Russell's Real Paradox: The Wise Man Is a Fool

Review of Bertrand Russell: A Life
By Caroline Moorehead
Viking, New York, 1993, 596 pages, $30.00

Writing to Lucy Donnelly in 1940 after an anxious moment, Bertrand Russell confessed: I used, when excited, to calm myself by reciting the three factors of a3 + b3 + c3 — 3abc; I must revert to this practice. I find it more effective than thoughts of the Ice Age or the goodness of God.

Reading this confession in Moorehead's excellent biography, I wondered just where Russell had picked up these factors. Was it as a young student, cramming for admission to Cambridge? I wondered whether he knew that this expression is the determinant of the 3 x 3 circulant matrix whose first row is [a, b, c]? And did he know that the factors, linear in a, b, c are the three eigenvalues of the matrix? Did he know that this factorization was historically the seed that, watered by Frobenius, grew into the great subject of group representation theory?

I conjecture that he did not. To Russell, the algebraic expression was a mantra. He saw mathematics as the stabilizing force in the universe; it was the one place where absolute certainty reigned. In search of this certainty, groping for it, he said, as one might grope for religious faith, he devoted the first half of a very long life to an attempt to establish the identity of mathematics and logic.

I first heard of Russell as an undergraduate. I did a chapter of Principia Mathematica, his masterwork, written (1910-1913) with Alfred North Whitehead, as a reading period assignment in a course in mathematical logic. At that time Russell was a celebrity, front-page news, having left the dots and epsilons and the "if-thens" of logic far behind. He had been appointed to a professorship of philosophy at CCNY in 1940, and almost immediately a charge of immorality was laid against him. It hit the papers. I, together with most undergraduates, sided with John Dewey, Albert Einstein, and Charlie Chaplin as they rushed in to defend Russell's right to teach epistemology.

When I learned more about the man, I admired and tried to emulate his courage to stand alone, his compassion for humanity, his sharp wit, his graceful literary style and willingness to write comprehensibly. I admired the fact that he had the guts to quit working on a topic that had exhausted him.

To readers of SIAM News, Russell is probably best known as the author (1901) of Russell's paradox: What is the existential status of the set of sets that are not members of themselves? Contradiction and inconsistency lurk in the shadow of this innocent construction. Russell bit his nails over it for years. In an attempt to deal with it, he came up with his Theory of Types, a cluttered arrangement that satisfied few people. Even today, people tinker with the Zermelo-Fraenkel axioms of set theory in an attempt to deal with the quandary.

But let me make up my own Who's Who entry.

Bertrand Arthur William Russell, 3rd Earl Russell (1872-1970), was the grandson of Lord John Russell, twice prime minister of England. He was a mathematician, a logician, an agnostic, and a sometime Cambridge don. Abandoning mathematics in 1914, he became a philosopher, a pacifist, an advocate of free love, a social activist, a soapbox orator, a pamphleteer, a theoretician of education, and the operator of the Beacon Hill School for children (1927-32). He wrote dozens of books, ranging from deep technical stuff to easily understood popularizations of science, philosophy, and politics, and treatises on how to live one's life sensibly. When he was jailed in 1918 for pacifism, my wife's aunt brought him fresh peaches. In 1940, the threat of his dismissal from the faculty at CCNY on the immorality charge became a cause celebre.

In 1967, he masterminded the International War Crimes Tribunal that met in Stockholm and "tried" Lyndon Johnson and other Americans for their brutal treatment of the Vietnamese. He was named a Fellow of the Royal Society in 1908 and received the Order of Merit in 1948 and the Nobel Prize for Literature in 1950.

So much for the externals. As regards the internals, I believe that no one has summed them up better than Beatrice Webb did in 1902:

Bertrand Russell's nature is pathetic in its subtle absoluteness: faith in an absolute logic, absolute ethic, absolute beauty and all of the most refined and rarified type. . . the uncompromising way in which he applies these frightens me for his future and for the future of those who love him and whom he loves. Compromise, mitigation, mixed motive, phases of health of body and mind, qualified statements, uncertain feelings, all seem unknown to him.

Russell was 30 when Beatrice Webb wrote these words. He lived to be 98. His life was strewn with hearts he broke and relationships he terminated. Former wives, children, friends, lovers were "dropped down an oubliette," someone said, and left there to rot. Webb's description, I think, remained valid throughout his life.

We have in Russell a verification of the comic, prototypical mathematician of the movies. If one puts credence in the theories of hemispheric specialization of the brain, one could say confidently that Russell was half-brained: The left hemisphere worked overtime producing symbols by the bushel, while the right hemisphere seemed to be atrophied. The famous art critic and historian Bernard Berenson, his sometime brother-in-law, remarked on how little the visual image meant to Russell. Aldous Huxley stressed this imbalance in his novel Crome Yellow (1922). Huxley's character Scogan, a bitter caricature of Russell, says of himself:

While I have a certain amount of intelligence, I have no aesthetic sense; while I possess the mathematical faculty, I am wholly without the religious emotion. . . . Life would be richer, warmer, brighter, altogether more amusing, if I could feel them. . . . I must have gone on looking at pictures for ten years before I would honestly admit to myself that these merely bored me.

Most people are fusions of contradictions, but surely Russell, the man who sought a contradiction-free mathematics, was himself the "Rolls-Royce" of contradictions. The extent to which this is true, as emphasized by Caroline Moorehead in her book, has become crystal clear with the availability of archival material.

Russell was a pacifist who hated other pacifists for thinking that human nature was essentially benign. He thought at an early age that "human actions like planetary motions could be calculated if we had sufficient skill" but resisted being "tracked" himself: "I don't think I could stand the academic atmosphere now [1919], having tasted freedom and the joys of a dangerous life."

While he claimed that the "world of pure reason knows no compromise, no practical limitations," Russell exhibited, at least in the minds of many people who seemed to be reasonable, the chaos of unreason. According to a friend of mine who tried to make a video profile of him at 92, Russell asserted that reason had its limitations: "Beware of rational argument," he said; "you need only one false premise in order to prove anything you please by logic."

He was a skeptic who claimed to have found the truth. As late as 1937, when he completed a second edition of Principles of Mathematics (to be distinguished from Principia Mathematica), he still asserted that logic and mathematics were identical. (In sympathy, but not in defense, I agree that it must be a bitter experience to have reached three score and ten years only to find that one's lifelong vision has collapsed.)

Russell was an aristocrat who was frequently broke. He was a liberal who expressed his ethnic and class prejudices openly and made no attempt to rid himself of them. He loved humanity in the abstract and championed individual rights, but he had absolutely no idea that real individuals had feelings.

He wrote books on how to live properly, although he totally mucked up three marriages. (The documents relating to his divorce from Dora Black, his second wife, occupy five large boxes in the archives.) His last and fourth marriage, undertaken at the age of 80, seems to have fared much better. He hated America and all it represented, but at one point talked of settling here permanently. He excoriated American professors for their shallowness and then sent his children to American universities. He milked America for its daughters, its accolades, its salary offers, its royalties, and for the security it afforded him from Hitler's army from 1938 to 1944. In political cartoons, he was depicted as the wisest man on earth who was at the same time the complete fool.

Russell's impact on mathematics was enormous. Principia Mathematica, a key technical advance between the work of Frege and Gödel, led later to much of the field of theoretical computer science, including AI. It established that most--possibly all--mathematical proofs could be expressed as the formal manipulation of symbols, verifiable, at least in theory, automatically. The only way to establish this was actually to have done it for some large collection of proofs. Gödel's incompleteness theorem, for example, would hardly have carried much punch without a previous demonstration that the techniques now shown to be incomplete seemed to suffice for all known mathematical proofs.

By establishing that many mathematical concepts can be reduced to set theory, Principia has had a great impact on the presentation of mathematics. The tendency of mathematical writers to give definitions in the often questionable form "a garble is a six-tuple $" is surely traceable to Principia.

Caroline Moorehead wisely avoided deep explanations of Russell's work in logic and in philosophy. Yet, from reading her book, I gained confirmation of a long-held feeling that mathematics derives from specific people, and that the particular psychological flavor of a particular individual can infuse that individual's mathematical creations. My own antiplatonic, antilogicist soul, nourished independently of Gödel's theorem, laughed heartily at Moorehead's reproduction of a proof sheet from Principia:

"Additional Errata to Volume I," and in the margins, in Russell's handwriting, errata to the errata to the errata.

In the social line, Russell's so-called immoralities have regrettably become standard operating procedures in the Western world, and legal practice, at least in the U.S., has favored individual freedom at the expense of social cohesion. What with computers, social and economic algorithms, and the counterclaim that happiness comes through the lack of constraints, we are living for better and for worse, with wisdom and with folly, in a world that Russell helped make.

Philip J. Davis, emeritus professor of applied mathematics at Brown University, is an independent writer, scholar, and lecturer based in Providence, Rhode Island. His e-mail address is

Reprinted from SIAM NEWS Volume 27-6, July 1994 (C) 1994 by Society for Industrial and Applied Mathematics All rights reserved.