May 2006

Incompleteness: The Proof and Paradox of Kurt Goedel

By Rebecca Goldstein

Publisher: Norton

Pages: 296

Price: $22.95

Review Author: Gary Mar

In its honor roll of the top 100 scientists and thinkers of the century, Time magazine lists Kurt Goedel. Yet the general public, and even much of the mathematical community, remains unaware of Goedel’s achievements. In a paper that established him as the greatest logician of the 20th century, Goedel published his famous Incompleteness Theorems (1931), which logically demonstrated that the dream pursued by mathematicians of mechanizing all mathematical truth in terms of theoremhood in an axiomatic system was impossible. Three new popular science books bring the paradoxical life and intellectual legacy of Kurt Goedel to the general public.

In Incompleteness: The Life and Logic of Kurt Goedel, Goldstein uses her skills not only as a philosopher to provide a lucid informal exposition of Goedel’s famous Incompleteness Theorems, but also as a novelist to portray the paradoxes of Goedel’s personal life. Goldstein, for example, recounts her experience as a new graduate student at Princeton when she “had looked up the name of the town’s most dazzling mind, the reigning, if reclusive, god at the fabled Institute for Advanced Study.” Bicycling to Goedel’s house was a “surreal” experience for Goldstein: “I stared in disbelief…. How could a man who had produced one of the most exquisite masterpieces of human thought have planted a pink flamingo on his front lawn?” The remainder of the chapter titled “Goedel’s Incompleteness” gives us insight into the life of this reclusive genius.

More precisely, Goedel’s theorems state that in any consistent formal system adequate for arithmetic there exists a true but unprovable formula (Goedel’s first incompleteness theorem), and moreover, the consistency of such a formal system cannot be proved within the system (Goedel’s second incompleteness theorem). Goedel’s theorems were revolutionary because they showed, using methods of utmost mathematical rigor, that one of the significant open problems of the century proposed by the great mathematician David Hilbert in 1900 was impossible to resolve.

As Goldstein notes, Goedel’s theorems, and Einstein’s Theory of Relativity, are sometimes cited as among the most compelling reasons for modernity’s rejection of the “myth of objectivity.” Yet quite the opposite is true. Although Goedel had attended the meetings of the exclusive Vienna Circle, he distanced himself from its atheism and logical positivism. In his popular manifesto of this philosophy, Language, Truth and Logic, A. J. Ayer argued that the statement “God exists” is cognitively meaningless because there could be no empirical experiences by which one could verify the existence of God. Goedel, however, regarded this atheism as a “prejudice of the times” and noted, “I don’t consider my work a ‘facet of the intellectual atmosphere of the early 20th century,’ but rather the opposite.” Indeed it was Goedel’s realism — in contrast to the implicit verificationism of the other foundational schools — that freed him to see clearly the profound consequences of the distinction between truth and provability.

Yourgrau’s A World Without Time is a philosophical account of Goedel’s contributions to relativistic cosmology and a personal account of his friendship with Einstein. During the last 15 years of his life at that institution, Einstein sought Goedel’s company. Einstein and Goedel were intellectual giants at odds with their Zeitgeist. Both were realists at a time when the paradoxes of quantum mechanics were pointing intellectuals in the direction of anti-realism. Einstein famously resisted quantum mechanics because “God doesn’t play dice,” and Goedel regarded quantum mechanics as the denial of “the possibility of knowledge…as the end of all theoretical science in the usual sense.” Their unfashionable realism was, moreover, supported by an even more unfashionable theism. Whereas Einstein’s theism was Spinozistic, Goedel described his religious beliefs as “theistic rather than pantheistic, following Leibniz rather than Spinoza.” Indeed, among Goedel’s unpublished papers was an original Leibnizian ontological proof for the existence of God.

In The One True Platonic Heaven, Casti gives an account of the philosophy of science and the academic politics at the Institute for Advanced Studies, circa 1946, the dawn of the computer age. The dramatis personae in this historically based work of “scientific fiction” include not only Einstein and Goedel, but also such luminaries as J. Robert Oppenheimer, the father of the atomic bomb, and John von Neumann, the mathematical polymath.

In Casti’s historical reconstruction, Goedel is the “Grand Exalted Ruler” of “the one, true platonic heaven.” A fundamental issue in the philosophy of mathematics is whether mathematics is an activity of discovery or one of invention. Goedel believed in Platonism — the view that mathematicians discover the objective properties of abstract mathematical objects, which, like the concrete objects of the physical world, possess a true and independent existence of the human mind. Goedel’s remarkable contributions to set theory — his proof of the consistency of the Axiom of Choice and Cantor’s Continuum Hypothesis (1937) with the axioms of Zermelo-Fraenkel Set Theory — lend credibility, in the minds of many mathematicians, to Goedel’s unfashionable Platonism.

Casti also gives us a glimpse into the unexalted realm of academic politics at the Institute for Advanced Studies. Von Neumann found it a scandal that his colleagues were reluctant to promote Goedel to Professor. This was due to their (largely justified) fears of Goedel’s mental instability and his obsessively legalistic frame of mind. In studying for his citizenship test, for example, Goedel had discovered a loophole in the U.S. Constitution that allowed for a dictatorship to legally arise. It took the Olympian efforts of his friends Einstein and Oskar Morgenstern to prevent Goedel from subverting his citizenship interview by discussing his “discovery.” Championing Goedel’s promotion, von Neumann asked, “How can any of us call ourselves ‘Professor’ if Goedel can not?”

It turns out that Goedel’s paper for Einstein provided a solution to the problem of Goedel’s academic advancement. Oppenheimer, then Director of the Institute for Advanced Studies, was on the committee for creating the Einstein Award, and suggested that the first award be shared by Harvard physicist Julian Schwinger for his work in quantum electrodynamics and by Goedel. On his 70th birthday on March 15, 1951, Einstein personally handed out the awards, saying to Schwinger “you deserve it” and to Goedel “you don’t need it.” Politically, this award was very much needed by Goedel. At this occasion, it was fitting that von Neumann, perhaps the first mathematician to grasp the revolutionary significance of Goedel’s Incompleteness Theorems 21 years earlier, would give tribute to Goedel’s work, hailing it as “a landmark which will remain visible far in space and time.”

The Einstein Award was the first formal academic honor Goedel ever received. It would become the first of many. Eventually, the mathematicians at the Institute for Advanced Studies solved their Goedel problem by promoting him to Professor but prohibiting him from serving on any committee but one, the committee having strictly to do with mathematical logic, of which Goedel was the sole member.

Die Welt is vernünftig (“The world is rational”), according to Goedel, in which “every chaos is merely a wrong appearance.” From a contemporary point of view, to seek rationality in all things may seem profoundly irrational. Yet the profound theoretical results of Einstein and Goedel were inspired by an optimistic faith in the power of reason and the conviction that the universe, rationally ordered by God, was comprehensible to the human mind.