Penrose : Shadows of the mind

Shadows of the mind is an attempt by Roger Penrose to prove that a computing machine can never achieve human like intelligence; the implication being that human brains are something more than “mere” computing machines.

I have been skimming through this book the last few days and I find it very well written. Though, that is its very downfall! The central argument is clearly presented and hence it is easy (for an average mind like mine) to see it’s fallacy.

I will try to show the fallacy here in informal terms. The central argument in the book is just about a page long and tries to combine the halting problem with Godel’s theorem. Expressed informally, it goes like this:

Assume that all algorithms can be arranged into an (infinite) ordered list, C0, C1, C2….

Let A(n) be an algorithm that halts if Cn does not halt. Now A(n) itself has to be listed in the above list, say as Ck. The question is “will A(k) ever halt”? This kind of self-reference obviously leads to a paradox and Penrose uses it to argue that humans are somehow better than computing machines.

However, we can construct a very similar argument with “humans” instead of algorithms:

Assume that all humans can be arranged into a (finite) ordered list, H0, H1, H2..

Let O be a human with oracle like powers that, upon hearing n, winks if Hn doesn’t wink. Now, this oracle herself has to be listed in the above list, say as Hk. The question is “will O wink if she hears k”?

Here, the human O will crumble just as easily as the algorithm A. The real problem was not with O or A, but with the paradoxical question(s) that they were asked.

This book has been a major disappointment in this regard. However, there are some interesting sections that deal with general theory of relativity and quantum physics, which are a good introductory read.

Update: While writing this post, I did a quick search on the web, and there is a similar formal refutation of Penrose’s argument here.

Update: An even better refutation is here, which deals with “systems of reasoning” and their limitations.  A parody of Penrose’s own dialog here.