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November 25, 2006
Unreasonable effectiveness (part 3)
A little more than twenty years after Hamming's essay, the computer scientist Bernard Chazelle penned an essay on the importance of the algorithm, in which he offers his own perspective on the unreasonable effectiveness of mathematics.
Mathematics shines in domains replete with symmetry, regularity, periodicity -- things often missing in the life and social sciences. Contrast a crystal structure (grist for algebra's mill) with the World Wide Web (cannon fodder for algorithms). No math formula will ever model whole biological organisms, economies, ecologies, or large, live networks.
Perhaps this, in fact, is what Hamming meant by saying that much of physics is logically deducible, that the symmetries, regularities, and periodicities of physical nature constrain it in such strong ways that mathematics alone (and not something more powerful) can accurately capture its structure. But, complex systems like organisms, economies and engineered systems don't have to, and certainly don't seem to, respect those constraints. Yet, even these things exhibit patterns and regularities that we can study.
Clearly, my perspective matches Chazelle's, that algorithms offer a better path toward understanding complexity than the mathematics of physics. Or, to put it another way, that complexity is inherently algorithmic. As an example of this kind of inherent complexity through algorithms, Chazelle cites Craig Reynolds' boids model. Boids is one of the canonical simulations of "artificial life"; in this particular simulation, a trio of simple algorithmic rules produce surprisingly realistic flocking / herding behavior when followed by a group of "autonomous" agents [1]. There are several other surprisingly effective algorithmic models of complex behavior (as I mentioned before, cellular automata are perhaps the most successful), but they all exist in isolation, as models of disconnected phenomenon.
So, I think one of the grand challenges for a science of complexity will be to develop a way to collect the results of these isolated models into a coherent framework. Just as we have powerful tools for working with a wide range of differential-equation models, we need similar tools for working with competitive agent-based models, evolutionary models, etc. That is, we would like to be able to write down the model in an abstract form, and then draw strong, testable conclusions about it, without simulating it. For example, imagine being able to write down Reynolds' three boids rules and deriving the observed flocking behavior before coding them up [2]. To me, that would prove that the algorithm is unreasonably effective at capturing complexity. Until then, it's just a dream.
Note: See also part 1 and part 2 of this series of posts.
[1] This citation is particularly amusing to me considering that most computer scientists seem to be completely unaware of the fields of complex systems and artificial life. This is, perhaps, attributable to computer science's roots in engineering and logic, rather than in studying the natural world.
[2] It's true that problems of intractability (P vs NP) and undecidability lurk behind these questions, but analogous questions lurk behind much of mathematics (Thank you, Godel). For most practical situations, mathematics has sidestepped these questions. For most practical situations (where here I'm thinking more of modeling the natural world), can we also sidestep them for algorithms?
posted November 25, 2006 01:19 PM in Things to Read | permalink