might ultimately help us, as humans, to behave less like machines.
You may ask why a mathematician is offering to take you on this journey. The simple answer is that AI, machine learning, algorithms and code are all mathematical at heart. If you want to understand how and why the algorithms that control modern life are doing what they do, you need to understand the mathematical rules that underpin them. If you don’t, you will be pushed and pulled around by the machines.
AI is challenging us to the core as it reveals how many of the tasks humans engage in can be done equally well, if not better, by machines. But rather than focus on a future of driverless cars and computerised medicine, this book sets out to explore whether these algorithms can compete meaningfully with the power of the human code. Can computers be creative? What does it mean to be creative? How much of our emotional response to art is a product of our brains responding to pattern and structure? These are some of the things we will explore.
But this isn’t just an interesting intellectual challenge. Just as the artistic output of humans allows us to get some insight into the complex human code that runs our brains, we will see how the art generated by computers provides a surprisingly powerful way to understand how the code is working. One of the challenges of code emerging in this bottom-up fashion is that the coders often don’t really understand how the final code works. Why is it making that decision? The art it creates may provide a powerful lens through which to gain access to the subconscious decisions of the new code. And it may also reveal limitations and dangers that are inherent in creating code that we don’t fully understand.
There is another, more personal, reason for wanting to go on this journey. I am going through a very existential crisis. I have found myself wondering, with the onslaught of new developments in AI, if the job of mathematician will still be available to humans in decades to come. Mathematics is a subject of numbers and logic. Isn’t that what a computer does best?
Part of my defence against the computers knocking on the door of the department, wanting their place at the table, is that as much as mathematics is about numbers and logic, it is a highly creative subject, involving beauty and aesthetics. I want to argue in this book that the mathematics we share in our seminars and journals isn’t just the result of humans cranking a mechanical handle. Intuition and artistic sensitivity are important qualities for making a good mathematician. Surely these are traits that can never be programmed into a machine. Or can they?
This is why, as a mathematician, I am attentive to how successful the new AI is being in gaining entry to the world’s galleries, concert halls and publishing houses. The great German mathematician Karl Weierstrass once wrote: ‘a mathematician that is not something of a poet will never be a true mathematician.’ As Ada Lovelace perfectly encapsulates, you need a bit of Byron as much as Babbage. Although she thought machines were limited, Lovelace began to realise the potential of these machines of cogs and gears to express a more artistic side of its character:
It might act upon other things besides number … supposing, for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent.
Yet she believed that any act of creativity would lie with the coder, not the machine. Is it possible to shift the weight of responsibility more towards the code? The current generation of coders believes it is.
At the dawn of AI, Alan Turing famously proposed a test to measure intelligence in a computer. I would now like to propose a new test: the Lovelace Test. To pass the Lovelace Test, an algorithm must originate a creative work of art such that the process is repeatable (i.e. it isn’t the result of a hardware error) and yet the programmer is unable to explain how the algorithm produced its output. This is what we are challenging the machines to do: to come up with something new, surprising and of value. For a machine to be deemed truly creative requires one extra step: its contribution should be more than an expression of the coder’s creativity or that of the person who built the data set. That is the challenge Ada Lovelace believed was insurmountable.
The chief enemy of creativity is good sense.
Pablo Picasso
The value placed on creativity in modern times has led to a range of writers and thinkers trying to articulate what it is, how to stimulate it, and why it is important. It was while sitting on a committee at the Royal Society assessing what impact machine learning was likely to have on society in the coming decades that I first encountered the theories of the cognitive scientist Margaret Boden. Her ideas on creativity struck me as the most relevant when it came to addressing or evaluating creativity in machines.
Boden is an original thinker who over the decades has managed to fuse many different disciplines: philosopher, psychologist, physician, AI expert and cognitive scientist. In her eighties now, with white hair flying like sparks and an ever-active brain, she is enjoying engaging enthusiastically with the prospect of what these ‘tin cans’, as she likes to call computers, might be capable of. To this end, she has identified three different types of human creativity.
Exploratory creativity involves taking what is already there and exploring its outer edges, extending the limits of what is possible while remaining bound by the rules. Bach’s music is the culmination of a journey Baroque composers embarked on to explore tonality by weaving together different voices. His preludes and fugues push the boundaries of what is possible before breaking the genre open and entering the Classical era of Mozart and Beethoven. Renoir and Pissarro reconceived how we could visualise nature and the world around us, but it was Claude Monet who really pushed the boundaries, painting his water lilies over and over until his flecks of colour dissolved into a new form of abstraction.
Mathematics revels in this type of creativity. The classification of finite simple groups is a tour de force of exploratory creativity. Starting from the simple definition of a group of symmetries – a structure defined by four simple axioms – mathematicians spent 150 years producing a list of every conceivable element of symmetry, culminating in the discovery of the Monster Symmetry Group, which has more symmetries than there are atoms in the Earth and yet fits into no pattern of other groups. This form of mathematical creativity involves pushing the limits while adhering to the rules of the game. It is like the explorer who thrusts into the unknown but is still bound by the limits of our planet.
Boden believes that exploration accounts for 97 per cent of human creativity. This is the sort of creativity that computers excel at: pushing a pattern or set of rules to the extremes is perfect for a computational mechanism that can perform many more calculations than the human brain. But is it enough? When we think of truly original creative acts, we generally imagine something more utterly unexpected.
The second sort of creativity involves combination. Think of how an artist might take two completely different constructs and seek to combine them. Often the rules governing one world will suggest an interesting new framework for the other. Combination is a very powerful tool in the realm of mathematical creativity. The eventual solution of the Poincaré Conjecture, which describes the possible shapes of our universe, was arrived at by applying very different tools to understand flow over surfaces. It was the creative genius of Grigori Perelman which realised that the way a liquid flows over a surface could unexpectedly help to classify the possible surfaces that might exist.
My
