Provenly Unprovable

Solomon Feferman

  • Incompleteness: The Proof and Paradox of Kurt Gödel by Rebecca Goldstein
    Norton, 224 pp, US $13.95, February 2006, ISBN 0 393 32760 4

Like Heisenberg’s uncertainty principle, Gödel’s incompleteness theorem has captured the public imagination, supposedly demonstrating that there are absolute limits to what can be known. More specifically, it is thought to tell us that there are mathematical truths which can never be proved. These are among the misconceptions that proliferate around Gödel’s theorem and its consequences. Incompleteness has been held to show, for example, that there cannot be a Theory of Everything, the so-called holy grail of modern physics. Some philosophers and mathematicians say it proves that minds can’t be modelled by machines, while others argue that they can be modelled but that Gödel’s theorem shows we can’t know this. Postmodernists have claimed to find support in it for the view that objective truth is chimerical. And in the Bibliography of Christianity and Mathematics (there really is such a publication) it is asserted that ‘theologians can be comforted in their failure to systematise revealed truth because mathematicians cannot grasp all mathematical truths in their systems either.’ The incompleteness theorem is also held to imply the existence of God, since only He can decide all truths.

Even Rebecca Goldstein’s book, whose laudable aim is to provide non-technical expositions of the incompleteness theorems (there are two) for a general audience and place them in their historical and biographical context, makes extravagant claims and distorts their significance. As Goldstein sees it, Gödel’s are ‘the most prolix theorems in the history of mathematics’ and address themselves ‘to the central question of the humanities: what is involved in our being human?’ – since they are concerned with ‘such vast and messy issues as the nature of truth and knowledge and certainty’. Unfortunately, these weighty claims disintegrate under closer examination, while the book is marred by a number of conceptual and historical errors.

On the face of it, Goldstein would appear an ideal choice to present Gödel’s work: she has taught philosophy of science and philosophy of mind at several universities and has also written successful novels and short stories, including The Mind-Body Problem and Properties of Light: A Novel of Love, Betrayal and Quantum Physics.

What she does very well is to provide a vivid biographical picture of Gödel, beginning midstream with his touching relationship with Einstein at the Institute for Advanced Study in Princeton, where, over the 15 years before Einstein’s death in 1955, they were often seen together. They were an odd-looking couple: ‘One of the men, dapperly dressed in a white linen suit with a matching white fedora, is still in his thirties, while the other, in baggy pants held up by old-world-style suspenders, is approaching 70.’ (Thus Gödel and Einstein in a widely reproduced summer photo. More often, even when it was warm, Gödel was bundled up in a heavy black overcoat.) In personality too they were opposites: Gödel was slight to the point of frailty, hypochondriac and buttoned up; Einstein was hearty and gregarious. But they had similar cultural backgrounds and each pursued central problems in his field. Both were German-speaking émigrés (‘exiles’, as Goldstein has it): Einstein was brought from Germany to the institute in 1933 by its founder, Abraham Flexner, as one of the first two permanent members (the other was the American mathematician Oswald Veblen, brother of Thorstein); Gödel fled Austria at the last minute in 1940 to avoid conscription and came to the institute as an ‘ordinary’ member, only later becoming a permanent one.

Gödel and Einstein had both already done their most important work, achieved in each case over a single, remarkable decade. In Einstein’s annus mirabilis of 1905 he published three seminal papers, including one on the special theory of relativity; the general theory of relativity was put forward in 1915. After that, he was increasingly sidelined in theoretical physics, which came to be dominated by the probabilistic theories of quantum mechanics. Until the end of his life, Einstein sought instead a deterministic unified theory of gravitation and electromagnetism, believing that ‘God does not play dice.’ For his part, Gödel’s three fundamental results were the completeness theorem for the first-order logic of predicates (in his PhD thesis of 1929); the incompleteness theorems a year later; and his proof of the consistency of two problematic hypotheses with the axioms of set theory in 1939. In the early 1940s, he worked on attempts to prove the independence of the first of these, the so-called Axiom of Choice, with only limited success; after that he devoted himself almost entirely to the philosophy of mathematics. Goldstein goes on to describe Gödel’s gradual descent via a surfeit of rationality into paranoia. Terrified of being poisoned, he died in January 1978 – according to the death certificate, of ‘malnutrition and inanition caused by personality disturbance’.

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