Alfred Tarski | |
|---|---|
 Tarski in 1968 | |
| Born | Alfred Teitelbaum (1901-01-14)January 14, 1901 |
| Died | October 26, 1983(1983-10-26) (aged 82) |
| 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 (/ˈtɑːrski/;Polish:[ˈtarskʲi]; bornAlfred Teitelbaum;[1][2][3] January 14, 1901 – October 26, 1983) was a Polish-American[4]logician andmathematician.[5] A prolific author best known for his work onmodel theory,metamathematics, andalgebraic logic, he also contributed toabstract algebra,topology,geometry,measure theory,mathematical logic,set theory,type theory, andanalytic philosophy.
Educated in Poland at theUniversity of Warsaw, and a member of theLwów–Warsaw school of logic and theWarsaw school of mathematics, he immigrated to the United States in 1939 where he became a naturalized citizen in 1945. Tarski taught and carried out research in mathematics at theUniversity of California, Berkeley, from 1942 until his death in 1983.[6]
His biographersAnita Burdman Feferman andSolomon 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 oftruth and the theory of models."[7]
Alfred Tarski was born Alfred Teitelbaum (Polish spelling: "Tajtelbaum"), to parents who werePolish Jews in comfortable circumstances. He first manifested his mathematical abilities while in secondary school, at Warsaw'sSzkoła Mazowiecka.[8] Nevertheless, he entered theUniversity of Warsaw in 1918 intending to studybiology.[9]
After Poland regained independence in 1918, Warsaw University came under the leadership ofJan Łukasiewicz,Stanisław Leśniewski andWacław 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.[9] Henceforth Tarski attended courses taught by Łukasiewicz, Sierpiński,Stefan Mazurkiewicz andTadeusz Kotarbiński, and in 1924 became the only person ever to complete a doctorate under Leśniewski's supervision. His thesis was entitledO wyrazie pierwotnym logistyki (On the Primitive Term of Logistic; published 1923). Tarski and Leśniewski soon grew cool to each other, mainly due to the latter's increasing anti-semitism.[7] However, in later life, Tarski reserved his warmest praise for Kotarbiński, which was reciprocated.
In 1923, Alfred Teitelbaum and his brother Wacław changed their surname to "Tarski". The Tarski brothers also converted toRoman Catholicism, Poland's dominant religion. Alfred did so even though he was an avowedatheist.[10][11]
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 thePolish–Soviet War. They had two children; a sonJan Tarski, who became a physicist, and a daughter Ina, who married the mathematicianAndrzej Ehrenfeucht.[14]
Tarski applied for a chair of philosophy atLwów University, but onBertrand Russell's recommendation it was awarded toLeon Chwistek.[15] In 1930, Tarski visited theUniversity of Vienna, lectured toKarl Menger's colloquium, and metKurt 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 theUnity of Science movement, an outgrowth of theVienna Circle. Tarski's academic career in Poland was strongly and repeatedly impacted by his heritage. For example, in 1937, Tarski applied for a chair atPoznań University but the chair was abolished to avoid assigning it to Tarski (who was undisputedly the strongest applicant) because he was a Jew.[16] 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 atHarvard University. Thus he left Poland in August 1939, on the last ship to sail from Poland for the United States before the German and Sovietinvasion of Poland and the outbreak ofWorld War II. Tarski left reluctantly, because Leśniewski had died a few months before, creating a vacancy which Tarski hoped to fill. Oblivious to theNazi threat, he left his wife and children in Warsaw. He did not see them again until 1946. During the war, nearly all his Jewish extended family were murdered at the hands of the German occupying authorities.
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 aGuggenheim Fellowship, theInstitute for Advanced Study inPrinceton (1942), where he again met Gödel. In 1942, Tarski joined the Mathematics Department at theUniversity 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 ofAndrzej Mostowski,Bjarni Jónsson,Julia Robinson,Robert Vaught,Solomon Feferman,Richard Montague,James Donald Monk,Haim Gaifman,Donald Pigozzi, andRoger Maddux, as well asChen Chung Chang andJerome Keisler, authors ofModel Theory (1973),[22] a classic text in the field.[23][24] He also strongly influenced the dissertations ofAdolf Lindenbaum,Dana Scott, andSteven Givant. Five of Tarski's students were women, a remarkable fact given that men represented an overwhelming majority of graduate students at the time.[24] However, he had extra-marital affairs with at least two of these students. After he showed another of his female student's[who?] work to a male colleague[who?], the colleague published it himself, leading her to leave the graduate study and later move to a different university and a different advisor.[25]
Tarski lectured atUniversity College, London (1950, 1966), theInstitut Henri Poincaré in Paris (1955), theMiller Institute for Basic Research in Science in Berkeley (1958–60), theUniversity of California at Los Angeles (1967), and thePontifical Catholic University of Chile (1974–75). Among many distinctions garnered over the course of his career, Tarski was elected to theUnited States National Academy of Sciences, theBritish Academy and theRoyal Netherlands Academy of Arts and Sciences in 1958,[26] receivedhonorary degrees from the Pontifical Catholic University of Chile in 1975, fromMarseille'sPaul Cézanne University in 1977 and from theUniversity of Calgary, as well as the Berkeley Citation in 1981. Tarski presided over theAssociation for Symbolic Logic, 1944–46, and the International Union for the History and Philosophy of Science, 1956–57. He was also an honorary editor ofAlgebra Universalis.[27]
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 onset theory, a subject to which he returned throughout his life.[29] In 1924, he andStefan Banach proved that, if one accepts theAxiom of Choice, aball 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 theBanach–Tarski paradox.[30]
InA decision method for elementary algebra and geometry, Tarski showed, by the method ofquantifier elimination, that thefirst-order theory of thereal numbers under addition and multiplication isdecidable. (While this result appeared only in 1948, it dates back to 1930 and was mentioned in Tarski (1931).) This is a very curious result, becauseAlonzo Church proved in 1936 thatPeano arithmetic (the theory ofnatural numbers) isnot decidable. Peano arithmetic is also incomplete byGödel's incompleteness theorem. In his 1953Undecidable theories, Tarski et al. showed that many mathematical systems, includinglattice theory, abstractprojective geometry, andclosure algebras, are all undecidable. The theory ofAbelian 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 taughtgeometry.[31] Using some ideas ofMario Pieri, in 1926 Tarski devised an originalaxiomatization for planeEuclidean geometry, one considerably more concise thanHilbert's.[32]Tarski's axioms form a first-order theory devoid of set theory, whose individuals arepoints, and having only two primitiverelations. 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 Euclideansolid geometry could be recast as a second-order theory whose individuals arespheres (aprimitive notion), a single primitive binary relation "is contained in", and two axioms that, among other things, imply that containmentpartially orders the spheres. Relaxing the requirement that all individuals be spheres yields a formalization ofmereology far easier to exposit thanLesniewski's variant. Near the end of his life, Tarski wrote a very long letter, published as Tarski and Givant (1999), summarizing his work on geometry.[33]
Cardinal Algebras studied algebras whose models include the arithmetic ofcardinal numbers.Ordinal Algebras sets out an algebra for the additive theory oforder types. Cardinal, but not ordinal, addition commutes.
In 1941, Tarski published an important paper onbinary relations, which began the work onrelation algebra and itsmetamathematics that occupied Tarski and his students for much of the balance of his life. While that exploration (and the closely related work ofRoger Lyndon) uncovered some important limitations of relation algebra, Tarski also showed (Tarski and Givant 1987) that relation algebra can express mostaxiomatic set theory andPeano arithmetic. For an introduction torelation algebra, see Maddux (2006). In the late 1940s, Tarski and his students devisedcylindric algebras, which are tofirst-order logic what thetwo-element Boolean algebra is to classicalsentential logic. This work culminated in the two monographs by Tarski, Henkin, and Monk (1971, 1985).[34]
Tarski's student,Robert Lawson Vaught, has ranked Tarski as one of the four greatest logicians of all time — along withAristotle,Gottlob Frege, andKurt Gödel.[7][35][36] However, Tarski often expressed great admiration forCharles Sanders Peirce, particularly for his pioneering work in thelogic of relations.
Tarski produced axioms forlogical consequence and worked ondeductive systems, the algebra of logic, and the theory of definability. His semantic methods, which culminated in the model theory he and a number of his Berkeley students developed in the 1950s and 60s, radically transformed Hilbert's proof-theoretic metamathematics.Around 1930, Tarski developed an abstract theory of logical deductions that models some properties of logical calculi. Mathematically, what he described is just a finitary closure operator on a set (the set ofsentences). Inabstract algebraic logic, finitary closure operators are still studied under the nameconsequence operator, which was coined by Tarski. The setS represents a set of sentences, a subsetT ofS a theory, and cl(T) is the set of all sentences that follow from the theory. This abstract approach was applied to fuzzy logic (see Gerla 2000).
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 asIntroduction to Logic and to the Methodology of Deductive Sciences.[39]
Tarski's 1969 "Truth and proof" considered bothGödel's incompleteness theorems andTarski's undefinability theorem, and mulled over their consequences for the axiomatic method in mathematics.
In 1933, Tarski published a very long paper in Polish, titled "Pojęcie prawdy w językach nauk dedukcyjnych",[40] "Setting out a mathematical definition of truth for formal languages." The 1935 German translation was titled "Der Wahrheitsbegriff in den formalisierten Sprachen", "The concept of truth in formalized languages", sometimes shortened to "Wahrheitsbegriff". An English translation appeared in the 1956 first edition of the volumeLogic, Semantics, Metamathematics. This collection of papers from 1923 to 1938 is an event in 20th-centuryanalytic philosophy, a contribution tosymbolic logic,semantics, and thephilosophy of language. For a brief discussion of its content, seeConvention T (and alsoT-schema).
A philosophical debate examines the extent to which Tarski's theory of truth for formalized languages can be seen as acorrespondence 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:
(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 adeflationary theory of truth or as embodyingtruth as a more substantial property (see Kirkham 1992).
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 modernmodel-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 differentcardinalities).[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]
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 inBuffalo; it was edited without his direct involvement byJohn Corcoran. It became the most cited paper in the journalHistory 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 theErlangen program of the 19th-century German mathematicianFelix Klein. Mautner (in 1946), and possibly[clarification needed] an article by the Portuguese mathematicianJosé 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 apolygon from anannulus (ring with a hole in the centre), but do not allow us to distinguish two polygons from each other.[citation needed]
Tarski's proposal[which?] was to demarcate the logical notions by considering all possible one-to-one transformations (automorphisms) of a domain onto itself. By domain is meant theuniverse of discourse of a model for the semantic theory of logic. If one identifies thetruth value True with the domain set and the truth-value False with the empty set, then the following operations are counted as logical under the proposal:
In some ways the present proposal is the obverse of that of Lindenbaum and Tarski (1936), who proved that all the logical operations ofBertrand Russell's andWhitehead'sPrincipia Mathematica are invariant under one-to-one transformations of the domain onto itself. The present proposal is also employed in Tarski andGivant (1987).[44]
Solomon Feferman andVann McGee further discussed Tarski's proposal[which?] in work published after his death. Feferman (1999) raises problems for the proposal and suggests a cure: replacing Tarski's preservation by automorphisms with preservation by arbitraryhomomorphisms. In essence, this suggestion circumvents the difficulty Tarski's proposal has in dealing with a sameness of logical operation across distinct domains of a given cardinality and across domains of distinct cardinalities. Feferman's proposal results in a radical restriction of logical terms as compared to Tarski's original proposal. In particular, it ends up counting as logical only those operators of standard first-order logic without identity.[citation needed]
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]
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