The Creativity Code. Marcus du Sautoy

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СКАЧАТЬ To receive their PhDs, doctoral candidates in mathematics have to create a new mathematical construct. They have to come up with something that has never been done before. I am tasked with teaching them how to do that. Of course, at some level they have been training to do this to a certain extent already. Solving problems involves personal creativity even if the answer is already known.

      That training is an absolute prerequisite for the jump into the unknown. By rehearsing how others have come to their breakthroughs you hope to provide the environment to foster your own creativity. And yet that jump is far from guaranteed. I can’t take anyone off the street and teach them to be a creative mathematician. Maybe with ten years of training we could get there, but not every brain seems to be able to achieve mathematical creativity. Some people appear to be able to achieve creativity in one field but not another, yet it is difficult to understand what makes one brain a chess champion and another a Nobel Prize-winning novelist.

      Margaret Boden recognises that creativity isn’t just about being Shakespeare or Einstein. She distinguishes between what she calls ‘psychological creativity’ and ‘historical creativity’. Many of us achieve acts of personal creativity that may be novel to us but historically old news. These are what Boden calls moments of psychological creativity. It is by repeated acts of personal creativity that ultimately one hopes to produce something that is recognised by others as new and of value. While historical creativity is rare, it emerges from encouraging psychological creativity.

      My recipe for eliciting creativity in students follows the three modes of creativity Boden identified. Exploration is perhaps the most obvious path. First understand how we’ve come to the place we are now and then try to push the boundaries just a little bit further. This involves deep immersion in what we have created to date. Out of that deep understanding might emerge something never seen before. It is often important to impress on students that there isn’t very often some big bang that resounds with the act of creation. It is gradual. As Van Gogh wrote: ‘Great things are not done by impulse but by small things brought together.’

      Boden’s second strategy, combinational creativity, is a powerful weapon, I find, in stimulating new ideas. I often encourage students to attend seminars and read papers in subjects that don’t appear to connect with the problem they are tackling. A line of thought from a disparate bit of the mathematical universe might resonate with the problem at hand and stimulate a new idea. Some of the most creative bits of science are happening today at the junctions between the disciplines. The more we can come out of our silos and share our ideas and problems, the more creative we are likely to be. This is where a lot of the low-hanging fruit is to be found.

      At first sight transformational creativity seems hard to harness as a strategy. But again the goal is to test the status quo by dropping some of the constraints that have been put in place. Try seeing what happens if we change one of the basic rules we have accepted as part of the fabric of our subject. These are dangerous moments because you can collapse the system, but this brings me to one of the most important ingredients needed to foster creativity – and that is embracing failure.

      Unless you are prepared to fail, you will not take the risks that will allow you to break out and create something new. This is why our education system and our business environment, both realms that abhor failure, are often terrible environments for fostering creativity. It is important to celebrate the failures as much as the successes in my students. Sure, the failures won’t make it into the PhD thesis, but we learn so much from failure. When I meet with my students I repeat again and again Beckett’s call to ‘Fail, fail again, fail better.’

      Are these strategies that can be written into code? In the past the top-down approach to coding meant there was little prospect of creativity in the output of the code. Coders were never too surprised by what their algorithms produced. There was no room for experimentation or failure. But this all changed recently: because an algorithm, built on code that learns from its failures, did something that was new, shocked its creators, and had incredible value. This algorithm won a game that many believed was beyond the abilities of a machine to master. It was a game that required creativity to play.

      It was news of this breakthrough that triggered my recent existential crisis as a mathematician.

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       READY, STEADY, GO

       We construct and construct, but intuition is still a good thing.

      Paul Klee

      People often compare mathematics to playing chess. There certainly are connections, but when Deep Blue beat the best chessmaster the human race could offer in 1997, it did not lead to the closure of mathematics departments. Although chess is a good analogy for the formal quality of constructing a proof, there is another game that mathematicians have regarded as much closer to the creative and intuitive side of being a mathematician, and that is the Chinese game of Go.

      I first discovered Go when I visited the mathematics department at Cambridge as an undergraduate to explore whether to do my PhD with the amazing group that had helped complete the classification of finite simple groups, a sort of Periodic Table of Symmetry. As I sat talking to John Conway and Simon Norton, two of the architects of this great project, about the future of mathematics, I kept being distracted by students at the next table furiously slamming black and white stones onto a large 19×19 grid carved into a wooden board.

      Eventually I asked Conway what they were doing. ‘That’s Go. It’s the oldest game that is still being played to this day.’ In contrast to the war-like quality of chess, he explained, Go was a game of territory. Players take it in turn to place white and black pieces or stones onto the 19×19 grid. If you manage to surround a collection of your opponent’s stones with your own, you capture your opponent’s stones. The winner is the player who has captured the most stones by the end of the game. It sounded rather simple. The subtlety of the game, Conway explained, is that as you try to surround your opponent, you must avoid having your own stones captured.

      ‘It’s a bit like mathematics: simple rules that give rise to beautiful complexity.’ It was while watching the game evolve between two experts as they drank coffee in the common room that Conway discovered that the endgame was behaving like a new sort of number that he christened ‘surreal numbers’.

      I’ve always been fascinated by games. Whenever I travel abroad I like to learn and bring back the game locals like to play. So when I got back from the wild outreaches of Cambridge to the safety of my home in Oxford I decided to buy Go from the local toy shop to see what it was that was obsessing these students. As I began to explore the game with one of my fellow students in Oxford, I realised how subtle it was. It was hard to identify a clear strategy that would help me win. And as more stones were laid down on the board, the game seemed to get more complicated, unlike chess, where as pieces are gradually removed the game starts to simplify.

      The American Go Association estimates that it would take a number with 300 digits to count the number of games of Go that are legally possible. In chess the computer scientist Claude Shannon estimated that a number with 120 digits (now called the Shannon number) would suffice. These are not small numbers in either case, but they give you a sense of the wide range of possible permutations.

      I had played a lot of chess as a kid. I enjoyed working through the logical consequences of a proposed move. It appealed to the mathematician that was growing inside me. The tree of possibilities in chess branches in a controlled manner, making it manageable for a computer and even a human to СКАЧАТЬ