Note: It’s been almost two years since my last post. We had a baby, and life has been more full with two little boys. It’s been lots of fun! But I kept coming back to this post, so eventually I was able to finish it. No guarantee on when the next post will be...

Recently, I have been really excited about the Free Will Theorem by John Conway and Simon Kochen. It states that if we have free will then elementary particles have some degree of free will too¹. If you believe we can make free decisions, then this theorem has profound implications. In particular, it challenges that dichotomy between inanimate matter and us humans with our minds, emotions, and free will. Perhaps electrons and galaxies also have some degree of internal life?

But first do we have free will? I believe we do have a limited ability to choose between different options and that our decisions have the potential to change the course of the unfolding universe. But this feels like a religious belief, because I can’t prove it. When I look at the laws of physics, things look to be deterministic or to be driven by randomness. Fundamentally, I’m a bunch of electrons, protons, and neutrons that obey the laws of physics, so how does my free will fit into this picture if it exists at all?

These questions about free will are core to the mind-body problem. One way to state the problem is, since we are matter where does our subjective inner life come from? I want to write about some of my grappling with the mind-body problem, and about how the Free Will Theorem helps.

Wondering about the mind body problem

I’ve struggled with what you could call the mind-body problem since I was a young. I remember feeling uncomfortable when people described the human brain as “just a bag of chemicals”. It didn’t make sense to me. How could a bag of chemicals learn, make decisions, and love? But we are are made from matter, so I was left with an unsettled feeling, that maybe I was wrong. In college, I considered the idea that maybe inner life doesn’t exist at all. I didn’t know it at the time but this is called eliminative materialism². To me that is an incredibly dark idea. Fascinating that you could have a whole universe without any awareness, but also deeply lonely and disturbing. But I could not accept this idea because it would mean denying my own inner life.

I also considered the idea that perhaps I am the only one that experiences inner life. What if the people I’m talking to are just responding according to the electrical signals in their brain that have been trained by their experiences and by evolution? What if they have no subjective experience? Perhaps them talking about it is just a quirk of evolution like peacock tails. I also didn’t know this at the time but this describes solipsism³. It didn’t make sense to me because I’m probably not that special, so most likely other people have an inner life as well.

After graduate school, I entered a period of questioning the foundations of my Catholic faith, and in the process learned about Pierre Teilhard de Chardin. He writes in “The Phenomenon of Man” about his theory that all things have a inner subjective experience⁴. I found this idea to be profound. I had already believed that animals have some type of consciousness, but now I was invited to consider even electrons to have some small amount of inner life. Teilhard’s view of the world and the mind is a form of panpsychism, specifically panexperientialism⁵. This idea suggests that all matter has some degree of mentality from humans down to electrons.

But say we and everything else in the universe has this inner experience, does it matter? Do our thoughts change anything or are they fully determined by our evolving physical configuration? Is free will an illusion? I am definitely biased is this area. I can’t shake the idea that our choices matter, that we are not entirely determined by our past. I do admit that I am heavily influenced by my environment, but still I believe that at least in a small way I can choose between different paths.

With these types of questions there are no easy answers, but the Free Will Theorem gives a useful perspective. It describes free will as the ability to take an action that can not be predicted by any previous information in the universe. If this type of free will exists then it can change the course of the universe; the story is still being written.

The Free Will Theorem

John Conway and Simon Kochen are both mathematicians but their Free Will Theorem has implications in physics and philosopy. Conway gave a series of lectures about this theorem, and acknowledged that you have to be a little crazy to make a big statement outside your field⁶. But they had something important to say.

The setup

To make their proof, they imagined a situation where two spin-1 particles (for example certain atoms) are first “twinned” or entangled (in a singleton state) and then one of them is moved to Mars while the other remains on Earth. There is an experimenter A on Earth and an experimenter B on Mars and each experimenter can measure the spin of their respective particle. If they measure the squared component of the spin along a certain direction, then they will get a 1 or a 0. It turns out that for spin 1 particles, if one experimenter measures the spin in three orthogonal directions then result will be two 1’s and one 0 in some order. This is derivable from quantum mechanics (I don’t know how!), but they take it as an axiom which they call the SPIN axiom.

The second key piece of the experiment is that the particles are entangled. Even though one particle is on Earth and the other is on Mars, when each experimenter measures the spin in the same direction they will measure the same value for the spin. This a strange effect. We don’t see this type of behavior in daily life, but it is what we observe experimentally for entangled electrons and other very small particles in entangled states. Conway and Kochen call this effect the TWIN axiom.

The final setup for the theorem is the limit on the speed that information can travel between the two experimenters. From relativity theory, we know that no signals carrying information can travel faster than the speed of light. Since the experimenters are so far apart this means that each has time to perform an experiment before the other can influence it by sending a signal. They call this the FIN axiom. With these three axioms they derive the following theorem.

The theorem

This paper has set up a situation that relates the experimenter’s choice of measurement direction to the response of the particles. The theorem shows that if the experimenters can freely choose the direction to measure a particle then the particle’s response is free in the same way. This is a strange, but intriguing idea. Here is how they state the theorem:

The Free Will Theorem (assuming SPIN, TWIN, and FIN)].

If the choice of directions in which to perform spin 1 experiments is not a function of the information accessible to the experimenters, then the responses of the particles are equally not functions of the information accessible to them.

John Conway and Simon Kochen

The measured spin does not exist before the measurement

We are used to properties existing before you measure them. For instance, we don’t expect that measuring a box will change its dimensions. But small particles in the quantum regime are different. They can be in a state where the measurement outcome is random. It turns out for this specific example of spin 1 particles, you can prove that the measured spin does not exist before the measurement. If it did then there would be a deterministic function that gives the spin based on the direction of measurement. Such a function, called a 101-function, does not exist. The proof is call the Kochen-Specker paradox and depends on a geometric puzzle with no solution⁷. That’s good, because if the measured value already existed then the particles would definitely not have any choice in how they responded to the measurement.

Breaking down the proof

Not only does the measured spin not exist until the experiment, the Free Will Theorem proves that it can’t be predicted. The proof works by contradiction. First it assumes the opposite, that it is possible to predict the measured spin of the particle from information available to the particle. After a series of deductions John Conway and Simon Kochen find a contradiction. If the opposite of the Free Will Theorem is false, then the Free Will Theorem is true.

So how do they find the contradiction? Here are the steps they follow:

1. First they assume that there actually are functions that could predict the responses of the particles given any information available to the particle before the direction of measurement was chosen by the experimenter.
2. Then they use the TWIN axiom, that the particles are entangled, to show that the function for particle a is equal to the function for particle b if the measurement directions match.
3. Next they note that the information available to particle a before the measurement does not include the direction of the measurement of particle b, because according to relativity there are some frames of reference where particle b is measured before particle a. This allows them to define a function that returns the measured spin values using only the measurement directions as input.
4. Finally they observe that this new function is a 101-function, but such a function does not exist. So by contradiction they prove the Free Will Theorem

I keep having to re-read the theorem to follow through the logic, but each time it makes sense. If you’re interested, I definitely recommend you read it yourself. As far as mathematical papers, it is relatively accessible.

Implications of the Free Will Theorem

At the low level, the Free Will Theorem is just talking about whether it is possible to predict the outcome of a particular experiment, but the conclusion has deep implications. If the experimenters could pick the directions for the experiment then the particles could pick their response. How can particles pick a response? Do they have some degree of consciousness? It strange to think about everything in the universe having some small amount of inner life, but if it is true then it points to where our own free will comes from. Perhaps the free choices of the particles that make us up don’t completely cancel out, which would leave us with some ability to make free choices ourselves. In their philisophical remarks, John Conway and Simon Cochen write:

The authors strongly believe, however, that there is a way our brains prevent some of this cancellation, so allowing us to integrate what remains and producing our own free will

John Conway and Simon Cochen

It is not clear at all how that emergence of our free will comes from the free wills of our constituent particles. But it makes a lot of sense to me. From this perspective, there is no need for a magical life force that animates the body. Matter itself is animated and has some limited ability to determine its own destiny.

Of course, there is no requirement to believe in free will for people or for sub-atomic particles. The Many Worlds interpretation of quantum mechanics is a deterministic interpretation that is gaining popularity⁸. The idea is that the wavefunction of the universe is what is ultimately real. It evolves deterministically, but contains projections that correspond to different versions of reality. Each projection is a parallel world. Every time there is what appears to be a free choice of a particle or a person, the world divides with one world for each possible outcome. There is a certain beauty to this theory because everything can be explained mathematically, in principle.

I find this Many Worlds interpretation to be unsatisfying, because I have such difficulty letting go of the idea that I can make choices and that they could make a difference. But I want my beliefs to match reality as much as possible. The Free Will theorem provides a coherent picture of how free will works if it exists. I find it to be an expansive and exciting view of the world. I like how John Conway and Simon Cochen end their paper:

Einstein could not bring himself to believe that “God plays dice with the world,” but perhaps we could reconcile him to the idea that “God lets the world run free.”

John Conway and Simon Cochen

A world where all the protons and electrons are running free sounds great to me. I hope it’s true. Until I find evidence that disproves it, I’ll take it as an article of faith,

---

1. Conway, John and Kochen, Simon. “The Free Will Theorem”. URL=Arxiv.orghttps://arxiv.org/abs/quant-ph/0604079v1
2. Ramsey, William, “Eliminative Materialism”, The Stanford Encyclopedia of Philosophy (Summer 2020 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/sum2020/entries/materialism-eliminative/
3. Stephen P. Thornton. “Solipsism and the Problem of Other Minds”, Internet Encyclopedia of Philosophy, URL=https://iep.utm.edu/solipsis/#H1 accessed 12/30/2022.
4. Pierre Teilhard de Chardin, The Phenomenon of Man, p56-57
5. Kind, Amy. “Panexperientialism, Cognition, and the Nature of Experience”, Psyche 2006: VOLUME 12 ISSUE 5. URL=http://journalpsyche.org/files/0xab06.pdf2021-05-27 09:05 Panpsychism
6. Conway, John. “Free Will and Determinism in Science and Philosophy”, lecture series, URL=https://mediacentral.princeton.edu/media/Free+Will+and+Determinism+in+Science+and+Philosophy/1_h7r12hbc
7. Kochen, S., Specker, E., J. of Math. and Mech. 17 (1967), 59–87. URL=http://www.iumj.indiana.edu/IUMJ/fulltext.php?artid=17004&year=1968&volume=17
8. Sean Carroll. “Reality as a Vector in Hilbert Space”, 2021. URL=https://arxiv.org/abs/2103.09780