The dual biography of the fathers of modern mathematics. In 1931 the mathematician and logician Kurt Godel proved that within a formal system questions exist that are neither provable nor disprovable on the basis of the axioms that define the system. This is known as Godel's Undecidability Theorem. He also showed that in a sufficiently rich formal system in which decidability of all questions is required, there will be contradictory statements. This is known as his Incompleteness Theorem. In establishing these theorems Godel showed that there are problems that cannot be solved by any set of rules or procedures; instead for these problems one must always extend the set of axioms. This disproved a common belief at the time that the different branches of mathematics could be integrated and placed on a single logical foundation. Alan Turing later provided a constructive interpretation of Godel's results by placing them on an algorithmic foundation: there are numbers and functions that cannot be computed by any logical machine. His work formed the blueprint for the computer and cracked the infamous Enigma encryption.
Janna Levin is a theoretical physicist and the shadows of Godel and Turing loom large in her life. Her obsession with these troubled geniuses, and her own grappling with their mathematical legacies, have led her to write this unique mixture of biography, history and memoir.
Janna Levin is a professor of mathematics at Barnard College and Columbia University. She has published widely in academic journals and her work has been featured in science magazines and newspapers worldwide. She lives in New York City.
How the Universe Got Its Spots (Bfmt Apr 03)