Incompleteness

Rebecca Goldstein

One can (assuming the consistency of classical mathematics) even give examples of propositions (and indeed of such a type as Goldbach and Fermat) which are really contextually true but unprovable in the formal system of classical mathematics.

"The more I think about language, the more it amazes me that people ever understand each other."

Kurt Godel is the greatest logician since Aristotle. He starved himself to death and is largely forgotten. The only things less understood than the man are his theories. I read this book because I had heard of Godel in various other books. I got my hands on a copy of the proofs for his two incompleteness theorems but couldn't understand them and so turned to a biography in the hopes that it would explain the theorems in slightly less technical language. This Goldstein does, even if she sacrifices some of the mathematical beauty and precision in order to do so.
Perhaps the most wonderful aspect of reading about Godel is the sense of awe you will come away with. There is an eerie mystical haze that hangs about the man--such incredible discoveries, discoveries which have implications on some of the deepest levels of human life--and yet both the man and his discoveries are little known and less discussed.
I have heard Godel compared to Descartes--what Descartes did to geometry with coordinates, Godel does to mathematical logic with Godel numbering. I did not appreciate the importance of this seemingly arbitrary semantic distinction of his--I still do not, but I have an inkling from Goldstein's book of the depth and power that it contains.
Reading about Godel is essential if you believe in logic. If you have heard of such characters as Wittgenstein and Russel and Frege, if you have come across people who would use logic as the perfect and COMPLETE system with which to explain the world, Godel is a necessity. It will change the way you view logic, change the way you view metaphysics--it will shake your certainty in the usefulness of logic as well as relativism. It is ironic, as long as you don't think about it, that a man would almost prove objective truth by proving the incompleteness and therefore incapacity of logic to explain and understand reality. Bring your brain to this one though, it will prove a challenging read. Oh, and that first quote at the top of the page here, that is the phrase that Godel used to tell the world he had proven logic inadequate.