Incompleteness.

My final top 100 draft prospect ranking for 2015 is up for Insiders, and my latest review for Paste covers the Temple of Elemental Evil boardgame.

Rebecca Goldstein’s Incompleteness: The Proof and Paradox of Kurt Gödel, part of the same “Great Discoveries” series that includes David Foster Wallace’s Everything and More: A Compact History of Infinity, takes the abstruse topic of Gödel’s two incompleteness theorems and folds it into a readable, compelling biography of both the man and his ideas. Using the logician’s friendship with Albert Einstein as a hook, Goldstein gives us about as intimate a portrait of the intensely private Gödel as we can get, while also laying the groundwork so the non-metamathematicians among us can understand the why and how of Gödel’s theorems.

There’s very little actual math involved in Incompleteness, because the two theorems in question revolve around the nature of mathematics, notably arithmetic, as an axiomatic system that mathematicians and philosophers of the early 1900s were trying to use as a basis to describe arithmetic as a formal system – that is, one that can be fully described via a system of symbols and syntax, used to construct well-formed formulas (wffs, not to be confused with wtfs), some of which are the axioms that define the whole system. If such axioms can describe the entire system in a way that an algorithm (which is probably easiest to conceive as a computer program) can use those axioms and only those axioms to prove all truths or assertions about the natural numbers, then arithmetic would be considered a complete system, something Gödel proved was impossible in his first theorem. His second theorem went even further, showing that such a system of arithmetic also could not be consistent within itself by demonstrating that a statement of the system’s consistency would generate an internal contradiction.

Gödel imself was an incomplete figure, a hypochondriac who degenerated into outright paranoia later in life, and a socially awkward man who would likely have been diagnosed in today’s world of psychiatry and medicine as a depressive or even somewhere on the autism spectrum. He formed few lasting friendships, feared that no one could understand him (and given the meandering paths of his mind, I don’t doubt this was true), and was often shockingly aloof to what was happening around him. He fled his native Austria before World War II, even though he wasn’t Jewish, because his association with the secular Jewish scholars of the Vienna Circle (most of whom were logical Positivists, arguing that anything that could not be empirically proven could not be considered true) cost him his university position. Yet he remained unaware of the state of affairs in his native country, once (according to Goldstein) asking a Jewish emigré who had fled Nazi persecution for the United States, “What brings you to America?”

Goldstein does a superb job setting the scene for Gödel’s emergence as a logician/metamathematician and as a figure of adulation and controversy. The Vienna Circle was, in Goldstein’s depiction, somewhat insular in its unwavering acceptance of Positivism, and as such certain that Hilbert’s second problem, asking for a proof that arithmetic is internally consistent, would be proven true. Gödel’s second theorem showed this was not the case, leading to his own intellectual isolation from thinkers who were heavily invested in the Positivist view of mathematics, a bad outcome for a man who was already prone to introversion and diffidence. G&omul;del found it difficult to express himself without the use of logic, and while his station at Princeton’s illustrious Institute for Advanced Study – a sort of magnet think tank that became a home for great scholars in math and the sciences – put him in contact with Albert Einstein, it also deepened his solitude by limiting his orbit … although that may have been inevitable given his late-life delusions of persecution. Goldstein did encounter Gödel once at a garden party at Princeton, only to see him at his most gregarious, holding court with a small group of awestruck graduate students, only to disappear without a trace when his spell of socializing was over. He was, in her description, a phantom presence around campus, especially once his daily walks with Einstein ended with the latter’s death in 1955.

While giving a basic description of Gödel’s theorems and proofs, focusing more on their implications than on the underlying math, Goldstein does send readers to the 1958 Gödel’s Proof, a 160-page book that attempts to give readers a more thorough understanding of the two theorems even if those readers lack any background in higher math. Incompleteness focuses instead on the man as much as it does on his work, producing a true narrative in a story that wouldn’t otherwise have had one, making it a book that I could recommend to anyone who can stand Goldstein’s occasional use of a $2 word (“veridical”) when a ten-cent one (“truthful”) would have done.