Alfred Tarski facts for kids
Quick facts for kids
Alfred Tarski
|
|
|---|---|
 |
|
| Born |
Alfred Teitelbaum
January 14, 1901 Warsaw, Congress Poland
|
| Died | October 26, 1983 (aged 82) |
| Nationality | Polish, American |
| Education | University of Warsaw (Ph.D., 1924) |
| Known for |
|
| Scientific career | |
| Fields | Mathematics, logic, formal language |
| Institutions |
|
| Thesis | O wyrazie pierwotnym logistyki (On the Primitive Term of Logistic) (1924) |
| Doctoral advisor | Stanisław Leśniewski |
| Doctoral students |
|
| Other notable students | Evert Willem Beth |
| Influences | |
| Influenced |
|
Alfred Tarski (born Alfred Teitelbaum; January 14, 1901 – October 26, 1983) was a famous Polish-American logician and mathematician. He wrote many books and papers, and is best known for his work on model theory, which studies how mathematical ideas can be represented. He also helped develop metamathematics (the study of mathematics itself) and algebraic logic.
Tarski studied at the University of Warsaw in Poland. He was part of important groups of thinkers, like the Lwów–Warsaw school of logic. In 1939, he moved to the United States and became an American citizen in 1945. From 1942 until his death in 1983, Tarski taught and did research at the University of California, Berkeley.
Experts like Anita Burdman Feferman and Solomon Feferman say that Tarski, along with Kurt Gödel, changed how people thought about logic in the 20th century. This was especially true for his ideas about truth and model theory.
Contents
Who Was Alfred Tarski?
His Early Life and Education
Alfred Tarski was born Alfred Teitelbaum in Warsaw, Poland. His parents were Polish Jews and were quite well-off. He showed a talent for mathematics early on, even in high school. However, when he started at the University of Warsaw in 1918, he first planned to study biology.
After Poland became independent in 1918, the University of Warsaw became a leading place for studying logic and math. A professor named Stanisław Leśniewski saw Tarski's math skills and encouraged him to switch from biology. Tarski then took classes from famous mathematicians and logicians. In 1924, he earned his doctorate, becoming the only person to get a Ph.D. under Leśniewski.
In 1923, Alfred Teitelbaum and his brother changed their last name to "Tarski." They also became Roman Catholics, which was the main religion in Poland. Alfred did this even though he was an atheist (someone who does not believe in God).
His Career and Move to America
After getting his doctorate, Tarski taught logic and mathematics at the University of Warsaw. He also worked as an assistant to a professor named Jan Łukasiewicz. Because these jobs didn't pay much, Tarski also taught math at a high school in Warsaw. Before World War II, it was common for smart thinkers in Europe to teach high school. So, from 1923 until he left for the United States in 1939, Tarski wrote many important papers and textbooks while mostly earning a living by teaching high school.
In 1929, Tarski married Maria Witkowska, who was also a teacher. They had two children: a son named Jan, who became a physicist, and a daughter named Ina.
Tarski applied for a teaching position at Lwów University, but it went to someone else. In 1930, he visited the University of Vienna and met Kurt Gödel, another very famous logician. In 1935, Tarski returned to Vienna and then went to Paris to share his ideas about truth.
Tarski's connections to a group called the "Unity of Science" likely saved his life. He was invited to speak at a conference at Harvard University in September 1939. He left Poland in August 1939, on the last ship to sail for the United States before the German and Soviet invasion of Poland and the start of World War II. Tarski left his wife and children in Warsaw, not knowing the danger they faced. He didn't see them again until 1946. During the war, nearly all his Jewish relatives were killed by the German authorities.
Once in the United States, Tarski worked at several universities, including Harvard and the City College of New York. In 1942, he joined the Mathematics Department at the University of California, Berkeley. He stayed there for the rest of his career. Tarski became an American citizen in 1945. Even after he officially retired in 1968, he continued to teach until 1973 and guided students working on their Ph.D.s until he died.
Tarski guided 24 students who were getting their Ph.D.s. Some of his notable students included Julia Robinson and Solomon Feferman. He also influenced the work of many others. It's worth noting that five of his students were women, which was unusual at a time when most graduate students were men.
Tarski gave lectures at many universities around the world and received many honors. He was elected to the United States National Academy of Sciences and the British Academy. He also received honorary degrees from several universities.
Tarski's Work in Mathematics
Tarski was interested in many different areas of mathematics. His collected writings fill about 2,500 pages! His former student, Solomon Feferman, wrote a good summary of Tarski's math and logic achievements.
Tarski's first paper, written when he was just 19, was about set theory, which is the study of collections of objects. He continued to work on this topic throughout his life.
The Banach–Tarski Paradox
In 1924, Tarski and another mathematician named Stefan Banach proved a very surprising result. They showed that if you accept a certain idea in math called the Axiom of Choice, you can take a ball (like a solid sphere) and cut it into a few pieces. Then, you can put those pieces back together to make a ball that is much larger than the original, or even two new balls, each the same size as the original! This strange idea is called the Banach–Tarski paradox.
Deciding Math Problems
In a book published in 1948, Tarski showed that it's possible to create a method to decide if certain math statements about real numbers (numbers that can have decimals) are true or false. This was a very interesting discovery because, in 1936, another mathematician named Alonzo Church proved that it's not possible to do this for statements about natural numbers (counting numbers like 1, 2, 3...).
Tarski and his colleagues also showed in 1953 that many other math systems are "undecidable," meaning there's no general method to figure out if their statements are true or false.
Geometry and Relations
In the 1920s and 30s, Tarski often taught high school geometry. He created a simpler way to describe Euclidean geometry (the geometry you learn in school) using just a few basic ideas. He proved that this system was "decidable."
Tarski also did important work on binary relations, which are ways to describe how two things are connected (like "is taller than" or "is a part of"). This led to the study of relation algebra. He showed that this algebra could describe much of axiomatic set theory and Peano arithmetic (a system for natural numbers).
Tarski's Work in Logic
Tarski is considered one of the four greatest logicians of all time, along with Aristotle, Gottlob Frege, and Kurt Gödel. He greatly admired Charles Sanders Peirce for his early work in the logic of relations.
Tarski created rules for "logical consequence" and worked on deductive systems (ways to reason step-by-step). His ideas about semantics (the study of meaning) and model theory changed how people thought about logic.
Around 1930, Tarski developed an abstract idea of logical deductions. This helps us understand how conclusions follow from a set of statements. This way of thinking is still used today in abstract algebraic logic.
In 1936, Tarski wrote an article called "On the concept of logical consequence." In it, he argued that a conclusion logically follows from its starting ideas if every situation where the starting ideas are true also makes the conclusion true. He also wrote a classic short book called Introduction to Logic and to the Methodology of Deductive Sciences, which explains logic and how to use it in science.
What is Truth in Formal Languages?
In 1933, Tarski published a very long and important paper about how to define truth in formal languages (like the languages used in math and computer science). This paper, often called "The concept of truth in formalized languages," was a big deal in 20th-century analytic philosophy and symbolic logic.
Tarski's theory of truth is for these special, carefully built languages. It says that for any statement "p" in such a language, the definition of truth must include:
- "p" is true if and only if p.
This means, for example, that the statement "Snow is white" is true if and only if snow is actually white. This idea helps us understand what it means for a statement to be true in a precise, mathematical way.
What are Logical Ideas?
Another interesting idea from Tarski is found in his paper "What are Logical Notions?" (1986). In this paper, Tarski tried to figure out what makes some ideas "logical" and others "non-logical."
He used an idea from a 19th-century mathematician named Felix Klein, who classified different types of geometry (like Euclidean geometry or topology) based on what they kept the same when shapes were transformed. For example, in Euclidean geometry, the size and shape of objects stay the same when you move them. In topology, you can stretch or bend objects, but you can't tear them.
Tarski suggested that logical ideas are those that stay the same no matter how you transform a "domain" (the set of all possible things we are talking about). If an idea stays true or holds true no matter how you rearrange or transform the things in your world, then it's a logical idea.
For example, basic logical operations like "and," "or," and "not" are considered logical. Also, ideas like "all" (universal quantifier) and "there exists" (existential quantifier) are logical. Tarski also discussed whether set membership (the idea of something belonging to a group) is logical. He said it depends on how you build your set theory.
Tarski's ideas about logical notions have been further discussed by other logicians, like Solomon Feferman and Vann McGee, who have built upon his original thoughts.
Selected Publications
- Anthologies and collections
- 1986. The Collected Papers of Alfred Tarski, 4 vols. Givant, S. R., and McKenzie, R. N., eds. Birkhäuser.
- Givant Steven (1986). "Bibliography of Alfred Tarski". Journal of Symbolic Logic 51 (4): 913–41. doi:10.2307/2273905.
- 1983 (1956). Logic, Semantics, Metamathematics: Papers from 1923 to 1938 by Alfred Tarski, Corcoran, J., ed. Hackett. This collection includes translations of some of Tarski's most important early papers, like The Concept of Truth in Formalized Languages.
- Original publications of Tarski
- 1930 Une contribution à la théorie de la mesure. Fund Math 15 (1930), 42–50.
- 1930. (with Jan Łukasiewicz). "Untersuchungen uber den Aussagenkalkul" ["Investigations into the Sentential Calculus"], Comptes Rendus des seances de la Societe des Sciences et des Lettres de Varsovie, Vol, 23 (1930) Cl. III, pp. 31–32 in Tarski (1983): 38–59.
- 1931. "Sur les ensembles définissables de nombres réels I", Fundamenta Mathematicae 17: 210–239 in Tarski (1983): 110–142.
- 1936. "Grundlegung der wissenschaftlichen Semantik", Actes du Congrès international de philosophie scientifique, Sorbonne, Paris 1935, vol. III, Language et pseudo-problèmes, Paris, Hermann, 1936, pp. 1–8 in Tarski (1983): 401–408.
- 1936. "Über den Begriff der logischen Folgerung", Actes du Congrès international de philosophie scientifique, Sorbonne, Paris 1935, vol. VII, Logique, Paris: Hermann, pp. 1–11 in Tarski (1983): 409–420.
- 1936 (with Adolf Lindenbaum). "On the Limitations of Deductive Theories" in Tarski (1983): 384–92.
- 1937. Einführung in die Mathematische Logik und in die Methodologie der Mathematik. Springer, Wien (Vienna).
- 1994 (1941). Introduction to Logic and to the Methodology of Deductive Sciences. Dover.
- 1941. "On the calculus of relations", Journal of Symbolic Logic 6: 73–89.
- 1944. "The Semantical Concept of Truth and the Foundations of Semantics," Philosophy and Phenomenological Research 4: 341–75.
- 1948. A decision method for elementary algebra and geometry. Santa Monica CA: RAND Corp.
- 1949. Cardinal Algebras. Oxford Univ. Press.
- 1953 (with Mostowski and Raphael Robinson). Undecidable theories. North Holland.
- 1956. Ordinal algebras. North-Holland.
- 1965. "A simplified formalization of predicate logic with identity", Archiv für Mathematische Logik und Grundlagenforschung 7: 61-79
- 1969. "Truth and Proof", Scientific American 220: 63–77.
- 1971 (with Leon Henkin and Donald Monk). Cylindric Algebras: Part I. North-Holland.
- 1985 (with Leon Henkin and Donald Monk). Cylindric Algebras: Part II. North-Holland.
- 1986. "What are Logical Notions?", Corcoran, J., ed., History and Philosophy of Logic 7: 143–54.
- 1987 (with Steven Givant). A Formalization of Set Theory Without Variables. Vol.41 of American Mathematical Society colloquium publications. Providence RI: American Mathematical Society. ISBN: 978-0821810415. Review
- 1999 (with Steven Givant). "Tarski's system of geometry", Bulletin of Symbolic Logic 5: 175–214.
- 2002. "On the Concept of Following Logically" (Magda Stroińska and David Hitchcock, trans.) History and Philosophy of Logic 23: 155–196.
See Also
In Spanish: Alfred Tarski para niños
- History of philosophy in Poland
- Cylindric algebra
- Interpretability
- Weak interpretability
- List of things named after Alfred Tarski
