Emergence and the edge of chaos

A rule on a postcard

Take a long row of squares, each black or white. Now fix a rule for the next row, and make it as plain as possible: each square’s new color depends only on its own color and the colors of its left and right neighbors. Three squares, each black or white, make eight possible neighborhoods, and a rule says what each becomes. Eight yes-or-no answers make 2⁸ = 256 possible rules. That is the whole system. It is called an elementary cellular automaton, and in the early 1980s the physicist Stephen Wolfram began to run all 256 of them, one row under the next, to see what happens.

Many do what you would expect: they die out, or settle into stripes. Rule 30, the thirtieth in his numbering, does not. Start it with one black square, and below it grows a triangle whose left edge is regular and whose inside looks like noise. By Wolfram’s tests, its center column looks random. Wolfram has written that the Wolfram Language (Mathematica) used rule 30 to generate random numbers “for many years” before replacing it with a slightly more efficient rule. Nothing in it is random: the same start always gives the same picture. A rule that fits on a postcard produces what looks like chance, which is the lesson of Lesson 2 once more, in another form.

Four kinds of universe

Wolfram looked at a great many of these pictures and sorted them, by eye, into four kinds. (Popular accounts often speak of three kinds; Wolfram’s papers of 1983 and 1984 have four.)

• Class 1: everything settles into one uniform state, all black or all white.
• Class 2: everything settles into a few stable or repeating patterns, stripes or blinkers.
• Class 3: chaos, with a look of noise (rule 30 is the standard example).
• Class 4: a mixture of order and surprise, with localized structures that travel, collide, and sometimes survive collisions.

Class 4 is rare. Among the 256 rules there are only 88 that differ once you allow mirror images and swapping black for white, and in one published tabulation only four of those 88 (rules 41, 54, 106, and 110) count as class 4. The best known is rule 110. It makes tangled triangles and little gliders that pass through each other, and in 2004 the mathematician Matthew Cook published a proof that it is universal: with the right starting pattern, it can carry out any computation that any computer can, though painfully slowly. Wolfram had conjectured this in the mid-1980s, and Cook’s main steps date from the 1990s. A one-line rule can compute anything. (Class 4 universality in general is a conjecture, not a theorem; only rule 110 has the proof.)

There is an older, more famous example. In October 1970 Martin Gardner described, in his Scientific American column, the “Game of Life,” invented by the Cambridge mathematician John Conway. Cells on a grid are alive or dead. A live cell with two or three live neighbors stays alive; a dead cell with exactly three live neighbors comes to life; every other cell dies or stays dead. Nothing in these rules mentions motion. Yet a five-cell pattern, the glider, walks across the grid, one cell diagonally every four steps. Life, too, is universal: working computers built inside it date from the early 2000s.

What emerges

The glider is a good case for pinning down the word. A pattern, in the physicist’s sense (the Chinese word is 斑图, bāntú), is a regularity at the level of the whole. A flock’s shape, an ant trail, a traffic jam, the stripes on a fish: each is described with words that none of the parts has. A single cell in Life cannot move. A single car cannot be a jam. This is emergence: a property or pattern of a whole that is not a property of any of its parts, and that arises from how the parts interact. The idea is old. The English philosopher G. H. Lewes introduced the word “emergent” in 1875, following John Stuart Mill; Aristotle made the point with a syllable. In the Metaphysics he says that a syllable is not merely its letters: the syllable “ba” is something besides the sounds b and a, since if you take the letters apart the syllable no longer exists. (The common slogan, “the whole is more than the sum of its parts,” is a later paraphrase.)

A pattern like this can also float free of its parts, much as software floats free of hardware. At a mass display, thousands of people hold up colored cards and make a picture; the same number “60” can be made from cards, from the pixels of a screen, or from lighted windows in a tall building. A glider can be built from the cells of any Life grid, whether it is drawn on paper, run on a laptop, or acted out by people. The pattern is independent of what it is made of. This is why complexity scientists study patterns as objects in their own right.

Real patterns of this kind appear in living things. In 1952 the mathematician Alan Turing showed that two chemicals that react and diffuse at different speeds can turn a uniform mixture into spots or stripes, with no blueprint of the pattern anywhere. In 1995 Shigeru Kondo and Rihito Asai reported that the stripes on a marine angelfish, Pomacanthus, keep rearranging as the fish grows in just the way a Turing-type model predicts. (Their paper says that this “suggests” the mechanism, not that it proves it, and a 2010 review notes how long such models were debated.) You met a physical cousin of this in Course 5: the hexagonal cells that form in a pan of liquid heated from below.

Weak and strong emergence

Philosophers distinguish two strengths. Weak emergence, in the definition of the philosopher Mark Bedau (1997), is a pattern that follows from the micro-level rules but that you can discover only by running the system step by step, since there is no shortcut. Rule 30, Life’s gliders, and the boids’ flock are examples. Weak emergence is not controversial. It is the claim that the whole is hard to predict from the parts, even though it is wholly explained by them.

Strong emergence is a much bigger claim: that the whole has causal powers that cannot, even in principle, be derived from the parts, so that the whole can push back on its own parts (“downward causation”). It is often put vividly. My thought can make me slap my arm, and in doing so I kill some of the cells that gave rise to the thought. Aristotle’s “formal cause,” the shape or pattern of the whole, is invoked as the cause. Most scientists would grant the everyday facts: of course a thought moves an arm, and a flock’s shape changes what each bird does. What is disputed is whether such cases are anything more than the parts’ own physics, organized in a particular way. The philosopher Jaegwon Kim argued that they are not: if every physical event already has a sufficient physical cause, there is no room left for a macro-level cause to add anything. The philosopher David Chalmers calls consciousness the only clear candidate for strong emergence, and treats everything else as weak. Others, including the physicist George Ellis, defend real top-down causation. In this course the safe course is to treat weak emergence as established and strong emergence as an open and disputed question.

One more distinction matters. It is sometimes suggested that patterns exist only when someone sees them, with a face in a cloud as the example. A face in a cloud really is in the observer: the tendency to see faces in random shapes has a name, pareidolia. The “Face on Mars,” a hill photographed by the Viking 1 orbiter on 25 July 1976, looked like a face at about 50 meters per pixel, and like a plain mesa at about 4 meters per pixel in 1998 and about 2 meters in 2001. A flock’s order is a different kind of thing. It can be measured, with no human eyes, as a number (the agreement of headings), and it feeds back on the birds, since each one’s motion depends on the others. A useful test of a pattern worth studying: does it make a difference to the parts, and could an instrument find it?

The edge of chaos

Rule 30 is chaotic, rule 4 is dull, and rule 110 sits between them. Is there a dial that moves a system from one to the other? In 1990 the computer scientist Christopher Langton proposed one for cellular automata with more colors: take a rule table, and count what fraction of its entries lead to a non-blank cell. Call that fraction λ (lambda). At low λ, nearly everything dies, which is order. At high λ, the pictures are noise, which is chaos. In between, Langton suggested, are the complex ones, in a narrow band that resembles a phase transition, like water between ice and steam. The phrase for the band, “the edge of chaos,” is associated with Norman Packard (1988), Langton, and the biologist Stuart Kauffman, and it reached a wide audience in two 1992 books on the Santa Fe Institute. A related picture says that if you plot some measure of “complexity” from perfect order to perfect disorder, it rises and then falls: a repeating pattern is simple, a random one is simple to describe statistically, and the interesting structure lies between.

It is an attractive picture, and it has been used to explain many things: why a flock of birds looks alive, why a team works best between rigid control and free-for-all, why a child learns best on a stage with edges but no script. In one popular reading, a flock’s three rules pull two ways: alignment and cohesion pull the birds toward order, separation toward disorder, and the lifelike behavior lives in the tug of war. That is a nice way to read the model. It is an analogy, though, and not a result.

The honest state of the science is mixed. Some evidence supports a version of the idea: for example, the logistic map of Lesson 2 shows its most intricate structure at the border between regular and chaotic behavior. But the strong form, that complex computation sits at a particular λ and that evolution drives systems there, did not hold up when tested. In 1993 Melanie Mitchell, Peter Hraber, and James Crutchfield repeated Packard’s experiment, in which a genetic algorithm evolves cellular automata to perform a computation. The evolved rules did not cluster near the critical values that Langton’s idea predicts; they clustered at about 0.43 and 0.57, and they concluded that Packard’s interpretation was wrong. Their 1994 follow-up found “no evidence for a generic relationship” between λ and computational ability. Nor is the inverted-U curve universal: a 2008 study by David Feldman, Carl McTague, and James Crutchfield states that there is not a universal complexity-entropy curve, and that its shape depends on which measure of complexity you choose.

Other fields have borrowed the phrase freely. Some neuroscientists have argued that the cortex works near a critical point, based on “neuronal avalanches” of activity in cultures and slices of rat cortex (Beggs and Plenz, 2003), and the starlings of Lesson 3 show correlations that span the whole flock. Critics have shown that similar power laws can be produced by systems that are not critical at all, so this remains an active and disputed research program. Management writers have advised firms to stay “at the edge of chaos,” and a four-year study of one organization questioned the assumption that organizations behave this way. These are analogies from a physical idea that is itself unsettled.

Four points, checked

This course ends with a four-point summary of its own, and each point can be marked by how far the evidence goes.

1. Local, simple interactions can produce complex behavior. Boids, ant trails, rule 30, and Life are all examples. This is well supported.
2. The complexity comes from the pattern of the interactions, not from complicated parts. Also well supported: a flock is made of simple birds.
3. The interesting complexity lies between order and chaos. An attractive idea with mixed support, and not a law.
4. A different way of thinking: favor emergence over design, allow nonlinearity, treat collective patterns as things in their own right, and expect causes to run in both directions. The first three are good working advice; the strong form of the last (that wholes cause things beyond their parts’ physics) is disputed.