Monday, October 29, 2007

It's complicated ...

It has been noted that when trying to explain almost anything, I have a habit of declaring—"It's complicated ..." And of course, it is (whatever it is). But is that just a cop-out?

Complexity is particularly challenging when decisions have to be made. There may be many factors beyond our control, that we understand poorly, or that we're not even aware of. And how these factors interact is often unclear. Complexity brings with it uncertainty, and uncertainty is always unsettling. One solution is to do nothing, and sometimes that's the best choice, as expressed in the aphorism "first do no harm". But, as in the case of climate change, the decision to do nothing ("more studies are needed") is often a poor choice.

Coping with complexity

Faced with complexity, we simplify. That is, we employ models that make it easier to think about the situation. This comes so naturally to us that we're often completely unaware that's what we're doing. For example, when we are "behind on sleep", we need to "catch up". This metaphor could be called SLEEP IS A RACE. The complexities of human sleep requirements are reduced to simple addition and subtraction, represented in terms of a racetrack (essentially a number line). This is an example of a mental model.

Mental models are more slippery than scientific models because they tend to be much less explicit. Scientific models are published and debated and put to empirical test. But we are embedded in our mental models, and like embedded journalists, our objectivity is profoundly compromised. And yet we can't do without mental models, any more than we can do without scientific models.

A double-edged razor

A well-known principle in science is Occam's razor, which argues for simplicity in modeling. But simplicity can go too far. Albert Einstein's take on this was:
Everything should be made as simple as possible, but no simpler.
An extreme case of simplification is polarized thinking. George W. Bush's "You're either with us or against us" is an apt example. Thinking like this both justifies and perpetuates violent conflict. Bush's statement is an example of what I might call a polarized moral or perhaps theological model.

Logic's limits

Even when we're aware of our models, it's easy to mistake them for reality. This applies not just to the elements of a model, but also to deductive inferences (which I'll refer to simply as logic) obtained in the context of the model. Logic has a seductive appeal: it offers certainty provided we observe some elementary rules, known since at least the time of Aristotle. There's just one hitch: our model has to be correct.

In fairly trivial situations, it may be possible to know our model is correct. One example is when we're analyzing data from a computer simulation. We can know we're using the correct (or incorrect) model because we wrote the program that generated the data!

But generally our model is a simplified representation (sometimes called an idealization) of a more complicated reality. In the context of the model, logic is infallible, but that may not translate back to the real world.

The apparent certainty of logic may encourage polarized thinking—a trap I've fallen into on, ahem, one or two occasions. When all propositions are simply TRUE or FALSE, everything is so easy, so tidy. But so misleading.

I'm not discounting the value of logic, but I am pointing out that we have to be very careful not to make it something it's not.

Questioning our models

Rene Descartes famously questioned all of his assumptions, arguing that:
If you would be a real seeker after truth, it is necessary that at least once in your life you doubt, as far as possible, all things.
Finally, he was left with no beliefs that he felt he could justify but the fact of his own existence. Although this approach seems a bit extreme, the idea that we should subject our models to careful examination is of paramount importance.

In order to honestly critique our models, we need to "think outside the box". A model is indeed a kind of box, and often an opaque one at that. We can grow very comfortable inside our models, to the point where we can scarcely conceive of another approach.

In the field of statistics, there is a large literature on model selection. The simplest case is that of nested models, where one model is an extension of another. But, writes Malcolm Forster:
Models belonging to different theories, across a revolutionary divide, are usually non-nested. A typical example involves the comparison of Copernican and Ptolemaic models of planetary motion. It is not possible to obtain a sun-centered model from a earth-centered model by adding circles. Cases like this are the most puzzling, especially with respect to the role of simplicity.
This is closely related to what Thomas Kuhn called the incommensurability of scientific paradigms.

The fine print

Of course there's more to say. After all, it's complicated ...

First, I want to point out that I'm using the term "models" in a very general sense. I'm including scientific models, theories, paradigms, mathematical models, statistical models, mental models, metaphors, assumptions, beliefs, moral/ethical models, and even theological models. And probably a bunch of other types of models I haven't thought of. Some people use the term in a much narrower sense. But my point here has been that several issues around simplicity are important for many different kinds of models.

Models serve a variety of purposes. Models can be descriptive, explanatory, causal, predictive, or normative. (And again, I'm sure this list should be longer.) It would be interesting to examine how simplicity plays a role in these different cases. Of course I haven't really defined simplicity, and that's a whole other area.

Finally, here's an entertaining list of
rules of mathematical modeling

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