliness in mathematics - with, I think, agreement that a good proof is about understanding, and a proof which doesn't tell you why something is true is inherently unsatisfactory.
Artists in the audience asked some of the most interesting questions. One was about symbols - how do mathematicians choose their symbols? Villani's answer made me aware just how important choice of symbols can be - they permit a level of automatic checking and most symbols carry meaning for the reader which helps comprehension. (I remember struggling with the German letters symbolising group varieties in an algebra book: I couldn't immediately tell which was which and, not surprisingly, I never got on top of the material. My fault, not the authors: I should have spent time copying the letters until I could read them at a glance!)
The final question, again an insight into how the practice mathematics relates to that of the visual artist, was about the "extraordinary mark-making" mathematicians do on blackboards, and whether this can survive in the digital age. Atiyah's answer agreed on the importance of the hand in doing maths, and that communication of mathematics involves body movement, props, blackboards - the garnish that makes something palatable. Villani, noting that blackboards are now not always available in lecture rooms, felt that the blackboard is unrivalled: much more than with a computer, one can improvise, erase, keep some material and lose other bits; it forces you not to overflow, and when you're frustrated, you can bang your head on it!
This was an amazing afternoon, rich in provocative ideas. The mathematicians' passion came across; there were insights about mathematics, art and music. If these notes are garbled, it was because there was so much to think about! I do hope the recording will be made available, even though it will probably show that I have misunderstood and misinterpreted a lot of what was said. I have rarely spent such a stimulating afternoon in a gallery (and that is a very strong statement).