Quora Answer : Is complexity theory a science, a research agenda, a philosophical standpoint, or what?
I'd say it's mainly a branch of mathematics - a collection of models / modelling techniques - that has some interesting scientific / philosophical (especially philosophy of science) consequences.
The most interesting consequence, philosophically, is this : these models all demonstrate what used to be called "sensitive dependency on initial conditions" or sensitivity to small details. Unlike more traditional models, they are "fractal" or "scale free" ... differences that are initially very small, spiral up to become very large, sooner or later.
That has a weird effect from a philosophy of science perspective. You can't really claim that you are making a specific model of a specific bit of reality.
For example, if you use Newton's Laws (which don't have much "sensitive dependency") to model the movement of Neptune, you can really claim that your model is a model of Neptune. You can say it's making predictions about Neptune. And if Neptune starts deviating from those predictions, you might start questioning your model.
In the case of the "complexity" models, though. Whether of weather systems, or economic systems, or biological evolution or learning in the brain, you can't really plausibly claim that you've pinned down the details sufficiently that this is a model of any particular weather system. Or this actual economy. Or this actual person learning to solve a task. Etc.
I mean, you can claim that. And you might get away with it over very short time-scales. But you pretty well know that, longer term, the model's behaviour is going to diverge wildly from the real thing.
So you are obliged to think about your models in a different way. These are not models of particulars in the world. Making predictions about those particulars. Instead, these models are representatives of certain kinds of situations in the real world. They aren't going to make predictions. Rather, they're "concept demonstrators" in which the same "sorts of things" might happen. You might see hurricanes or economic downturns or extinction events. But there's no sense in which these correspond to the actual instances of these events in the world.
That's quite tricky, because quite a lot of philosophy of science (famously, the Popperian idea of science, which a lot of scientists latched onto) sees science as being ALL about making predictions about real things. The hallmark of "real science" is making predictions which observations of real things can falsify.
So what happens when your "science" is looking at models that can demonstrate plausible analogues to types of phenomena, but can't make models or falsifiable predictions about actual instances of the phenomena?
Some people will say "well, this can't really be science then, can it? If your models aren't falsifiable.". And they'll label it "simulation" ... a kind of maths.
Others will accept being pushed back to a much weaker notion of testing which is "does the model keep looking like the real thing?" : does it produce hurricanes or housing bubbles that have the "characteristic" of the real economy? Where "characteristic might be some higher-order statistical property like "frequency" or "distribution of sizes" or similar.
In summary, "complexity theory" is a class of models that force you to rethink your philosophy of science notion of what counts as "valid observation". And puts practical constraints on the kind of data you can hope to collect.