Spontaneous order revisited

Back in September, I wrote a post critiquing the responses of Tim Swanson and Brian Doss (of Catallarchy fame) to Stephen Strogatz’ book Sync: How Order Emerges from Chaos in the Universe, Nature, and Daily Life. Those responses an be found in Brian’s initial post and the ensuing comments thread and suggest that those studying spontaneous order would be best served by following Alfred Marshall’s advice to “burn the mathematics”. My post (which apparently lost the first paragraph or two in the switch to a new domain back in December), is called “Spontaneous Order” and also spawned a response from Neil.

Okay, with the citations out of the way, the issue of the day is whether I still agree with what I said back in September, now that I’ve just finished reading the book; and the answer is largely “yes”. In fact, I don’t think I went nearly far enough in my criticism of the notion that those studying spontaneous order should avoid mathematical formalism. For example, the following was my conclusive summary of the post:

My point is not to demonstrate that the study of spontaneous order is a mathematical discipline, nor that it should be. Rather, I just want to make the point that it has certain similarities to mathematics and, of course, will necessarily need to use mathematical tools in many instances. In fact, though I admit to not knowing nearly enough to be able to have any insights, it seems like mathematics, especially areas of study like graph theory and networks, might be able to shed some light on some of the applications of spontaneous order mentioned by Strogatz

Not very bold, right? Well, I hadn’t actually read the book yet. Now that I have, it’s abundantly clear that anybody who has actually read Strogatz’ book already knows that it is, at heart, a math book; the fact that it’s not publicized as such has more to do with the irrational fear most people have of mathematics. Virtually every result Strogatz cites is a result in pure or applied mathematics, with all the usual deduction and separation from empirical strategies that that entails. Many of these are proofs about idealized models of coupled oscillators, results which probably help explain how, for example, Thai fireflies flash in sync, but building an experiment to actually test this is so difficult that it apparently hasn’t been done to any degree of satisfaction yet (or, if it has, Strogatz doesn’t mention it). The same holds for, say, three-dimensional synchronicity, which has applications to cardiac arrhythmia, but which is discussed in the book purely in terms of mathematics and chemical reactions in a very special kind of fluid.

The point is this: right now, the mathematics of spontaneous order is both several steps ahead of and well behind the real world. It’s several steps ahead in the sense that mathematical explanations of synchronous processes seems, in large measure, to be ahead of the capability of experimental science to confirm (or deny, of course). On the other hand, mathematics is obviously very far behind the real world, as we can’t yet accurately model the spontaneous order that occurs between nerve cells to make our hearts beat, let alone the presumably much more complicated processes occurring within our brains. Whatever the case, reading Strogatz’ book confirmed my suspicion that, in fact, mathematics is essential to the emerging field of spontaneous order (as a side note, both Doss and Swanson, in the original Catallarchy post linked above, seem to reject mathematics because it conflicts with the principles of Austrian economics and the Austrians’ rejection of empirical economics is well-known; so my question is this: if Austrians reject empiricism as well as mathematics (i.e. deduction), how, exactly, do they advocate gaining knowledge? (Of course, I know the answer, but the Austrian-sympathizers would do well, in my opinion, to keep this question in mind)).

This all having been said, Sync was a bit too devoid of mathematical content for my taste, in the sense that, although almost everything in the book boiled down to mathematics, Strogatz explained most of the actual mathematical machinery in terms of analogies to runners on a track or audiences clapping or whatever, whereas I would have liked to see greater mathematical rigor (not necessarily the equations themselves, which are almost certainly too complicated to mean very much to the uninitiated, but rather a more rigorous argument, with references to actual mathematical principles, theorems, etc.). For example, when Strogatz says “[u]sing a theorem from topology, Winfree proved that a twisted scroll ring was impossible, at least as a solitary entity”, it would have been nice if he had explained what theorem, exactly, even if only in the endnotes. Or, as Neil says,

In fact, as [Strogatz] chronicled the mathematical history of sync as an abstract study, I found myself wanting various symbols and equations

As an ego-boost, I’ll point out that in the above quotation from my September post, my suggestion that graph theory and networks “might be able to shed some light on some of the applications of spontaneous order mentioned by Strogatz” was right on, as Chapter Nine of Sync is titled and devoted to “Small-World Networks”.

Now, a couple of quotations to think about (I’ve omitted one interesting and very extended passage, because I want to dedicate and entire, separate post to it, hopefully some time this weekend):

In other words, a dumb rule (majority rule) running on a smart architecture (a small world) achieved performance that broke the world record.

— pg. 251 (Here, Strogatz is talking about the density classification problem for one-dimensional binary automata, where he and one of his students decided to re-wire the binary automata as a small-world network — where most of the connections between automata (think lightbulbs) are locally clustered, but a few are long-distance — and almost immediately, using the simplest algorithm imaginable, were able to solve the problem more consistently than the best algorithm using “dumb architecture”)

Barabási and his team pointed out that scale-free networks [like the Internet or protein interactions in yeast] also embody a compromise bearing the stamp of natural selection: They are inherently resistant to random failures, yet vulnerable to deliberate attack against their hubs. Given that mutations occur at random, natural selection favors designs that can tolerate haphazard insults. By their very geometry, scale-free networks are robust with respect to random failures, because the vast majority of nodes have few links and are therefore expendable. Unfortunately, this evolutionary design has a downside. When hubs are selectively targeted (something that random mutation could never do), the integrity of the network degrades rapidly—the size of the giant component collapses and the average path length swells, as nodes become isolated, cast adrift on their own little islands.

— pg. 257

Helbing and Huberman computed the long-term traffic patterns under a variety of different conditions. When there were only a few vehicles on the road, all the cars sailed past the slower-moving trucks without ever decelerating, while the trucks lumbered along at their maximum safe speed of 55 miles an hour. At higher but still moderate densities of traffic, some unlucky cars found themselves trapped behind trucks for a long time, with no room to pass or switch lanes.

At a critical density of traffic—about 35 vehicles in each lane per mile of road—all the cars and trucks spontaneously synchronized, traveling down the highway like a solid block. Remarkably, out of pure competition, with no coordinator or central authority, a large group of selfish individuals ended up in a cooperative state that was optimal for all of them. (Adam Smith would approve.) This state was optimal in the sense that the flux of traffic was as high as it could be: The number of cars and trucks passing through a given stretch of highway per hour was maximized. It was also the safest way for traffic to flow, because the drivers had no opportunities to change lanes or pass (the maneuvers associated with most accidents). Helbing and Huberman tested their model against data taken from a two-lane Dutch highway and found evidence of the predicted state. At the critical density, the car speeds were at their most stable, as measured by their velocity fluctuations, and lane changing and passing were minimized. Unfortunately—and as the model also predicted—the crystalline state proved to be delicate. At densities just above critical, it melted into a disorganized liquid state, which created opportunities for passing again, leading to unsteady, stop-and-go traffic.

— pgs. 269-70

“In individuals, insanity is rare, but in groups, parties, nations, and epochs it is the rule.”

— Nietzsche, cited on pg. 273