Four Ways of Thinking

Introduction

Science and mathematics are, in large part, about finding better ways of reasoning. This book describes four ways of getting nearer to the truth.
Stephen Wolfram hypothesized that every process, biological or physical, personal or social, natural or artificial, lies in one of only four classes of behavior:
- Stable systems - Are those that reach and stay at an equilibrium, like a ball rolling to rest.
- Periodic systems - Are those that exhibit repeating patterns, like walking or cycle or our daily routines
- Chaos - Is unpredictable, life the weather or the roll of a dice or the flip of a coin
- Complexity - Is like the rise and fall of civilizations, the structure of governments or multinationals
Sumpter suggests there are only two types of worthwhile arguments:
- Class 1, which reach a stable resolution and class 4, where important ideas are discussed but may never be resolved.
- Class 2, recurrent bickering over the same point and class 3, chaotic back and forths, should be avoided
Classes 1-3 are about solving everyday problems. Class 4, complex thinking is more focused on introspection and self-reflection.

Statistical Thinking

American researchers after the war invented the methods for handling, processing and understanding information, and now we are using our skills to feed entropy to the masses.
While the mean is the average of a set of values, the median is the middle value in the set.
Statistical thinking about our health works.
There is just one correct way of making measurements and treating data, hidden within infinitely many incorrect ways.
We cannot know, from cross-country comparison data alone, which factors cause happiness or merely happen to correlate with it. We don't know if better healthcare or better social support causes an increase in happiness, or if nations in which people have developed a more positive outlook on life build better healthcare and social support. What we do know is that people in more stable, more prosperous countries with greater social support tend to describe themselves as happier.
Statistical significance is a measure of the probability that results of a study would have arisen by chance. Small differences can be explained by chance.
Effect size is the impact of the identified difference. If I buy this am I significantly happier? If not, then the effect size is small.
All three aspects - causality, statistical significance, and effect size must hold, for a study to have value.
In any good scientific research group or community, a balance is needed between the contrarians and the less individualistic majority who push for consensus. We want our hypotheses to be challenged, but we don't want to be paralyzed by uncertainty; We want to get as close as possible to the truth, given the limited time and resources we have to collect data and conduct experiments.
Many of the inspirational ideas which permeate our collective consciousness have only very limited application to you as an individual. Positive psychology interventions - like asking participants to write down all the good things that happened to them during a day - can be helpful for some, but they explain only around 1% of the variance between people.
Numbers are essential to understanding humanity, but they are not enough if we want to know about ourselves and those around us as individuals.
The way to find connections is to change our point of view. Instead of seeing the world from above, as if we were all-powerful and all-knowing, we should see it from below. We should realize there is more than one way of thinking about the world.

Interactive Thinking

Lotka's reaction is a reasonable model of how predators, like foxes, affect the population of a prey, like rabbits:
- R -> 2R (rabbits breed)
- R + F -> 2F (foxes eat rabbits and create more foxes)
- F -> D (foxes die)
The model goes round in cycles:
- Rabbits increase until there are too many foxes eating them.
- Then foxes continue to increase while rabbits decline
- Then foxes haven't enough to eat and both decline
- Until there are few enough foxes that rabbits can increase again and so the cycle repeats
They never reach stability. The interactions between the species take us on an endless cycle.
In systems of interactions, cycles are just as common as stability, and we can find them all around us.
The world is not stable. Interactions produce patterns that are more than the sum of their parts.
X + 2Y -> 3Y The "It takes two" rule - It takes two smilers to infect one non-smiler to make three smilers
The interactive view of the world is written in terms of chemical reactions.
Interactive thinking is more individual and personal than statistical thinking. It relies less on data and more on thinking through the consequences of our actions. It captures how people make the same choices and decisions as their friends, how conformity spreads through groups, and how our moods swing up and down. But it is no less scientific than the forms of stable, statistical thinking - it can even provide much more comprehensive answers to some of the most important questions in our society.
In epidemics, initial exponential growth leads to a large number of infections, but after some time the infectives start to recover: I -> R
As a result, the growth of infectives reaches a peak and starts to fall.
This is the SIR model (Susceptible, Infectives, Recovered), which leads to herd immunity.
We don't rely so much on data, but build instead on reasoning.
Culture, ideas, jokes, behavior and fashion are all contagious.
In a typical epidemic curve, the majority of infections occur in the middle. There is a fast-growing start, a middle where many people are infected, and a tail where the last people are infected. In any epidemic, you are much more likely to be in the middle section than at either extreme
I + R -> 2R - When people infected with a fad meet a recovered individual, they themselves recover more quickly, so social epidemics are different from virus epidemics.
Social contagion can be a force for good. We are continually sensing out each other's approval in order to find the right thing to do.
Divorce is possibly the most extreme form of social "recovery": we are more likely to end the most important relationship in our lives just because our friends have done the same.
Letting ourselves drift in the comings and goings of our interactions isn't irrational. What is irrational is a belief that things are better when they are stable.
For ant colonies finding food, small colonies always fail, large colonies always succeed, and for medium-sized colonies, success depends on how many find the food to start with.
Tipping points involve two stable states - one where almost no-one engages in a behavior and one where lots of people do.
The model of ants shows that multiple stable states, separated by tipping points, result from the "it takes two" infection reaction
If we want to make a change for the better, we need to increase the intensity of our interactions. It isn't enough to try something once, we need to build momentum within a group. Once we have the momentum, when we have reached the stable state where everyone is involved, then it will be easier to keep going.
The dream is to find a system, a way of capturing the essence of what we see. One set of reactions. One list of simple rules.
The second law of thermodynamics is the reason that all real-world chemical reactions eventually reach equilibrium: over time the reactions will become balanced and the molecules will become evenly dispersed. But for Lotka the second law did not hold for living systems, which were characterized by perpetual creation of new plants and animals.
Lotka wrote that his fellow humans were "accelerating the circulation of matter through the lifecycle, both by enlarging the wheel and by causing it to spin faster. Are humans driving some as yet unknown physical quantity toward a maximum? This is now made to appear probable, and it is found that the physical quantity in question is of the dimensions of power, or energy per unit time."
The second law tells us that in physical systems, things get more and more disorganized and random over time.
Elementary cellular automata are rules which tells us how to convert one string of bits into another string.
Rules can be deterministic (always doing the same thing) or probabilistic. Humans don't always do the same thing in the same situation. We are fundamentally unpredictable. The probabilistic rule captures some of our unpredictability.
Ultimately the only person that we can truly change is ourselves. If you change how your respond to others, the underlying rules of your interactions, then you will also change the outcomes of those interactions.
Top-down (statistical) thinking is top-down, starting with a theory and then looking at how well that theory explains the data.
Bottom-up (interactive) thinking starts with observations of how we think the world is, and generalizes these observations to a set of rules.
We always bring our own subjectivity to any question, in deciding which data to plot and which to ignore. This is why we need interactive thinking - to start from our own understanding and work forward and upward using logical reasoning.

Chaotic Thinking

Results reproduced from memory are always fallible. They can't be relied upon because the parts don't necessarily need to fit together. Memorized steps don't support each other. But if each step follows on from another in a logical manner, then errors become impossible. It's possible to be lazy (by avoiding memorizing the proof) and to never make a mistake. In maths, if you know the underlying logic then you are in control.
The 1950s engineer controls the world to make it more predictable and stable.
The opposite of positive feedback is regulatory feedback. This occurs when we suddenly decide to curb our consumption.
Mathematicians have found sensitivity to initial conditions for a wide range of rules in which small numbers multiply and large numbers become small. Whenever feedback is followed by a sharp regulation, chaos can occur.
Gradual, carefully planned changes succeed, where drastic measures fail.
Chaos is neither stable, settling down at a single point, nor is it periodic, repeating the same pattern over and over.
There are 10bn galaxies, and each contains 100bn stars. We will soon be 10bn people on the planet and we each have 100bn neurons in our brain. Each star, when it flickers, is a neuron firing, connecting to another neuron. Yet most of us never come up to the mountains to look into our own minds."
Margaret Hamilton pushed for simplicity, repeatability, and understandability. She coined the term "software engineering".
In Chinese philosophy:
- The yin is chaotic. It is passive, and allows itself to be dragged around by whims and desires.
- The yang is order. It is active and aims to control the future.
Order and chaos are intimately intertwined. The key is to recognize that trying to control the long term leads to over-regulation and even more chaos, but neglecting to control the short term leads to insecurity and even less order. Getting the balance right isn't easy, but recognizing that neither order nor chaos can live without the other is a good start.
Hamilton provided us with a yang, that of controlled engineering stability, but we still need a mathematical yin, the secrets of entropy.
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. Entropy is a measure of the amount of information in a string.
In general, the more unpredictable the string is, the longer the binary string needed to represent it. It is in this sense that randomness is information: random strings require longer binary encodings, because they contain more information.
There was no way for me to guess, just by looking at the bits which came previously, whether the next bit in this sequence should be a 0 or a 1. This meant that the entropy of this binary string was maximal, the middle column was completely unpredictable.
The trick to finding the right number as quickly as possible is to keep dividing the numbers into equal-sized groups at each step. When playing 20 questions, it is, in principle, possible to identify one out of over 1m different objects.
The number of questions we need to ask to guess a number is the same as the length of the binary encoding of that number, which is given by the entropy. The entropy of a number 1 to 20 is therefore 4.4.
Entropy measures:
- The number of questions needed to establish an outcome.
- The number of bits needed to encode a message about that outcome
- The randomness of that outcome.
The more uncertain or unpredictable a situation, the more questions we need to ask before reaching a conclusion.
Entropy always increases or remains constant.
The longer we wait between observations, the harder we have to work in order to find out the current state of affairs.
Everything we do, every day, as time goes by the entropy increases. No matter how well we know ourselves today, we cannot know what the future has in store for us.
The uniform distribution - has more or less even possible outcomes
The normal distribution - is a bell shape with more possible outcomes clustering in the middle.
A long-tailed distribution - trails off slowly to the right
The Poisson distribution - Goals in football occur infrequently during a match and the fact that a team has scored in the A7th minute does not influence the probability that they will score in any other minute. Goals are both rare and random.
How to measure the entropy of the English language? About half of what we write is predictable and redundant, but half of it remains unpredictable and random. It is here the information lies.
When things go wrong it is easy to blame those near to us who were involved in the decision. Chaotic thinking teaches us that there are always going to be large aspects of our world that are random, that cannot be predicted. Entropy is always with us, created by chaos. We can't predict it and there is nothing we can do about it.
How to think:
- The first step is statistical thinking. Know the numbers. But we also need to realize that the numbers don't usually tell us how we ought to act or interact with others.
- Interactive thinking helps us think through how your actions affect others and how you let their actions affect you. Understand why you are stuck in a cycle of doing things you don't want to do by looking at the underlying rules of your behavior
- Chaotic thinking helps us decide which aspects of our lives we aim to control and which we want to let go. Get ready to let go!

Complex Thinking

What Esther is describing is the Matrix, the unfathomably large nature of data in the modern world.
The noise of all the sports channels. The world will become increasingly dominated by that type of noise. Our concentration levels will drop, as we flip from one activity to another, unable to discern between the important and the trivial. News, sports, politics, gaming, opinions, facts, and fiction will all combine to one almost infinite source of entropy.
A true theory of complexity, which sat somewhere between randomness and chaos on one side and order and stability on the other, and which took into account interactions but moved past the simpler examples of predator-prey cycles and susceptible-infected-recovered models of disease. Such a theory needs a change of perspective, where we acknowledge that we live in a world of a trillion dimensions and make a determined effort to find a new path through them.
Kolmogorov: a pattern is as complex as the length of the shortest description that can be used to produce it.
Complexity depends on how good we are at explaining it. By proposing the theory of gravity, Newton replaced innumerable complicated explanations with a small series of concise mathematical equations to describe the motion of objects - which would accurately describe a wide range of future observations too.
Without the people trying to do the explanations, nothing is complex or simple. Science is the process of finding progressively shorter explanations for the phenomena around us. As scientists find these explanations, apparently complex things suddenly become simple. It is the things that are hard to explain which are complex.
In 1933, Kolmogorov proposed three axioms for probability:
- Events cannot have negative probability
- At least one event must have a probability of 100%
- If two or more events are mutually exclusive, then the probability that at least one of them occurs is the sum of the probability of each of them occurring.
Complexity is not a property of the output itself. It is the length of the program that generates or describes the output.
Saying that 170k people in London are homeless is not enough. It is a short explanation, but too short.
The secret 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.
Emergent patterns. Kolmogorov didn’t know that there were phenomena that looked complex but were in fact (according to his definition) simple. The whole of life itself could be the outcome of cellular automata-like rules. Biological life, with all its twisting, turning, dynamic structures, might be nothing more than the output of a simple updating rule. Even our own minds and consciousness might have emerged out of such simplicity.
Complex patterns lie on the edge of total randomness and order. Complexity is created at the edge of chaos and order.
Cellular automata were suggested by Von Neumann in the 1940, then by John Conway in the 1970s, with the game of life, were cells are in a 2D grid, and during each step:
- A living cell with one live neighbor dies (death by loneliness)
- A living cell with 2-3 living neighbors remains alive
- A living cell with 4+ living neighbors dies (death by overcrowding)
- A dead cell with 3 living neighbors becomes alive (reproduction)
- A dead cell with any other number of live neighbors remains dead
The model captures something of the essence of life: Isolation, overcrowding, and reproduction, and following these rules, we see structures:
- Die out
- Become stable
- Oscillate
- Move
There are only two states, living and dead, but living organisms can have more states.
Von Neumann wanted to find a self-reproducing automata, a system that produces children which then go on to have their own children. He saw this as the hallmark of biological life, and that computers could help explain the roots of biological organization.
There are programs with a Kolmogorov limit of 280 characters that can generate lifelike videos containing trees, mountainous forests, clouds, waves, and ancient and modern cities.
By describing systems from the bottom up in terms of local interactions between the individual units we can explain more complex patterns at a higher level. A bottom-up description of individual birds helps explain the apparent complexity of the movement of the flock. A bottom-up description of cancerous cells helps explain the growth of a tumour.
We are all part of a bottom-up system. Each of us follows our individual rules of interaction and out of that emerges the complexity of our society.
The global seating pattern in the reading room, a socially-distanced frozen state, is an emergent property of our social rules of interaction. Although we avoid sitting next to strangers, no one explicitly said that library seating should look like a slightly deformed chequerboard.
These physical structures we create - often for reasons of ease or efficiency - can form social boundaries that often feel much more entrenched than the people who are part of them intended.
individuals in a group where a third or less of people are the same sex as they are will move to a randomly chosen group where the majority have the same sex as them. This is a relatively weak preference on the part of the individual - everyone is happy to be in a minority - yet after everyone has moved, the majority of people are in all-male or all-female groups.
The way social structures emerge from individual interactions lets us cut through the complexity. The individuals are often behaving in a much simpler (or even unconscious or unthinking) way than it appears when we look at the structure as a whole. Once we have established the rules of interactions, we have reduced the apparent complexity to an understandable simplicity.
The way we interact, inadvertently and collectively, creates hard edges between us. It is your responsibility to see where they lie and to soften them.
A man who enjoys talking about football starts to search out all-male groups, a girl who is left out on the walk home starts to define herself as a loner, students who can’t sit still in a library conclude that studying isn’t for them, the person stuck in a traffic jam starts to see all other drivers as idiots. Ubuntu emphasizes that we are shaped as individuals by the systems we find ourselves in. We are people, through other people.
Individuals in crowds change the bottom-up rules that they follow, precisely because of how they feel inside the crowd. A concert or a pilgrimage is not just a physical mass of moving bodies but also a new sense of social identity, a connection to others.
People who identify with each other want to be closer to each other and are happier.
Our job as scientists is to use models to help us. Find short explanations of how the parts in the system interact. That way, you are getting to the essence of complexity.
An answer can be more important because it is funny and steps outside the system. It emerges unexpectedly from the way you pose the problem and tells us something we haven’t thought about before.
Wittgenstein: « My propositions serve as elucidations in the following way: anyone who understands me eventually recognizes them as nonsensical, when he has used them - as steps to climb beyond them (He must, so to speak, throw away the ladder after he has climbed up it. He must transcends these propositions, and then he will see the world aright. »
Once we have seen how a pattern emerges, we have transcended our understanding of the phenomena we are studying. We see the joke that the world was playing on us. Now the complex has become simple.
But complexity has not disappeared. Its form has just shifted with our new viewpoint. It is not the model itself that is important, it is the insight we gain from it. Once we have found some new understanding, we need to 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. Only once we realize the sheer scale of complexity can we start to navigate a way through its deepest secrets.
Maybe Richard has become sanctimonious as he points out how others can’t control their urges? Maybe Jennifer’s study of social structures made her feel a new sense of detachment, where she sees her fellow human beings as simply interchangeable points of larger, indiscriminate wholes?. Dealing with complexity is not just about finding the right model. It is also about a willingness to step out of that model and see the way it has changed us.
A repeating string, or any string with a pattern that can be described using an algorithm has a low value of complexity which can be referred to as K. The most complex string is a string that could only be described by itself. But how many of all the different strings we can write down cannot be given a simpler explanation that simply writing down the string itself?
Once the strings get long enough, the probability that you will find a pattern becomes zero. Complexity is the rule, rather than the exception.
It is, quite simply, impossible to imagine how many different ways your brain can be configured. It is true that our brain experiences only a tiny fraction of its possible configurations, but even so, we would need a very long string of numbers to represent all our experiences and brain states.
Mathematicians call the length of the string its dimension. Each and every one of us is a many-billion-dimensional object.
Personality tests show Extroversion vs introversion, intuition vs observation, thinking vs feeling, judging vs prospecting, assertiveness vs turbulence, etc, but this is far short of our true, billion-dimensional complexity.
Kolmogorov believed that maths was a balance between the trivial and the impossible. His ability to change his point of view was why his peers viewed him as a unique talent or genius.
He showed how increasing the speed of the boat led to a transition from stability through periodicity to chaos.
The complexity of a person does not lie in the abstract or the infinite, but in the finite description of what they do with their life. The more complex a person is, the harder we have to work to describe them.
1996 - The Sokal hoax
We can summarize people in a way that is much more accurate than the list of numbers (J is always learning, N is a control freak, B is an inquisitive listener). We should strive for a simple idea that allows a larger idea to emerge. Untimately, however we can’t know if the way we describe them is the best way. These descriptions change with context. We can never fully understand another person (or ourselves).
There is a thinking based on:
- Numbers - Do your research, collect the evidence
- Interactions - How do you respond to each other? Find a way of breaking the negative cycle
- Chaos - Is it better to let go or to take control of a situation? If you let go, then embrace the randomness. If you take control, then prepare your strategy as if you are landing on the moon
- Complexity - Remember that we are all part of a much larget social system: family, work, and society. And we all have our own innermost feelings that we can often never fully understand. Try to see everyone as an individual by finding the words that best capture who they are.
Kolmogorov had found that the meaning of life, the one that he had found, involved enriching his own internal thoughts and engaging with the complex internal lives of others. This was what made his life worthy. Words spoken sincerely to each other. And everything helped him in his movement towards clarity.