Logicomix: an Epic Search for Truth by Apostolos Doxiadis and Christos Papadimitriou

This is an entertaining, fast-moving graphic novel (a very thick comic book) that tells the story of Bertrand Russell’s attempt to provide logical foundations for mathematics and also find a path to absolute certainty about the world. It is presented in the form of a public lecture given by Russell, in which Russell talks about his life and the recent history of mathematics and logic.ย ย 

The authors admit that this is a work of fiction, since some of the history has been changed for dramatic purposes or ease of exposition. But the work of great mathematicians, logicians and philosophers, including Cantor, Frege, Poincare, Hilbert, Wittgenstein, Godel and Turing, is accurately summarized. One theme in the novel is the apparent association between logical and mathematical skill and insanity.ย 

There are also interludes that feature the authors and artists working on the book — an act of self-reference that fits very nicely with the main theme of the novel — and attending performances of Greek tragedy in Athens.ย ย 

In general, the writing is better than the artwork, in particular because the characters’ facial expressions lack subtlety.ย ย 

There is also a helpful addendum that describes some of the main characters and concepts (one of the authors is a professor of computer science at Berkeley).ย This is how Godel’s proof of the Incompleteness Theorem is summarized: “Godel proved his Incompleteness Theorem by creating … a statement that … essentially says, in the language of arithmetic, ‘this statement isย unprovable’. Any consistent axiomatic theory in which one can formulate such a statement must be necessarily incomplete: for either this statement isย false,ย in which case it is both falseย andย provable, contradicting theย consistencyย of the axiomatic system, orย true, in which case it is both trueย andย unprovable, establishing itsย incompleteness“. ย (4/15/11)

Incompleteness: The Proof and Paradox of Kurt Godel by Rebecca Goldstein

This is the story of Kurt Godel, considered to be the greatest logician since Aristotle and the second greatest mind of the 20th century after Einstein. The book includes discussion of the Vienna Circle, whose meetings Godel sometimes attended; Godel’s differences with Wittgenstein, whose views impressed the Vienna Circle but not Godel; and Godel’s friendship with Einstein, when both of them were at the Institute for Advanced Studies in Princeton.

Godel is most famous for proving that that any formal system that is rich enough to contain arithmetic or number theory must contain a true statement that can neither be proven nor disproven, in other words, that such a system is necessarily incomplete. A corollary of the incompleteness theorem is that any such system cannot be proven to be consistent within the system itself. Goldstein suggests that the incompleteness theorem demonstrates that there is a mathematical reality beyond the reach of any human-made formal system.

Goldstein steps through the proof of incompleteness, but I didn’t or couldn’t follow the whole proof.ย  The proof seems to rely on the strange consequences that result from sentences that refer to themselves, such as “This sentence is false”. Russell ruled out such sentences in his Theory of Types, and that seems like a good idea to me. It may be arbitrary, but it seems right to say that sentences that refer to themselves and sets that contain themselves should not be allowed in any formal system, given the paradoxical results that follow. ย (5/11/10)