Wednesday, March 23, 2005

The Process - Part II

This is the mid-morning post, running a little later than I'd like, but not much. Usually I try to post again somewhere between 10:00 and 11:30. For this second post I have a number of options. I can go back to Townhall, and if I saw two interesting articles there, I often will I'll pick the easier article to write on for the morning post, and let my mind digest the other one a bit more. Alternatively I will go to a couple of other sources. I might visit Commondreams.Org or Working For Change if I want a liberal article, or I might visit the New York Times Editorial Page or the Village Voice to see if they have anything interesting. Haven't been to the Village Voice in a while come to think of it. Alternatively, I might just stumble across something interesting at another website, and post on that.

For example, this morning I read an interesting discussion of Kurt Gödel over at Salon. I try to keep links to Salon to a minimum. I think it's a brilliant website, but you have to either pay to watch it (as I have) or watch a brief ad in order to gain access to their content. In this case I think the article is strong enough to warrant a mention.

It is actually a book review of a recent book on Gödel's work, which seeks to place said work in the context of the times. Gödel (along with Einstein and Heisenberg) are often credited with having proved the unknowability of the universe. In fact, Gödel would probably have been uncomfortable with that interpretation of his work, which can be seen as a reaction and critique of the Vienna Circle and the Logical Positivists or Formalists.
If formalism were correct, then it followed that mathematics could also be overhauled so that every part of it was "consistent" and the entire system was "complete." It could be boiled down to a set of rules or axioms and procedures so basic and ironclad that a machine -- the computers that were just beyond the historical horizon -- could perform it. It could be finally purged of the paradoxes that had been plaguing the field for hundreds of years. Mathematical intuition, the source of ideas that can't be formally proven but possessing what Goldstein calls, "the urgent cogency that compels belief," has no place in such a system.

Gödel's theorem used the rules of formalism itself to demonstrate that the formalist project could never be achieved. In what Goldstein calls "one of the most astounding pieces of mathematical reasoning ever produced," he demonstrated that in the kind of system that the formalists aspired to, it was possible to make a statement that was both unprovable and yet also true. This works a little like the famous "liar's paradox," in which the statement "This statement is false," can only be true if it is also false and vice versa. But Gödel's theorem was not a paradox, precisely because it pointed to the difference between what could be proven and what was true.
I'm not sure I understand that completely, but I thought it was interesting.

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