# Do dice play god?

Stewart considers six ages of uncertainty. Clearly people have always been fascinated with the future and have tried in many ways to remove its inherent uncertainty. So there was an age of belief in external powers such as gods, oracles, horoscopes and reading the future from the bowels of a slaughtered goat.

With the development of mathematical instruments like Newton's laws of motion it was possible to predict the trajectories of the planets and to describe the dynamical behaviour of objects on earth. This triggered the conviction that we were living in a deterministic mechanical world organized like a clockwork and that everything was predictable provided that we could measure all initial conditions and all the parameters involved. Uncertainty still existed but only as the consequence of our inability to measure everything with sufficient precision.

Gambling is also something of all ages, but since around the sixteenth century, patterns were observed and ideas of frequencies of random events and the probability of how often they can be expected became useful tools for gamblers and they were successfully applied. Cardano, Fermat, Pascal, Huygens, and Jakob Bernoulli developed the basics of the theory and coin tossing and rolling dice became common instruments to generate random sequences. Observation errors were also considered to be a random phenomenon. Their analysis showed the bell shape of the normal distribution when sufficiently many are accumulated. Linear regression and least squares fitting were born. Quetelet started to apply this kind of analysis, originally used by astronomers and physicists, to social and other data, and this became the origin of expectations and an abstract, non-existing, "average person" was distilled from the data. This is how gradually statistics came about. But there were problems since probabilities seemed to change depending on prior knowledge leading to fallacies and paradoxes contradicting common sense. Bayes eventually formalized all this with formulas. Ardent discussions about Bayesian versus frequentist interpretations were the result. Still today many counter-intuitive results can be the origin of a lot of Fake News.

The fourth age started at the beginning of the twentieth century when mathematicians thought to have uncertainty well under control. But then nature forced quantum mechanical mysteries upon the experimental physicists, and uncertainty became an inherent property of the world we live in.

But also in mathematics, uncertainty was reintroduced when mathematicians started to model nonlinear dynamical systems. These are deterministic, but they can be supersensitive to tiny perturbations, a phenomenon popularized as the butterfly effect. Thus in this fifth age even deterministic systems became unpredictable.

The sixth age is the age we are living in today. Since uncertainty is not going to go away, mathematicians and scientists are trying to manage uncertainty. Sometimes we can even use it to our advantage, but there are still many open problems to solve.

It is clear that there is no crisp boundary between these periods. There are for example even today still people believing they can read the future from tea leaves or they think they get messages from "the other side". Therefore also Stewart cannot separate and treat these six ages in a strict chronological order. The eighteen chapters are more thematic and some topics may require to go way back in time to trace the origins. However, as we read on, we see how insights into uncertainty is growing and how we can bring it somewhat under control.

Stewart has written many books already and knows better than anyone else how to bring a story about mathematics to a broad audience. So the mathematics are painlessly made crystal clear. What is most interesting here are the side tracks. Among these I count the fact that physically throwing a dice or tossing a coin is not as random as one would expect. There are also these seemingly impossible results like if a family has two children and you know one of them is a girl, what is the probability that they have two girls, which is very different from the problem where you know that the eldest is a girl. Stewart also gives some confronting examples of people found guilty in court based on wrong statistics.

On a more theoretical side there is a good discussion of difficult concepts like entropy, information and the arrow of time. Of course quantum physics is more difficult and requires a rather extensive discussion. Entropy as well as quantum theory is still today subject to different interpretations and Stewart adds his own vision to the discussion. He is also explicit about Bell's theorem (1964) which shows that the EPR (Einstein-Podolsky-Rosen) paradox is inconsistent with the theory. Stewart explains some loopholes in Bell's theorem that have been raised and adds some of his own.

That known uncertainties are most influential on our modern society is illustrated with other examples. Strange attractors and the dynamics of weather forecasting are explained, and how climate change cannot be denied, and what mechanism is playing in the consequential disasters, the so called extreme events. We are still suffering from the consequences of the 2008 crisis in the bank sector, and there are heated discussions by people objecting against vaccination. These are two other important topical issues of our society today. Finally it is shown that for simulations it is important to generate random sequences, but it is a complicated problem to generate one that is "as random as can be". And how should randomness be measured anyway?

So this is far from a dull introduction to probability theory and statistics. It is a lively story with historical roots but with many relevant references to how managing uncertainty is important for our everyday life, as well as for the big challenges that our society is facing today.

**Submitted by Adhemar Bultheel |

**25 / Nov / 2019