February 4, 2024. My recent riffs on chaoplexity (see here and here) got me brooding over The Game of Life and its relevance to free quantum mechanics and free will. Here’s the backstory:

The Game of Life is a cellular automaton, a grid of cells whose states depend on the states of neighboring cells, as determined by preset rules. Scientific American’s legendary math columnist Martin Gardner introduced The Game of Life to the world in 1970 after getting a letter about it from its inventor, mathematician John Conway.

The Game of Life, often abbreviated to 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 neighbors are alive. Otherwise, the cell dies or remains dead.

So simple! And yet Life, when a computer iterates its rules, yields endlessly varied patterns. These include quasi-animated clusters of cells known as “longboats,” “gliders,” “spaceships” and my favorite, “Speed Demonoids.”

Like the famed fractal the Mandelbrot set, The Game of Life inspired researchers in chaos and complexity, fields so similar that I lump them together as chaoplexity. Just as Life’s odd digital fauna and flora stem from straightforward, simple rules, so might many hideously complex real-world things, like brains and stock markets. Of course, science has already found simple rules underpinning nature, rules embodied by the second law of thermodynamics, evolutionary theory and general relativity. But chaoplexologists hope—so far in vain—to find rules that apply even more broadly.

So what does The Game of Life have to do with free will and quantum mechanics? Here’s what. Conventional cellular automata, including Life, are deterministic, meaning that each state of the automaton dictates the next state. The automata lack the randomness, or wiggle room, that might allow for choices, free will.

Conventional cellular automata are also local, in the sense that what happens in one cell depends on what happens in neighboring cells. But according to quantum mechanics, nature seethes with nonlocal “spooky actions.” Remote, apparently disconnected things can be “entangled,” influencing each other in mysterious ways as if via ghostly filaments. Moreover, there is a random element to nonlocal effects. Wiggle room!

Two questions: Can cellular automata incorporate nonlocal, random entanglements? And if so, might these cellular automata lend support to free will? The answers to these questions are “Yes” and “Maybe,” respectively. Researchers have created cellular automata that incorporate quantum effects, including nonlocality. There are even quantum versions of The Game of Life. But experts disagree on whether these models bolster the case for free will.

Physicist and Nobel laureate Gerard ‘t Hooft says quantum cellular automata models rule out free will. In his 2015 monograph The Cellular Automaton Interpretation of Quantum Mechanics, ‘t Hooft argues that a quantum cellular automaton eliminates the randomness of quantum mechanics. ‘t Hooft postulates that “hidden variables” underpin apparently random quantum behavior. He embraces a position called “superdeterminism,” which eliminates any hope for free will; our fates are fixed from the big bang on.

But another authority on cellular automata, Stephen Wolfram, 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. 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 will do; you can only wait to see what happens. This unpredictability is compatible with free will--or so Wolfram suggests.

John Conway, Life’s creator, also defended free will, sort of, before his death four years ago. In his 2009 paper “The Strong Free Will Theorem,” Conway and his fellow mathematician Simon Kochen argue that quantum mechanics provides 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 many different ways, which are unconstrained by the preceding state of the universe. Similarly, the particles’ spin, as measured by the physicists, is not predetermined.

Conway and Kochen conclude that physicists observing particles possess free will—and so do the particles they are observing. “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.”

Having met Conway [see Postscript], I suspect his free-will paper might have been a joke. And to be honest, “proofs” of free will seem as dubious to me as denials. Proofs tend to equate free will with randomness and unpredictability. My choices, at least important ones, are neither random nor unpredictable, 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 fear deterministic views of human nature undermine efforts to combat sexism, racism and militarism.

If pressed, I’ll concede that ‘t Hooft could be right. I might be a mortal, 3-D, analog version of the “Speed Demonoid,” plodding from square to square, my thoughts and actions determined by rules beyond my ken.

But I hate to accept that grim worldview. Life without free will lacks meaning, and hope. Especially in dark times, my faith in free will consoles me, and makes me feel less bullied by the deadly, all-too-real game of life.

Postscript: In 1993 I took a train to Princeton to interview a famous mathematician, When I poked my head into his office, he 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, the man yelled without turning, What’s your birthday! Uh, June 23, I said. Year! he shouted. Year! 1953, I replied. A split second later the man blurted out, Tuesday! He tapped his keyboard, stared at the screen and exulted, Yes! Finally facing me, he explained that he belongs to a global group of people who calculate the day of the week of any date, past or present, as quickly as possible. He, the man informed me with a manic grin, is one of the world’s fastest day-of-the-week calculators. That was my introduction to John Conway.

Self-plagiarism Alert: This is a revised version of a paywalled column posted on ScientificAmerican.com a few years ago. That content wants to be free!

Further Reading:

The Chaoplexity Delusion

Mitchell Feigenbaum and the End of Chaoplexity

Free Will and the Sapolsky Paradox

Free Will and the Could-You-Have-Chosen-Otherwise Gambit

Farts, Boners and Free Will. Seriously

Free Will, War and the Tolstoy Paradox

I also get deep into quantum mechanics and free will in my free online book My Quantum Experiment.