The Emperor’s New Mind

The Emperor’s New Mind by the very eminent physicist Roger Penrose is one of my favourite science books and yet its goal is to argue for a very false hypothesis of Penrose’s. Like Asimov’s New Guide to Science and 1, 2, 3, Infinity which I covered in a previous post, the book’s strength comes from its very wide coverage. This book is a lot heavier than the other two and because it’s pretty rigorous it’s a slight struggle for a layman.

The book sets out to bolster Penrose’s claim that the human mind is not a computer (and therefore the AI project is not viable) because it does more than than “mere” computation. The second part of that claim is that a proper explanation of mind will be at the quantum level: once quantum physics is properly married to general relativity and a theory of gravity, and once we tack on the arrow of time everything will become clear. I’ll explain a bit later why I think he’s Extremely Wrong. But in the meantime, it’s still a wonderful premise for a book. Because to get to his wrong argument he must first explain a bit about:

• Algorithms, Turing machines, computability
• Mathematics and the nature of proof
• Classical mechanics
• General and special relativity
• Quantum mechanics
• Entropy, cosmology, the arrow of time, the origins of the universe

This takes about 80% of the book so even if the other 20% is wrong argument it’s still a great intro into the subjects — being Penrose he is very rigorous beyond most treatments.

Penrose clears a few misconceptions that laypeople who’ve only read popular books may have. For instance, many other books describe classical mechanics as something based simply on Newton’s laws (such as F = ma). However, these are very rarely used in their original formulation. Post-Newtonian classical mechanics is more about Hamiltonian spaces where a system of particles is taken to be a multi-dimensional point (with dimensions for each particle’s position and momentum). The computation of a classical system — the world of billard balls colliding with each other — therefore becomes a computation in many (sometimes an infinite amount of) dimensions. Penrose also gives a few reasons why the classical world might not even be computable, so that if you knew the position and momentum of all particles you still might not be able to work out exactly what will happen even in principle. Many philosophical debates (eg. free will, the mind and the existence of gods) often involve arguments about classical mechanics. One service this book performs is that it undercuts the notion that the classical world is somehow simple and “easily” computable.

Onto where Penrose is wrong. Bizarely, he seems to think the human mind is not a computer because of Godel’s Incompleteness Theorem. Let’s outline briefly (and I welcome corrections). Godel’s Incompleteness Theorem says this: take any useful mathematical system. By useful I mean it doesn’t contain a contradiction and is at least powerful enough to express the arithmetic of whole numbers. So we can use arithmetic as an example. Given the rules of this system (ie. elements, formal assumptions and rules of inference), we can construct a statement that’s true but is not provable from the system itself. I won’t mention Godel’s ingenious way of generating this statement. Although it’s the main part of the theorem, it’s very hard to understand and isn’t relevant to Penrose. But basically let’s just take it for granted that this sentence (call it X) can be made.

Here’s Penrose’s argument: the [human] mind of the mathematician “knows” that X is true. However a computer which has the system of arithmetic programmed into it can never prove X is true true and will therefore never “know” X is true! Let’s give the computer a helping hand and just add “X is true” to its list of axioms. Not so fast! says Penrose. This new system has its own Godel sentence (X’) which IT can’t prove. So again the computer doesn’t “know” X’ is true while I Penrose, possessor of a human mind, “know” X’ is true. So for any computer there’s a sentence it can provably never “know” is true, and yet a human mind “knows” all of them are true! So a human mind isn’t a computer.

I’ve made fun of this argument but I do recommend the actual book heartily. As for why Penrose is wrong, I might post on this next year (when I plan to cover a lot more about consciousness and mind). But for now I’ll leave it as an exercise for the reader. After all, this is a time of laziness, overeating and overdrinking so you might want a little brain workout. If you want a hint, it’s in my annoying and repetitive punctuation of the paragraph above…