This is a series on the book Gödel, Escher, Bach: An eternal golden braid by Douglas Hofstadter.
Earlier diaries are here
Today, we will examine Jumping out of the System p. 465-479.
UPDATE We've got a ways to go with GEB, but I am already thinking about the NEXT book we should read together. So, make suggestions in the comments; I will collect them over the weeks, and then, when we are near the end of GEB, start doing polls.
From the overview:
The repeatability of Godel's argument is shown, with the implication that TNT is not only incomplete, but "essentially incomplete". The fairly notorious argument by J.R. Lucas, to the effect that Godel's theorem says that human thought cannot in any sense be "mechanical" is analyzed and found wanting.
There are two parts to this chapter. In the first, DH shows that an argument very like the Cantor diagonal argument can be applied to Godel's theorem and TNT, and that it can be applied as many times as you want, as in Birthday Cantatatata. This is important, but I have little to say about it. At the end of this part he defines "essential incompleteness" - by the nature of formal systems, you can always add another axiom, but that doesn't get around Godel.
On p. 470, DH likens formal systems to fissionable material. Each can reach "critical mass" and blow up. I don't think DH really proves that there is nothing right at the edge of this - but I think Godel does.
The second part is more interesting to me. DH fairly well demolishes the Lucas argument given on p. 471 ... namely, that since people can Godelize, and computers cannot Godelize, people must be smarter. But he does not go far enough; computers can clearly do things that people cannot, and some of these have something to do with what we tend to think of as 'intelligent'. Does that mean computers are smarter than people? Some people cannot Godelize, and, until Godel came along, no one could Godelize, so, were the people who existed before Godel only as smart as computers? Did people become smarter after Godel?
The real problem is defining intelligence. Since we can't even agree on what intelligence is among humans, how can we do it between humans and animals, much less humans and computers?