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The Man Who Defined Truth

Martin Davis

Alfred Tarski: Life and Logic. Anita Burdman Feferman and Solomon Feferman. vi + 425 pp. Cambridge University Press, 2004. $35.

All people use logic in making their way through the world. Scientists in particular use it in reasoning about relations among experimental results and theoretical explanations. The first attempt to make logic itself the subject of rational inquiry was by Aristotle in his theory of the syllogism. The 19th-century English mathematician George Boole saw how to transform Aristotelian syllogistics into mathematics—specifically, a kind of algebra. The German mathematician Gottlob Frege, followed by Bertrand Russell, turned things around, making logic primary and seeing all of mathematics as based on logic. During the 20th century, as logic developed into a vibrant branch of mathematics, tension remained between those two aspects of the field: logic as a part of mathematics and logic as the foundation of mathematics.

Alfred TarskiClick to Enlarge Image

Alfred Tarski, who became one of the great logicians of the 20th century, was born in 1901 in Warsaw into a middle-class Jewish family named Teitelbaum. Poland, one of the new states formed after World War I, tended to treat its large Jewish population like an alien presence. As an ambitious young man with a powerful mathematical ability, Alfred found it necessary to shed his Jewish identity to have any hope of an academic career. He changed his name to Tarski and converted to Catholicism, the religion of the majority. September 1939 found him in the United States for a conference, and the German invasion of Poland separated him from his wife and children. He accepted a position at the University of California, Berkeley, although California seemed like the end of the world to him. After the war his wife and children joined him, and there he remained for the rest of his long, eventful life.

Tarski was a charismatic teacher who charmed his students, but he demanded perfection and could be devastatingly abusive to those who failed to measure up. His extensive amorous involvements, which were hardly discreet, were apparently accepted by his loyal wife. He was particularly attracted to clever, lively women and was not at all inhibited in offering his attentions to his female students. When a lover and former student arrived in Berkeley from Poland for a year's sabbatical leave, Tarski thought nothing of installing her in his home with his wife and two growing children, something his son found hard to forgive. On another occasion, his wife did move out, providing his latest conquest with details of house management.

His was a fascinating life, and the new biography Alfred Tarski: Life and Logic covers it all. The authors are exceptionally well qualified to tell his story: Solomon Feferman, who was one of Tarski's doctoral students in the 1950s and is now on the faculty at Stanford, is a superb logician in his own right, and independent scholar and writer Anita Burdman Feferman, who is the author of a biography of logician Jean van Heijenoort, also knew Tarski well. The Fefermans (who are husband and wife) were personally acquainted with many of the people they write about here, and they have obtained some remarkably intimate information. The book is beautifully written and a pleasure to read on a number of levels.

The atmosphere in which the young Tarski became a mathematician was heady. The massive three volumes of Alfred North Whitehead and Bertrand Russell's Principia Mathematica, which claimed to provide a logical foundation for all of mathematics, had been published in 1910, 1912 and 1913. By the 1920s the abstract theory of sets was playing an increasingly central role in certain branches of mathematics. Mathematics in Poland was heavily influenced by the confluence of those developments.

By the time of his arrival in the United States in 1939, Tarski had some very significant accomplishments to his credit. Perhaps most notorious was the Banach-Tarski "paradox," which is described in the first of a series of six "Interludes" that explain the background of Tarski's achievements. This theorem, which Tarski and another Polish mathematician, Stefan Banach, arrived at independently, states that a solid ball of any given size (a pea, for example) can be decomposed into pieces that can be reassembled to produce another ball of any other desired size (such as the Sun). Because one would expect that the volumes of the pieces would add up to the size of the original ball no matter how they were rearranged, this seems flatly impossible. The "catch" that furnishes a way out of the impasse is that the pieces are such bizarrely complicated structures that no numerical volume can be attached to them: They are "nonmeasurable" sets.

Another early achievement (only published much later) was a decision procedure for sentences written in the language of the arithmetic of real numbers. These are sentences that can be written using variables ranging over the real numbers, using symbols for the operations of addition and multiplication and for the relations of equality and order (<), and finally, using the logical operations of not, and, or, if … then, and exists. Tarski produced an algorithm that could determine of any such sentence whether or not it is true. This feat is particularly striking because it has turned out that if the variables are permitted to range over only the integers (instead of all real numbers), no such decision procedure is possible.

For philosophers, Tarski's great achievement was his audacious assault on the notion of truth. He was able, under suitable conditions, to give a mathematically precise definition of what it means to say that a given sentence of a language is true. One of these conditions was that the language in question be completely formalized—its syntax had to be so clearly specified that one could say precisely just which utterances are legitimate sentences. The other was that the sentence had to have a well-defined semantics—the meaning of the individual components of the sentence had to be given. The metalanguage in which this truth definition is developed is, in general, separate from the language whose true sentences are being identified. As Kurt Gödel had shown, it is possible for a language to function as its own metalanguage. But for this case, Tarski was able to prove his famous theorem on the nondefinability of truth: Under very general conditions, the notion of "truth" of the sentences of a language cannot be defined in that same language.

When logic is developed with the mercilessly rigorous syntax needed for foundational purposes, the details are a bit ungainly and not at all appealing to one who seeks elegance in mathematics. Tarski always seemed to work on formulations of logical matters that could hope to have the same elegance found elsewhere in mathematics. Some of his early publications tried to frame logical inference in an abstract general setting, hiding the messy details. In later years, he went back to the algebraic roots of 19th-century logic; his cylindrical algebras were to do for 20th-century logic what Boole's algebra had done for more restricted logical systems.

A crucial aspect of Tarski's legacy is his influence as a teacher. He was notoriously hard on his doctoral students. They were expected to come to his house, stay into the morning hours in a closed, smoke-filled room and help him with his research programs and proposals. His students took longer to finish than most, because Tarski seemed never to be convinced that enough had been done for a dissertation. Nevertheless, his 24 students are a remarkable group of outstanding, highly productive logicians. Despite the obstacles they faced and surmounted, most of them have been greatly appreciative of what they learned from Tarski and view him with real affection.

By any standards, Tarski was a great mathematician whose work has influenced very many researchers. But perhaps his greatest achievement was convincing the administration of the University of California, Berkeley, that logic was important enough to justify the resources he was demanding—resources that made it possible to assemble a group of scholars who made the Department of Mathematics at Berkeley a great world center for logic, a status it retains today.

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