Alfred Tarski
Alfred Tarski was a Polish-American logician and mathematician. A prolific author best known for his work on model theory, metamathematics, and algebraic logic, he also contributed to abstract algebra, topology, geometry, measure theory, mathematical logic, set theory, type theory, and analytic philosophy.
Educated in Poland at the University of Warsaw, and a member of the Lwów–Warsaw school of logic and the Warsaw school of mathematics, in 1939 he immigrated to the United States, where in 1945 he became a naturalized citizen. Tarski taught and carried out research in mathematics at the University of California, Berkeley, from 1942 until his death in 1983.
His biographers Anita Burdman Feferman and Solomon Feferman state that, "Along with his contemporary, Kurt Gödel, he changed the face of logic in the twentieth century, especially through his work on the concept of truth and the theory of models."
Life
Early life and education
Alfred Tarski was born Alfred Teitelbaum to Polish-Jewish parents in comfortable circumstances. He first showed mathematical ability in secondary school, at Warsaw's Szkoła Mazowiecka. Nevertheless in 1918 he entered the University of Warsaw intending to study biology.After Poland regained independence in 1918, Warsaw University came under the leadership of Jan Łukasiewicz, Stanisław Leśniewski, and Wacław Sierpiński and quickly became a world-leading research institution in logic, foundational mathematics, and the philosophy of mathematics. Leśniewski recognized Tarski's potential as a mathematician and encouraged him to abandon biology.
Tarski attended courses taught by Łukasiewicz, Sierpiński, Stefan Mazurkiewicz, and Tadeusz Kotarbiński and in 1924 became the only person ever to complete a doctorate under Leśniewski's supervision. His thesis, O wyrazie pierwotnym logistyki, was published in 1923.
Tarski and Leśniewski soon grew cool to each other, mainly due to Leśniewski's growing anti-semitism. In later life, however, Tarski expressed the warmest praise for Kotarbiński.
In 1923, Alfred Teitelbaum and his brother Wacław changed their surname to “Tarski”. The brothers also converted to Roman Catholicism, Poland's dominant religion, even though Alfred was an avowed atheist.
Career
After becoming the youngest person ever to complete a doctorate at Warsaw University, Tarski taught logic at the Polish Pedagogical Institute, mathematics and logic at the university, and served as Łukasiewicz's assistant. Because these positions were poorly paid, from 1925 Tarski also taught mathematics at the Third Boys’ Gymnasium of the Trade Union of Polish Secondary-School Teachers. Before World War II it was not uncommon for European intellectuals of research caliber to teach secondary school. Until his departure for the United States in 1939, Tarski wrote several textbooks and many papers – a number of them ground-breaking – while supporting himself mainly by teaching secondary-school mathematics.In 1929 Tarski married fellow teacher Maria Witkowska, a Catholic Pole who had served during the Polish–Soviet War as a courier for the Polish Army. They had two children: son Jan Tarski, who became a physicist; and daughter Ina, who married mathematician Andrzej Ehrenfeucht.
Tarski applied for a chair of philosophy at Lwów University, but on Bertrand Russell's recommendation it was awarded to Leon Chwistek. In 1930 Tarski visited the University of Vienna, lectured to Karl Menger's colloquium, and met Kurt Gödel. Thanks to a fellowship, he was able to return to Vienna during the first half of 1935 to work with Menger's research group. From Vienna he traveled to Paris to present his ideas on truth at the first meeting of the Unity of Science movement, an offshoot of the Vienna Circle.
Tarski's academic career in Poland was affected by his heritage. In 1937, when he applied for a chair at Poznań University, the chair was abolished to avoid assigning it to Tarski – indisputably the strongest applicant – because he was Jewish.
Tarski's ties to the Unity of Science movement likely saved his life, because they resulted in his being invited to address the Unity of Science Congress held in September 1939 at Harvard University. He left Poland in August 1939, on the last ship to sail from Poland for the United States before the German and Soviet invasion of Poland and the outbreak of World War II. Tarski left reluctantly, because Leśniewski had died a few months before, creating a vacancy which Tarski hoped to fill. Oblivious to the Nazi threat, he left his wife and children in Warsaw, not to see them again until 1946. During the war, nearly all his Jewish extended family were murdered by the German occupiers.
In the United States, Tarski held a number of temporary teaching and research positions: at Harvard University, at City College of New York, and thanks to a Guggenheim Fellowship, at the Institute for Advanced Study at Princeton, where he again met Gödel. In 1942 Tarski joined the mathematics department at the University of California, Berkeley, where he spent the rest of his career. In 1945 he became an American citizen. Although emeritus from 1968, he taught until 1973 and supervised Ph.D. candidates until his death. At Berkeley, Tarski acquired a reputation as an astounding and demanding teacher, as noted by many observers:
File:20070206 uw buw hall glowny biblioteki.jpg|thumb|Atop four pillars at entrance to Warsaw University Library stand statues of Lwów-Warsaw School philosophers Kazimierz Twardowski, Jan Łukasiewicz, Alfred Tarski, Stanisław Leśniewski.
Tarski supervised twenty-four Ph.D. dissertations, including those of – in chronological order – Andrzej Mostowski, Bjarni Jónsson, Julia Robinson, Robert Vaught, Solomon Feferman, Richard Montague, James Donald Monk, Haim Gaifman, Donald Pigozzi, and Roger Maddux, as well as Chen Chung Chang and Jerome Keisler, authors of Model Theory, a classic in the field. He also strongly influenced the dissertations of Adolf Lindenbaum, Dana Scott, and Steven Givant.
Five of Tarski's students were women, remarkably so, given that men were then the overwhelming majority of graduate students. He had extramarital affairs with at least two of the women. After he showed the work of another female student to a male colleague, the colleague published it himself, causing the woman to break off her graduate studies and later move to a different university.
Tarski lectured at University College, London, the Institut Henri Poincaré in Paris, the Miller Institute for Basic Research in Science at Berkeley, the University of California at Los Angeles, and the Pontifical Catholic University of Chile.
Among many distinctions garnered over his career, Tarski was elected to the United States National Academy of Sciences, the British Academy, and the Royal Netherlands Academy of Arts and Sciences in 1958; and received honorary degrees from the Pontifical Catholic University of Chile in 1975, from Marseille's Paul Cézanne University in 1977, and from the University of Calgary, as well as a Berkeley Citation in 1981.
Tarski presided over the Association for Symbolic Logic, 1944-46, and the International Union for the History and Philosophy of Science, 1956-57; and was an honorary editor of Algebra Universalis.
Work in mathematics
Tarski's mathematical interests were exceptionally broad. His collected papers run to about 2,500 pages, most of them on mathematics, not logic. For a concise survey of Tarski's mathematical and logical accomplishments by his former student Solomon Feferman, see "Interludes I-VI" in Feferman and Feferman.Tarski's first paper, published when he was 19 years old, was on set theory, a subject to which he returned throughout his life. In 1924, he and Stefan Banach proved that, if one accepts the Axiom of Choice, a ball can be cut into a finite number of pieces, and then reassembled into a ball of larger size, or alternatively it can be reassembled into two balls whose sizes each equal that of the original one. This result is now called the Banach–Tarski paradox.
In A decision method for elementary algebra and geometry, Tarski showed, by the method of quantifier elimination, that the first-order theory of the real numbers under addition and multiplication is decidable. This is a very curious result, because Alonzo Church proved in 1936 that Peano arithmetic is not decidable. Peano arithmetic is also incomplete by Gödel's incompleteness theorem. In his 1953 Undecidable theories, Tarski et al. showed that many mathematical systems, including lattice theory, abstract projective geometry, and closure algebras, are all undecidable. The theory of Abelian groups is decidable, but that of non-Abelian groups is not.
While teaching at the Stefan Żeromski Gimnazjum in the 1920s and 30s, Tarski often taught geometry. Using some ideas of Mario Pieri, in 1926 Tarski devised an original axiomatization for plane Euclidean geometry, one considerably more concise than Hilbert's. Tarski's axioms form a first-order theory devoid of set theory, whose individuals are points, and having only two primitive relations. In 1930, he proved this theory decidable because it can be mapped into another theory he had already proved decidable, namely his first-order theory of the real numbers.
In 1929 he showed that much of Euclidean solid geometry could be recast as a second-order theory whose individuals are spheres, a single primitive binary relation "is contained in", and two axioms that, among other things, imply that containment partially orders the spheres. Relaxing the requirement that all individuals be spheres yields a formalization of mereology far easier to exposit than Lesniewski's variant. Near the end of his life, Tarski wrote a very long letter, published as Tarski and Givant, summarizing his work on geometry.
Cardinal Algebras studied algebras whose models include the arithmetic of cardinal numbers. Ordinal Algebras sets out an algebra for the additive theory of order types. Cardinal, but not ordinal, addition commutes.
In 1941, Tarski published an important paper on binary relations, which began the work on relation algebra and its metamathematics that occupied Tarski and his students for much of the balance of his life. While that exploration uncovered some important limitations of relation algebra, Tarski also showed that relation algebra can express most axiomatic set theory and Peano arithmetic. For an introduction to relation algebra, see Maddux. In the late 1940s, Tarski and his students devised cylindric algebras, which are to first-order logic what the two-element Boolean algebra is to classical sentential logic. This work culminated in the two monographs by Tarski, Henkin, and Monk.