On Monday, Sir Michael Atiyah spoke about beauty in mathematics. Shigefumi Mori’s talk on Thursday started with an interesting meditation on similarities between mathematics, design, and impressionist paintings. Where Atiyah’s talk was very general, Mori drew specific parallels between the process of creating math and creating art.
Mori described both art and mathematics as trying to “capture an object and create a piece of work.” This language of “capturing” a property of an object made the similarities between math and impressionism most clear. Impressionists were not trying to depict objects realistically but to capture “unpaintables” such as atmosphere and light. “They wanted to draw light, but you can’t draw it,” he said. “Instead, you draw the affect of light on objects.” Claude Monet often painted the same scene at different times of the day or year as a way to capture light and atmosphere. Mori compared Monet’s haystack and Rouen cathedral series to this diagram from David Mumford’s book Lectures on Curves on an Algebraic Surface.
This diagram is not meant to be self-explanatory. After all, we’re just here for an impression. (Mumford won the 2007 Steele Prize for mathematical exposition, so his books are probably a good place to get more information about schemes. There is also a blog post you can check out about this and other charts from Mumford’s books.) Mori’s point was that we get information about the general case we’re interested in, marked “0” on the diagram above, by looking at what happens in the cases marked 2, 3, 5, 7, and so on. These cases are like the different times of day when Monet painted his haystacks.
Mori also compared algebraic stacks to Picasso’s Portrait of Marie-Therese. The painting is a picture of a woman drawn from two different perspectives, and he thinks of stacks as algebraic figures drawn in different ways. And in fact, his appreciation of the math contributed to his appreciation of the art. He said, “before I noticed the similarity between Picasso and algebraic stacks, I didn’t much like the painting.”
I’m going to end this post with two quotes that Mori presented towards the beginning of his talk that really struck a chord with me. Sometimes inspiration just strikes out of the blue, but more often, people who have great ideas have them because they have been doing hard work and turning problems over in their minds for a long time.
First, he quoted Buckminster Fuller, a great designer and architect, who said, “When I am working on a problem I never think about beauty. I only think about how to solve the problem. But when I have finished, if the solution is not beautiful, I know it is wrong.”
Then Mori mentioned a visit French mathematician André Weil made to Japan in the 1950s. In a question and answer session, a Japanese mathematician asked him to whom mathematical ideas occur. Weil replied, “to people who can continue working without ideas.”