Puzzling and fuzzy ensembles

We are no longer surprised that mathematics tells us interesting and useful things about the universe. Some of us suspected as much a long time ago, notably the Pythagoreans with their curious mix of solid mathematics and number mysticism. Sir Isaac Newton drove the message home when he discovered simple mathematical rules - "laws of nature" - for motion and gravity. In his line of development the role of mathematics is clear and direct: it provides a precise description of certain aspects of reality. We learn that we live in a deterministic universe, one that runs according to fixed mathematical rules. Our understanding of how that description works out may often involve simplifications and idealisations, such as replacing cratered planets by uniform spheres, but that is just a matter of technique.

About 150 years ago, a second and very different paradigm for the application of mathematics to nature came into being: probability theory and statistics. The viewpoint here is at the opposite extreme from the tidy determinism of Newton and his successors. It is the idea that random chance has its own patterns, too - patterns that become visible "in the long run" or "on average". The patterns that lurk in random events have a slightly fuzzy feel to them - they are not sharp-edged and clear-cut like Newtonian rules. They are patterns like this: "If you toss a coin a large number of times, then the numbers of heads and tails will almost always be pretty close together." What is going on here? Deborah J. Bennett's Randomness is a short book in the tradition and physical format established by Longitude. I read it from cover to cover on the plane between Birmingham and Edinburgh, a 50-minute flight.

The book can most easily be described as a brief history of chance; it goes at a gentle pace, treating basic probability theory and not much else. I can cheerfully recommend it to anyone who is a total beginner when it comes to probability, what it means, why it is desperately puzzling, and what it can do for us despite that. The style is smooth and elegant, but the writing is slightly detached in an academic sort of way - calm, simple, but lacking in contrast. It is at its strongest on the history of probability, and at its weakest when it tries to teach the reader instead of just maintaining the momentum of the story. But it is fascinating to read about the pioneers of probability, such as Pierre Simon de Laplace with his "normal distribution" - now more familiar as the notorious bell curve - and Adolphe Quetelet, perhaps the first to realise that there are statistical patterns in human behaviour. And I applaud the blunt reminder that when it comes to the real world the "normal" distribution is actually highly abnormal, so a lot of the allegedly basic statistics found in textbooks is a lot less useful than it is cracked up to be.

My main criticism: it left me wanting more. A sequel, please: one that comes to grips with the essence of chance and explains why its patterns, though expressed in terms that are just as precise as those of Newton, seem fuzzy because of the way we are forced to interpret them. Probabilities are patterns not of single events, but of "ensembles" of events, to borrow the physicists' term - lists of things that might have happened, not just whatever it was that did happen. We interpret the mathematics of those ensembles by sampling some of their members, and that is where the apparent fuzziness creeps in. The statement "a fair coin throws heads or tails with equal probability" can be phrased with mathematical precision, and used to predict with exquisite accuracy the properties of ensembles of coin tosses. What it cannot achieve is an equally precise prediction of a single coin toss.

The Jungles of Randomness is a very different book - a collection of topics from the frontiers of today's mathematical research, many of them loosely linked by the theme of randomness, but others unashamedly included for other reasons. The style is breezy and journalistic. We are told, for instance, about the Tilt-A-Whirl, a fairground ride whose first customers were flung in random circles at White Bear Lake in 1926. Random? Not really. The Tilt-A-Whirl is a machine, and cannot therefore be truly random: it must be obeying those rigid laws that Newton wrote down. But it is random enough that its riders cannot predict which way it will turn next. Forty years ago nobody would even have recognised an interesting question here, but today we can give an instant answer: the Tilt-A-Whirl is chaotic. It obeys deterministic rules, yet it behaves unpredictably. That is one aspect of chance that the founding fathers of probability theory did not anticipate.

Here is another. A few paragraphs back I trotted out the standard philosophy that probabilities are really properties of ensembles, not of individual events. Along with that goes the idea that randomness is a property of a process, not of the events that the process produces. For example, the lottery machine can generate an ordered sequence like 1, 2, 3, 4, 5, 6, or a disordered one like 6, 11, 33, 35, 36, 44, with the process of generation being equally random both times. The IBM mathematician Gregory Chaitin has a different view: his theory can distinguish between an ordered sequence and a random one. A sequence of numbers is random if the information in it is "incompressible": it cannot be generated by a computer algorithm that can be described by a shorter sequence. This "algorithmic information theory" has spectacular implications for computability, and challenges the statistical view of randomness. For example, the digits of p are not random by Chaitin's definition, despite exhibiting no clear pattern. Why? Because the phrase "the digits of p", turned into an algorithm to find those digits, is a highly compressed description of the sequence. Randomness gives the classical view of probability: it is rooted in the past. The Jungles of Randomness opens up entire new vistas on the nature of chance and unpredictability: its sights are set firmly on the future. Both books are well worth buying, but they repay the reader in different coinage.

Ian Stewart is director, Mathematics Awareness Centre, University of Warwick.