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!