The source
The exact words
“Either mathematics is incompletable in this sense, that its evident axioms can never be comprised in a finite rule, that is to say, the human mind (even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems of the type specified.”
“Some basic theorems on the foundations of mathematics and their implications”. Josiah Willard Gibbs Lecture, American Mathematical Society, Brown University, Providence, 26 December 1951. First printed in Gödel's Collected Works, Volume III (1995).
A note on the wording. The short line on the collage is how the idea circulates today. It is a compression, not Gödel's sentence: his own wording is the longer, more careful disjunction above.
The story
Why Gödel said it
Twenty years after his incompleteness theorems, Gödel was invited to give the Gibbs Lecture, the American Mathematical Society's most prestigious public talk. By 1951 everyone was asking whether the new electronic computers could think; Alan Turing had published “Computing Machinery and Intelligence” the year before. Gödel used the lecture to say exactly what his 1931 theorems did and did not settle about that question.
His answer was an either–or. Any machine that proves theorems follows a finite rule, and the incompleteness theorem says that such a rule can never capture all of mathematical truth. So either the human mind does something no finite machine can do, or there are perfectly concrete arithmetic questions that no one, human or machine, will ever be able to answer. Gödel insisted that the disjunction itself was a proven fact, not an opinion.
Which half did he believe? Privately, the first. He thought the mind was not a machine and that mathematical objects were as real as stones. But he was too careful to claim he had proved it, and the lecture was never published in his lifetime. It sat among his papers until the Collected Works appeared in 1995.
Gödel spent those years at the Institute for Advanced Study in Princeton, where his daily walks with Einstein became legendary. The story of that friendship is in When Einstein Walked with Gödel.
Logician
Who was Kurt Gödel?
- Born
- 28 April 1906, Brünn
- Died
- 14 January 1978, Princeton
- Known for
- Incompleteness theorems (1931)
- Said it
- Gibbs Lecture, 1951
Kurt Gödel was born in Brünn, then in Austria-Hungary, now Brno in the Czech Republic. In 1931, at 25, he published the incompleteness theorems: any consistent formal system strong enough for arithmetic contains true statements it cannot prove, and it cannot prove its own consistency.
He left Vienna for Princeton in 1940 and stayed at the Institute for Advanced Study for the rest of his life. He also proved the consistency of the continuum hypothesis with set theory and found rotating-universe solutions to Einstein's equations that allow time travel. His fear of being poisoned grew with age; he died in 1978, having refused to eat.
