Zeno's Paradox
February 02, 2004 - Back in Business

Well, I'm hoping that the post-holidays chaos is now finished. I spent last week at an educational technologies conference in San Diego before going out to the Bay Area and showcasing some of AT's new technology to the likes of Yahoo!, Creative Commons, and the Internet Archive. It was a successful trip, and I managed to knock out a few chapters of GEB. (Though I haven't made any headway on the discrete mathematics since last writing.)

I enjoyed the chapter devoted to fully illustrating how Gödel's Incompleteness Theorem works and the step-by-step methods used to illustrate the proof. While I need to go back and re-read it (especially the sections about using Cantor's diagonal trick), I think that I'm that much closer to understanding it beyond the descriptions I've read thus far in layman's terms. The more that I think about it, the more that I've been thinking of things in such contexts. For instance, the idea that I would find the hoopla surrounding Euclid's fifth postulate (the one about diagonal lined) and its consequences fascinating, would have been unthinkable two years ago. (Then again, I do have a decent track record deciding that I'll never be interested in one thing or another and having that bite me later on, when I find out that it's actually pretty cool.)

However, I may have found the most interesting chapter of the book so far -- the one describing BLOOP, FLOOP, and GLOOP. I've programmed in a number of languages so far and the idea that something as simple as a for loop give your language the ability to compute one class of problems, while a while loop opens up a whole new world, and that we've not found anything after the while loop to make our languages solve any new classes of problems, is really interesting to me. Of course, I'm not doing the description justice, so please read it yourself if interested.

One of the things that I've been trying to do with this blog is to put out some questions I have while doing this type of reading. I haven't really made any headway on the "why not divide by zero" one, but here's my new one: Are Gödel's Incompleteness Theorem, the Turing machine halting problem, and Heisenberg's Uncertainty Principle all different manifestations of some greater natural law in the way that electricity and magnetism are different facets of the same thing? There are interesting similarities between all these that seem to suggest there is some barrier to complete empirical and logical knowledge.

Anyways, I'd better get back to work. Hopefully a lighter schedule will allow me the luxury of posting here more often.

Posted by br284 at 11:44 AM

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