I was going to write this up as a proper argument. I probably will at some point. However, for the time being, here is a sketch of an argument which appears to conclusively demonstrate the non-physical nature of the mind. Please let me know if you spot a flaw.

First, a definition. By ‘physical’ I mean that which is amenable to the science of physics. Contemporary physics is essentially a body of mathematical theorems and models applied to empirical data. The ‘physical world’ consists of everything in the world that is amenable to mathematical treatment. That which is not mathematical is deemed not to exist, or to have some sort of lesser existence.

Now, if anything physical is essentially mathematical, it could be simulated on a computer. Of course, in practice, we lack the hardware necessary to simulate anything large or complex realistically. We use approximations, we take short-cuts and so on. But, in theory, we could write a computer program to simulate a physical system completely.

Therefore, if the mind is physical it could be simulated on a computer. It is not necessary to identify the mind with the brain but, for the sake of example, let’s do that. Then all we would have to do is to map the brain accurately enough — maybe with some new scanning technique, or nanobots, or whatever — and use the data obtained to construct a software model. That model would then be functionally equivalent to the brain, hence it would be a mind.

However, all computer programs can be represented in the lambda calculus. And anything in the lambda calculus is equivalent, via a Curry-Howard isomorphism, with a logical proof — specifically, a proof in first-order logic.

But first-order logic is incapable of expressing things that the mind can express. Famously, Gödel proved that any first-order system will contain sentences that are not provable within that system yet are obviously true. That is to say, a human mind can see that they are true but they cannot be proved within the system itself. J.R. Lucas explains this well: take a look at his article Minds, Machines and Gödel for a more detailed exposition of this bit of the argument. (Also note that, according to Tarski, truth itself cannot be expressed within a logical system.)

Lucas says, “Gödel’s theorem seems to me to prove that Mechanism is false, that is, that minds cannot be explained as machines.” However, based on what I have said above, I wish to go further:

1. If minds are physical then they can be explained as machines.

2. Gödel’s theorem shows that minds cannot be explained as machines.

3. Therefore, minds are not physical.

This does not necessarily entail dualism. All it means is that minds are not physical in the sense that they cannot be completely described by physical science. However, that does mean that the physical world is not the whole world.

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