I just finished reading Complexity: A Guided Tour by Melanie Mitchell. The book is meant to be an introduction to complexity theory for the general reader.

The book works as a lucid review of many interesting topics in science and mathematics. I’d read Mitchell’s book on genetic algorithms, and she’s a gifted writer. Here she explores (among other things) dynamical systems, chaos, information theory, genetic algorithms, cellular automata, analogical reasoning, and network theory. She does a great job explaining difficult concepts in a clear manner.

But I’m afraid the book as a whole didn’t hang together for me. It didn’t help any that Mitchell immediately provoked a pet peeve of mine in the preface. Here’s her opening paragraph:

Reductionism has been the dominant approach to science since the 1600s. Rene Descartes, one of reeductionism’s earliest proponents, described his own scientific method thus: “to divide all the difficulties under examination into as many parts as possible, and as many as were required to solve them in the best way” and “to conduct my thoughts in a given order, beginning with the simplest and most easily understood objects, and gradually ascending, as it were step by step, to the knowledge of the most complex.”

She then goes on to basically say that reductionism is dead, and that complexity theory has slain it. I suppose I take offense at this because I consider myself and pretty much every working scientist a reductionist, although not in the straw man way that the term is most often employed. To understand any system, from galaxies to bacterium to brains, we try to decompose the system into constituent parts and understand the actions between them. Often, the hardest part is figuring out how to partition, at what level, to yield the most progress. But throughout the whole book, with the provocative statement that reductionism is dead, I vainly waited for a description of an alternative way to carry out scientific investigation.

Yes, chaos theory has demonstrated that there are systems which are incredibly sensitive to initial conditions. Cellular automata demonstrate surprisingly complex behavior given very simple units following very simple policies. But how do these examples invalidate the reductionist approach?

If we want to understand how weather systems, economies, ant colonies, gene regulatory networks, and brains all work, and reductionism is a flawed way of going about it, then what’s the right way? I’m afraid I never got that answer from the book. Maybe that wasn’t the thrust of the book, but from the opening I expected something along those lines. After all, I’m interested in understanding how neural systems work, and if I’m going about it in some misguided fashion, I’d like to know how to correct my course.

Perhaps she’s just scapegoating pre-20th-century scientists for thinking the universe was much simpler than it is. I’m not sure how constructive that is, or even the extent to which it’s true. Mitchell does a good job of showing how discoveries in the 20th century threw several curveballs at science, and that in many ways the universe is more weird, surprising, and difficult to understand than we might have thought before. But I don’t see how that invalidates the general approach. And at this point in the history of science I think it would be naive to not expect further revelations that demonstrate an even greater depth of weirdness and complexity in the world. In fact, I think that’s one of the exciting aspects of science.

Also, I’m not sure what to make of the apparent dichotomy between complex systems, and those that are apparently not complex. The notion of emergence, the idea that the whole is more than the sum of its parts, seems to hold for nearly any system. A simple lever has properties that the parts unassembled do not. Is this emergence?

So, I give the book a thumbs up as a review of many interesting and important ideas, but a thumbs down for its ideology.