Logic’s Tower of Babel

Abstract

Late in his life, Jung speculated that the natural numbers, the integers, “contain the whole of mathematics and everything yet to be discovered in this field.” This article presents the attempts by mathematicians to address this question in their terms; that is, whether arithmetic (the mathematics of the natural numbers) was complete and consistent.Early in the twentieth century, mathematicians began to seek a formalism that could provide a solid foundation for mathematics. The first important product of this new formalism was Giuseppe Peano’s Postulates: five axioms from which the full arithmetic of the natural numbers or integers (i.e., 0, 1, 2, 3, …) can be derived. Inspired by Peano’s achievement, philosopher and mathematician Bertrand Russell began a project to show that mathematics could be reduced to logic. His overweening aim was to eventually show that all science could be reduced to logic.Logician Kurt Gödel realized that the goal of the formalists and logicians was impossible. He produced a logically impeccable proof that no system at least as complex as arithmetic could be proved both complete and consistent within the system. In essence, he proved that the core of mathematical discovery must be intuitive: direct perception of reality, which then clothes itself in mathematical garb. This accords closely with Jung’s own insight, which was based on the idea that each number is qualitatively different from every other number. To this day, Gödel’s proof stands unchallenged.