Paul Joseph Salomon Benacerraf (born 1931) 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. McDonnell Distinguished University Professor of Philosophy.
Benacerraf is perhaps best known for his two papers What Numbers Could Not Be (1965) and Mathematical Truth (1973), and for his anthology on the philosophy of mathematics, co-edited with Hilary Putnam. He was elected a Fellow of the American Academy of Arts and Sciences in 1998.
In What Numbers Could Not Be, he argues against a Platonist view of mathematics, and for structuralism, on the ground that what is important about numbers is the abstract structures they represent rather than the objects that number words ostensibly refer to. In particular, this argument is based on the point that Zermelo and von Neumann give distinct, and completely adequate, identifications of natural numbers with sets.
In Mathematical Truth, he argues that no interpretation of mathematics offers a satisfactory package of epistemology and semantics; it is possible to explain mathematical truth in a way that is consistent with our syntactico-semantical treatment of truth in non-mathematical language, and it is possible to explain our knowledge of mathematics in terms consistent with a causal account of epistemology, but it is in general not possible to accomplish both of these objectives simultaneously. He argues for this on the grounds that an adequate account of truth in mathematics implies the existence of abstract mathematical objects, but that such objects are epistemologically inaccessible because they are causally inert and beyond the reach of sense perception. On the other hand, an adequate epistemology of mathematics, say one that ties truth-conditions to proof in some way, precludes understanding how and why the truth-conditions have any bearing on truth.
- Benacerraf, Paul (1960) Logicism, Some Considerations, Princeton, Ph.D. Dissertation, University Microfilms.
- Benacerraf, Paul (1965) What Numbers Could Not Be, The Philosophical Review, 74:47-73.
- Benacerraf, Paul (1967) God, the Devil, and Gödel, The Monist, 5l: 9-33.
- Benacerraf, Paul (1973) Mathematical Truth, The Journal of Philosophy, 70: 661-679.
- Benacerraf, Paul (1981) Frege: The Last Logicist, The Foundations of Analytic Philosophy, Midwest Studies in Philosophy, 6: l7-35.
- Benacerraf, Paul (1985) Skolem and the Skeptic, Proceedings of the Aristotelian Society, Supplementary Volume 56: 85-ll5.
- Benacerraf, Paul and Putnam, Hilary (eds.) (1983) Philosophy of Mathematics : Selected Readings 2nd edition, Cambridge University Press: New York.
- Benacerraf, Paul (1996) Recantation or Any old ω-sequence would do after all, Philosophia Mathematica, 4: 184-189.
- Benacerraf, Paul (1996) What Mathematical Truth Could Not Be - I, in Benacerraf and His Critics, A. Morton and S. P. Stich, eds., Blackwell's, Oxford and Cambridge, pp 9–59.
- Benacerraf, Paul (1999) What Mathematical Truth Could Not Be - II, in Sets and Proofs, S. B. Cooper and J. K. Truss, eds., Cambridge University Press, pp. 27–51.
Books about Benacerraf
- Zimmermann, Manfred (1995) Wahrheit und Wissen in der Mathematik. Das Benacerrafsche Dilemma, 1. Auflage, Transparent Verlag, Berlin.
- Gupta, Anoop K. (2002) Benacerraf's Dilemma and Natural Realism for Mathematics. Ph.D. Dissertation, Ottawa University.
Papers about Benacerraf
- Hilton, P. What 'What Numbers Could Not Be', by Paul Benacerraf', is.
- Ebert, Philip A. (2007) What Mathematical Knowledge Could Not Be, in St Andrews Undergraduate Philosophy Society Journal, 1 (1) pp. 46–70.
- Hale, Bob and Wright, Crispin (2002) Benacerraf's Dilemma Revisited European Journal of Philosophy, Issue 10:1.
- Lucas, J. R. (1968) Satan stultified: a rejoinder to Paul Benacerraf, The Monist, vol.52, No.1, pp. 145–158.
Writings on Benacerraf
- Benacerraf Interview by The Dualist and the Stanford Philosophy Department
- "Whatever I am now, it happened here" by Caroline Moseley
- Paul Benacerraf's homepage at Princeton
- Paul Benacerraf at the Mathematics Genealogy Project
- Paul Benacerraf's lecture series at UNMSM (Lima, Peru)[dead link]