By the end of this article, I promise you will understand what scientists usually mean when they talk about ‘complexity’ which is also what we mean by complexity when we talk about it on this website. By the way, as I will explain, it is not the same as 'chaos theory' but is related.
So, where to begin? Complexity is a big subject and includes a lot of different ideas. It can be confusing because people often mean different things by the term ‘complexity’. I will explain here what ‘complexity science’ means to most scientists which is a bit different to what ‘complexity’ means in an everyday sense. I should note however, that this is my approach to explaining complexity science. Some of these things are still disputed because complexity science evolves and involves disagreement too.
So the way I would explain it is this: Scientists understand lots of things with theories that seem complex but in a sense are simple. Things are simple to a scientist when they can understand the system they are studying using mathematics that is hard to learn but once understood explains very concisely how a system behaves. In a way, complexity to a scientist is actually not that different from what people normally think about when they refer to complexity in the everyday sense.
Before computers, natural laws or theories of nature had to be understood and represented by human minds alone, aided only by pen and paper, chalk and blackboard. Even if that meant understanding difficult mathematics like calculus, complicated geometry or statistics, it didn’t require computers for humans to make progress. In that sense, all of science was still 'simple' rather than ‘complex’. This approach was extraordinarily successful for a long time. It turned out that lots of systems in nature can be represented by some pretty short equations (e.g. E = MC squared) and so scientists made lots of progress with 'simple' theories. Here are some examples: Newton and the law of gravity, Darwin and the theory of evolution, Einstein and the laws of relativity. This 'simple' approach even includes the physicists and mathematicians who initially developed quantum theory and theories of sub-atomic particles. All this (amazing!) scientific progress was made by working with equations that were hard to develop and discover but could be easily calculated and understood by human minds alone to successfully predict how nature behaved.
However, when computing began in the 1950s some people began to realise that we could now use computers to try to explore problems that previously were impossible for human beings to calculate. This is because there were lots of equations where to even begin to make enough calculations to be able to say what the system or aspect of nature they were interested might be doing just would take far too long for a human to do the calculations.
For example, one of the first uses of computers was to improve weather prediction. Before computers, people had to look at the current weather and compare it to a catalogue of previous weather charts. They methodically and laboriously went through their files and predicted the weather by looking to see which of their thousands of historical charts looked most similar to today's weather. This was despite the fact that most of the physics and mathematical equations of the weather was very well understood. The problem was that although the equations were known, they were impossible for even a huge team of people to calculate before the weather changed and all that work was irrelevant! As many people now know, the weather is 'chaotic'. Chaos maths uses well-understood equations such as those used to model the weather, but where any tiny change in the values you put into those equations can often change the prediction completely. It was only because of computers that the equations that are so sensitive to the values put into them were discovered. The fact that weather is actually chaotic was therefore discovered because of computer programs and so the mathematics of chaos can be viewed as closely related to the science of complexity. They both come out of starting to use computers to make calculations about nature.
Nowadays, complexity goes far beyond chaos theory and is a subject in its own right. It refers in general to the study of systems in nature that are best modelled by simple or complex computer programs and algorithms. So what are these systems like? They are complex to a scientist because human minds can't make progress on understanding them without the aid of computer programs. They are often described as systems at 'the edge of chaos and order' or messy problems for scientists to deal with. Often there is no well understood mathematical equation that can model the system at all (unlike weather), only computer programs of varying complexity.
Complexity scientists have words to describe the kinds of systems that are like this. They use concepts like 'emergence', 'path dependence', and refer to processes that are 'far from thermodynamic equilibrium'. The meaning of these ideas is beyond this 'Complexity 101' article, but basically, mean 'messy' and 'complicated' relative to the mathematical equations we can easily calculate and understand once the maths is understood. Therefore, for a scientist, a 'complex system' just means a system where short and concise mathematical equations can't neatly capture how the system behaves. Even relatively simple looking systems can display complexity. James White's art is one of these kinds of systems because his artwork displays a complexity that probably cannot be captured by short mathematical equations. However, computers which are programmed appropriately can probably roughly and concisely mimic the behaviour of James' physical system (the artwork) as it develops and James guides it. His paintings and practice demonstrate intuitively and concisely what complexity often looks like in other systems. Mathematical concepts like 'emergence' and 'path dependence' can, therefore, be seen visually in James' work. Some of the same patterns can also be seen in systems which are undoubtedly complex, such as in biological organisms or computer networks and other kinds of network.
In general, what systems might display the properties of complexity? It's a broad topic because lots of different things can't be understood very well by concise equations that we know about. It includes the brain, economic systems, biological systems, and presumably the 'primordial soup' at the beginning of life on earth. These problems are definitely not the ‘low hanging fruit’ of the science tree. They are not easy to describe using conventional, known mathematics. Therefore complexity scientists try to use computers to help them to get at these difficult subjects and to model how they work. They sometimes do this by exploring the 'digital universe' of possible computer programs to find ones which look and behave in similar ways to the things they are trying to understand.
So now, hopefully, you understand a little about what complexity means to a scientist and how complexity science seeks to make progress in understanding the world around us. Complexity is not the same as chaos but is closely related. Perhaps, have another look at James' work which demonstrates the science of 'complexity' in a visual way. Complexity scientists have found they can often judge whether a system is likely to be complex just by looking at it with a trained eye.