Simplifying popular mathematics: finding the right balance of equations. Too few means the ideas aren’t fully expressed, and too many means it’s too complex for non-experts. In her book, Incompleteness Rebecca Goldstein manages to strike that balance, taking Gödel’s work on incompleteness beyond academia and presenting it in an engaging narrative. Incompleteness introduces the key players who’ve grappled with the philosophical implications of Gödel’s work over the past seventy-five years.
We meet Kurt Gödel, the protagonist, and his idea of mathematical Platonism. Then, we encounter Ludwig Wittgenstein, the antagonist, with his idea of linguistic relativism. Wittgenstein convinces others to join his cause, but Gödel, the lone genius, stands against them, supported by his friendship with Albert Einstein.
Gödel faces the logical positivists, who challenge his ideas. Despite his laconic style, his first incompleteness result proves significant. However, the intellectuals misinterpret his theorems as support for Wittgenstein and the positivists. Goldstein sets the record straight, presenting Gödel’s interpretation of his work as vindication for mathematical Platonism.
Understanding Gödel’s philosophical interpretation is crucial in unraveling the broader implications of his theorems. While Goldstein sympathizes with Gödel, she doesn’t give the same consideration to Wittgenstein. Perhaps her biased telling is due to her personal connection to the subject matter.
Incompleteness also delves into the set theoretic paradoxes, both historically and logically. However, the description of Russell’s paradox as “grievous” seems exaggerated, considering its role in the development of Gödel’s result.