The newly-published book *Four Ways of Thinking* by David Sumpter has a subtitle “*Statistical, Interactive, Chaotic and Complex*” which gives a big clue as to its contents and ideas. These four philosophies essentially map onto the four types of cellular automata identified by Stephen Wolfram, and it’s a fun ride seeing why this is the case, with plenty of interesting historical anecdotes and lessons for life along the way.

The best-known cellular automaton is John Conway’s Game of Life, where patterns following simple rules can combine together to give dazzling and constantly changing structures, and it has long been an exciting area of research to understand how this can happen, but it’s probably less well mapped out in the popular literature how these constructions relate to topics like chaos theory.

First, to get it out of the way: yes, David Sumpter kindly blurbed my own book *Numbercrunch*. However, this review is absolutely not one of those log-rolling *keiretsu*-type things where people “spontaneously” recommend one another’s books without mentioning that they share a publisher or an agent. In fact, David (@soccermatics on Twitter) was one of the first people I had professional interactions with on that site, which led to him hosting me on a visit to Uppsala back in 2015. I’ve long admired the work of his research group, and enjoyed all his previous books, and he (like all the other people we approached, including

With that disclaimer done, onto the book! Beyond its division into four parts, *Four Ways of Thinking* has a somewhat unusual (though not unwelcome) structure, where the mathematical material is layered with fictional components. These come through stories about a group of friends in London applying the lessons of the book to their own lives, and an account of David first learning this material at the Santa Fe Summer School in 1997. Personally, I enjoyed this - though some technically-minded people might feel it gets in the way of the meatier material of the book, there’s plenty of detail for those people to get their teeth into too, and a more general reader will appreciate seeing practical applications of the theory.

Each of the four ways of thinking is partly described through the lives of some of the people who pioneered it. I’ve already written on this Substack about Claude Shannon and Andrei Kolmogorov, so it’s great to see them in a starring role. But there are also nice explanations of the work of Ronald Fisher (in the statistics part), Alfred Lotka (in the interactive part) and Margaret Hamilton (in the chaotic part), among others.

There’s a sense in which the book builds up hierarchically and historically across the four topics. So for example, the section on statistical thinking comes first, and describes the classical work of Fisher in dealing with static, “independent identically distributed” type problems. But of course statistics is a living subject that has adapted to the more challenging problems of today, and which underpins machine learning and AI - modern statistics has plenty to say about situations which are interactive, chaotic and complex as well! However, within the structure of the book, I think this presentation restricting itself to older work makes perfect sense.

Having dealt with Fisher and statistics, the book moves onto interactive systems - those governed by evolution equations such as Lotka’s models of ecological populations and the SIR equations which underpin epidemic modelling. There’s a nice account here of phase portraits (which long-term Twitter followers will know that I’m a big fan of), which naturally leads into the third way of thinking - chaos. The book draws a nice connection between the chaotic evolution of processes studied by scientists like Edward Lorenz, and the work of Margaret Hamilton in trying to eliminate unpredictability from NASA’s code for the Apollo missions.

It’s fair to say that David’s own heart leans towards the final way of thinking, the complex systems paradigm. This is not to say that he underplays the value of the other approaches, but simply that his own research work, whether on the emergence of animal swarm behaviour from simple rules or on tracking the patterns in Barcelona’s midfield, is inspired by this fourth philosophy. I have to admit to being somewhat more ambivalent. This may be because this review of Wolfram’s *A New Kind of Science* is one of my all-time favourite academic hatchet jobs:

it is my considered, professional opinion that A New Kind of Science shows that Wolfram has become a crank in the classic mold, which is a shame, since he's a really bright man, and once upon a time did some good math, even if he has always been arrogant.

Or perhaps it’s because of the (ironic) use of simple linear extrapolations by certain complex systems researchers during the COVID and monkeypox outbreaks giving the subject a bad name:

Nonetheless, Sumpter himself is a great advocate for the complex systems approach, and gives a clear account of the emergence of complicated behaviour from simple rules. In particular, he draws a fascinating parallel to the work of Kolmogorov, who sought to quantify the complexity of patterns in terms of the length of the shortest computer programme which can generate them. This idea, which has something of the flavour of data compression about it, builds on Shannon’s own work and has always fascinated me, and it’s great to see a popular account of it here. In this way, the fact that one of Wolfram’s simple cellular automata can act as a universal computer (in the sense described by Alan Turing) can be seen as a great pinnacle of complex systems research.

Overall then, I learned a lot from *Four Ways of Thinking, *which* *gives a very readable account of a large body of modern mathematical research and is a worthy addition to David Sumpter’s catalogue of excellent popular books, and I think anyone who reads this Substack would enjoy it too.