Before I get to the serious stuff, a quick story about John Conway, a.k.a. the “mathematical magician.” I met him in 1993 in Princeton while working on “The Death of Proof.” When I poked my head into his office, Conway was sitting with his back to me staring at a computer. Hair tumbled down his back, his sagging pants exposed his ass-cleft. His office overflowed with books, journals, food wrappers and paper polyhedrons, many dangling from the ceiling. When I tentatively announced myself, he yelled without turning, What’s your birthday! Uh, June 23, I said. Year! Conway shouted. Year! 1953, I replied. After a split second he blurted out, Tuesday! He tapped his keyboard, stared at the screen and exulted, Yes! Finally facing me, Conway explained that he belongs to a group of people who calculate the day of the week of any date, past or present, as quickly as possible. He, Conway informed me with a manic grin, is one of the world’s fastest day-of-the-week calculators.
This encounter came back to me recently as I read a wonderful New York Times tribute to Conway, felled by COVID-19 last year at the age of 82. The Times focuses on the enduring influence of the Game of Life, a cellular automaton invented by Conway more than a half century ago. Scientific American’s legendary math columnist Martin Gardner introduced the Game of Life, sometimes just called Life, to the world in 1970 after receiving a letter about it from Conway. The Times riff on Life got me thinking anew about old riddles. Like, Does free will exist?
Some background. A cellular automaton is a grid of cells whose states depend on the states of neighboring cells, as determined by preset rules. The Game of Life is a two-dimensional cellular automaton with square cells that can be in one of two states, alive or dead (often represented by black or white). *A given cell’s state depends on the state of its eight immediate neighbors. A dead cell comes to life if three of its neighbors are alive, and a live cell stays alive if two or three of its neighbors are alive. Otherwise, the cell dies or remains dead. So simple!* And yet Life, when the rules are applied over and over, ideally by a computer, yields endlessly varied patterns, including quasianimated clusters of cells known as “longboats,” “gliders,” “spaceships” and my favorite, “Speed Demonoids.”
Like the Mandelbrot set, the famous fractal icon, the Game of Life inspired the fields of chaos and complexity, which are so similar that I lump them together under a single term: chaoplexity. Chaoplexologists assume that just as Life’s odd digital fauna and flora stem from straightforward rules, so do many real-world things. With the help of computer simulations, chaoplexologists hoped to discover the rules, or algorithms, underpinning stuff that has long resisted conventional scientific analysis, from immune systems and brains to stock markets and whole civilizations. (The “big data” movement has recycled the hope, and hype, of chaoplexology.)
Of course, the Game of Life can be interpreted in different ways. It resembles a digital, animated Rorschach test upon which scholars project their biases. For example, philosopher Daniel Dennett, commenting on Conway’s invention in the Times, points out that Life’s “higher-order patterns” emerge from processes that are “completely unmysterious and explicable.... No psionic fields, no morphic resonances, no élan vital, no dualism.”
Dennett’s comment annoyed me at first; Life just gives him an excuse to reiterate his defense of hard-core materialism. But Life, Dennett goes on to say, shows that deterministic rules can generate “complex adaptively appropriate structures” capable of “action” and “control.” Yes! I thought, my own bias coming into play. Dennett clearly means that deterministic processes can spawn phenomena that transcend determinism, like minds with free will.
Then another thought occurred to me, inspired by my ongoing effort to understand quantum mechanics. Conventional cellular automata, including Life, are strictly local, in the sense that what happens in one cell depends on what happens in its neighboring cells. But quantum mechanics suggests that nature seethes with nonlocal “spooky actions.” Remote, apparently disconnected things can be “entangled,” influencing each other in mysterious ways, as if via the filaments of ghostly, hyperdimensional cobwebs.
I wondered: Can cellular automata incorporate nonlocal entanglements? And if so, might these cellular automata provide even more support for free will than the Game of Life? Google gave me tentative answers. Yes, researchers have created many cellular automata that incorporate quantum effects, including nonlocality. There are even quantum versions of the Game of Life. But, predictably, experts disagree on whether nonlocal cellular automata bolster the case for free will.
One prominent explorer of quantum cellular automata, Nobel laureate Gerard ‘t Hooft, flatly rules out the possibility of free will. In his 2015 monograph The Cellular Automaton Interpretation of Quantum Mechanics, ‘t Hooft argues that some annoying features of quantum mechanics—notably its inability to specify precisely where an electron will be when we observe it—can be eliminated by reconfiguring the theory as a cellular automaton. ‘t Hooft’s model assumes the existence of “hidden variables” underlying apparently random quantum behavior. His model leads him to a position called “superdeterminism,” which eliminates (as far as I can tell; ‘t Hooft’s arguments aren’t easy for me to follow) any hope for free will. Our fates are fixed from the big bang on.
Another authority on cellular automata, Stephen Wolfram, creator of Mathematica and other popular mathematical programs, proposes that free will is possible. In his 2002 opus A New Kind of Science, Wolfram argues that cellular automata can solve many scientific and philosophical puzzles, including free will. He notes that many cellular automata, including the Game of Life, display the property of “computational irreducibility.” That is, you cannot predict in advance what the cellular automata are going to do, you can only watch and see what happens. This unpredictability is compatible with free will, or so Wolfram suggests.
John Conway, Life’s creator, also defended free will. In a 2009 paper, “The Strong Free Will Theorem,” Conway and Simon Kochen argue that quantum mechanics, plus relativity, provide grounds for belief in free will. At the heart of their argument is a thought experiment in which physicists measure the spin of particles. According to Conway and Kochen, the physicists are free to measure the particles in dozens of ways, which are not dictated by the preceding state of the universe. Similarly, the particles’ spin, as measured by the physicists, is not predetermined.
Their analysis leads Conway and Kochen to conclude that the physicists possess free will—and so do the particles they are measuring. “Our provocative ascription of free will to elementary particles is deliberate,” Conway and Kochen write, “since our theorem asserts that if experimenters have a certain freedom, then particles have exactly the same kind of freedom.” That last part, which ascribes free will to particles, threw me at first; it sounded too woo. Then I recalled that prominent scientists are advocating panpsychism, the idea that consciousness pervades all matter, not just brains. If we grant electrons consciousness, why not give them free will, too?
To be honest, I have a problem with all these treatments of free will, pro and con. They examine free will within the narrow, reductionistic framework of physics and mathematics, and they equate free will with randomness and unpredictability. My choices, at least important ones, are not random, and they are all too predictable, at least for those who know me.
For example, here I am arguing for free will once again. I do so not because physical processes in my brain compel me to do so. I defend free will because the idea of free will matters to me, and I want it to matter to others. I am committed to free will for philosophical, ethical and even political reasons. I believe, for example, that deterministic views of human nature make us more likely to accept sexism, racism and militarism. No physics model—not even the most complex, nonlocal cellular automaton--can capture my rational and, yes, emotional motives for believing in free will, but that doesn’t mean these motives lack causal power.
Just as it cannot prove or disprove God’s existence, science will never decisively confirm or deny free will. In fact, ‘t Hooft might be right. I might be just a mortal, 3-D, analog version of the Speed Demonoid, plodding from square to square, my thoughts and actions dictated by hidden, superdeterministic rules far beyond my ken. But I can’t accept that grim worldview. Without free will, life lacks meaning, and hope. Especially in dark times, my faith in free will consoles me, and makes me feel less bullied by the deadly Game of Life.
I obsess over free will and related riddles in my two most recent books: Pay Attention: Sex, Death, and Science, and Mind-Body Problems: Science, Subjectivity & Who We Really Are.
*Editor's note: The passage between the asterisks was revised because the description in the original version was incorrect.