My latest Insider post covers eight top 100 prospects who took a step back this year. I’ll also hold a Klawchat here at 1 pm ET.
I read Rebecca Goldstein’s biography of Kurt Gödel, Incompleteness, last summer, and I believe it was within her book that I read about James Newman and Ernest Nagel’s book Gödel’s Proof that attempts to explain the Austrian logician’s groundbreaking findings. The 114-page volume does a great job of building up to the final proof, but I have to concede that the 19-page section near the end that reveals the fatal blow Gödel delivered to Bertrand Russell, David Hilbert, and others who believed in the essential completeness of mathematical systems lost me in its nested language and ornate symbols. (The newest edition includes a foreword by Douglas Hofstadter, who wrote about the proof in Gödel, Escher, Bach, which won the Pulitzer Prize for Non-fiction.)
Gödel was himself a fascinating figure, a philosopher, mathematician, and logician who wrote a paper with two theorems at age 25 that stunned the world of mathematics in their method and conclusions, proving that any axiomatic system of arithmetic that is consistent cannot be complete. Completeness here means that every true formula that can be expressed within the system can be proven within the system. Gödel’s trick was to create an entire system of expressing logical formulas via what is now called Gödel numbering, and then to craft a formula that says itself that it is unprovable within the system. His proof further stated that even if you could add an axiom to this system of mathematics to cover this new exception, the formula could always be rephrased to pose a new exception, and thus the system is essentially incomplete.
All of which is to say that I just couldn’t follow the nested statements that constitute Nagel and Newman’s explanation of Gödel’s proof. I haven’t read Gödel’s original paper, because it is a truth universally acknowledged that you’ve got to have some serious math background to understand it, so I will accept the claim that Nagel and Newman made it much easier to grasp … but I still only get this at a superficial level. When the authors compare this to Richard’s Paradox, an earlier device that Gödel cited in his paper, I could understand it; these are all descendants of the “This statement is false” type of logical trick that causes an inherent contradiction. Gödel appears to have done the same thing for arithmetic. I just couldn’t quite get to the mental finish line on this one. I guess you could say my understanding of the topic remains ….
…incomplete.
Next up: I finished and will review Laurent Binet’s HHhH, and have begun Clifford Simak’s Hugo-winning novel Way Station.