Siobhan Roberts has a piece in the current New Yorker on Kurt Gödel:
This all reminded me of my own experience with Gödelian thought decades ago...
When I was around 20 years-old someone showed me Kurt Gödel's "proof of the existence of God" (a simplified version of course), and explained that Gödel, who I'd never heard of, was regarded as one of the most brilliant men of the century. I couldn't believe it! I looked over the proof, thought it was one of the most ridiculous academic pieces I'd ever seen, refutable by any intelligent 8th grader, and promptly filed the name "Gödel" away in my mind as a silly individual to be ignored.
During those same years I was, on my own, playing around with ideas about self-reference in language, and what I later found to be known of as "recursion." Intuitively, I felt these were crucial concepts for comprehending the way the brain worked -- but ironically it also implied that we COULDN'T understand how the brain worked, because that would require the brain to analyze itself, and the point was, that I didn't think such a recursive process was possible -- a device cannot fully explore its own processes. My ideas were purely intuitive though, and I couldn't find a way to verbalize or refine them. So they just sort of hovered there loosely in the back of my mind.
Some years passed and, in general, I deliberately ignored Gödel whenever his name arose, but at some point was reading a popular math volume by someone I respected, and it included a chapter on Gödel and this thing I'd not heard of called "the incompleteness theorem." Reluctantly, I read what this author I regarded highly, had to say about Gödel's work. And, I was blown away! Kurt Gödel was saying, and indeed being hailed for rigorously proving, what I could only barely sense, but never put into empirical words, that certain knowledge could never be proven using human logic, that any system of thought can only be fully known by another system outside or beyond that system. It was a tour de force. I suddenly wanted to read much more about Gödel, the man I'd earlier discarded to the intellectual junkbin. Gödel's life was of course fascinating, and he did have many somewhat 'crazy' thoughts (and I STILL regard his God proof as one of them!**), but simultaneously he generated some of the most profound ideas ever derived by human brains, changing the face of philosophy and mathematics forever, as few have done.
Then, in 1979, a young whippersnapper named Doug Hofstadter came out of nowhere to write "Gödel, Escher, Bach," one of the most acclaimed nonfiction books of the 20th century, winning numerous awards, and setting its youthful author on the road to fame and... well, more fame. And again I thought, "WOW!" someone figured out a clever, creative, insightful way to take all this self-reference/recursion stuff and put it into a book form that boggles readers' minds. (I had always hoped that one of Hofstadter's followups, "I Am a Strange Loop," might really blow the doors wide open on this subject area, but unfortunately it did not.)
There are so many good reads on Gödel and his work out there these days that I won't even favor any by singling them out; you can find plenty on him just by googling. And we've already had fine movies about John Nash, Alan Turing, and Ramanujan. Perhaps a film version of Gödel is not far off.
I will close out however with this quote (I've used before as a 'Sunday reflection') from Freeman Dyson in "The Scientist As Rebel":
"Gödel's theorem shows conclusively that in pure mathematics reductionism does not work. To decide whether a mathematical statement is true, it is not sufficient to reduce the statement to marks on paper and to study the behavior of the marks. Except in trivial cases, you can decide the truth of a statement only by studying its meaning and its context in the larger world of mathematical ideas.----------------------------------------
"It is a curious paradox that several of the greatest and most creative spirits in science, after achieving important discoveries by following their unfettered imaginations, were in their later years obsessed with reductionist philosophy and as a result became sterile. Hilbert was a prime example of this paradox. Einstein was another…
"Science in its everyday practice is much closer to art than to philosophy. When I look at Gödel's proof of his undecidability theorem, I do not see a philosophical argument. The proof is a soaring piece of architecture, as unique and as lovely as Chartres Cathedral… The proof is a great work of art. It is a construction, not a reduction. It destroyed Hilbert's dream of reducing all mathematics to a few equations, and replaced it with a greater dream of mathematics as an endlessly growing realm of ideas. Gödel proved that in mathematics the whole is always greater than the sum of the parts. Every formalization of mathematics raises questions that reach beyond the limits of the formalization into unexplored territory."
** [just to be clear, my own view of God aligns with Martin Gardner's "fideist" view (a form of theism), as a concept so far beyond human comprehension or definition, let alone "proof," that it almost defies discussion, which too often turns God into what I call the 'Santa Claus' version that predominates Western religion (...even while people often try denying that is their version)]