Alfred Tarski and the Undefinability of Truth

Alfred Tarski (1901-1983)

Alfred Tarski (1901-1983)

On January 14, 1902, Polish-American mathematician and logician Alfred Tarski was born. A prolific author he is 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, and analytic philosophy. For my annual Semantic Web Technologies lecture series I always introduce my students to model-theoretic semantics as a means to enable a formal representation of meaning for languages. I guess, they don’t like the mathematical overhead. But nevertheless, you will need it to make sense of any logical expression. But, let’s get back to Alfred Tarski.

Born as Alfred Teitelbaum into a family of Polish Jews of comfortable circumstances, Tarski first manifested his mathematical abilities while in secondary school, at Warsaw’s Szkoła Mazowiecka. Nevertheless, he entered the University of Warsaw in 1918 intending to study biology. After Poland regained independence in 1918, Warsaw University quickly became a world-leading research institution in logic, foundational mathematics, and the philosophy of mathematics. Famous Mathematician Stanisław Leśniewski recognized Tarski’s potential as a mathematician and encouraged him to abandon biology. Henceforth Tarski attended courses taught by Jan Łukasiewicz, Wacław Sierpiński, and became the only person ever to complete a doctorate under Leśniewski’s supervision.

In 1923, Alfred Teitelbaum and his brother Wacław changed their surname to Tarski and also converted to Roman Catholicism, Poland’s dominant religion, even though Tarski was an avowed atheist. After becoming the youngest person ever to complete a doctorate at Warsaw University, Tarski served as Łukasiewicz’s assistant. Between 1923 and his departure for the United States in 1939, Tarski not only wrote several textbooks and many papers, a number of them ground-breaking, but also did so while supporting himself primarily by teaching high-school mathematics at Warsaw secondary school, because of the small salary at Warsaw University. In 1930, Tarski visited the University of Vienna, lectured to Karl Menger’s colloquium, and met Kurt Gödel.[4] Due to an invitation from Harvard University, Tarski was able to leave 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. He did not see them again until 1946. During the war, nearly all his extended family died at the hands of the German occupying authorities. Thanks to a Guggenheim Fellowship, Tarski visited the Institute for Advanced Study in Princeton in 1942, where he again met Gödel, who also had fled from Nazi Germany. Subsequently, he joined the Mathematics Department at the University of California, Berkeley, where he spent the rest of his career until he became emeritus in 1968.

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.[3] He is recognised as one of the four greatest logicians of all time, the other three being Aristotle, Frege, and Gödel. Of these Tarski was the most prolific as a logician and his collected works, excluding his books, runs to 2500 pages. Tarski made important contributions in many areas of mathematics: set theory, measure theory, topology, geometry, classical and universal algebra, algebraic logic, various branches of formal logic and metamathematics. He produced axioms for ‘logical consequence‘, worked on deductive systems, the algebra of logic and the theory of definability. He can be considered a mathematical logician with exceptionally broad mathematical interests.[1]

One example of his achievements is 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 .[3]

Another great achievement was his assault on the notion of truth. Tarski 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 syntax of the language in question be formally well-defined, i.e. one could say precisely just which expressions are legitimate sentences and which not. Moreover, a sentence had to have a well-defined semantics, i.e. the meaning of the individual components of the sentence had to be (formally) given. Now, 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 previously 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 “undefinability theorem“: Under very general conditions, the notion of “truth” of the sentences of a language cannot be defined in that same language.[3] Thus, Tarski radically transformed Hilbert’s proof-theoretic metamathematics. He destroyed the borderline between metamathematics and mathematics by his objection to restricting the role of metamathematics to the foundations of mathematics

At yovisto, you can learn more about the history of mathematical logics in the lecture of Prof Christos H. Papadimitriou on the Graphic Novel ‘Logicomix: An Epic Search for Truth‘.

References and Further Reading

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