Alfred Tarski
Alfred Tarski | |
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Born | Alfred Teitelbaum January 14, 1901 |
Died | October 26, 1983 | (aged 82)
Nationality | Polish, American |
Education | University of Warsaw (Ph.D., 1924) |
Known for |
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Scientific career | |
Fields | Mathematics, logic, formal language |
Institutions |
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Thesis | O wyrazie pierwotnym logistyki (On the Primitive Term of Logistic) (1924) |
Doctoral advisor | Stanisław Leśniewski |
Doctoral students | |
Other notable students | Evert Willem Beth |
Alfred Tarski (.
Educated in Poland at the
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."[7]
Life
Early life and education
Alfred Tarski was born Alfred Teitelbaum (
After Poland regained independence in 1918, Warsaw University came under the leadership of
In 1923, Alfred Teitelbaum and his brother Wacław changed their surname to "Tarski". The Tarski brothers also converted to
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, Tarski also taught mathematics at the Third Boys’ Gimnazjum of the Trade Union of Polish Secondary-School Teachers (later the Stefan Żeromski Gimnazjum), a Warsaw secondary school, beginning in 1925.[12] Before World War II, it was not uncommon for European intellectuals of research caliber to teach high school. Hence until 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.[13] In 1929 Tarski married fellow teacher Maria Witkowska, a Pole of Catholic background. She had worked as a courier for the army in the Polish–Soviet War. They had two children; a son Jan Tarski, who became a physicist, and a daughter Ina, who married the mathematician Andrzej Ehrenfeucht.[14]
Tarski applied for a chair of philosophy at
Once in the United States, Tarski held a number of temporary teaching and research positions: Harvard University (1939), City College of New York (1940), and thanks to a Guggenheim Fellowship, the Institute for Advanced Study in Princeton (1942), 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. Tarski became an American citizen in 1945.[17] Although emeritus from 1968, he taught until 1973 and supervised Ph.D. candidates until his death.[18] At Berkeley, Tarski acquired a reputation as an astounding and demanding teacher, a fact noted by many observers:
His seminars at Berkeley quickly became famous in the world of mathematical logic. His students, many of whom became distinguished mathematicians, noted the awesome energy with which he would coax and cajole their best work out of them, always demanding the highest standards of clarity and precision.[19]
Tarski was extroverted, quick-witted, strong-willed, energetic, and sharp-tongued. He preferred his research to be collaborative — sometimes working all night with a colleague — and was very fastidious about priority.[20]
A charismatic leader and teacher, known for his brilliantly precise yet suspenseful expository style, Tarski had intimidatingly high standards for students, but at the same time he could be very encouraging, and particularly so to women — in contrast to the general trend. Some students were frightened away, but a circle of disciples remained, many of whom became world-renowned leaders in the field.[21]
Tarski supervised twenty-four Ph.D. dissertations including (in chronological order) those of
Tarski lectured at
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.[28]
Tarski's first paper, published when he was 19 years old, was on
In A decision method for elementary algebra and geometry, Tarski showed, by the method of
While teaching at the Stefan Żeromski Gimnazjum in the 1920s and 30s, Tarski often taught
In 1929 he showed that much of Euclidean
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
Work in logic
Tarski's student,
Tarski produced axioms for logical consequence and worked on
In [Tarski's] view, metamathematics became similar to any mathematical discipline. Not only can its concepts and results be mathematized, but they actually can be integrated into mathematics. ... Tarski destroyed the borderline between metamathematics and mathematics. He objected to restricting the role of metamathematics to the foundations of mathematics.[37]
Tarski's 1936 article "On the concept of logical consequence" argued that the conclusion of an argument will follow logically from its premises if and only if every model of the premises is a model of the conclusion.[38] In 1937, he published a paper presenting clearly his views on the nature and purpose of the deductive method, and the role of logic in scientific studies.[29] His high school and undergraduate teaching on logic and axiomatics culminated in a classic short text, published first in Polish, then in German translation, and finally in a 1941 English translation as Introduction to Logic and to the Methodology of Deductive Sciences.[39]
Tarski's 1969 "Truth and proof" considered both Gödel's incompleteness theorems and Tarski's undefinability theorem, and mulled over their consequences for the axiomatic method in mathematics.
Truth in formalized languages
In 1933, Tarski published a very long paper in Polish, titled "Pojęcie prawdy w językach nauk dedukcyjnych",
A philosophical debate examines the extent to which Tarski's theory of truth for formalized languages can be seen as a correspondence theory of truth. The debate centers on how to read Tarski's condition of material adequacy for a true definition. That condition requires that the truth theory have the following as theorems for all sentences p of the language for which truth is being defined:
- "p" is true if and only if p.
(where p is the proposition expressed by "p")
The debate amounts to whether to read sentences of this form, such as
"Snow is white" is true if and only if snow is white
as expressing merely a deflationary theory of truth or as embodying truth as a more substantial property (see Kirkham 1992).
Logical consequence
In 1936, Tarski published Polish and German versions of a lecture, “On the Concept of Following Logically",[41] he had given the preceding year at the International Congress of Scientific Philosophy in Paris. A new English translation of this paper, Tarski (2002), highlights the many differences between the German and Polish versions of the paper and corrects a number of mistranslations in Tarski (1983).[41]
This publication set out the modern model-theoretic definition of (semantic) logical consequence, or at least the basis for it. Whether Tarski's notion was entirely the modern one turns on whether he intended to admit models with varying domains (and in particular, models with domains of different cardinalities).[citation needed] This question is a matter of some debate in the philosophical literature. John Etchemendy stimulated much of the discussion about Tarski's treatment of varying domains.[42]
Tarski ends by pointing out that his definition of logical consequence depends upon a division of terms into the logical and the extra-logical and he expresses some skepticism that any such objective division will be forthcoming. "What are Logical Notions?" can thus be viewed as continuing "On the Concept of Logical Consequence".[citation needed]
Logical notions
Tarski's "What are Logical Notions?" (Tarski 1986) is the published version of a talk that he gave originally in 1966 in London and later in 1973 in Buffalo; it was edited without his direct involvement by John Corcoran. It became the most cited paper in the journal History and Philosophy of Logic.[43]
In the talk, Tarski proposed demarcation of logical operations (which he calls "notions") from non-logical. The suggested criteria were derived from the Erlangen program of the 19th-century German mathematician Felix Klein. Mautner (in 1946), and possibly[clarification needed] an article by the Portuguese mathematician José Sebastião e Silva, anticipated Tarski in applying the Erlangen Program to logic.[citation needed]
The Erlangen program classified the various types of geometry (Euclidean geometry, affine geometry, topology, etc.) by the type of one-one transformation of space onto itself that left the objects of that geometrical theory invariant. (A one-to-one transformation is a functional map of the space onto itself so that every point of the space is associated with or mapped to one other point of the space. So, "rotate 30 degrees" and "magnify by a factor of 2" are intuitive descriptions of simple uniform one-one transformations.) Continuous transformations give rise to the objects of topology, similarity transformations to those of Euclidean geometry, and so on.[citation needed]
As the range of permissible transformations becomes broader, the range of objects one is able to distinguish as preserved by the application of the transformations becomes narrower. Similarity transformations are fairly narrow (they preserve the relative distance between points) and thus allow us to distinguish relatively many things (e.g., equilateral triangles from non-equilateral triangles). Continuous transformations (which can intuitively be thought of as transformations which allow non-uniform stretching, compression, bending, and twisting, but no ripping or glueing) allow us to distinguish a polygon from an annulus (ring with a hole in the centre), but do not allow us to distinguish two polygons from each other.[citation needed]
Tarski's proposal[
- Truth-functions: All truth-functions are admitted by the proposal. This includes, but is not limited to, all n-ary truth-functions for finite n. (It also admits of truth-functions with any infinite number of places.)
- Individuals: No individuals, provided the domain has at least two members.
- Predicates:
- the one-place total and null predicates, the former having all members of the domain in its extension and the latter having no members of the domain in its extension
- two-place total and null predicates, the former having the set of all ordered pairs of domain members as its extension and the latter with the empty set as extension
- the two-place identity predicate, with the set of all order-pairs <a,a> in its extension, where a is a member of the domain
- the two-place diversity predicate, with the set of all order pairs <a,b> where a and b are distinct members of the domain
- n-ary predicates in general: all predicates definable from the identity predicate together with disjunction and negation(up to any ordinality, finite or infinite)
- Quantifiers: Tarski explicitly discusses only monadic quantifiers and points out that all such numerical quantifiers are admitted under his proposal. These include the standard universal and existential quantifiers as well as numerical quantifiers such as "Exactly four", "Finitely many", "Uncountably many", and "Between four and 9 million", for example. While Tarski does not enter into the issue, it is also clear that polyadic quantifiers are admitted under the proposal. These are quantifiers like, given two predicates Fx and Gy, "More(x, y)", which says "More things have F than have G."
- Set-Theoretic relations: Relations such as of the domain are logical in the present sense.
- Set membership: Tarski ended his lecture with a discussion of whether the set membership relation counted as logical in his sense. (Given the reduction of (most of) mathematics to set theory, this was, in effect, the question of whether most or all of mathematics is a part of logic.) He pointed out that set membership is logical if set theory is developed along the lines of type theory, but is extralogical if set theory is set out axiomatically, as in the canonical Zermelo–Fraenkel set theory.
- Logical notions of higher order: While Tarski confined his discussion to operations of first-order logic, there is nothing about his proposal that necessarily restricts it to first-order logic. (Tarski likely restricted his attention to first-order notions as the talk was given to a non-technical audience.) So, higher-order quantifiers and predicates are admitted as well.[citation needed]
In some ways the present proposal is the obverse of that of Lindenbaum and Tarski (1936), who proved that all the logical operations of Bertrand Russell's and Whitehead's Principia Mathematica are invariant under one-to-one transformations of the domain onto itself. The present proposal is also employed in Tarski and Givant (1987).[44]
Vann McGee (1996) provides a precise account of what operations are logical in the sense of Tarski's proposal in terms of expressibility in a language that extends first-order logic by allowing arbitrarily long conjunctions and disjunctions, and quantification over arbitrarily many variables. "Arbitrarily" includes a countable infinity.[45]
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. S2CID 44369365.
- 1983 (1956). Logic, Semantics, Metamathematics: Papers from 1923 to 1938 by Alfred Tarski, Corcoran, J., ed. Hackett. 1st edition edited and translated by J. H. Woodger, Oxford Uni. Press.[46]This collection contains translations from Polish of some of Tarski's most important papers of his early career, including The Concept of Truth in Formalized Languages and On the Concept of Logical Consequence discussed above.
- 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).[47][48] 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.[49]
- 1949. Cardinal Algebras. Oxford Univ. Press.[50]
- 1953 (with Raphael Robinson). Undecidable theories. North Holland.[51]
- 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
- History of philosophy in Poland
- Cylindric algebra
- Interpretability
- Weak interpretability
- List of things named after Alfred Tarski
- Timeline of Polish science and technology
References
- ^ Alfred Tarski, "Alfred Tarski", Encyclopædia Britannica.
- ^ School of Mathematics and Statistics, University of St Andrews, "Alfred Tarski", School of Mathematics and Statistics, University of St Andrews.
- ^ "Alfred Tarski". Oxford Reference.
- ^ Gomez-Torrente, Mario (March 27, 2014). "Alfred Tarski - Philosophy - Oxford Bibliographies". Oxford University Press. Retrieved October 24, 2017.
- ^ Alfred Tarski, "Alfred Tarski", Stanford Encyclopedia of Philosophy.
- ^ Feferman A.
- ^ a b c Feferman & Feferman, p.1
- ^ Feferman & Feferman, pp.17-18
- ^ a b Feferman & Feferman, p.26
- ^ Feferman & Feferman, p.294
- ^ "Most of the Socialist Party members were also in favor of assimilation, and Tarski's political allegiance was socialist at the time. So, along with its being a practical move, becoming more Polish than Jewish was an ideological statement and was approved by many, though not all, of his colleagues. As to why Tarski, a professed atheist, converted, that just came with the territory and was part of the package: if you were going to be Polish then you had to say you were Catholic." Anita Burdman Feferman, Solomon Feferman, Alfred Tarski: Life and Logic (2004), page 39.
- MR 3307383.
- ^ McFarland, McFarland & Smith 2014, p. 319.
- ^ Feferman & Feferman (2004), pp. 239–242.
- ^ Feferman & Feferman, p. 67
- ^ Feferman & Feferman, pp. 102-103
- ^ Feferman & Feferman, Chap. 5, pp. 124-149
- ^ Robert Vaught; John Addison; Benson Mates; Julia Robinson (1985). "Alfred Tarski, Mathematics: Berkeley". University of California (System) Academic Senate. Retrieved 2008-12-26.
- ^ Obituary in Times, reproduced here
- ^ Gregory Moore, "Alfred Tarski" in Dictionary of Scientific Biography
- ^ Feferman
- ^ Chang, C.C., and Keisler, H.J., 1973. Model Theory. North-Holland, Amsterdam. American Elsevier, New York.
- ^ Alfred Tarski at the Mathematics Genealogy Project
- ^ a b Feferman & Feferman, pp. 385-386
- ^ Feferman & Feferman, pp. 177–178 and 197–201.
- ^ "Alfred Tarski (1902 - 1983)". Royal Netherlands Academy of Arts and Sciences. Retrieved 17 July 2015.
- ^ O'Connor, John J.; Robertson, Edmund F., "Alfred Tarski", MacTutor History of Mathematics Archive, University of St Andrews
- ^ Feferman & Feferman, pp. 43-52, 69-75, 109-123, 189-195, 277-287, 334-342
- ^ a b "Alfred Tarski". mathshistory.st-andrews.ac.uk. Retrieved 28 April 2023.
- arXiv:2108.05714 [math.HO].
- ^ McFarland, McFarland & Smith 2014, Section 9.2: Teaching geometry, pp. 179–184.
- ^ Adam Grabowski. "Tarski's Geometry and the Euclidean Plane in Mizar" (PDF). ceur-ws.org. Retrieved 28 April 2023.
- S2CID 18551419.
- ^ "Tarski's convention-T and inductive definition?". goodmancoaching.nl. 22 May 2022. Retrieved 28 April 2023.
- S2CID 27153078.
- ^ Restall, Greg (2002–2006). "Great Moments in Logic". Archived from the original on 6 December 2008. Retrieved 2009-01-03.
- S2CID 28783841.
- S2CID 13217777.
- ^ "Introduction To Logic And To The Methodology Of Deductive Sciences". archive.org. Retrieved 28 April 2023.
- ^ Alfred Tarski, "POJĘCIE PRAWDY W JĘZYKACH NAUK DEDUKCYJNYCH", Towarszystwo Naukowe Warszawskie, Warszawa, 1933. (Text in Polish in the Digital Library WFISUW-IFISPAN-PTF) Archived 2016-03-04 at the Wayback Machine.
- ^ S2CID 120956516.
- ISBN 978-1-57586-194-4.
- ^ "History and Philosophy of Logic".
- JSTOR 2274990. Retrieved 28 April 2023.
- JSTOR 1523019. Retrieved 28 April 2023.
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Further reading
- Biographical references
- ISBN 978-0-19-512800-0.
- OCLC 54691904.
- Frost-Arnold, Greg (2013). Carnap, Tarski, and Quine at Harvard: Conversations on Logic, Mathematics, and Science. Chicago: Open Court. ISBN 9780812698374.
- Givant Steven (1991). "A portrait of Alfred Tarski". Mathematical Intelligencer. 13 (3): 16–32. S2CID 122867668.
- Patterson, Douglas. Alfred Tarski: Philosophy of Language and Logic (Palgrave Macmillan; 2012) 262 pages; biography focused on his work from the late-1920s to the mid-1930s, with particular attention to influences from his teachers Stanislaw Lesniewski and Tadeusz Kotarbinski.
- Logic literature
- The December 1986 issue of the Journal of Symbolic Logic surveys Tarski's work on real closed fields (Lou Van Den Dries), decidable theory (Doner and Wilfrid Hodges), metamathematics (Blok and Pigozzi), truth and logical consequence (John Etchemendy), and general philosophy(Patrick Suppes).
- Blok, W. J.; Pigozzi, Don, "Alfred Tarski's Work on General Metamathematics", The Journal of Symbolic Logic, Vol. 53, No. 1 (Mar., 1988), pp. 36–50
- Chang, C.C., and Keisler, H.J., 1973. Model Theory. North-Holland, Amsterdam. American Elsevier, New York.
- Corcoran, John, and Sagüillo, José Miguel, 2011. "The Absence of Multiple Universes of Discourse in the 1936 Tarski Consequence-Definition Paper", History and Philosophy of Logic 32: 359–80. [1]
- Corcoran, John, and Weber, Leonardo, 2015. "Tarski's convention T: condition beta", South American Journal of Logic. 1, 3–32.
- ISBN 1-57586-194-1
- .
- Kluwer Academic Publishers.
- Grattan-Guinness, Ivor, 2000. The Search for Mathematical Roots 1870-1940. Princeton Uni. Press.
- Kirkham, Richard, 1992. Theories of Truth. MIT Press.
- Maddux, Roger D., 2006. Relation Algebras, vol. 150 in "Studies in Logic and the Foundations of Mathematics", Elsevier Science.
- JSTOR 2371821.
- McGee Van (1996). "Logical Operations". Journal of Philosophical Logic. 25 (6): 567–80. S2CID 32381037.
- Popper, Karl R., 1972, Rev. Ed. 1979, "Philosophical Comments on Tarski's Theory of Truth", with Addendum, Objective Knowledge, Oxford: 319–340.
- Sinaceur H (2001). "Alfred Tarski: Semantic shift, heuristic shift in metamathematics". Synthese. 126: 49–65. S2CID 28783841.
- Smith, James T., 2010. "Definitions and Nondefinability in Geometry", American Mathematical Monthly117:475–89.
- Wolenski, Jan, 1989. Logic and Philosophy in the Lvov–Warsaw School. Reidel/Kluwer.
External links
- Stanford Encyclopedia of Philosophy:
- Tarski's Truth Definitions by Wilfred Hodges.
- Alfred Tarski by Mario Gómez-Torrente.
- Algebraic Propositional Logic by Ramon Jansana. Includes a fairly detailed discussion of Tarski's work on these topics.
- Tarski's Truth Definitions by
- Tarski's Semantic Theory on the Internet Encyclopedia of Philosophy.