On the Quantum God of the Gaps

I recently listened to an episode of the Atheistically Speaking podcast on the subject of Einstein's and Gödel's belief (or lack thereof) in God.  The podcast overall is quite enjoyable and going strong after more than 200 episodes; I've only started listening in the past few weeks and have been alternating between listening to new and archived episodes. At a point in this particular show, the host (Thomas) and his guest (Kurt) quite literally say: "Hey, if there's a physicist out there listening to this, let us know what you think..." So, this post is the result of taking them at their word, given that so much of the show had me shaking my head throughout. I should also mention that Thomas was kind enough to invite me to write to him directly, and the following is a slightly polished version of the e-mail I sent him. Here we go:

Quantum Mechanics (QM) is part of a core of courses that all physicists take, no matter what their field. Obviously, people who specialize in QM will get into it a lot deeper than those that don't. There are some complicated areas of research going on right now, and there's plenty of debate going on--but the "basics", if we could call them that, are more than enough to refute just about everything Kurt said on episode 242.

First, the easy point: if you're going  to take the approach "some smart guy believes X, so X is true" your're already in trouble because, as Bertrand Russell said, if you rely on an authority for your argument, there will always be other authorities who disagree. So if we were to tally the John Polkinghornes of the world as evidence that there could be a god hiding somewhere in QM--or free will, or the soul, or whatever--, then we would have to be intellectually honest and tally all the other guys who say he's full of shit as evidence that there's no such thing, which is everybody else. People in this latter camp include Sean Carroll, Lawrence Krauss, Richard Feynman, Tim Maudlin, Steven Weinberg, and many others. Polkinghorne is a theologian, so he's intellectually compromised by definition. This is not an ad-hominem: this is a simple statement of the fact that he can't be relied upon to be intellectually honest. If he were, he wouldn't be a theologian.

Now, on to actual QM: Just as in high-school algebra we had some complicated equation with numbers and letters, and we were asked to solve for "x", so too in QM there's an equation to solve, only mere algebra won't quite do the trick. This is the Schrödinger equation, which looks something like this: \[ i \hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V\Psi. \]
It may look complicated, but the idea is the same as in high school algebra: solve for the wave function, \(\Psi\) (greek letter "Psi").  After you do that, what you will get is an equation for \(\Psi\) that tells you what its value is in terms of x (position) and t (plain old time), plus some other physical constants and the specifics of your system (these are included in the V term above). Just as "x" was uniquely determined by all the other variables in high-school algebra, so too is \(\Psi\) determined--fully, for all time--by all the other squiggles in the Schrödinger equation. This is a second-degree partial differential equation and, with the right input, is fully deterministic.

The subtlety arises in the fact that \(\Psi\) is not a number, but a probability density, and that's where laymen get mixed up. Here is where an analogy will help and, if you were to remember any part of this post as particularly useful, this would be it. Imagine you have a fair die that you can throw. For any given throw, it's pretty hard to know what number is going to come up. Naively, one would think that the die coughs up a random number between 1 and 6 each time. However, if you throw the die many times, eventually you'll see a pattern: each number comes up roughly 1/6 of the time. That 1/6 is determined by the geometry of the die. No matter what individual result you look at--1,4,2,5,3,etc.--the 1/6 is always the same for every throw. That 1/6, roughly, is \(\Psi\). The geometry of the die is everything else in the Schrödinger equation. So even though any given throw is undetermined, the overall 1/6 is fixed for as long as the geometry is fixed.

Every experiment ever done confirms that the above equation, or some version of it, is true. If Kurt  (or Polkinghorne, or whoever) wants to say that god is hiding somewhere in \(\Psi\), then they're making a claim that god can do no better that to show up as what probability theory would predict anyway without him, which seems like a pretty lame god to me.

As if that weren't enough, there's an entire field of research on hidden variable theory--the idea that there is a more fundamental, fully deterministic physics underneath the probabilistic character of the wave function. So the statement by Kurt, that it's just "a brute fact that QM is this way" is not obviously true either. Anyway, on to Gödel:

First, the necessary caveat: I'm a physicist, not a mathematician. With that said, Gödel's incompleteness theorems state that, for any given mathematical system based on arithmetic and axioms, there will be one of two inevitable outcomes: 1) there will be some theorems that will be true but unprovable, or 2) all theorems will be provable, but some will contradict others.

As an example, consider the sentence

"This statement is true".

Each word there is well-defined and fits the usual syntax to create a meaningful idea. You can think of each one of these words as an axiom (a basic assumption that doesn't need to be proven), and the syntax is the logic that is used to put them together and derive a theorem, which is a conclusion--the meaning of the statement itself. However, we can follow the same axioms and rules to construct a statement like this:

"This statement is false."

Each word there is well-defined, and all the words are arranged properly, but we can't decide on the meaning of the statement because the content of the statement refutes its logical construction!

Now, even though this is certainly important for the philosophy of mathematics, there is no way that you can get from it to the claim made by Kurt that some things are, and always will be, unknowable. That's a non-sequitur at worst, and a trivial argument from ignorance at best. A key distinction to keep in mind is that unprovable does not mean unknowable: remember, item 1) for the incompleteness theorems says "true, but not provable". In a way, this is part of what theoretical physics is about: there are some things we know are true (thanks to experiment) and we work to explain them--that is, to prove why it must be the case that they are true.

In any case, when people speak of god, they always mess up and fall into contradictions in the axioms themselves.  Their conclusions are unsound because their premises are gibberish and the arguments never get off the ground--there are no theorems to be derived at all! This happens to all definitions of god that include omnipotence, for example. Once they include that as a basic assumption, the rest is white noise.

Thomas covered other many other areas of disagreement with Kurt on the show itself, and some commenters on the episode site have done so as well, so I'll leave it here for now. As a conclusion, I would say that people aren't smuggling contraband into QM (god, free will, the soul, consciousness) as much as they're smuggling it into their misunderstanding of QM. Sometimes, these people even have PhD's and published peer-reviewed papers. Bad ideas in science are weeded out eventually, but someone has to actually get on the ground and do it. This must be done by the experts, but it can be a slow, thankless process and so most of them just stay out of it and focus on their research; only a few jump in and get down and dirty. In the meantime, one has to be patient and wait it out.