>>Jerry,
>>
>>< bg > No they're not. I recall thinking as I was reading the Preface, "this is the most profound book I've ever read." The only problem I had with it was that reading it made me very tired from ducking as the concepts went over my head.:-)
>
>Take Godel, for instance:
>"Gödel's results were a landmark in 20th-century mathematics, showing that mathematics is not a finished object, as had been believed. It also implies that a computer can never be programmed to answer all mathematical questions."
>http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Godel.html
>
>Especially his 2nd Theorem:
> "The proof of Gödel's Incompleteness Theorem is so simple, and so sneaky, that it is almost embarassing to relate. His basic procedure is as follows:
>
>
>1) Someone introduces Gödel to a UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.
>2) Gödel asks for the program and the circuit design of the UTM. The program may be complicated, but it can only be finitely long. Call the program P(UTM) for Program of the Universal Truth Machine.
>3) Smiling a little, Gödel writes out the following sentence: "The machine constructed on the basis of the program P(UTM) will never say that this sentence is true." Call this sentence G for Gödel. Note that G is equivalent to: "UTM will never say G is true."
>4) Now Gödel laughs his high laugh and asks UTM whether G is true or not.
>5) If UTM says G is true, then "UTM will never say G is true" is false. If "UTM will never say G is true" is false, then G is false (since G = "UTM will never say G is true"). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements.
>6) We have established that UTM will never say G is true. So "UTM will never say G is true" is in fact a true statement. So G is true (since G = "UTM will never say G is true").
>7) "I know a truth that UTM can never utter," Gödel says. "I know that G is true. UTM is not truly universal."
>Think about it - it grows on you ...
>
>With his great mathematical and logical genius, Gödel was able to find a way (for any given P(UTM)) actually to write down a complicated polynomial equation that has a solution if and only if G is true. So G is not at all some vague or non-mathematical sentence. G is a specific mathematical problem that we know the answer to, even though UTM does not! So UTM does not, and cannot, embody a best and final theory of mathematics ...
>
>Although this theorem can be stated and proved in a rigorously mathematical way, what it seems to say is that rational thought can never penetrate to the final ultimate truth ... But, paradoxically, to understand Gödel's proof is to find a sort of liberation. For many logic students, the final breakthrough to full understanding of the Incompleteness Theorem is practically a conversion experience. This is partly a by-product of the potent mystique Gödel's name carries. But, more profoundly, to understand the essentially labyrinthine nature of the castle is, somehow, to be free of it. "
>
>http://www.miskatonic.org/godel.html
>
>It's an amazing piece of logic... so simple yet it remained hidden for two hundred years. I am suprised that Newton, Gauss or LaPlace didn't think of it.
>JLK

Jerry,

As I recall, Hofstadter in GEB summed the above up very nicely and succinctly. Basically, he said to take a piece of paper and on one side write: "The statement on the other side is true." On the other side write: "The statement on the other side is false.":-)