“And the Sun stood still, and the Moon stopped, until the nation took vengeance on their enemies.” This verse is a translation of a text in the Book of Joshua, the sixth book in the Bible and the Christian Old Testament. In this story, Joshua led the Israelites across the Jordan River to the promised land of Canaan. After the crossing, he prayed that the sun and moon would stop until the Israelites took revenge on their enemies. His prayers were heard. The sun and the moon stopped with what they normally do: shining. There was a total darkness. Contemporary scientists believe this verse must be related to a solar eclipse. Calculations indicate that the only annular solar eclipse visible from Canaan between 1500 and 1050 BC occurred on the afternoon of October 30, 1207 BC [1].

A mathematical model for solar eclipses
A scientific model provides background on this biblical passage. When we combine various historical records of solar eclipses with our understanding of the Earth’s orbit and the moon, it is possible to create a model which calculates that there is a cycle of 18 years (6585,3 days to be precise) in which the Sun, the Moon and planet Earth are in the same position with respect to each other. And because the distance from the Moon to Earth varies, there are different types of solar eclipses: total, annular and hybrid solar eclipses. By incorporating this information into a mathematical model, we now have a formula which enables us to accurately determine when solar eclipses occurred throughout history and predict when they will happen in the future, thousands of years from now. This is an example of a mathematical model for a very stable system.

Another model

You can design a mathematical model to predict solar eclipses. Another model, portrayed in the diagram above, explains why sunlight does not reach the earth during an eclipse. Both the diagram and the mathematical model that calculates when an eclipse will occur or has occurred are simplifications of reality but with different objectives.

Models are essentially instruments to study or control reality; they are tools made for specific tasks. And because the conclusions that are derived from them are essentially simplifications, they don’t suffice to explain the whole, only parts of it. Yet, models are appealing. They can be applied to explain the past and predict future events. Furthermore, models can be used to understand phenomena that occur in a different location or different context. The example of the solar eclipse is very stable. However, not everything is as stable as our solar system.

Stable and unstable

A cyclist is asked to cycle from A to B at a speed of 16 feet per second.

This information can be used as the basis for a model, a graph that helps you calculate the distance the cyclist has covered and predicts how many feet and miles the cyclist will cover. After 1 second the cyclist has covered 16 feet, after 2 seconds he or she covered 32 feet, and so on. These calculations are valid in the Netherlands and Australia, at night and during the day, now and a 100 years from now, regardless of who is riding the bike, and whether the cyclist is going uphill or downhill. The model provides predictability, except when the cyclist is being obstructed by traffic or when he or she gets tired.

This example proves how natural processes are often influenced by a large number of sometimes unknown factors. These factors affect the course of processes, complicating our ability to calculate their outcomes by using models.

A gymnast weighing 55 kg jumps from a height of 1 meter. Calculate the forces that her knees must transmit when landing. The same gymnast makes a 2 meter jump. Now calculate the strain on the gymnast’s knees again. What happens when the gymnast makes a jump from 10 meters? She is likely to get injured. Hence, you cannot design a model that can be endlessly extended to all jumping heights.

A patient with heart failure shows a promising improvement in his heart function when administered a daily dosage of 0.5 mg digoxin (a medicine to increase the pumping function of the heart). Will this improvement double when 1 mg of digoxin per day is administered? Probably not. And when 10 mg per day is administered the treatment will cause a real risk of death. This shows us that dose-response relationships are often non-linear.

Edward Lorenz’s butterfly effect is a well-known metaphor for processes that have a high degree of complexity and are difficult to calculate. Lorenz was a mathematician and a meteorologist. With the advent of computers, Lorenz saw the chance to combine mathematics and meteorology. He set out to construct a mathematical model of the weather using a set of differential equations representing changes in temperature, pressure, wind velocity and other meteorological variables. When he repeated his calculations, he was surprised to see that the computer came up with a completely different result.

At first Lorenz assumed that a vacuum tube in his computer had broken down, but much to his surprise, there had been no malfunction. The hitch lay in the numbers he had inserted. His Royal McBee, an extremely slow and crude computer by today’s standards, had only six decimal places stored in its memory. To save space on the printout, only three appeared: Lorenz had entered the shorter, rounded-off numbers assuming that the difference was inconsequential. But as it turned out, in Lorenz’s particular system of equations, such small deviations have enormous ramifications, leading to a completely different weather forecast. Lorenz subsequently dubbed his discovery “the butterfly effect”: the nonlinear equations that govern the weather have such an incredible sensitivity to initial conditions, that a butterfly flapping its wings in Brazil could set off a tornado in Texas [2].

In other words: a small change in the initial conditions can result in large differences at a later stage. Please note, however, that this example of the butterfly effect only focusses on rounding of numbers. We’ll see later in this book that things get staggeringly crazier when you look at the truly emergent effects, the subject of this book. We will then look at interrelations rather than calculating with numbers.

An experiment has been conducted [3] in which 29 research teams consisting of 61 analysts, were asked to investigate the question whether soccer referees are more likely to give red cards to dark-skin-toned players than to white-skin-toned players. All research teams were provided with the same dataset (over 2000 soccer players in four national leagues at the highest level) to perform the analysis and were asked to design their own research model to demonstrate a possible correlation. Twenty teams (69%) found a statistically significant positive effect and nine teams (31%) were unable to demonstrate a correlation. This experiment suggests that the way in which a (research) model is designed partly determines its outcome. Even with experts with honest intentions subjective analytic choices are easily made. It is difficult, if not impossible, to rule out bias.

What is 2x sadness or 3x gratitude? How do you measure 2x the taste of cinnamon? Can the beauty of Mona Lisa be multiplied by 3?

In the real world, where irregularities lurk around every corner, you have to tread carefully when using models, because their reliability is innately limited. And in some cases models are simply not fit for the job to begin with.