Determined, page 14
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]
An actual shell on the left, a computer-generated pattern on the right
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.
6
Is Your Free Will Chaotic?
The Age of Chaos
The upheaval in the early 1960s caused by chaos theory, strange attractors, and sensitive dependence on initial conditions was rapidly felt throughout the world, fundamentally altering everything from the most highfalutin philosophical musings to the concerns of everyday life.
Actually, not at all. Lorenz’s revolutionary 1963 paper was mostly met with silence. It took years for him to begin to collect acolytes, mostly a group of physics grad students at UC Santa Cruz who supposedly spent a lot of time stoned and studied things like the chaoticism of how faucets drip.[*] Mainstream theorists mostly ignored the implications.
Part of the neglect reflected the fact that chaos theory is a horrible name, insofar as it is about the opposite of nihilistic chaos and is instead about the patterns of structure hidden in seeming chaos. The more fundamental reason for chaoticism getting off to a slow start was that if you have a reductive mindset, unsolvable, nonlinear interactions among a large number of variables is a total pain to study. Thus, most researchers tried to study complicated things by limiting the number of variables considered so that things remained tame and tractable. And this guaranteed the incorrect conclusion that the world is mostly about linear, additive predictability and nonlinear chaoticism was a weird anomaly that could mostly be ignored. Until it couldn’t be anymore, as it became clear that chaoticism lurked behind the most interesting complicated things. A cell, a brain, a person, a society, was more like the chaoticism of a cloud than the reductionism of a watch.[1]
By the eighties, chaos theory had exploded as an academic subject (this was around the time that the pioneering generation of renegade stoner physicists began to be things like a professor at Oxford or the founder of a company using chaos theory to plunder the stock market). Suddenly, there were specialized journals, conferences, departments, and interdisciplinary institutes. Scholarly papers and books appeared about the implications of chaoticism for education, corporate management, economics, the stock market, art and architecture (with the interesting idea that we find nature to be more beautiful than, say, modernist office buildings, because the former has just the right amount of chaos), literary criticism, cultural studies of television (with the observation that, like chaotic systems, television “dramas are both complex and simple at the same time”), neurology and cardiology (in both of which, interestingly, too little chaoticism was appearing to be a bad thing[*]). There were even scholarly articles about the relevance of chaos theory to theology (including one with the wonderful title “Chaos at the Marriage of Heaven and Hell,” in which the author wrote, “Those of us who seek to engage modern culture in our theological reflection cannot afford to overlook chaos theory”).[2]
Meanwhile, interest in chaos theory, accurate or otherwise, burst into the general public’s consciousness as well—who could have predicted that? There were the ubiquitous wall calendars of fractals. Novels, books of poetry, multiple movies, TV episodes, numerous bands, albums, and songs commandeered strange attractor or the butterfly effect in their titles.[*] According to a Simpsons fandom site, in one episode during her baseball-coaching period, Lisa is seen reading a book called Chaos Theory in Baseball Analysis. And as my favorite, in the novel Chaos Theory, part of the Nerds of Paradise Harlequin romance series, our protagonist has her eyes on handsome engineer Will Darling. Despite his unbuttoned shirt, six-pack, and insouciant bedroom eyes, it is understood that Will must still be a nerd, since he wears glasses.[3]
The growing interest in chaos theory generated the sound of a zillion butterfly wings flapping. Given that, it was inevitable that various thinkers began to proclaim that the unpredictable, chaotic cloud-ness of human behavior is where free will runs free. Hopefully, the material already covered, showing what chaoticism is and isn’t, will help show how this cannot be.
The giddy conclusion that chaoticism proves free will takes at least two forms.
Wrong Conclusion #1: The Freely Choosing Cloud
For free-will believers, the crux of the issue is lack of predictability—at innumerable junctures in our lives, including highly consequential ones, we choose between X and not-X. And even a vastly knowledgeable observer could not have predicted every such choice.
In this vein, physicist Gert Eilenberger writes, “It is simply improbable that reality is completely and exhaustively mappable by mathematical constructs.” This is because “the mathematical abilities of the species Homo sapiens are in principle limited because of their biological basis. . . . Because of [chaoticism], the determinism of Laplace[*] cannot be absolute and the question of the possibility of chance and freedom is open again!” The exclamation mark at the end is Eilenberger’s; a physicist means business if he’s putting exclamation marks in his writing.[4]
Biophysicist Kelly Clancy makes a similar point concerning chaoticism in the brain: “Over time, chaotic trajectories will gravitate toward [strange attractors]. Because chaos can be controlled, it strikes a fine balance between reliability and exploration. Yet because it’s unpredictable, it’s a strong candidate for the dynamical substrate of free will.”[5]
Doyne Farmer weighs in as well in a way I found disappointing, given that he was one of the faucet-drip apostles of chaos theory and should know better. “On a philosophical level, it struck me [that chaoticism was] an operational way to define free will, in a way that allowed you to reconcile free will with determinism. The system is deterministic, but you can’t say what it’s going to do next.”[6]
As a final example, philosopher David Steenburg explicitly links the supposed free will of chaos with morality: “Chaos theory provides for the reintegration of fact and value by opening each to the other in new ways.” And to underline this linkage, Steenburg’s paper wasn’t published in some science or philosophy journal. It was in the Harvard Theological Review.[7]
So a bunch of thinkers find free will in the structure of chaoticism. Compatibilists and incompatibilists debate whether free will is possible in a deterministic world, but now you can skip the whole brouhaha because, according to them, chaoticism shows that the world isn’t deterministic. As Eilenberger summarizes, “But since we now know that the slightest, immeasurably small differences in the initial state can lead to completely different final states (that is, decisions), physics cannot empirically prove the impossibility of free will.”[8] In this view, the indeterminism of chaos means that, although it doesn’t help you prove that there is free will, it lets you prove that you can’t prove that there isn’t.
But now to the critical mistake running through all of this: determinism and predictability are very different things. Even if chaoticism is unpredictable, it is still deterministic. The difference can be framed a lot of ways. One is that determinism allows you to explain why something happened, whereas predictability allows you to say what happens next. Another way is the woolly-haired contrast between ontology and epistemology; the former is about what is going on, an issue of determinism, while the latter is about what is knowable, an issue of predictability. Another is the difference between “determined” and “determinable” (giving rise to the heavy-duty title of one heavy-duty paper, “Determinism Is Ontic, Determinability Is Epistemic,” by philosopher Harald Atmanspacher).[9]
Experts tear their hair out over how fans of “chaoticism = free will” fail to make these distinctions. “There is a persistent confusion about determinism and predictability,” write physicists Sergio Caprara and Angelo Vulpiani. The first name–less philosopher G. M. K. Hunt of the University of Warwick writes, “In a world where perfectly accurate measurement is impossible, classical physical determinism does not entail epistemic determinism.” The same thought comes from philosopher Mark Stone: “Chaotic systems, even though they are deterministic, are not predictable [they are not epistemically deterministic]. . . . To say that chaotic systems are unpredictable is not to say that science cannot explain them.” Philosophers Vadim Batitsky and Zoltan Domotor, in their wonderfully titled paper, “When Good Theories Make Bad Predictions,” describe chaotic systems as “deterministically unpredictable.”[10]
Here’s a way to think about this extremely important point. I just went back to that fantastic pattern in the last chapter, on page 138, and estimated that it is around 250 rows long and 400 columns wide. This means that the figure consists of about 100,000 boxes, each now either open or filled. Get a hefty piece of graph paper, copy the row 1 starting state from the figure, and then spend the next year sleeplessly applying rule 22 to each successive row, filling in the 100,000 boxes with your #2 pencil. And you will have generated the same exact pattern as in the figure. Take a deep breath and do it a second time, same outcome. Have a trained dolphin with an extraordinary capacity for repetition go at it, same result. Row eleventy-three would not be what it is because at row eleventy-two, you or the dolphin just happened to choose to let the open-or-filled split in the road depend on the spirit moving you or on what you think Greta Thunberg would do. That pattern was the outcome of a completely deterministic system consisting of the eight instructions comprising rule 22. At none of the 100,000 junctures could a different outcome have resulted (unless a random mistake occurred; as we’ll see in chapter 10, constructing an edifice of free will on random hiccups is quite iffy). Just as the search for an uncaused neuron will prove fruitless, likewise for an uncaused box.
Let’s frame this in the context of human behavior. It’s 1922, and you’re presented with a hundred young adults destined to live conventional lives. You’re told that in about forty years, one of the hundred is going to diverge from that picture, becoming impulsive and socially inappropriate to a criminal extent. Here are blood samples from each of those people, check them out. And there’s no way to predict which person is above chance levels.
