Ian Stewart's "Do Dice Play God?"

Ian Stewart was born in 1945 and educated at Cambridge (MA) and Warwick (PhD). He is an Emeritus Professor in the Mathematics Department at Warwick University, where he divides his time equally between research into nonlinear dynamics and furthering public awareness of mathematics. He is also an Emeritus Professor of Gresham College, London. He has held visiting positions in Germany, New Zealand, and the USA. He has five honorary doctorates (Open University, Westminster, Louvain, Kingston, and Brighton) and is an honorary wizard of Unseen University on Discworld.

Stewart is best known for his popular science writing—mainly on mathematical themes.

He applied the “Page 99 Test” to his new book, Do Dice Play God?: The Mathematics of Uncertainty, and reported the following:

The top half of page 99 finishes off a discussion of two main current approaches to probability and statistics: Frequentist and Bayesian. In the Frequentist approach, which until recently was dominant, the probability of an event quantifies the proportion of times it occurs, in the long run, after many repeated trials. In the Bayesian view, which goes back to the Presbyterian Minister Thomas Bayes in 1736, probability is a measure of how strongly we (should) believe that something is true.

The bottom half of page 99 looks at one area where Bayesian methods have a serious impact on people’s lives: criminal trials. More to the point, the misuse of Bayesian methods — notably conditional probability, the likelihood of some event occurring, given that something else has already occurred. The Prosecutor’s Fallacy confuses the probability of some event occurring to a randomly chosen person with the probability of the accused being guilty, given that the event has occurred.

The text here reads:

A court of law might seem an unlikely test ground for mathematical theorems, but Bayes’s Theorem has important applications to criminal prosecutions. Unfortunately, the legal profession largely ignores this, and trials abound with fallacious statistical reasoning. It’s ironic — but highly predictable — that in an area of human activity where the reduction of uncertainty is vital, and where well-developed mathematical tools exist to achieve just that, both prosecution and defence prefer to resort to reasoning that is archaic and fallacious. Worse, the legal system itself discourages the use of the mathematics. You might think that applications of probability theory in the courts should be no more controversial than using arithmetic to decide how much faster than the speed limit someone is driving. The main problem is that statistical inference is open to misinterpretation, creating loopholes that both prosecution and defence lawyers can exploit.

The page 99 text doesn’t work very well for Do Dice Play God?, even though the chapter concerned covers a very important topic. The reason is that no one-page selection, except perhaps from the opening chapter which outlines the contents, can convey the broad scope of this particular book. The one feature that does generalize to the rest of the book is the link between mathematical theory and human impact; in this case, that a court can convict an innocent person of a serious crime — such as a mother murdering her own children, or a nurse murdering dozens of hospital patients — on the basis of flawed mathematics, even when there is absolutely no other corroborative evidence.

The book is about a much broader topic: uncertainty. It concentrates on the many different mathematical techniques that have been developed, over the ages, to help us manage uncertainty, reduce it, remove it, or exploit it. Probability and statistics represent only one of six ‘Ages of Uncertainty’ that provide a loose organisational structure to a widely ranging discussion. Page 99 gives a false impression because its scope is too limited.

In the first Age of Uncertainty, we were at the mercy of the natural world, subject to fires, floods, earthquakes, famine, hurricanes, and tsunamis — not to mention the unpredictable ravages of other people, such as an invading army. Unable to control these things, an evolving priesthood invented belief systems, attributing such events to the will of the gods. The priests claimed the ability to predict what the gods would do, or even to influence their decisions, based on methods such as examining the liver of a sacrificed animal.

This first age is still with us, perhaps in more sophisticated forms, but for most practical purposes it has given way to the second Age of Uncertainty: the scientific method. Planets don’t wander about the sky according to godly whim: they follow regular elliptical orbits, aside from tiny disturbances that they inflict on each other. Uncertainty is merely temporary ignorance. With enough effort and thought, we can work out the underlying laws and predict what once was hidden from human knowledge.

Science forced us to find an effective way to quantify how certain or uncertain an event is, and how errors affect observations. This opened up a new branch of mathematics: probability theory. The theory grew from the needs and experiences of gamblers, who wanted a better grasp of ‘the odds’, and astronomers, who wanted to obtain accurate observations from imperfect telescopes. Probability, and its applied arm of statistics, dominated the third Age of Uncertainty, and led to a revolution: the application of statistics to large-scale human behaviour.

The fourth Age of Uncertainty arrived at the start of the 20th century. Until then, it was assumed that uncertainty reflected human ignorance. If we were uncertain about something, it was because we didn’t have the information needed to predict it. New discoveries in fundamental physics forced us to revise that view. According to quantum theory, sometimes the information we need simply isn’t available, because even Nature doesn’t know it.

The fifth Age of Uncertainty emerged when mathematicians and scientists realised that even when you know the exact laws that govern some system, it can still be unpredictable, because unavoidable errors in observations can grow exponentially and swap the true prediction. This is ‘chaos theory’, and it explains such things as why weather is so unpredictable, even though we understand the basic physics that it involves.

We have now entered the sixth Age of Uncertainty, characterised by the realisation that uncertainty comes in many forms, each being comprehensible to some extent. We now possess an extensive mathematical toolkit to help us make sensible choices in a world that’s still horribly uncertain. ‘Big data’ is all the rage, although right now we’re better at collecting it than we are at doing anything useful with it. Our mental models can now be augmented with computational ones.

The story of these six ages spans a wide range of human activity, and many branches of science. In particular, quantum uncertainty is still not properly understood, mainly because we don’t really know how to model an observation of a quantum system. I cover this ground in two chapters: first the orthodox story, then the unorthodox alternatives currently emerging. The topics in the book range from reading entrails to SatNav, from gambling with dice to fake news, from statistical regularities in human behaviour to the widely misunderstood difference between weather and climate.

We’re beginning to recognise that the world is much more complex than we like to imagine, and everything is interconnected. Every day brings new discoveries about uncertainty, in its many different forms and meanings, and new methods to help us deal with it. The science of uncertainty is the science of the future.

Visit Ian Stewart's website.

--Marshal Zeringue