Mathematical Necessity, Nature and the bed of Procrustes

This is a repost of an article I published few years ago. I have decided to publish it here following the conversation I had on Twitter space with @RokoMijic (Twitter handle) on AI risk. The issue of whether or not the human mind is Turing computable occurred. Here are my thoughts about this complex issue.

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In It’s been proved, Brett Hall holds the claim that “the brain is a computer” has been proved. I disagree with the fact this statement has been proved. Note however, that I don’t make here any judgment on the claim “the brain is a computer” itself (whether it is true or false at the end of the day).

What I do claim however is that exactly one of the following claim is true:

1. Either the statement “the brain is a computer” is true. Then, a proof has to be a constructive proof, meaning that you have to provide a positive evidence (a software piece) for which you can experimentally prove that it behaves like a brain, whatever you intend behind the words “behave” and “experimentally” in this context (what experiences you can set up in order to validate/invalidate the claim).

2. Or “the brain is not like a computer”. In this case, I claim that the Church-Turing thesis implies that one will never have a scientific proof of that impossibility. Indeed, if one would have such a proof it would mean that one would have a working definition/theory of what is “a brain” in order to prove that no Turing-Machine could simulate it: otherwise if you don’t have such a definition you can’t say you can’t do something you cannot even define ! But having a working definition (that is a theory in which one can compute how states evolve from one to another) means having a Turing-Machine level description of what a brain is and thus it would contradict the hypothesis that the brain is not like a computer.

The main problem in Brett Hall’s article is the conflation of proofs of different nature. Proof in physics and proof in mathematics are not epistemically equivalent.

• A proof in a mathematical/theoretical context is a sentence that obeys to some syntactic rules. You can check a proof without understanding anything about what the proof is about. Ultimately you just have to check that the sentence that you call “a proof” is well built: usually it is a syntactical tree in which nodes are inference rules and leafs are axioms. The root of the tree being the proved sentence. The correctness property insures that every proved sentence is true (the converse, that every true sentence is provable is called completeness and is not verified as soon as the logic becomes powerful enough, there are the famous Gödel’s incompleteness theorems).

• A proof in a physic context is different. It is statement in a theory for which one can provide a decidable way to find counter-examples in reality. As long as no counter-examples have been found the statement is “proved”. The theory has not to explain every natural phenomenon : one has simply to include correct range of applications. For instance one can say that, within proper limits, Newtonian physics has been proved. Relativity may expand those limits and be proved as well.

Note that there is a huge difference between the two kind of proofs: in the mathematical world a proof is a syntactic property and its relation with truth lies in the facts that a) the theory is consistent (otherwise anything is provable in an inconsistent theory and there is no point to talk about proofs anymore) and b)the correctness of the provability procedure. In the physics world a proof is an experimental link between a theory and experiments in the real world.

Now, let us consider the following statement for Brett Hall’s article :

So when we say “David Deutsch proved the Turing Principle(1) in 1985 that “every finitely realizable physical system can be simulated to arbitrary precision by a universal model computing machine operating by finite means” (2) we mean that beginning with some uncontroversial assumptions from mainstream quantum theory along with what was already known from classical computing as discovered by Turing, he was able to reach that conclusion.

what is referred to is the following thesis

Every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means

that is stated in “Quantum theory, the Church-Turing principle and the universal quantum computer (1985)”.

What I claim is that this does not prove that the brain is a computer. Indeed, for that one would have to prove that a brain is a “finitely realizable physical system”. The subtle point is that a brain is not a machine, or an experiment, that has been built following a plan. Where I do agree with this statement is that any experiment built will follow this Deutch-Church-Turing principle.

Indeed, in order to set up an experiment one has to write a text in order to precisely define it. Moreover, if one wants to check his theory vs reality one has to compute in his pet theory what the result should be, and check it against the experimental results. But because of this use of a peculiar technology (the language with which you describe your theory) one will be limited in the range of possible experiments. You can only talk about experiments that can be described using language. Those limitations do not necessarily apply to the physical world.

To think that the reality has to perfectly fit into any theory of the world is not without recalling the story of Procustres:

In the Greek myth, Procrustes was a son of Poseidon with a stronghold on Mount Korydallos at Erineus, on the sacred way between Athens and Eleusis.[1] There he had a bed, in which he invited every passer-by to spend the night, and where he set to work on them with his smith’s hammer, to stretch them to fit. In later tellings, if the guest proved too tall, Procrustes would amputate the excess length;

So the problem is not that Deutch’s proof is false, but that the Deutch-Church-Turing thesis does not entail that the brain is a computer.

As far as I know there is no proof of the fact the brain is a computer, though it could change: someone could provide a constructive proof by exhibiting a software, a testable definition of what a brain does, and experimental proofs that the software match its specification. But what I do know is that noone will ever find a proof that a the brain is not like a computer. In that, it is very similar to the semi-decidability status of the halting problem.