It seems nobody has mentionned Gregory Chaitin[1] yet, and in my opinion it's kind of mandatory when discussing this very topic. So now it's done :-). I like his work a lot. His not really working on this subject anymore (now he's developping a new field called metabiology which is also fascinating but not very relevant here) but he used to work on Algorithmic Information Theory and he wrote a lot of books and essays (you can find almost everything on his webpage) on the subject we're discussing here. _The Limits of Mathematics_[2] might be the most relevant to Hacker News users.
It seems nobody has mentionned Gregory Chaitin[1] yet
With good reason. The logician Torkel Franzen said it well in "Gödel's Theorem: An Incomplete Guide" [Chapter 8, p 148]: ``Chaitin's claim to have discovered "the amazing fact that some mathematical statements are true for no reason" has no apparent content, but seems rather to be based on the general associations surrounding the word "random".''
[1] http://www.cs.umaine.edu/~chaitin/ [2] http://news.ycombinator.com/item?id=1725936