# On Determinism

Back in 1814, Pierre-Simon Laplace was mulling over the implications of Newtonian mechanics, and realized something profound. If there were a vast intelligence — since dubbed Laplace’s Demon — that knew the exact state of the universe at any one moment, and knew all the laws of physics, and had arbitrarily large computational capacity, it could both predict the future and reconstruct the past with perfect accuracy. While this is a straightforward consequence of Newton’s theory, it seems to conflict with our intuitive notion of free will. Even if there is no such demon, presumably there is some particular state of the universe, which implies that the future is fixed by the present. What room, then, for free choice? What’s surprising is that we still don’t have a consensus answer to this question. Subsequent developments, most relevantly in the probabilistic nature of predictions in quantum mechanics, have muddied the waters more than clarifying them.

Massimo Pigliucci has written a primer for skeptics of determinism, in part spurred by reading (and taking issue with) Alex Rosenberg’s new book *The Atheist’s Guide to Reality*, which I mentioned here. And Jerry Coyne responds, mostly to say that none of this amounts to “free will” over and above the laws of physics. (Which is true, even if, as I’ll mention below, quantum indeterminacy can propagate upward to classical behavior.) I wanted to give my own two cents, partly as a physicist and partly as a guy who just can’t resist giving his two cents.

Echoing Massimo’s structure, here are some talking points:

* There are probably many notions of what determinism means, but let’s distinguish two. The crucial thing is that the universe can be divided up into different moments of time. (The division will generally be highly non-unique, but that’s okay.) Then we can call “global determinism” the claim that, if we know the exact state of the whole universe at one time, the future and past are completely determined. But we can also define “local determinism” to be the claim that, if we know the exact state of some part of the universe at one time, the future and past of a certain region of the universe (the “domain of dependence”) is completely determined. Both are reasonable and relevant.

* It makes sense to be interested, as Massimo seems to be, in whether or not the one true correct ultimate set of laws of physics are deterministic or not. He argues that we don’t know, and that’s obviously right, since we don’t know what the final theory is. But that’s a rather defeatist attitude all by itself; we can look at the theories we do understand and try to draw lessons from them.

* Classical mechanics, which you might have thought was deterministic if anything was, actually has some loopholes. We can think of certain situations where more than one future obeys the equations of motion starting from the same past. This is discussed a bit in the Stanford Encyclopedia of Philosophy article on causal determinism. But I personally don’t find the examples that impressive. For one thing, they are highly non-generic; you have to work really hard to find these kinds of solutions, and they certainly aren’t stable under small perturbations. More importantly, classical mechanics isn’t right; it’s just an approximation to quantum mechanics, and these finely-tuned classical solutions would be dramatically altered by quantum effects.

* General relativity is a classical theory, so it’s also not correct, but we don’t have the final theory of quantum gravity so it’s worth a look. As Massimo points out, there are good examples in GR where traditional global determinism breaks down; naked singularities would be an example. (Basically, determinism breaks down when information can in principle “flow in” from a singularity or boundary that isn’t included in “the whole universe at one moment of time.”) We might sidestep this problem by arguing that naked singularities aren’t physical, which is quite reasonable. But there are much more benign examples, such as anti-de Sitter space — a maximally symmetric spacetime with a negative cosmological constant. This universe has no singularities, but does have a boundary at infinity, so a single moment of time only determines part of the universe, not the whole thing. On the other hand, like the classical-mechanics examples alluded to above, this seems like a technicality that can be cleared up with a slight change of definition, e.g. by imposing some simple boundary condition at infinity.

Much more importantly, these kinds of GR phenomena are very far away from our everyday lives; there’s really no relevance to discussions of free will. GR violates global determinism in the strict sense, but certainly obeys local determinism; that’s all that should be required for this kind of discussion.

* Quantum mechanics is where things get interesting. When a quantum state is happily evolving along according to the Schrödinger equation, everything is perfectly deterministic; indeed, more so than classical mechanics, because the space of states (Hilbert space) doesn’t allow for the kind of non-generic funny business that let non-deterministic classical solutions sneak in. But when we make an observation, we are unable to deterministically predict what its outcome will be. (And Bell’s theorem at least suggests that this inability is not just because we’re not smart enough; we never will be able to make such predictions.) At this point, opinions become split about whether the loss of determinism is real, or merely apparent. This is a crucial question for both physicists and philosophers, but not directly relevant for the question of free will.

The traditional (“Copenhagen”) view is that QM is truly non-deterministic, and that probability plays a central role in the measurement process when wave functions collapse. Unfortunately, this process is extremely unsatisfying, not just because it runs contrary to our philosophical prejudices but because what counts as a “measurement” and the quantum/classical split are extremely ill-defined. Almost everyone agrees we should do better, despite the fact that we still teach this approach in textbooks. Someone like Tom Banks would try to eliminate the magical process of wave function collapse, but keep probability (and thus a loss of determinism) as a central feature. There is a whole school of thought along these lines, which treats the quantum state as a device for tracking probabilities; see this excellent post by Matt Leifer for more details.

The other way to go is many-worlds, which says that the ordinary deterministic evolution of the Schrödinger equation is all that ever happens. The problem there is comporting such a claim with the reality of our experience — we see Schrödinger’s cat to be alive or dead, not ever in a live/dead superposition as QM would seem to imply. The resolution is that “we” are not described by the entire quantum state; rather, we live in one branch of the wave function, which also includes numerous other branches where different outcomes were observed. This approach (which I favor) restores determinism at the level of the fundamental equations, but sacrifices it for the observational predictions made by real observers. If I were keeping a tally, I would certainly put this one in the non-determinism camp, for anyone interested in questions of free will.

* Then there is the question of whether or not the lack of determinism in QM plays any role at all in our everyday lives. When we flip a coin or play the lottery, one might think that the relevant probabilities are “purely classical” — i.e. they stem from our lack of knowledge about the state of the muscles and nerves in my hand and the wind and the coin that is about to be flipped, but if I knew all of those things I could make a perfectly deterministic prediction about what would happen to the coin. (Indeed, a well-trained magician can flip a coin and get whatever result they want.)

This is actually a tricky problem, to which the answers aren’t clear. Yes, there may be a level of classical description in terms of a probability distribution; but where does that probability distribution come from? Physicists disagree about whether or not quantum mechanics plays a crucial role here. Since I have friends in high places, this weekend I emailed Andy Albrecht, who answered and brought David Deutsch into the conversation. They both argue — plausibly, although I’m not really qualified to pass judgment — that essentially

allclassical probabilities can ultimately traced down to the quantum wave function. And indeed, that this reasoning provides the only sensible basis for talking about probabilities at all! (David mentions that Lev Vaidman seems to disagree, so it’s not uncontroversial by any means.) They are both, in other words, firmly anti-Bayesian in their view on probability. A good Bayesian thinks that probabilities are always statements about our fundamental ignorance concerning what is “really” going on. Albrecht and Deutsch would argue that’s not true, probabilities are ultimately always statements about the wave function of the universe. If they’re right — and again, it looks plausible, but I need to think about it more — then QM effects are indeed of crucial importance in accounting for our inability to predict the future in the everyday world.

* I should say something about chaos, which always comes up in these discussions. In classical mechanics, even when the underlying model is perfectly deterministic, it can often be the case that a small uncertainty in our knowledge of the initial state can lead to large uncertainty in the future/past evolution. (E.g. for the tumbling of Hyperion.) This is sometimes brought up as if it causes problems for determinism: “since tiny mistakes propagate, you couldn’t realistically predict the future anyway.” This is about as irrelevant as it is possible to be irrelevant. The Laplacian viewpoint was always that if you had

perfectinformation, you could predict the past and future. But that was always a statement of principle, not of practice. Of course, in practice, you have nowhere near enough information to make the kinds of calculation that Laplace’s vast intellect likes to do. That was perfectly obvious long before the advent of chaos theory. The correct statement is “in a classical deterministic system, with perfect information and arbitrary computing power you can predict the future in principle, but not in practice,” and that statement is completely unaltered by an understanding of chaos.

So where does that leave us? My personal suspicion is that the ultimate laws of physics will embody something like the many-worlds philosophy: the underlying laws are perfectly deterministic, but what happens along any specific history is irreducibly probabilistic. (In a better understanding of quantum gravity, our notion of “time” might be altered, and therefore our notion of “determinism” might be affected; but I suspect that there will still be some underlying equations that are rigidly obeyed.) But that’s just a suspicion, not anything worth taking to the bank. For everyday-life purposes, we can’t get around the fact that quantum mechanics makes it impossible to predict the future robustly.

Of course, this is all utterly irrelevant for questions of free will. (I’m sure Massimo knows this, but he didn’t discuss it in his blog post.) We can imagine four different possibilities: determinism + free will, indeterminism + free will, determinism + no free will, and indeterminism + no free will. All of these are logically possible, and in fact beliefs that some people actually hold! Bringing determinism into discussions of free will is a red herring.

It matters, of course, how one defines “free will.” The usual strategy in these discussions is to pick your own definition, and then argue on that basis, no matter what definition is being used by the person you’re arguing with. It’s not a strategy that advances human knowledge, but it makes for an endless string of debates.

A better question is, if we choose to think of human beings as collections of atoms and particles evolving according to the laws of physics, is such a description accurate and complete? Or is there something about human consciousness — some strong sense of “free will” — that allows us to deviate from the predictions that such a purely mechanistic model would make?

If *that’s* your definition of free will, then it doesn’t matter whether the laws of physics are deterministic or not — all that matters is that there are laws. If the atoms and particles that make up human beings obey those laws, there is no free will in this strong sense; if there is such a notion of free will, the laws are violated. In particular, if you want to use the lack of determinism in quantum mechanics to make room for supra-physical human volition (or, for that matter, occasional interventions by God in the course of biological evolution, as Francis Collins believes), then let’s be clear: you are not making use of the rules of quantum mechanics, you are simply *violating* them. Quantum mechanics doesn’t say “we don’t know what’s going to happen, but maybe our ineffable spirit energies are secretly making the choices”; it says “the probability of an outcome is the modulus squared of the quantum amplitude,” full stop. Just because there are probabilities doesn’t mean there is room for free will in that sense.

On the other hand, if you use a weak sense of free will, along the lines of “a useful theory of macroscopic human behavior models people as rational agents capable of making choices,” then free will is completely compatible with the underlying laws of physics, whether they are deterministic or not. That is the (fairly standard) compatibilist position, as defended by me in Free Will is as Real as Baseball. I would argue that this is the most useful notion of free will, the one people have in mind as they contemplate whether to go right to law school or spend a year hiking through Europe. It is not so weak as to be tautological: we could imagine a universe in which there were simple robust future boundary conditions, such that a model of rational agents would not be sufficient to describe the world. E.g. a world in which there were accurate prophesies of the future: “You will grow up to marry a handsome prince.” (Like it or not.) For better or for worse, that’s not the world we live in. What happens to you in the future is a combination of choices you make and forces well beyond your control — make the best of it!