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Alfred Tarski died on October 27, 1983, at the age of 82. He was one of the great mathematicians of the century and one of the leading figures in the entire history of logic. In addition to his own remarkable creative work in over 300 publications, he encouraged and stimulated others to an exceptional extent in their creative efforts. Not only his students, colleagues, and scientific collaborators, but many other mathematicians and philosophers throughout the world regarded Tarski as a key influence in their own scholarly development.
Alfred Tarski was born in Warsaw in 1901. He began publishing papers in 1921 and received the Ph.D. in mathematics from the University of Warsaw in 1924. In a short time he was a key member of the famous "Polish school" (about 1910-1938) of mathematicians and philosophers. Tarski's teachers in logic and philosophy were philosophers Kotarbinski, Lukasiewicz, and Lesniewski. His teachers in mathematics included Banach and Sierpinski. His thesis was written under Lesniewski, who bequeathed to Tarski his interest in the theory of definition. By 1924, Tarski had already exhibited his interest in many parts of mathematics. With Stefan Banach, then the leader of the Polish mathematicians, he published in that year the famous "Banach-Tarski paradox" (that a solid sphere can be taken apart and reassembled into two spheres, each the same size as the original).
He was already working in nearly all of the wide group of fields in which he continued working all his life--logic, set theory, both ordinary algebra and universal algebra, topology and measure theory, and geometry. Some great mathematicians work here and there over the whole subject. Tarski's area of interest was not that broad but still very broad indeed, and it was all one connected, unbroken piece!
In 1926-28, Tarski held a seminar at Warsaw University on a relatively new topic in logic, called the method of eliminating quantifiers. This seminar was Tarski's beginning work on what 30 years later came to be called the theory of models, the branch of logic in which perhaps his greatest work lies. The studies in the seminar soon began to go in two directions. On the one hand, various new and more difficult "eliminations" were carried out. These led in particular to a well-known result of Presburger (1930) concerning addition of natural numbers. They also led to one of Tarski's most famous results, a decision method for all questions in the elementary theory of the real field and in elementary geometry (announced in 1931, published in full in 1948). He showed that theoretically a machine can be built to answer all such questions. This work has many applications in algebra; a well-known modern graduate textbook on algebra by Nathan Jacobson devotes a chapter to it.
The other direction in which the seminar studies led was this: Tarski saw that, in order to state the results of the seminar precisely, one needed to define the notions of truth and definability. Eventually (in 1933), Tarski published, in a philosophical journal, his famous paper on the theory of truth in formalized languages. He obtained two remarkable results: that truth can be precisely defined for essentially all formalized languages and, using a famous result of Gödel [1931], that the truth for a system cannot be defined within the system itself. Tarski's work on truth is valued highly by mathematicians. It also has been extremely influential in philosophy. The logical positivists had argued that to know the meaning of a sentence is to know the conditions under which it would be true; combining this idea with Tarski's method for defining truth for formalized languages, a number of more recent philosophers have concluded that, to understand the sentences of a language, is, in effect, to have, implicitly or explicitly, a Tarski-like definition of truth for that language. The implications of this view are being explored at the present time in the writings of a number of leading philosophers.
In the thirties, Tarski also made fundamental contributions to set theory and to the study of Boolean algebras (where his work interacted with that of the leading American mathematicians, Garret Birkhoff and Marshall Stone).
Because of anti-Semitism, Tarski taught during the whole period, 1924-1939, in a lycée. (He was also a docent or adjunct professor at the University of Warsaw.) But an infinitely worse period lay ahead in which the Nazis would rule Poland. Fortunately, Tarski had gone to America in 1939 for a lecture tour; he stayed perforce. His wife, Maria (born a Catholic), and children spent the war years in Warsaw. Thanks to Maria Tarski's courage, they survived and, in 1946, they finally came to America. During 1939-42, Tarski held positions at Harvard, the College of the City of New York, and the Institute for Advanced Study. Then in 1942, G. C. Evans of the Berkeley Mathematics Department brought him to the University of California. Tarski loved Berkeley and never left again for longer than a few months.
Tarski was already well known in America. (In 1952-53, he was chosen as the American Mathematical Society's Colloquium Lecturer.) At once he began to attract many able students. In Poland he had already had one doctoral student, Andrzej Mostowski, who became the leader of the Polish logicians and set theorists for several decades after the war. From 1946 to 1982, over 20 men and women obtained doctoral degrees from Tarski--each after an extremely friendly but rigorous period of apprenticeship. They now constitute a considerable fraction of the world's leading logicians.
Under Tarski's influence, a number of other logicians were brought to Berkeley in the mathematics and philosophy departments. He was also influential in the early stages of the now esteemed logic groups at Stanford and UCLA. In 1958, he took the lead in founding an interdisciplinary Ph.D. program at Berkeley in logic and the methodology of science, which brings together faculty and students interested in these subjects from many departments.
In the period 1945-70, Tarski made important research contributions in a number of fields. As a result of these contributions and also his consultations with logicians all over the world (many of whom visited Berkeley), he was the world leader in the foundations of mathematics in these years. The fields to which he contributed included: (1) decision methods--new positive results and a well-known book on obtaining negative results; (2) set theory, where he was the moving force for the great advances in 1960-70 concerning very large cardinal numbers; and (3) the whole subject called model theory, to which he continued to make important contributions. In the forties, his interest in universal algebra increased and it was henceforth one of his main interests. The fraction of scholars in this field today who consider themselves his disciples is quite comparable with that for logic.
In 1968, Tarski became professor emeritus, though he was recalled to teach for five more years. He was a member of the National Academy of Science, as well as the Royal Netherlands Academy of Sciences and Letters and the British Academy. He was the Shearman Memorial Lecturer at University College, London, in 1952 and again in 1965. He served as president of the Association for Symbolic Logic in 1944-46, and of the International Union for the History and Philosophy of Science in 1956-57. Tarski was Faculty Research Lecturer at Berkeley in 1962-63. He was awarded the Berkeley Citation in 1981 and received several honorary degrees.
Both in writing and lecturing, Alfred Tarski maintained an almost unimaginable level of clarity. His semi-popular book, Introduction to Logic, written before the war, is today still the best reading there is for anyone curious about mathematical logic. It has been translated into a dozen languages.
Alfred Tarski will be missed by many who counted him first as a warm and loyal friend. He had a wide range of interests, from national and world affairs to botany and gardening. All of these interests were nearly as avid as his interest in logic.
He is survived by his wife, Maria, of 54 years, by his son, Jan, his daughter, Kristina, and three grandchildren.
Robert Vaught
John Addison Leon
Henkin Benson Mates
Julia Robinson
Alfred Tarski was born in Warsaw in 1901. He began publishing papers in 1921 and received the Ph.D. in mathematics from the University of Warsaw in 1924. In a short time he was a key member of the famous "Polish school" (about 1910-1938) of mathematicians and philosophers. Tarski's teachers in logic and philosophy were philosophers Kotarbinski, Lukasiewicz, and Lesniewski. His teachers in mathematics included Banach and Sierpinski. His thesis was written under Lesniewski, who bequeathed to Tarski his interest in the theory of definition. By 1924, Tarski had already exhibited his interest in many parts of mathematics. With Stefan Banach, then the leader of the Polish mathematicians, he published in that year the famous "Banach-Tarski paradox" (that a solid sphere can be taken apart and reassembled into two spheres, each the same size as the original).
He was already working in nearly all of the wide group of fields in which he continued working all his life--logic, set theory, both ordinary algebra and universal algebra, topology and measure theory, and geometry. Some great mathematicians work here and there over the whole subject. Tarski's area of interest was not that broad but still very broad indeed, and it was all one connected, unbroken piece!
In 1926-28, Tarski held a seminar at Warsaw University on a relatively new topic in logic, called the method of eliminating quantifiers. This seminar was Tarski's beginning work on what 30 years later came to be called the theory of models, the branch of logic in which perhaps his greatest work lies. The studies in the seminar soon began to go in two directions. On the one hand, various new and more difficult "eliminations" were carried out. These led in particular to a well-known result of Presburger (1930) concerning addition of natural numbers. They also led to one of Tarski's most famous results, a decision method for all questions in the elementary theory of the real field and in elementary geometry (announced in 1931, published in full in 1948). He showed that theoretically a machine can be built to answer all such questions. This work has many applications in algebra; a well-known modern graduate textbook on algebra by Nathan Jacobson devotes a chapter to it.
The other direction in which the seminar studies led was this: Tarski saw that, in order to state the results of the seminar precisely, one needed to define the notions of truth and definability. Eventually (in 1933), Tarski published, in a philosophical journal, his famous paper on the theory of truth in formalized languages. He obtained two remarkable results: that truth can be precisely defined for essentially all formalized languages and, using a famous result of Gödel [1931], that the truth for a system cannot be defined within the system itself. Tarski's work on truth is valued highly by mathematicians. It also has been extremely influential in philosophy. The logical positivists had argued that to know the meaning of a sentence is to know the conditions under which it would be true; combining this idea with Tarski's method for defining truth for formalized languages, a number of more recent philosophers have concluded that, to understand the sentences of a language, is, in effect, to have, implicitly or explicitly, a Tarski-like definition of truth for that language. The implications of this view are being explored at the present time in the writings of a number of leading philosophers.
In the thirties, Tarski also made fundamental contributions to set theory and to the study of Boolean algebras (where his work interacted with that of the leading American mathematicians, Garret Birkhoff and Marshall Stone).
Because of anti-Semitism, Tarski taught during the whole period, 1924-1939, in a lycée. (He was also a docent or adjunct professor at the University of Warsaw.) But an infinitely worse period lay ahead in which the Nazis would rule Poland. Fortunately, Tarski had gone to America in 1939 for a lecture tour; he stayed perforce. His wife, Maria (born a Catholic), and children spent the war years in Warsaw. Thanks to Maria Tarski's courage, they survived and, in 1946, they finally came to America. During 1939-42, Tarski held positions at Harvard, the College of the City of New York, and the Institute for Advanced Study. Then in 1942, G. C. Evans of the Berkeley Mathematics Department brought him to the University of California. Tarski loved Berkeley and never left again for longer than a few months.
Tarski was already well known in America. (In 1952-53, he was chosen as the American Mathematical Society's Colloquium Lecturer.) At once he began to attract many able students. In Poland he had already had one doctoral student, Andrzej Mostowski, who became the leader of the Polish logicians and set theorists for several decades after the war. From 1946 to 1982, over 20 men and women obtained doctoral degrees from Tarski--each after an extremely friendly but rigorous period of apprenticeship. They now constitute a considerable fraction of the world's leading logicians.
Under Tarski's influence, a number of other logicians were brought to Berkeley in the mathematics and philosophy departments. He was also influential in the early stages of the now esteemed logic groups at Stanford and UCLA. In 1958, he took the lead in founding an interdisciplinary Ph.D. program at Berkeley in logic and the methodology of science, which brings together faculty and students interested in these subjects from many departments.
In the period 1945-70, Tarski made important research contributions in a number of fields. As a result of these contributions and also his consultations with logicians all over the world (many of whom visited Berkeley), he was the world leader in the foundations of mathematics in these years. The fields to which he contributed included: (1) decision methods--new positive results and a well-known book on obtaining negative results; (2) set theory, where he was the moving force for the great advances in 1960-70 concerning very large cardinal numbers; and (3) the whole subject called model theory, to which he continued to make important contributions. In the forties, his interest in universal algebra increased and it was henceforth one of his main interests. The fraction of scholars in this field today who consider themselves his disciples is quite comparable with that for logic.
In 1968, Tarski became professor emeritus, though he was recalled to teach for five more years. He was a member of the National Academy of Science, as well as the Royal Netherlands Academy of Sciences and Letters and the British Academy. He was the Shearman Memorial Lecturer at University College, London, in 1952 and again in 1965. He served as president of the Association for Symbolic Logic in 1944-46, and of the International Union for the History and Philosophy of Science in 1956-57. Tarski was Faculty Research Lecturer at Berkeley in 1962-63. He was awarded the Berkeley Citation in 1981 and received several honorary degrees.
Both in writing and lecturing, Alfred Tarski maintained an almost unimaginable level of clarity. His semi-popular book, Introduction to Logic, written before the war, is today still the best reading there is for anyone curious about mathematical logic. It has been translated into a dozen languages.
Alfred Tarski will be missed by many who counted him first as a warm and loyal friend. He had a wide range of interests, from national and world affairs to botany and gardening. All of these interests were nearly as avid as his interest in logic.
He is survived by his wife, Maria, of 54 years, by his son, Jan, his daughter, Kristina, and three grandchildren.
Robert Vaught
John Addison Leon
Henkin Benson Mates
Julia Robinson
This University of California obituary is available at THIS LINK