The departure point for mathematician Kurt Gödel’s incompleteness theorems is the brain-teasing liar’s paradox, which reads, “This sentence is a lie.” If the statement is true, then it must be false, and if it is false, then it must be true. How can it be both? Gödel wondered the same thing, and used a modification of the liar’s paradox to dismantle the logical underpinnings of mathematics and, by extension, much of science and philosophy as well. First he modified the liar’s paradox to read, “This sentence is not provable.” Then he translated it into mathematical language and went to work. Before he was done, he had dismantled a foundational work by Bertrand Russell and Alfred North Whitehead that attempted to establish a comprehensive logical basis for all mathematical truths. Gödel demonstrated that any logical system either contains contradictions like the liar’s paradox or axioms that cannot be proven, even if self-evidently true. In effect, Gödel proved there is more to the truth than anything that can be proven.
As a young man, Gödel was part of the Vienna Circle, a group of scientists, mathematicians and philosophers organized in the 1920s around a movement called logical positivism. They hoped to dispense with religion and metaphysics altogether by insisting that statements could only be considered true if they were mathematically proven or empirically verifiable. Gödel’s incompleteness theorem effectively pulled the rug out from under such assertions. Even worse, from their point of view, it left the door open for religious truths by demonstrating that no system based on logic or reason alone could provide all the answers.
Gödel settled in America after the Nazis marched into Austria, and he fell in with Albert Einstein, his colleague and fellow refugee at the Institute for Advanced Studies in Princeton, NJ. Both found themselves intellectually isolated from their peers, although for different reasons. Einstein had laid the groundwork for quantum physics in the early 20th century but balked at the direction it took under Werner Heisenberg and others. Gödel, of course, was on the outs with the materialists, but his incompleteness theorems may have been the least of his sins. What would his colleagues have thought had it become widely known that he was a religious believer?
It remains unclear how Gödel became an apostate to atheism. Raised as a freethinker, he seemingly fit right in with the Vienna Circle, at least at first. Einstein, his close friend, described himself as being a believer in “Spinoza’s God” (referring to the 17th-century Dutch philosopher Baruch Spinoza), by which he meant he did not believe in a God “who concerns himself with the fate and the doings of mankind.” By contrast, Gödel wrote, “My belief is theistic, not pantheistic, following Leibniz rather than Spinoza.” Gottfried Wilhelm Leibniz, Spinoza’s German contemporary, was a philosopher and co-discoverer (with Isaac Newton) of calculus. Gödel, a lifelong student of Leibniz, even took it upon himself to rework Leibniz’s ontological argument for the existence of God.
Ontological proofs have a long tradition in Western Christianity dating back to St. Anselm of Canterbury in the 11th century. Anselm argued by a series of logical steps that if God, defined as a “maximally great being,” can be conceived of in the mind, then he must exist in reality. Descartes, Spinoza and Leibniz all had a hand in fashioning their own variations on the same basic argument. Gödel’s contribution was to restate Leibniz’s proof in mathematical terms, using modal logic.
Gödel is sometimes characterized as the greatest logician since Aristotle. But in subjecting the existence of God to logical argument, he created the same vulnerabilities as any other logical argument. The logic may be unassailable, but the axioms supporting that argument are not. Indeed, his incompleteness theorems may be a better argument for the existence of God than his ontological proof. He had proven that no logical system is entirely self-contained. To the extent that the universe is self-contained, as materialists claim – that is to say, entirely explainable in terms of natural laws -- then there must be something outside the system to complete the picture. Once you put God in the system and try to prove his existence, the incompleteness theorems still apply. If there is a “maximally great being” whose existence can be proven, then there must be a maximally greater being whose existence cannot.
Bertrand Russell and Alfred North Whitehead, Principia Mathematica ·
Siobhan Roberts, “Waiting for Gödel,” The New Yorker (June 29, 2016)