George Boolos Explained

George Boolos should not be confused with George Boole.

Region:Western philosophy
Era:20th-century philosophy
George Boolos
Birth Name:George Stephen Boolos
Birth Date:4 September 1940
Birth Place:New York City, U.S.
Death Place:Cambridge, Massachusetts, U.S.
Education:Princeton University (AB)
Oxford University
Massachusetts Institute of Technology (PhD)
School Tradition:Analytic philosophy
Main Interests:Philosophy of mathematics, mathematical logic
Notable Ideas:Hume's principle
Nonfirstorderizability
The Hardest Logic Puzzle Ever
Doctoral Advisor:Hilary Putnam
Thesis Title:The Hierarchy of Constructible Sets of Integers
Thesis Year:1966

George Stephen Boolos (;[1] September 4, 1940 – May 27, 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.[2]

Life

Boolos was of Greek-Jewish descent.[3] He graduated with an A.B. in mathematics from Princeton University after completing a senior thesis, titled "A simple proof of Gödel's first incompleteness theorem", under the supervision of Raymond Smullyan.[4] Oxford University awarded him the B.Phil. in 1963. In 1966, he obtained the first PhD in philosophy ever awarded by the Massachusetts Institute of Technology, under the direction of Hilary Putnam. After teaching three years at Columbia University, he returned to MIT in 1969, where he spent the rest of his career.

A charismatic speaker well known for his clarity and wit, he once delivered a lecture (1994b) giving an account of Gödel's second incompleteness theorem, employing only words of one syllable. At the end of his viva, Hilary Putnam asked him, "And tell us, Mr. Boolos, what does the analytical hierarchy have to do with the real world?" Without hesitating Boolos replied, "It's part of it". An expert on puzzles of all kinds, in 1993 Boolos reached the London Regional Final of The Times crossword competition. His score was one of the highest ever recorded by an American. He wrote a paper on "The Hardest Logic Puzzle Ever"—one of many puzzles created by Raymond Smullyan.

Boolos died of pancreatic cancer on 27 May 1996.[5]

Work

Boolos coauthored with Richard Jeffrey the first three editions of the classic university text on mathematical logic, Computability and Logic. The book is now in its fifth edition, the last two editions updated by John P. Burgess.

Kurt Gödel wrote the first paper on provability logic, which applies modal logic—the logic of necessity and possibility—to the theory of mathematical proof, but Gödel never developed the subject to any significant extent. Boolos was one of its earliest proponents and pioneers, and he produced the first book-length treatment of it, The Unprovability of Consistency, published in 1979. The solution of a major unsolved problem some years later led to a new treatment, The Logic of Provability, published in 1993. The modal-logical treatment of provability helped demonstrate the "intensionality" of Gödel's Second Incompleteness Theorem, meaning that the theorem's correctness depends on the precise formulation of the provability predicate. These conditions were first identified by David Hilbert and Paul Bernays in their Grundlagen der Arithmetik. The unclear status of the Second Theorem was noted for several decades by logicians such as Georg Kreisel and Leon Henkin, who asked whether the formal sentence expressing "This sentence is provable" (as opposed to the Gödel sentence, "This sentence is not provable") was provable and hence true. Martin Löb showed Henkin's conjecture to be true, as well as identifying an important "reflection" principle also neatly codified using the modal logical approach. Some of the key provability results involving the representation of provability predicates had been obtained earlier using very different methods by Solomon Feferman.

Boolos was an authority on the 19th-century German mathematician and philosopher Gottlob Frege. Boolos proved a conjecture due to Crispin Wright (and also proved, independently, by others), that the system of Frege's Grundgesetze, long thought vitiated by Russell's paradox, could be freed of inconsistency by replacing one of its axioms, the notorious Basic Law V with Hume's Principle. The resulting system has since been the subject of intense work.

Boolos argued that if one reads the second-order variables in monadic second-order logic plurally, then second-order logic can be interpreted as having no ontological commitment to entities other than those over which the first-order variables range. The result is plural quantification. David Lewis employed plural quantification in his Parts of Classes to derive a system in which Zermelo–Fraenkel set theory and the Peano axioms were all theorems. While Boolos is usually credited with plural quantification, Peter Simons (1982) has argued that the essential idea can be found in the work of Stanislaw Leśniewski.

Shortly before his death, Boolos chose 30 of his papers to be published in a book. The result is perhaps his most highly regarded work, his posthumous Logic, Logic, and Logic. This book reprints much of Boolos's work on the rehabilitation of Frege, as well as a number of his papers on set theory, second-order logic and nonfirstorderizability, plural quantification, proof theory, and three short insightful papers on Gödel's Incompleteness Theorem. There are also papers on Dedekind, Cantor, and Russell.

Publications

Books

Articles

LLL = reprinted in Logic, Logic, and Logic.

FPM = reprinted in Demopoulos, W., ed., 1995. Frege's Philosophy of Mathematics. Harvard Univ. Press.

See also

References

External links

Notes and References

  1. https://www.youtube.com/watch?v=LKvjIsyYng8 "Can you solve the three gods riddle? – Alex Gendler"
  2. News: Van Gelder, Lawrence. Lawrence Van Gelder. George Boolos, 55, Philosopher. 30 May 1996. NY Times.
  3. Modern Logic . 6 . 3 . 304-310 . Project Euclid . Irving H. Anellis . GEORGE S. BOOLOS . July 1996.
  4. Book: Boolos, George Stephen. A simple proof of Gödel's first incompleteness theorem. 1961. Princeton, NJ: Department of Mathematics. English.
  5. Web site: Professor George Boolos Dead at 55 . MIT News . en . May 29, 1996.

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