Most of us understand the world by using common sense and intuition. To a large extent, this means assuming that most things behave in a roughly linear way: small changes in inputs lead to small changes in outputs, and if increasing an input variable leads to an increase in response, it will do so consistently.

Unfortunately, many of the systems we need to understand, such as the climate, ecosystems and financial markets, are complex. Complex systems often behave in very nonlinear ways, meaning that common sense and intuition fail.

A small change in an input can often lead to a dramatic change in the behaviour of a complex system: they have thresholds and breakpoints. Sometimes complex systems have negative feedback loops, in which the system essentially compensates for a change in an input variable. Sometimes, however, there is positive feedback and a small change leads to a cascade of effects resulting in a much bigger effect on the response. For example, if warming leads to melting of the permafrost, this will release methane, causing a further increase in temperature, leading to further melting permafrost.

Small changes may shift a system from one that returns to equilibrium when disturbed, into one that cycles, or even one that has unpredictable chaotic behaviour. Complex systems are also influenced by a large number of variables, often further confounded by random effects.

Using simple intuition based on linear assumptions to understand how these systems behave leads to serious misconceptions. Climate change “sceptics” often assert that because there has been a steady year-on-year increase in carbon dioxide concentration in the atmosphere but the temperature has not increased in a similarly steady fashion, this disproves the entire theory of anthropogenic climate change. In a complex system with random effects, one would not expect to see this simple relationship.

The only way to understand how complex systems behave is to use mathematical models. All of us use models all the time: the “common sense” intuition that systems are approximately linear is just a very simple form of implicit model. Because intuition fails with complex systems, the alternative is to use carefully constructed mathematical models.

Climate change deniers say “it’s only a model”, failing to recognise that they too have used a model, but a very simple and inappropriate one.

It is common to see the claim that, based on data, global warming has paused over the last 15 years. This statement is in fact based on a model: a linear regression model. The problem is that the model is inappropriate – you should not apply a linear regression to an autocorrelated time series. The way in which it is linked to data is inappropriate – you should not test for trend by starting at or near a maximum in the time series. And the interpretation is inappropriate – failing to reject a null hypothesis of no change is not evidence that no change has occurred.

Recently there have been enormous advances in constructing and analysing models of complex systems. Such models are behind everything from the design of the A380 super jumbo to the growth of your superannuation. Use of models relies on adequate knowledge of parameters – those numbers that specify how a change in one or more of the variables in a complex model affects how others change in response. Modern computing power has transformed the ways in which we can estimate parameters from data and how we can test whether the predictions made using models with these parameters are consistent with the real world.

This is not to say that current models are absolutely correct. As the statistician George Box put it, “all models are wrong, but some are useful”. Modelling of complex systems is a stepwise or iterative process, in which the results of one generation of models identify gaps in understanding, leading to improved models in the next generation. Our models will never be perfectly right, but they will become increasingly useful in managing the complex systems which affect all our lives.