George Boolos - AbsoluteAstronomy.com

George Stephen Boolos was a philosopher and a mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

ian who taught at the Massachusetts Institute of Technology

Massachusetts Institute of Technology

The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...

Life

Boolos graduated from Princeton University

Princeton University

Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....

in 1961 with an A.B.

Bachelor of Arts

A Bachelor of Arts , from the Latin artium baccalaureus, is a bachelor's degree awarded for an undergraduate course or program in either the liberal arts, the sciences, or both...

in mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

. Oxford University awarded him the B.Phil in 1963. In 1966, he obtained the first Ph.D.

Ph.D.

A Ph.D. is a Doctor of Philosophy, an academic degree.Ph.D. may also refer to:* Ph.D. , a 1980s British group*Piled Higher and Deeper, a web comic strip*PhD: Phantasy Degree, a Korean comic series* PhD Docbook renderer, an XML renderer...

in philosophy ever awarded by the Massachusetts Institute of Technology

Massachusetts Institute of Technology

, under the direction of Hilary Putnam

Hilary Putnam

Hilary Whitehall Putnam is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science...

. After teaching three years at Columbia University

Columbia University

Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...

, he returned to MIT in 1969, where he spent the rest of his career until his death from cancer

Cancer

Cancer , known medically as a malignant neoplasm, is a large group of different diseases, all involving unregulated cell growth. In cancer, cells divide and grow uncontrollably, forming malignant tumors, and invade nearby parts of the body. The cancer may also spread to more distant parts of the...

.

A charismatic speaker well-known for his clarity and wit, he once delivered a lecture (1994b) giving an account of Gödel's

Kurt Gödel

Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...

second incompleteness theorem, employing only words of one syllable. At the end of his viva, Hilary Putnam

Hilary Putnam

asked him, "And tell us, Mr. Boolos, what does the analytical hierarchy

Analytical hierarchy

In mathematical logic and descriptive set theory, the analytical hierarchy is a higher type analogue of the arithmetical hierarchy. It thus continues the classification of sets by the formulas that define them.-The analytical hierarchy of formulas:...

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
The Times
The Times is a British daily national newspaper, first published in London in 1785 under the title The Daily Universal Register . The Times and its sister paper The Sunday Times are published by Times Newspapers Limited, a subsidiary since 1981 of News International...

crossword

Crossword

A crossword is a word puzzle that normally takes the form of a square or rectangular grid of white and shaded squares. The goal is to fill the white squares with letters, forming words or phrases, by solving clues which lead to the answers. In languages that are written left-to-right, the answer...

competition. His score was one of the highest ever recorded by an American. He wrote a paper on "the hardest logic puzzle ever

The Hardest Logic Puzzle Ever

The Hardest Logic Puzzle Ever is a title coined by American philosopher and logician George Boolos in an article published in The Harvard Review of Philosophy in 1996 for the following Raymond Smullyan inspired logic puzzle:Boolos provides...

"—one of many puzzles created by Raymond Smullyan

Raymond Smullyan

Raymond Merrill Smullyan is an American mathematician, concert pianist, logician, Taoist philosopher, and magician.Born in Far Rockaway, New York, his first career was stage magic. He then earned a BSc from the University of Chicago in 1955 and his Ph.D. from Princeton University in 1959...

Work

Boolos coauthored with Richard Jeffrey

Richard Jeffrey

Richard C. Jeffrey was an American philosopher, logician, and probability theorist. He was a native of Boston, Massachusetts....

the first three editions of the classic university text on mathematical logic

Mathematical logic

, Computability and Logic. The book is now in its fifth edition, the last two editions updated by John P. Burgess

John P. Burgess

John Burgess is a John N. Woodhull Professor of Philosophy at Princeton University. He received his Ph.D. from UC Berkeley's Group in Logic and Methodology of Science. His interests include logic, philosophy of mathematics and metaethics...

.

Kurt Gödel

Kurt Gödel

wrote the first paper on provability logic

Provability logic

Provability logic is a modal logic, in which the box operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory, such as Peano arithmetic....

, which applies modal logic

Modal logic

Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

—the logic of necessity and possibility—to the theory of mathematical proof

Mathematical proof

In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...

, 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

Solomon Feferman

Solomon Feferman is an American philosopher and mathematician with major works in mathematical logic.He was born in New York City, New York, and received his Ph.D. in 1957 from the University of California, Berkeley under Alfred Tarski...

.

Boolos was an authority on the 19th-century German mathematician and philosopher Gottlob Frege

Gottlob Frege

Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...

. Boolos proved a conjecture due to Crispin Wright

Crispin Wright

Crispin Wright is a British philosopher, who has written on neo-Fregean philosophy of mathematics, Wittgenstein's later philosophy, and on issues related to truth, realism, cognitivism, skepticism, knowledge, and objectivity....

(and also proved, independently, by others), that the system of Frege's Grundgesetze, long thought vitiated by Russell's paradox

Russell's paradox

In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...

, could be freed of inconsistency by replacing one of its axioms, the notorious Basic Law V

Gottlob Frege

with Hume's Principle

Hume's principle

Hume's Principle or HP—the terms were coined by George Boolos—says that the number of Fs is equal to the number of Gs if and only if there is a one-to-one correspondence between the Fs and the Gs. HP can be stated formally in systems of second-order logic...

. 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

Second-order logic

In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory....

plurally

Plural quantification

In mathematics and logic, plural quantification is the theory that an individual variable x may take on plural, as well as singular values. As well as substituting individual objects such as Alice, the number 1, the tallest building in London etc...

, then second-order logic can be interpreted as having no ontological commitment

Ontology

Ontology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...

to entities other than those over which the first-order variables

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

range. The result is plural quantification

Plural quantification

. David Lewis

David Kellogg Lewis

David Kellogg Lewis was an American philosopher. Lewis taught briefly at UCLA and then at Princeton from 1970 until his death. He is also closely associated with Australia, whose philosophical community he visited almost annually for more than thirty years...

employed plural quantification in his Parts of Classes to derive a system in which Zermelo-Fraenkel set theory and the Peano axioms

Peano axioms

In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano...

were all theorems. While Boolos is usually credited with plural quantification

Plural quantification

, Peter Simons

Peter Simons

Peter Simons, FBA, is a professor of philosophy at Trinity College Dublin.He studied at the University of Manchester, and has held teaching posts at the University of Bolton, the University of Salzburg, where he is Honorary Professor of Philosophy, and the University of Leeds...

(1982) has argued that the essential idea can be found in the work of Stanislaw Lesniewski

Stanislaw Lesniewski

Stanisław Leśniewski was a Polish mathematician, philosopher and logician.Leśniewski went to a high school in Irkutsk...

.

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

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

, second-order logic

Second-order logic

and nonfirstorderizability

Nonfirstorderizability

In formal logic, nonfirstorderizability is the inability of an expression to be adequately captured in particular theories in first-order logic. Nonfirstorderizable sentences are sometimes presented as evidence that first-order logic is not adequate to capture the nuances of meaning in natural...

, plural quantification

Plural quantification

, proof theory

Proof theory

Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...

, and three short insightful papers on Gödel's Incompleteness Theorem. There are also papers on Dedekind

Richard Dedekind

Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...

, Cantor

Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...

, and Russell

Bertrand Russell

Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...

Books

1979. The Unprovability of Consistency: An Essay in Modal Logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

. Cambridge University Press.
1990 (editor). Meaning and Method: Essays in Honor of Hilary Putnam
Hilary Putnam
Hilary Whitehall Putnam is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science...

. Cambridge University Press.
1993. The Logic of Provability. Cambridge University Press.
1998 (Richard Jeffrey
Richard Jeffrey
Richard C. Jeffrey was an American philosopher, logician, and probability theorist. He was a native of Boston, Massachusetts....

and John P. Burgess
John P. Burgess
John Burgess is a John N. Woodhull Professor of Philosophy at Princeton University. He received his Ph.D. from UC Berkeley's Group in Logic and Methodology of Science. His interests include logic, philosophy of mathematics and metaethics...

, eds.). Logic, Logic, and Logic. Harvard University Press.
2007 (1974) (with Richard Jeffrey
Richard Jeffrey
Richard C. Jeffrey was an American philosopher, logician, and probability theorist. He was a native of Boston, Massachusetts....

and John Burgess
John Burgess
For other persons named John Burgess, see John Burgess .John William Burgess was a pioneering American political scientist...

). Computability and Logic, 4th ed. Cambridge University Press.

Articles

LLL = reprinted in Logic, Logic, and Logic.

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

1968 (with Hilary Putnam

Hilary Putnam

), "Degrees of unsolvability of constructible sets of integers," Journal of Symbolic Logic 33: 497-513.

1969, "Effectiveness and natural languages" in Sidney Hook

Sidney Hook

Sidney Hook was an American pragmatic philosopher known for his contributions to public debates.A student of John Dewey, Hook continued to examine the philosophy of history, of education, politics, and of ethics. After embracing Marxism in his youth, Hook was known for his criticisms of...

, ed., Language and Philosophy. New York University Press.

1970, "On the semantics of the constructible levels," ' 16: 139-148.

1970a, "A proof of the Löwenheim-Skolem theorem," Notre Dame Journal of Formal Logic 11: 76-78.

1971, "The iterative conception of set," Journal of Philosophy 68: 215-231. Reprinted in Paul Benacerraf
Paul Benacerraf
Paul Joseph Salomon Benacerraf is an American philosopher working in the field of the philosophy of mathematics who has been teaching at Princeton University since he joined the faculty in 1960. He was appointed Stuart Professor of Philosophy in 1974, and recently retired as the James S....

and Hilary Putnam
Hilary Putnam
Hilary Whitehall Putnam is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science...

, eds.,1984. Philosophy of Mathematics: Selected Readings, 2nd ed. Cambridge Univ. Press: 486-502. LLL

1973, "A note on Evert Willem Beth
Evert Willem Beth
Evert Willem Beth was a Dutch philosopher and logician, whose work principally concerned the foundations of mathematics.- Biography :...

's theorem," Bulletin de l'Academie Polonaise des Sciences 2: 1-2.

1974, "Arithmetical functions and minimization," Zeitschrift für mathematische Logik und Grundlagen der Mathematik 20: 353-354.

1974a, "Reply to Charles Parsons
Charles Parsons (philosopher)
Charles Dacre Parsons is a distinguished figure in the philosophy of mathematics.He is a son of social scientist Talcott Parsons. A specialist in the philosophy of mathematics and logic, Parsons earned his Ph.D. at Harvard University in 1961, under the direction of Burton Dreben and Willard Van...

' 'Sets and classes'." First published in LLL.

1975, "Friedman's
Harvey Friedman
Harvey Friedman is a mathematical logician at Ohio State University in Columbus, Ohio. He is noted especially for his work on reverse mathematics, a project intended to derive the axioms of mathematics from the theorems considered to be necessary...

35th problem has an affirmative solution," Notices of the American Mathematical Society 22: A-646.

1975a, "On Kalmar's consistency proof and a generalization of the notion of omega-consistency," Archiv für Mathematische Logik und Grundlagenforschung 17: 3-7.

1975a, "On second-order logic
Second-order logic
In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory....

," Journal of Philosophy 72: 509-527. LLL.

1976, "On deciding the truth of certain statements involving the notion of consistency," Journal of Symbolic Logic 41: 779-781.

1977, "On deciding the provability of certain fixed point statements," Journal of Symbolic Logic 42: 191-193.

1979, "Reflection principles and iterated consistency assertions," Journal of Symbolic Logic 44: 33-35.

1980, "Omega-consistency and the diamond," Studia Logica 39: 237-243.

1980a, "On systems of modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

with provability interpretations," Theoria 46: 7-18.

1980b, "Provability in arithmetic and a schema of Grzegorczyk," Fundamenta Mathematicae 106: 41-45.

1980c, "Provability, truth, and modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

," Journal of Philosophical Logic 9: 1-7.

1980d, Review of Raymond M. Smullyan, What is the Name of This Book? The Philosophical Review 89: 467-470.

1981, "For every A there is a B," Linguistic Inquiry 12: 465-466.

1981a, Review of Robert M. Solovay
Robert M. Solovay
Robert Martin Solovay is an American mathematician specializing in set theory.Solovay earned his Ph.D. from the University of Chicago in 1964 under the direction of Saunders Mac Lane, with a dissertation on A Functorial Form of the Differentiable Riemann–Roch theorem...

, Provability Interpretations of Modal Logic," Journal of Symbolic Logic 46: 661-662.

1982, "Extremely undecidable sentences," Journal of Symbolic Logic 47: 191-196.

1982a, "On the nonexistence of certain normal forms in the logic of provability," Journal of Symbolic Logic 47: 638-640.

1984, "Don't eliminate cut," Journal of Philosophical Logic 13: 373-378. LLL.

1984a, "The logic of provability," American Mathematical Monthly 91: 470-480.

1984b, "Nonfirstorderizability again," Linguistic Inquiry 15: 343.

1984c, "On 'Syllogistic inference'," Cognition 17: 181-182.

1984d, "To be is to be the value of a variable (or some values of some variables)," Journal of Philosophy 81: 430-450. LLL.

1984e, "Trees and finite satisfiability: Proof of a conjecture of John Burgess
John P. Burgess
John Burgess is a John N. Woodhull Professor of Philosophy at Princeton University. He received his Ph.D. from UC Berkeley's Group in Logic and Methodology of Science. His interests include logic, philosophy of mathematics and metaethics...

," Notre Dame Journal of Formal Logic 25: 193-197.

1984f, "The justification of mathematical induction
Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...

," PSA 2: 469-475. LLL.

1985, "1-consistency and the diamond," Notre Dame Journal of Formal Logic 26: 341-347.

1985a, "Nominalist Platonism," The Philosophical Review 94: 327-344. LLL.

1985b, "Reading the Begriffsschrift
Begriffsschrift
Begriffsschrift is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book...

," Mind 94: 331-344. LLL; FPM: 163-81.

1985c (with Giovanni Sambin), "An incomplete system of modal logic," Journal of Philosophical Logic 14: 351-358.

1986, Review of Yuri Manin, A Course in Mathematical Logic, Journal of Symbolic Logic 51: 829-830.

1986-87, "Saving Frege from contradiction," Proceedings of the Aristotelian Society 87: 137-151. LLL; FPM 438-52.

1987, "The consistency of Frege's Foundations of Arithmetic" in J. J. Thomson, ed., 1987. On Being and Saying: Essays for Richard Cartwright. MIT Press: 3-20. LLL; FPM: 211-233.

1987a, "A curious inference," Journal of Philosophical Logic 16: 1-12. LLL.

1987b, "On notions of provability in provability logic," Abstracts of the 8th International Congress of Logic, Methodology and Philosophy of Science 5: 236-238.

1987c (with Vann McGee), "The degree of the set of sentences of predicate provability logic that are true under every interpretation," Journal of Symbolic Logic 52: 165-171.

1988, "Alphabetical order," Notre Dame Journal of Formal Logic 29: 214-215.

1988a, Review of Craig Smorynski, Self-Reference and Modal Logic, Journal of Symbolic Logic 53: 306-309.

1989, "Iteration again," Philosophical Topics 17: 5-21. LLL.

1989a, "A new proof of the Gödel incompleteness theorem," Notices of the American Mathematical Society 36: 388-390. LLL. An afterword appeared under the title "A letter from George Boolos," ibid., p. 676. LLL.

1990, "On 'seeing' the truth of the Gödel sentence," Behavioral and Brain Sciences 13: 655-656. LLL.

1990a, Review of Jon Barwise
Jon Barwise
Kenneth Jon Barwise was an American mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used....

and John Etchemendy
John Etchemendy
John W. Etchemendy and of Basque descent is Stanford University's twelfth and current Provost. He succeeded John L. Hennessy to the post on September 1, 2000....

, Turing's World and Tarski's World, Journal of Symbolic Logic 55: 370-371.

1990b, Review of V. A. Uspensky, Gödel's Incompleteness Theorem, Journal of Symbolic Logic 55: 889-891.

1990c, "The standard of equality of numbers" in Boolos, G., ed., Meaning and Method: Essays in Honor of Hilary Putnam

Hilary Putnam

. Cambridge Univ. Press: 261-278. LLL; FPM: 234-254.

1991, "Zooming down the slippery slope," Nous 25: 695-706. LLL.

1991a (with Giovanni Sambin), "Provability: The emergence of a mathematical modality," Studia Logica 50: 1-23.

1993, "The analytical completeness of Dzhaparidze's polymodal logics," Annals of Pure and Applied Logic 61: 95-111.

1993a, "Whence the contradiction?" Aristotelian Society Supplementary Volume 67: 213-233. LLL.

1994, "1879?" in P. Clark and B. Hale, eds. Reading Putnam. Oxford: Blackwell: 31-48. LLL.

1994a, "The advantages of honest toil over theft," in A. George, ed., Mathematics and Mind. Oxford University Press: 27-44. LLL.

1994b, "Gödel's second incompleteness theorem explained in words of one syllable," Mind 103: 1-3. LLL.

1995, "Frege's theorem and the Peano postulates," Bulletin of Symbolic Logic 1: 317-326. LLL.

1995a, "Introductory note to *1951" in Solomon Feferman

Solomon Feferman

et al., eds., Kurt Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...

, Collected Works, vol. 3. Oxford University Press: 290-304. LLL. *1951 is Gödel’s 1951 Gibbs lecture, "Some basic theorems on the foundations of mathematics and their implications."

1995b, "Quotational ambiguity" in Leonardi, P., and Santambrogio, M., eds. On Quine. Cambridge University Press: 283-296. LLL

1996, "The hardest logical puzzle ever," Harvard Review of Philosophy 6: 62-65. LLL. Italian translation by Massimo Piattelli-Palmarini, "L'indovinello piu difficile del mondo," La Repubblica (16 April 1992): 36-37.

1996a, "On the proof of Frege's theorem" in A. Morton and S. P. Stich, eds., Paul Benacerraf
Paul Benacerraf
Paul Joseph Salomon Benacerraf is an American philosopher working in the field of the philosophy of mathematics who has been teaching at Princeton University since he joined the faculty in 1960. He was appointed Stuart Professor of Philosophy in 1974, and recently retired as the James S....

and his Critics. Cambridge MA: Blackwell. LLL.

1997, "Constructing Cantorian counterexamples," Journal of Philosophical Logic 26: 237-239. LLL.

1997a, "Is Hume

David Hume

David Hume was a Scottish philosopher, historian, economist, and essayist, known especially for his philosophical empiricism and skepticism. He was one of the most important figures in the history of Western philosophy and the Scottish Enlightenment...

's principle analytic?" In Richard G. Heck, Jr., ed., Language, Thought, and Logic: Essays in Honour of Michael Dummett
Michael Dummett
Sir Michael Anthony Eardley Dummett FBA D.Litt is a British philosopher. He was, until 1992, Wykeham Professor of Logic at the University of Oxford...

. Oxford Univ. Press: 245-61. LLL.

1997b (with Richard Heck), "Die Grundlagen der Arithmetik, §§82-83" in Matthias Schirn, ed., Philosophy of Mathematics Today. Oxford Univ. Press. LLL.

1998, "Gottlob Frege

Gottlob Frege

and the Foundations of Arithmetic." First published in LLL. French translation in Mathieu Marion and Alain Voizard eds., 1998. Frege. Logique et philosophie. Montréal and Paris: L'Harmattan: 17-32.

2000, "Must we believe in set theory

Set theory

?" in Gila Sher and Richard Tieszen, eds., Between Logic and Intuition: Essays in Honour of Charles Parsons
Charles Parsons (philosopher)
Charles Dacre Parsons is a distinguished figure in the philosophy of mathematics.He is a son of social scientist Talcott Parsons. A specialist in the philosophy of mathematics and logic, Parsons earned his Ph.D. at Harvard University in 1961, under the direction of Burton Dreben and Willard Van...

. Cambridge University Press. LLL.

External links

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Life

Work

Books

Articles

See also

External links