Friday, January 23, 2009

Gödel’s Proof by Ernest Nagel & James Newman

An excellent little book that explains what Gödel did in a manner that is probably as accessible as possible without sacrificing too much important information.
That said, various parts required re-reading, and other parts require still further re-reading for me to feel as if I fully understand (the gist of what) what Gödel did mathematically.
In English, “Gödel proved that it is impossible to establish the internal logical consistency of a very large class of deductive systems –number theory for example – unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that of the systems themselves.”

This was a very significant development in the 1930s as many thought that mathematical systems could be consistent and complex enough to do all the things one would want them to do. It is important to note that what is mainly discussed is the manipulation of symbols, not meaning. The authors quote “Russel’s famous epigram: pure mathematics is the subject in which we do not know what we are talking about, or whether what we are saying is true.”

How did Gödel manage to do all this? The main insight is that he realized that “typographical properties of long chains of symbols can be talked about in an indirect but perfectly accurate manner by instead talking about the properties of prime factorizations of large integers.”
Make sense? :P
I happily got to the point where I understand the above statement, but not the notions of the math (thus why re-readings are required). Further, the authors present a very simplified view of the whole thing which makes me feel even more mathematically ignorant and incompetent, as well as reinforce my decision to move away from esoteric math into the realm of people and ideas in my education.

I appreciated the discussion of mathematical vs. meta-mathematical statements and found the information about tautologies very useful (i.e., in Logic, a tautology is “defined as a statement that excludes no logical possibilities” and isn’t just some specious argument).

The book is also useful because the significance of what Gödel did, although valid, is often misplaced; far too many sophists have misappropriated the results and attempted to use Gödel as a way to invalidate math, science or A.I.

Finally, once again, I am impressed with Hofstadter, who edited and wrote a forward to this volume, as he first read Gödel’s Proof when he was 14. Then again, getting to talk to one of the original authors because your physicist father was friends with him would tend to give someone a leg up on understanding the complexities of Gödel’s Proof.


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