A Modern Introduction to Fuzzy Mathematics

BChaos and Vagueness

There are “simple” systems whose parts are governed by well‐understood physical laws, yet these systems behave unpredictably. If vagueness is indeed a fundamental property of this world, then what is its role in this unpredictable behavior?

B.1 Chaos Theory in a Nutshell

It is a fact that weather forecasting is not easy. It is also known that we cannot predict how the weather will be in 10 or 20 days from now. What is not widely known is why this happens. Moreover, it is not known whether we can make our predictions more accurate and if we can predict the weather for long periods.

Edward Norton Lorenz [195] was a mathematician and meteorologist who wanted to use computers to predict the weather. He created a simple model, but soon he realized that when he slightly changed the input parameters, he was getting completely different results. For example, he noticed that, when his computer simulation was fed with decimal numbers having three decimal digits instead of decimal numbers with six decimal digits, the results were not the same as he would expect. One does not need to reimplement his simulation, which involved 12 ordinary differential equations, instead, one can observe the same effect using a much simpler mathematical expression. For example, one can use an iteration of the quadratic expression with initial value and . If, at some point, the iteration ...