5

A Primer on Chaos

Suppose that just before you started reading this sentence, you reached to scratch an itch on your shoulder, noted that it’s becoming harder to reach that spot, thought of your joints calcifying with age, which made you vow to exercise more, and then you got a snack. Well, science has officially weighed in—each of those actions or thoughts, conscious or otherwise, and every bit of neurobiology underpinning it, was determined. Nothing just got it into its head to be a causeless cause.

No matter how thinly you slice it, each unique biological state was caused by a unique state that preceded it. And if you want to truly understand things, you need to break these two states down to their component parts, and figure out how each component comprising Just-Before-Now gave rise to each piece of Now. This is how the universe works.

But what if that isn’t? What if some moments aren’t caused by anything preceding them? What if some unique Nows can be caused by multiple, unique Just-Before-Nows? What if the strategy of learning how something works by breaking it down to its component parts is often useless? As it turns out, all of these are the case. Throughout the past century, the previous paragraph’s picture of the universe was overturned, giving birth to the sciences of chaos theory, emergent complexity, and quantum indeterminacy.

To label these as revolutions is not hyperbolic. When I was a kid, I read a novel called The Twenty-One Balloons,[*] about a utopian society on the island of Krakatoa built on balloon technology, destined to be destroyed by the famed 1883 eruption of the volcano there. It was fantastic, and the second I got to the end, I immediately flipped to the front to reread it. And it was then almost a quarter century before I immediately flipped to the front to reread a different book,[*] an introduction to one of these scientific revolutions.

Staggeringly interesting stuff. This chapter, and the five after it, reviews these three revolutions, and how numerous thinkers believe that you can find free will in their crevices. I will admit that the previous three chapters have an emotional intensity for me. I am put into a detached, professorial, eggheady sort of rage by the idea that you can assess someone’s behavior outside the context of what brought them to that moment of intent, that their history doesn’t matter. Or that even if a behavior seems determined, free will lurks wherever you’re not looking. And by the conclusion that righteous judgment of others is okay because while life is tough and we’re unfairly gifted or cursed with our attributes, what we freely choose to do with them is the measure of our worth. These stances have fueled profound amounts of undeserved pain and unearned entitlement.

The revolutions in the next five chapters don’t have that same visceral edge. As we’ll see, there aren’t a whole lot of thinkers out there citing, say, subatomic quantum indeterminacy when smugly proclaiming that free will exists and they earned their life in the top 1 percent. These topics don’t make me want to set up barricades in Paris, singing revolutionary anthems from Les Mis. Instead, these topics excite me immensely because they reveal completely unexpected structure and pattern; this enhances rather than quenches the sense that life is more interesting than can be imagined. These are subjects that fundamentally upend how we think about how complex things work. But nonetheless, they are not where free will dwells.

This and the next chapter focus on chaos theory, the field that can make studying the component parts of complex things useless. After a primer about the topic in this chapter, the next will cover two ways people mistakenly believe they’ve found free will in chaotic systems. First is the idea that if you start with something simple in biology and, unpredictably, out of that comes hugely complex behavior, free will just happened. Second is the belief that if you have a complex behavior that could have arisen from either of two different preceding biological states and there’s no way to ever tell which one caused it, then you can get away with claiming that it wasn’t caused by anything, that the event was free of determinism.

Back When Things Made Sense

Suppose that

X = Y + 1

If that is the case, then

X + 1 = ?

—and you were readily able to calculate that the answer is

(Y + 1) + 1.

Do X + 3 and you’ve instantly got (Y + 1) + 3. And here’s the crucial point—after solving X + 1, you were able to then solve X + 3 without first having to figure out X + 2. You were able to extrapolate into the future without examining each intervening step. Same thing for X + a gazillion, or X + sorta a gazillion, or X + a star-nosed mole.

A world like this has a number of properties:

• As we just saw, knowing the starting state of a system (for example, X = Y + 1) lets you accurately predict what X + whatever will equal, without the intervening steps. This property runs in both directions. If you’re given (Y + 1) + whatever, you know then that your starting point was X + whatever.

• Implicit in that, there is a unique pathway connecting the starting and ending states; it is also inevitable that X + 1 cannot equal (Y + 1) + 1 only some of the time.

• As shown dealing with something like “sorta a gazillion,” the magnitude of uncertainty and approximation in the starting state is directly proportional to the magnitude at the other end. You can know what you don’t know, can predict the degree of unpredictability.[1]

This relationship between starting states and mature states helped give rise to what has been the central concept of science for centuries. This is reductionism, the idea that to understand something complicated, break it down into its component parts, study them, add your insights about each component part together, and you will understand the complicated whole. And if one of those component parts is itself too complicated to understand, study its eensy subcomponent parts and understand them.

Reductionism like this is vital. If your watch, running on the ancient technology of gears, stops working, you apply a reductive approach to solving the problem. You take the watch apart, identify the one tiny gear that has a broken tooth, replace it, and put the pieces back together, and the watch runs. This approach is also how you do detective work—you arrive at a crime scene and interview the witnesses. The first witness observed only parts 1, 2, and 3 of the event. The second saw only 2, 3, and 4. The third, only 3, 4, and 5. Bummer, no one saw everything that happened. But thanks to a reductive mindset, you can solve the problem by taking the fragmentary component parts—each of the three witnesses’ overlapping observations, and combine them to understand the complete sequence.[*] Or as another example, in the first season of the pandemic, the world waited for answers to reductive questions like what receptor on the surface of a lung cell binds the spike protein of SARS-CoV-2, allowing it to enter and sicken that cell.

Mind you, a reductive approach doesn’t apply to everything. If there’s a drought, the sky dotted with puffy clouds that haven’t rained in a year, you don’t first isolate a cloud, study its left half and then its right half and then half of each half, and so on, until you find the itty-bitty gear in the center that has a broken tooth. Nonetheless, a reductive approach has long been the gold standard for scientifically exploring a complex topic.

And then, starting in the early 1960s, a scientific revolution emerged that came to be called chaoticism, or chaos theory. And its central idea is that really interesting, complicated things are often not best understood, cannot be understood, on a reductive level. To understand, say, a human whose behavior is abnormal, approach the problem as if this were a cloud that does not rain, rather than as a watch that does not tick. And naturally, humans-as-clouds generate all sorts of nearly irresistible urges for concluding that you are observing free will in action.

Chaotic Unpredictability

Chaos theory has its creation story. When I was a kid in the 1960s, inaccurate weather prediction was mocked with trenchant witticisms like “The weatherman on the radio [invariably, indeed, a man] said it’s going to be sunny today, so better bring an umbrella.” MIT meteorologist Edward Lorenz began using some antediluvian computer to model weather patterns in an attempt to increase prediction accuracy. Stick variables like temperature and humidity into the model and see how accurate the predictions became. See if additional variables, other variables, different weightings of variables,[*] improved predictability.

So Lorenz was studying a model on his computer using twelve variables. Time for lunch; halt the program in the middle of its cranking out a time course of predictions. Come back postlunch and, to save time, restart the program at a point before you stopped it, rather than starting all over. Punch in the values of those twelve variables at that time point, and let the model resume its predicting. That’s what Lorenz did, which is when our understanding of the universe changed.

One variable at that time point had a value of 0.506127. Except that on the printout, the computer had rounded it down to 0.506; maybe the computer hadn’t wanted to overwhelm this Human 1.0. In any case, 0.506127 became 0.506, and Lorenz, not knowing about this slight inaccuracy, ran the program with the variable at 0.506, thinking that it was actually 0.506127.

Thus, he was now dealing with a value that was a smidgen different from the real one. And we know just what should have happened now, in our supposedly purely linear, reductive world: the degree to which the starting state was off from what he thought it was (i.e., 0.506 rather than 0.506127) predicted how inaccurate his ending state would be—the program would generate a point that was only a smidgen different from that same point before lunch—if you superimposed the before- and after-lunch tracings, you’d barely see a difference.

Lorenz let the program, still depending on 0.506 instead of 0.506127, continue to run, and out came a result that was even more discrepant than he had expected from the prelunch run. Weird. And with each successive point, things got weirder—sometimes things seemed to have returned to the prelunch pattern but would then diverge again, with the divergences increasingly different, unpredictably, crazily so. And eventually rather than the program generating something even remotely close to what he saw the first time, the discrepancy in the two tracings was about as different as was possible.

This is what Lorenz saw—the pre- and postlunch tracings superimposed, a printout now with the status of a holy relic in the field (see figure on the next page).

Lorenz finally spotted that slight rounding error introduced after lunch and realized that this made the system unpredictable, nonlinear, and nonadditive.

By 1963, Lorenz announced this discovery in a dense technical paper, “Deterministic Non-periodic Flow,” in the highly specialized Journal of Atmospheric Sciences (and in the paper, Lorenz, while beginning to appreciate how these insights were overturning centuries of reductive thinking, still didn’t forget where he came from. Will it ever be possible to perfectly predict all of future weather? readers of the journal plaintively asked. Nope, Lorenz concluded; the chance of this is “non-existent”). And the paper has since been cited in other papers a staggering 26,000+ times.[2]

If Lorenz’s original program had contained only two weather variables, instead of the twelve he was using, the familiar reductiveness would have held—after a slightly wrong number was fed into the computer, the output would have been precisely as wrong at every step for the rest of time. Predictably so. Imagine a universe that consists of just two variables, the Earth and the Moon, exerting their gravitational forces on each other. In this linear, additive world, it is possible to infer precisely where they were at any point in the past and predict precisely where each will be at any point in the future;[*] if an approximation was accidentally introduced, the same magnitude of approximation would continue forever. But now add the Sun into the mix, and the nonlinearity happens. This is because the Earth influences the Moon, which means that the Earth influences how the Moon influences the Sun, which means that the Earth influences how the Moon influences the Sun’s influence on the Earth. . . . And don’t forget the other direction, Earth to Sun to Moon. The interactions among the three variables make linear predictability impossible. Once you’ve entered the realm of what is known as the “three-body problem,” with three or more variables interacting, things have inevitably become unpredictable.

When you have a nonlinear system, tiny differences in a starting state from one time to the next can cause them to diverge from each other enormously, even exponentially,[*] something since termed “sensitive dependence on initial conditions.” Lorenz noted that the unpredictability, rather than hurtling off forever into the exponential stratosphere, is sometimes bounded, constrained, and “dissipative.” In other words, the degree of unpredictability oscillates erratically around the predicted value, repeatedly a little more, a little less than predicted in the series of numbers you are generating, the degree of discrepancy always different, forever after. It’s like each data point you are getting is sort of attracted to what the data point is predicted to be, but not enough to actually reach the predicted value. Strange. And thus, Lorenz named these strange attractors.[*],[3]

So a tiny difference in a starting state can magnify unpredictably over time. Lorenz took to summarizing this idea with a metaphor about seagulls. A friend suggested something more picturesque, and by 1972 this was formalized into the title of a talk given by Lorenz. Here’s another holy relic of the field (see figure on the next page).

Thus was born the symbol of the chaos theory revolution, the butterfly effect.[*]^,[4]

Chaoticism You Can Do at Home

Time to see what chaoticism and sensitive dependence on initial conditions look like in practice. This makes use of a model system that is so cool and fun that I’ve even fleetingly wished that I could do computer coding, as it would make it easier to play with it.

Start off with a grid, like the one on a piece of graph paper, where the first row is your starting condition. Specifically, each of the boxes in the row can be in one of two states, either open or filled (or, in binary coding, either zero or one). There are 16,384 possible patterns for that row;[*] here’s our randomly chosen one:

Time now to generate the second row of boxes that are open or filled, that new pattern determined[*] by the pattern in row 1. We need a rule for how to do this. Here’s the most boring possible example: in row 2, a box that is underneath a filled box gets filled; a box underneath an open box remains open. Applying that rule over and over, using row 2 as the basis for row 3, 3 for 4, and so on, is just going to produce some boring columns. Or impose the opposite rule, such that if a box is filled, the one below it in the next row becomes open, while an open box spawns a filled one, and the outcome isn’t all that exciting, producing sort of a lopsided checkered pattern:

As the main point, starting with either of these rules, if you know the starting state (i.e., the pattern in row 1), you can accurately predict what a row anywhere in the future will look like. Our linear universe again.

Let’s go back to our row 1:

Now whether a particular row 2 box will be open or filled is determined by the state of three boxes—the row 1 box immediately above and the row 1 box’s neighbor on each side.

Here’s a random rule for how the state of a trio of adjacent row 1 boxes determines what happens in the row 2 box below: A row 2 box is filled if and only if one of the trio of boxes above it is filled in. Otherwise, the row 2 box will remain open.

Let’s start with the second box from the left in row 2. Here is the row 1 trio immediately above it (i.e., the first three boxes of row 1):

One of three boxes is filled, meaning that the row 2 box we’re considering will get filled:

Look at the next trio in row 1 (i.e., boxes 2, 3, and 4). Only one box is filled, so box 3 in row 2 will also be filled:

In the row 1 trio of boxes 3, 4, and 5, two boxes (4 and 5) are filled, so the next row 2 box is left open. And so on. The rule we are working with—if and only if one box of the trio is filled, fill in the row 2 box in question—can be summarized like this:

There are eight possible trios (two possible states for the first box of a trio times two possible for the second box times two for the third), and only trios 4, 6, and 7 result in the row 2 box in question being filled.

Back to our starting state, and using this rule, the first two rows will look like this:

But wait—what about the first and last boxes of row 2, where the box above has only one neighbor? We wouldn’t have that problem if row 1 were infinitely long in both directions, but we don’t have that luxury. What do we do with each of them? Just look at the box above it and the single neighbor, and use the same rule—if one of those two is filled, fill in the row 2 box; if both or neither of the two is filled, row 2 box is open. Thus, with that addendum in place, the first 2 rows look like this:

Now use the same rule to generate row 3:

Keep going, if you have nothing else to do.

Now let’s use this starting state with the same rule:

The first 2 rows will look like this:

Complete the first 250 or so rows and you get this:

Take a different, wider random starting state, apply the same rule over and over, and you get this:

Whoa.

Now try this starting state:

By row 2, you get this:

Nothing. With this particular starting state, row 2 is all open boxes, as will be the case in every subsequent row. Row 1’s pattern is snuffed out.

Let’s describe what we’ve learned so far in a metaphorical way, rather than using terms like input, output, and algorithm. With some starting states and the reproduction rule used to produce each subsequent generation, things can evolve into wildly interesting mature states, but you can also get some that go extinct, like that last example.

Why the biology metaphors? Because this world of generating patterns like this applies to nature (see figure on the next page).

We have just been exploring an example of a cellular automaton, where you start with a row of cells that are either open or filled, supply a reproduction rule, and let the process iterate.[*],[5]

The rule we’ve been following (if and only if one box of the trio above is filled . . .) is called rule 22 in the cellular automata universe, which consists of 256 rules.[*] Not all of these rules generate something interesting—depending on the starting state, some produce a pattern that just repeats for infinity in an inert, lifeless sort of way, or that goes extinct by the second row. Very few generate complex, dynamic patterns. And of the few that do, rule 22 is one of the favorites. People have spent their careers studying its chaoticism.

What is chaotic about rule 22? We’ve now seen that, depending on the starting state, by applying rule 22 you can get one of three mature patterns: (a) nothing, because it went extinct; (b) a crystallized, boring, inorganic periodic pattern; (c) a pattern that grows and writhes and changes, with pockets of structure giving way to anything but, a dynamic, organic profile. And as the crucial point, there is no way to take any irregular starting state and predict what row 100, or row 1,000, or row any-big-number will look like. You have to march through every intervening row, simulating it, to find out. It is impossible to predict if the mature form of a particular starting state will be extinct, crystalline, or dynamic or, if either of the latter two, what the pattern will be; people with spectacular mathematical powers have tried and failed. And this limit, paradoxically, extends to showing that you can’t prove that somewhere a few baby steps before reaching infinity, that the chaotic unpredictability will suddenly calm down into a sensible, repeating pattern. We have a version of the three-body problem, with interactions that are neither linear nor additive. You cannot take a reductive approach, breaking things down to its component parts (the eight different possible trios of boxes and their outcomes), and predict what you’re going to get. This is not a system for generating clocks. It’s for generating clouds.[6]

So we’ve just seen that knowing the irregular starting state gives you no predictive power about the mature state—you’ll just have to simulate each intervening step to find out.

Now consider rule 22 applied to each of these four starting states (see top figure on the next page).

Two of these four, once taken out ten generations, produce an identical pattern for the rest of time. I dare you to stare at these four and correctly predict which two it is going to be. It cannot be done.

Get some graph paper and crank through this, and you’ll see that two of these four converge. In other words, knowing the mature state of a system like this gives you no predictive power as to what the starting state was, or if it could have arisen from multiple different starting states, another defining feature of the chaoticism of this system.

Finally, consider the following starting state:

Which goes extinct by row 3:

Introduce a smidgen of a difference in this nonviable starting state, namely that the open/filled status of just one of the twenty-five boxes differs—box 20 is filled instead of open:

And suddenly, life erupts into an asymmetrical pattern (see figure on the next page).

Let’s state this biologically: a single mutation, in box 20, can have major consequences.

Let’s state this with the formalism of chaos theory: this system shows sensitive dependence on the initial condition of box 20.

Let’s state it in a way that is ultimately most meaningful: a butterfly in box 20 either did or didn’t flap its wings.

I love this stuff. One reason is because of the ways in which you can model biological systems with this, an idea explored at length by Stephen Wolfram.[*] Cellular automata are also inordinately cool because you can increase their dimensionality. The version we’ve been covering is one-dimensional, in that you start with a line of boxes and generate more lines. Conway’s Game of Life (invented by the late Princeton mathematician John Conway) is a two-dimensional version where you start with a grid of boxes and generate each subsequent generation’s grid. And produce absolutely astonishingly dynamic, chaotic patterns that are typically described as involving individual boxes that are “living” or “dying.” All with the usual properties—you can’t predict the mature state from the starting state—you have to simulate every intervening step; you can’t predict the starting state from the mature state because of the possibility that multiple starting states converged into the same mature one (we’re going to return to this convergence feature in a big way); the system shows sensitive dependence on initial conditions.[7]

(There’s an additional realm classically discussed when introducing chaoticism. I’ve sidestepped covering it here, however, because I’ve learned the hard way from my classrooms that it is very difficult and/or I’m very bad at explaining it. If interested, read up about Lorenz’s waterwheel, period doubling, and the significance of period 3 for the onset of chaos.)

With this introduction to chaoticism in hand, we can now appreciate the next chapter of the field—unexpectedly, the concepts of chaos theory became really popular, sowing the seeds for a certain style of free-will belief.