How to be Nonlinear

In the previous sections we have repeatedly indicated how linear concepts collapse under the weight of uninterpretable data and are replaced by nonlinear ones. Examples include the existence of non-integer exponents in the functional representation of data, long tails in the distribution functions, saturation phenomena and rhythmic behavior in open (biological) systems. Each has obliged us to seek a nonlinear representation of the underlying process. In each case we observed how a comfortable linear notion was inadequate to describe a particular phenomenon due to the existence of one or more nonlinear mechanisms and therefore had to be discarded. An example we considered is the amplification mechanism responsible for the log-normal distribution being replaced by a Pareto tail at the high end of the income distribution [cf. (§3.2)]. The log-normal distribution relies on a linear interpretation in terms of the logarithm of an individual’s income. The amplification mechanism at high incomes is nonlinear, however, and forces the replacement of the linear-based log-normal distribution with the nonlinear-based Pareto power law. The practical question remains: “How does one learn the techniques that are of value in one’s field without becoming burdened with perhaps interesting but nonetheless irrelevant mathematical baggage?”