Rebecca Goldstein’s Incompleteness: The Proof and Paradox of Kurt Gödel is another book in the Great Discoveries series of short books by noted authors about important moments in the history of science, and the people behind them. Previous volumes include Everything and More and A Force of Nature, both of which were excellent in their own way, and Incompleteness fits right in there with them.
As the subtitle makes clear, this is a book about Kurt Gödel’s famous Incompleteness Theorem, which shows that any formal logical system complex enough to describe arithmetic must allow the formation of statements that are true, but cannot be proved to be true within that system. If you’ve always wanted to know what all the fuss over Gödel, Escher, Bach was about, but can’t make it through, this book would be an excellent alternative. Goldstein does an excellent job of explaining the meaning of the theorem, putting it in historical context, and sketching out the unique way Gödel did the proof.
Of course, like all the other books in the series, the book also reflects the author’s tastes and professional inclination. Goldstein is a philosopher, and so she uses the book to make an argument about Gödel’s philosophy: that contrary to the common impression of the theorem as the work of some sort of mathematical postmodernist, Gödel was in fact a passionate and committed Platonist, firmly believing in the independent reality of mathematical ideas. To him, the important part of the theorem was that the unprovable statements were true, suggesting a wider and deeper mathematical universe than the formalist program would admit. He wasn’t out to destroy mathematical truth, but to confirm and in some sense ennoble it.
I’m not qualified to evaluate the accuracy of this claim, but Goldstein makes a convincing argument. She also does an excellent job of putting the theorem in historical context, sketching out both the philosophical circles of Vienna where Gödel cut his teeth, and the formalist program of David Hilbert (among others) that his famous result overthrew. And, of course, no biography of Gödel could hope to avoid his personal eccentricities, and the tragic descent into paranoia that led to his death.
This is a very well-done book, and if you have any interest in mathematics or the history and philosophy thereof, I recommend checking it out.