Kurt Gödel was a brilliant mathematician, logician and philosopher who made groundbreaking contributions to the foundations of mathematics and the philosophy of mind. He was born in 1906 in Brünn (now Brno), Austria-Hungary (now the Czech Republic) to a German-speaking family that owned a textile business. He showed an early interest in mathematics and languages, and entered the University of Vienna in 1924, where he studied under Hans Hahn, a member of the Vienna Circle of logical positivists. However, Gödel did not share their rejection of metaphysics and their faith in the empirical verification of all meaningful statements. He was more influenced by Plato, Leibniz and Kant, and developed a rationalist and realist view of mathematics and logic.
Gödel’s doctoral dissertation in 1929 proved the completeness theorem for first-order logic, which states that any logically valid formula can be derived from a given set of axioms using the rules of inference. This was a remarkable achievement that established his reputation as a leading logician. However, he soon shocked the mathematical world with his incompleteness theorems of 1931, which showed that any consistent formal system that is powerful enough to express arithmetic contains statements that are true but unprovable within the system. This meant that no formal system can capture all the truths of mathematics, and that some degree of undecidability and incompleteness is inevitable. Gödel’s incompleteness theorems had profound implications for the philosophy of mathematics, logic and mind, as they challenged the assumptions of Hilbert’s program to find a complete and consistent foundation for mathematics, and raised questions about the nature and limits of human reason.
Gödel also made important contributions to other areas of mathematics and logic, such as set theory, proof theory, model theory and recursion theory. He proved the relative consistency of the axiom of choice and the continuum hypothesis with the standard axioms of set theory (Zermelo-Fraenkel), which means that these statements cannot be disproved from the standard axioms, but neither can they be proved from them. He also discovered an unusual solution to Einstein’s field equations of general relativity, known as the Gödel metric, which describes a rotating universe that allows for the existence of closed timelike curves (paths in spacetime that return to their starting point). He also devised a formal proof of God’s existence based on modal logic, known as Gödel’s ontological argument.
A mathematician at IBM, Gregory Chaitin, says that Gödel’s work along with the uncertain, unpredictable elements in quantum physics demonstrates the limits of mathematics, in particular, and of scientific knowledge, in general. For everyday people, this means we should always be skeptical of “the research” as cited by so many pundits, politicians, and psychologists. More often than not, “the research” is weak not just because of core mathematical issues but also due to various types of bias, selectivity, and overgeneralization—not to mention the growing awareness of political interference, opportunism and fraud in science.
Gödel’s life was marked by personal difficulties and tragedies. He suffered from various health problems, such as depression, fear, and an eating disorder. He married Adele Nimbursky in 1938, despite his family’s opposition. He fled Nazi Germany in 1939 and moved to Princeton, New Jersey, where he joined the Institute for Advanced Study. There he became friends with Albert Einstein, who admired his work and shared his philosophical interests. However, he also became increasingly isolated and paranoid, fearing persecution and poisoning after the assassination of his close friend Moritz Schlick. He died in 1978 from starvation due to his refusal to eat anything that was not prepared by his wife, who had been hospitalized for several months.
Today Gödel would have been called Czech. Czech culture is often viewed as being on the edge with figures like Franz Kafka, Karel Čapek, and Alphonse Mucha. Sometimes that sharpness and innovation produce great creators who contribute immensely to society but other times it takes a decidedly dark turn. We cannot always know exactly what makes an otherwise exceptional person behave strangely or give in to self-harm. Sometimes it might be bad choices made along the way (e.g. an immigrant spies for a hostile power to advance their career). Other times, the darkness might arise merely by picking up the vibe of other people’s bad choices or situations and not being able to discern the difference.
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