I recently came across an announcement of a couple of talks by Greg Chaitin: http://www.youtube.com/watch?v=NnPspSp7AhQ&feature=mfu_in_order&list=UL (which is the first of six 15 minute segments) and http://www.cs.umaine.edu/~chaitin/sfi.html. The second, which is text, mentions favorably "Who Can Name the Bigger Number?" a paper by Scott Aaronson. I haven't had a chance to go through the Chaitin talks yet, but once I started the Aaronson paper it kept pulling me onward until I had finished it. He claims (among other things) that "one could define science as reason’s attempt to compensate for our inability to perceive big numbers." Our ability to express large numbers is not just a trivial pastime but marks our scientific progress. A very interesting paper.
The example he uses is the challenge to write down the largest number you can express on, say 1000 characters. Indirect references such as "the largest number on can write down in 1000 characters plus 1" or "the number of primitive particles in the universe" are not allowed. The number must be expressed directly. The issue becomes what notation can one use to express numbers. Exponentials, Ackermann's function, the busy beaver problem, etc. The conceptual machinery required to express ever larger numbers in a finite number of characters becomes quite challenging. What he doesn't require, of course, is that one explain the notation as part of the 1000 character limit. It's neat and relatively easy-to-read, a good Sunday evening paper.
-- Russ Abbott
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