This blog post is a quick summary of the book “Four ways of thinking” written by David Sumpter

Context

This is the first book that I read during 2024 winter holidays and I have thoroughly enjoyed it. When I first bought the book, I was not sure on whether the book will be worth the time. I thought it might be related to books like Debono’s Six thinking hats and might contain some generic guidelines to thinking. However given the author is a mathematician, the book is a treat for people who love stats and math or for those who love to read popular science books.

The key idea of the book is abundantly clear from the title of the book, i.e. there are four ways of thinking; statistical, interactive, chaotic and complex. The contents of the book are spread over four parts, each part dedicated to one specific type of thinking

The author makes the book more conversational by weaving all the concepts around a few fictional characters who are attending a workshop at Santa Fe. These fictional characters are attending a series of lectures on statistical, interactive, chaotic and complexity lectures at Santa Fe. It is in this setting that the author tries to explain all the important concepts around each of the ways of thinking. I think it is a pretty nice hack to make the book more accessible.

The author claims that the origin of the book came from Wolfram’s classification of patterns - stable, periodic, chaotic and complex. The book is an attempt to distill the key ideas behind the patterns in such a way that they can be applied to our own day to day lives.

Statistical Thinking

In this type of thinking model, one looks to get a grasp on summary statistics, sample estimates, sample distributions to get a broad sense of “what’s going on with the study population ?”. Asking relevant questions, conducting experiments, collating data, analyzing the data and communicating the data is at the heart of Statistical thinking.

The following are some of the main points mentioned in the first part of book:

• Importance of summary statistics such as mean, median and mode
• Maximum Likelihood principle explained a small sample surveyed who respond to their liking for gherkins
• Lady tasting tea experiment conducted by Fischer that highlights the importance of setting up rigorous statistical tests to validate hypothesis. The author takes a lot of pain to illustrate the key idea from Fischer and does it in a way that anyone with a little bit understanding of probability will understand it.
• Ronald Fischer contribution to the design of experiments, maximum likelihood estimation and other aspects of statistical theory. However there is as much to be learnt from the failures of Fischer as much as success stories of Fischer’s work. Fischer neglected causality and failed to consider the small effect size of the studies which supported his own world-view. Instead he used his skills in statistics to push us towards certain ways of seeing a complex world through an overly simplified lens. He presented his own prejudices - that smoking wasn’t bad for us and stupid people should not breed
• Statistical significance is a measure of probability that results of a study would have arisen by chance. This is not same as effect size. Effect size is that given there is a statistical significant factor, how much variation does it explain ? Also the test could report statistical significance, effect size but no causality
• Forest and the tree: The author talks about various ways in which statistics has been abused and communicated in a way that misinforms than informs. Fischer used statistics to propagate lies about smoking and cancer. He also used to justify a dangerous theory of eugenics. There is a also a mention of Angela Duckworth’s grit theory and Carol Dweck’s growth mindset in the same vein. Meta studies looking for the effect have not found significant variation explained by grit and growth mindset. Does that mean these studies are wrong ? The piece of advise from the author is that one must not confuse the forest from the tree. The findings apply to the forest and it might or might not apply to you, an individual tree in the forest.

Interactive Thinking

Interactive thinking is different from the statistical view of the forest of people. It is more individual. More personal. More related to our everyday experiences. It relies less on data and more on thinking through the consequences of our actions

In this type of thinking models, one needs to look at equations governing interaction patterns between various model constituents, frame the interactions in the form of a differential equation and solve the ODEs or PDEs to get to general and specific solutions based on initial conditions.

The author starts off with describing Alfred Lotka’s life as a undergraduate student saying that Lotka was pretty bored with chemical equations as the equations were always balanced and any experiments done in the lab lead to some stable condition. He could not reconcile this sort of chemistry with the theories propagated by Darwin and other philosophers whose work buzzed with variation, uncertainty, unending circle of life. In his work, he starts playing with equations describing various entities and ignore the most fundamental aspect of a chemical equation, i.e. equations being balanced.

Lotka had realized that by ignoring chemical balance, ignoring stability, he could produce the pattern he was looking for: he could produce the cyclical dynamics of life itself.

Lotka started off with a simple set of equations that were fundamentally unbalanced but could explain a few patterns like population of foxes and its prey, rabbits in a jungle.

In the words of the hypothetical professor, Parker, created by author, here is the key idea behind Lotka’s method

Parker explained that the key to Lotka’s method was describing how each individual component of the system affected the other components. In the example on Parker’s blackboard, the components were foxes and rabbits. When neuroscientists create a model of brains, the components are the neurons themselves and the chemical and electrical signals they send between them. In modelling insect swarms and bird flocks, the components are the animals. And in modelling our societies or economic systems, the components are us – individual humans.

Parker said that the father of modern economics, Adam Smith, had been wrong, because his stable thinking had convinced him that the market would reach and stay at equilibrium. But Smith’s thinking was, Parker said, reductionist. Accounting for our interactions, the way we also behaved like animal herds, showed that human society was anything but stable. We experience the same ups and downs as rabbit and fox populations. We are in constant flux.

Here are some of the points mentioned in this part of the book

• Susceptible, Infectives and Recovered Model (SIR) model used in epidemics
• Social epidemics are different from virus epidemics

• An example of social epidemic that we witness: A common public relations trick used by companies when viral ‘bad news’ appears about them is to place a follow-on story in the media that is similar to the original bad news but with a more positive slant. This new story not only presents the company’s point of view, it also exploits social recovery. When those infective people who have heard the original ‘bad news’ story tell it to those who heard the ‘positive slant’ version, they feel that they are sharing yesterday’s news. The story the infectives heard now appears less original than this new version, and they stop telling it. The trick to controlling the news agenda is to focus less on the infectives – those spreading the ‘bad news’ – and more on shifting susceptibles to recovered, thus dampening interest in the original story.
• The approach taken in those early days by Lotka, Ross and Hudson is widely applied across all types of biological systems
• Connection between cellular automata and Integrative Behavioural Couple Therapy (IBCT)
• Looking back, one hundred years on, even if there is no grand unifying ‘third law’, there is another way to see Lotka’s way of thinking – interactive and cyclic – as a success. Today, scientists use a variety of ways to talk about class I and class II thinking. Class I thinking is sometimes referred to as top-down. It starts with a theory and then looks at how well that theory explains data: does smoking explain cancer? Does life expectancy explain happiness? Class II thinking is more bottom-up. It starts with observations of how we think the world is – foxes eat rabbits, couples sometimes argue, it takes two people to start a health craze, we influence each other’s political opinions – and generalizes these observations to a set of rules. We then derive the consequences to create a theory. Using this approach, we don’t start with a cloud of data points from surveys – like we do when applying statistical thinking to health or happiness. Instead, we start by trying to understand the essence of a system: how it works, what are the key components, how do they fit together and when do they fail? From there we make predictions – predator‒prey cycles, bouts of shouting, tipping points in beard growing and exercising, political polarization. It is after we have made these predictions that we test them on data from the real world.
• Neither class I nor class II is always right or always wrong. We need to think in both of these ways.

Chaotic Thinking

Chaotic systems are characterized by three elements - positive feedback, regulatory feedback and small perturbations. The author provides a wonderful array of examples from our daily lives to show how chaos emerges from simple actions

The following are some of the points mentioned in the book

• Story of Margaret Hamilton and her work with Lorentz and NASA. She invented the term “Software engineering” at NASA.
• Bar problem that looks like a periodic system but is a chaotic system
• A guy tries to cut cake consumption by drastically reducing intake only to find that he is gobbling more cakes in the future than he would have anticipated. This behavior has got to do with the way chaotic sequence of cake consumption emerges. Instead he is better off choosing “moderation in small steps” as an alternative solution to his cake eating binge. A better solution is to aim to first stabilize and then slowly deflate behaviors we want to avoid.
• The irony is that it is our attempts to regulate ourselves that are creating the chaos. We all fall into this trap: we decide not to use social media for a week; we stop drinking alcohol completely for a month; or we decide to get out running and immediately set off at our top speed around the park. All these extreme responses are a form of regulation or control. But they are also exactly the type of control that generates chaos.By recognizing that regulation can create chaos, we can find an alternative method for getting things back on an even keel
• Edward Lorenz and his discovery of initial condition sensitiveness to the way a system evolves
• Butterfly effect
• The author introduces the yin and yang of life using the stories of Margaret Hamilton and a fictional character Lily-Rose. The former desires for control and the latter gives in to chaos. Using a story of fictional wedding planner and her chaotic husband, the author emphasizes that neither approach alone is sustainable. We need a health dose of order and disorder in our lives to make it meaningful and sustainable
• Claude Shannon’s information theory to mathematically characterize randomness
• A message where all letters occur equally often contains more information than a message with lots of repetitions of the same letter because we can’t find a shorter encoding for the former.
• The way to deal with randomness and chaos in our lives is the game of twenty questions
  • There is a direct correspondence between the coding created and the questions we ask. The more likely the letter, the fewer questions we need to ask in order to find out what it is.
  • The minimum number of questions needed to identify an object is bounded by Shannon entropy of the system
• The most efficient strategy in twenty questions is to ask questions that split the remaining possibilities into two roughly equal groups. This maximizes the information gained from each question. Similarly, in information theory, an optimal coding strategy minimizes entropy by designing codes that split the uncertainty in the system as evenly as possible
• Twenty questions is a practical embodiment of entropy and information gain showcasing how uncertainty can be systematically reduced through binary decisions
• Practical application of entropy and game of twenty questions: Those with more unusual problems, you have to listen more intently
• Teacher spending same amount of the time to every student does not mean he/she is fair. The fact that there will be certain students who are not typical and one needs to spend more time with them to uncover their problems is another example of information coding
• Chaos means that there is no point trying to work out the initial conditions or following the dynamics step by step oe even knowing how many steps have been taken. At that point we should simply ask questions.
• We cannot always live our lives as if we are sitting in mission control. We have to let go. And when we let go, entropy increases. But letting o produces a new possibility. A way of seeing the world not in terms of certainty but as something blurred out, as a distribution of possible outcomes.
• Randomness does produce reliable distribution of outcomes. It is not something that is totally unpredictable

Complex Thinking

The last part of the book is probably the most difficult part of the book to understand. I guess it goes with the spirit of the thinking style - complex.

The author starts by mentioning Kolmogorov, the Russian mathematician who was the first one to define complexity by stating

A pattern is as complex as the length of the shortest description that can be used to produce it.

In the previous thinking patterns described in the book, we had to break down the problems top down or bottom up or understanding that some of them were driven by randomness(to be again analyzed). The last part of the thinking essentially is saying that there are somethings that are difficult to simplify or break down. We might not be able to break it down but we could atleast measure it via Kolmogorov’s concepts and later developments

Through the eyes of fund raiser, the author says

The more our audience hears about a particular person, the more stuck they become in the details of that story. But we cannot expect others to create their own internal experience if the only seed they are given is numbers. The secret then, to capturing complexity, is to find relatable stories that are both personal and capture variety, then let them grow within the intended audience. We don’t need to tell every detail of the story we want to convey.

The following are some of the points mentioned in this part of the book:

• Using simple cellular automata rules, one can produce a pattern that is neither stable, periodic or random. Some of these lines and squiggly structures that come out of such initial conditions are known as emergent patterns
• Complexity is created at the edge of chaos and order
• Complex patterns, he said, can emerge from the simplest of rules
• The approach of describing local rules of interaction to help explain emergent patterns has during the last twenty-five years become an important part of every area of science.
• Examples of emergent behavior - people sitting in libraries, group of children walking home, people interacting in an office party
• Once we have found some new understanding, now look for further ways to climb through complexity. The nature of complex systems, with their many sides and deep intricacies, means that they always throw up new problems and new questions. We can’t ever satisfy ourselves with one view or one insight. We need, time after time, to restart the process, laugh at our previous attempts and start again.
• Complexity within ourselves
• Complexity is the rule, rather than an exception

Takeway

Loved the way the author describes various types of thinking. If you are a person who has built any sort of model, be it a physical model, computer model, statistical model, NLP model or XYZ model, you might have a far richer understanding of “what a model means ?” after reading this book. After all, one can easily interchange the word “model” with “thinking” as the former is a specification of the latter. Well worth my time in reading through the book. It reminds me of the book by Scott E Page that was equally a great read about various types of models.