To describe something as "nonlinear" is to describe it by what it is not. But let's be more direct. First, let's define "linear".

Simply stated, something is linear if its output is proportional to its input. If, when you're reading late at night, you want twice as much illumination (output) to see the book, then you double the number of light bulbs (input) by bringing over another similar lamp. If you want to buy twice as much buckwheat flour at the grocery store, you will pay twice as much.

Let's follow-up on this last example. Imagine that your store offers a bulk discount. Every additional pound of flour is 30% less that the previous pound. The incentive is to get you to buy more. It's a nonlinear incentive: the more you buy, the bigger the discount becomes.

A more realistic example comes from an ecology of animals that compete for food, but in which there is only a fixed amount of food available each day. As long as the population is small, all the animals get plenty of food. They grow and prosper, they reproduce and the population grows. But it can only grow so far. Once the population is beyond a balance with the available food, some animals do not get enough. Eventually they cannot reproduce and the population size decreases. In this ecology then, the population growth is a nonlinear function of the available food. At low populations, the growth is positive; at high populations, the growth is negative.

The concept of linearity is very closely related to that of reductionism. Reductionism is an approach to science that says that a system in nature can be understood solely in terms of how its parts work. How can this be? If the system is a linear composition of its parts this works great, since the system as a whole is proportional to each of its parts separately. But for many phenomena this doesn't work. For example, if you want to understand life, it is not possible to look only at the properties of the molecules in a living system. If the system is dismantled into all of the separate molecules, it is no longer alive. Life is nonlinear; death is linear.

Both linearity and reductionism fail, at least as general principles, for complex systems. In complex systems there are often strong interactions between system parts and these interactions often lead to the emergence of patterns and cooperation. That is, they lead to structures that are the properties of groups of parts, and not of the individual constituents.

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