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these transformational moments hinge on changing the rules of the game, or dropping an assumption that previous generations had been working under. The square of a number is always positive. All molecules come in long lines not chains. Music must be written inside a harmonic scale structure. Faces have eyes on either side of the nose. At first glance it would seem hard to program such a decisive break, and yet there is a meta-rule for this type of creativity. You start by dropping constraints and see what emerges. The art, the creative act, is to choose what to drop or what fresh constraint to introduce such that you end up with a new thing of value.

      If I were asked to identify a transformational moment in mathematics, the creation of the square root of minus one in the mid-sixteenth century would be a good candidate. This was a number that many mathematicians believed did not exist. It was referred to as an imaginary number (a derogatory term Descartes came up with to indicate that of course there was no such thing). And yet its creation did not contradict previous mathematics. It turned out it had been our mistake to exclude it. How can a computer come up with the concept of the square root of minus one when the data it is fed will tell it that there is no number whose square can be negative? A truly creative act sometimes requires us to step outside the system and create a new reality. Can a complex algorithm do that?

      The emergence of the Romantic movement in music is in many ways a catalogue of rule breaking. Instead of moving between close key signatures as Classical composers had done, new upstarts like Schubert chose to shift key in ways that deliberately broke expectations. Schumann left chords unresolved that Haydn or Mozart would have felt the need to complete. Chopin in turn composed dense moments of chromatic runs and challenged rhythmic expectations with his unusual accented passages and bending of tempos. The move from one musical movement to another: from Medieval to Baroque to Classical to Romantic to Impressionist to Expressionist and beyond is a story of breaking the rules. Each movement is dependent on the one before to appreciate its creativity. It almost goes without saying that historical context plays an important role in allowing us to define something as new. Creativity is not an absolute but a relative activity. We are creative within our culture and frame of reference.

      Can a computer initiate this kind of phase change and move us into a new musical or mathematical state? That seems a challenge. Algorithms learn how to act based on the data they interact with. Won’t this mean that they will always be condemned to producing more of the same?

      As Picasso once said: ‘The chief enemy of creativity is good sense.’ That sounds on the face of it very much against the spirit of the machine. And yet you can program a system to behave irrationally. You can create a meta-rule that will instruct it to change course. As we shall see, this is in fact something machine learning is quite good at.

      Can creativity be taught?

      Many artists like to fuel their own creation myth, appealing to external forces as responsible for their creativity. In Ancient Greece poets were said to be possessed by the muses, who breathed inspiration into the minds of men, sometimes sending them insane in the process. For Plato ‘a poet is holy, and never able to compose until he has become inspired, and is beside himself and reason is no longer in him … for no art does he utter but by power divine’. Ramanujan, the great Indian mathematician, likewise attributed his great insights to ideas he received in his dreams from his family goddess Namagiri. Is creativity a form of madness or a gift of the divine?

      One of my mathematical heroes, Carl Friedrich Gauss, was one of the worst at covering his creative tracks. Gauss is credited with creating modern number theory with the publication in 1798 of one of the great mathematical works of all time: Disquisitiones arithmeticae. When people tried to read the book to uncover where he got his ideas, they were mystified. The work has been described as a book of seven seals. Gauss seems to pull ideas like rabbits out of a hat, without ever really giving us an inkling of how he achieved this magic. Later, when challenged, he retorted that an architect does not leave up the scaffolding after the house is complete. Gauss, like Ramanujan, attributed one revelation to ‘the Grace of God’, saying he was ‘unable to name the nature of the thread which connected what I previously knew with that which made my success possible’.

      Yet the fact that an artist may be unable to articulate where their ideas came from does not mean that they followed no rules. Art is a conscious expression of the myriad of logical gates that make up our unconscious thought processes. There was of course a thread of logic that connected Gauss’s thoughts: it was just hard for him to articulate what he was up to – or perhaps he wanted to preserve the mystery, to fuel his image as a creative genius. Coleridge’s claim that the drug-induced vision of Kubla Khan came to him in its entirety belies all the preparatory material that shows the poet working on the ideas before that fateful day when he was interrupted by the person from Porlock. Of course, this makes for a good story. Even my own account of creation will focus on the flash of inspiration rather than the years of preparatory work I put in.

      We have an awful habit of romanticising creative genius. The solitary artist working in isolation is frankly a myth. In most instances what looks like a step change is actually a continuous growth. Brian Eno talks about the idea of ‘scenius’, not genius, to acknowledge the community out of which creative intelligence often emerges. The American writer Joyce Carol Oates agrees: ‘Creative work, like scientific work, should be greeted as a communal effort – an attempt by an individual to give voice to many voices, an attempt to synthesize and explore and analyze.’

      What does it take to stimulate creativity? Might it be possible to program it into a machine? Are there rules we can follow to become creative? Can creativity, in other words, be a learned skill? Some would say that to teach or program is to show people how to imitate what has gone before, and that imitation and rule following are both incompatible with creativity. And yet we have examples of creative individuals all around us who have studied and learned and improved their skills. If we study what they do, could we imitate them and ultimately become creative ourselves?

      These are questions I find myself asking every new semester. 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

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