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Understanding science: what we cannot know
Understanding science: what we cannot know

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2 Chaos

When rolling a die, you don’t know in advance how the die is going to land. You simply roll the die and hope for the best. As you’ve seen though, when you roll two or more dice, some outcomes are more likely than others.

But what if we could predict the way the die is going to land? It would cease to be a game of chance at all. For this to be possible, you would need to know everything about the die and the environment in which it is going to be rolled, as well as having the scientific theory and mathematics to predict its motion.

As it turns out, we do at least have the mathematics. This ‘differential calculus’ – which is concerned with the rate at which quantities such as position and speed change – has been successfully applied to many problems involving motion for several centuries. Such problems vary from calculating the rate of spread of infectious diseases to predicting the positions of the planets, and even designing video games. It was invented independently by both Isaac Newton and Gottfried Leibniz at the end of the 17th century.

This is a composite of two portraits, of Isaac Newton and Gottfried Leibniz.
Figure 5 (a) Isaac Newton (1643–1727), (b) Gottfried Leibniz (1646–1716)

Of course, each problem has its own particular complexities: the motion of a ball rolling down a smooth slope is quite different to the motion of a ball bouncing on rough ground. And it is one thing to know something in theory, and quite another to apply it in practice. Even getting to grips with the theory can take you in unexpected directions. One of the wonderful things about mathematics is that you can start out asking a question in one area of mathematics, and in the process of looking for a solution you end up somewhere else!

In this case, it turns out that trying to predict the motion of the dice will lead us from the 17th century mathematics of Newton and Leibniz to the 20th century mathematics of chaos theory.