Thinker · TH-0318
Paul Benacerraf
Paul Benacerraf was a philosopher of mathematics, born in Paris and raised partly in Venezuela, who spent his career at Princeton University. Two short…
1931 – 2025 · Paris, Caracas, Princeton (New Jersey)
Traditions: Analytic
Paul Benacerraf was a philosopher of mathematics, born in Paris and raised partly in Venezuela, who spent his career at Princeton University. Two short papers made him central to the field: 'What Numbers Could Not Be' argued that numbers cannot be identified with any particular sets, and 'Mathematical Truth' posed the dilemma that a good semantics for mathematics seems to require abstract objects that a good epistemology cannot explain our knowledge of.
Key ideas
- Benacerraf's dilemma: a uniform semantics treats mathematical truths as about abstract objects, but causally inert objects seem unknowable.
- Numbers cannot be particular sets, since different set-theoretic reductions work equally well and disagree.
- This identification problem motivates structuralism: arithmetic describes a structure, not specific objects.
- Any account of mathematical truth must fit with an account of mathematical knowledge.
Major works
- Philosophy of Mathematics: Selected Readings (1964, edited with Hilary Putnam)
- What Numbers Could Not Be (1965)
- Mathematical Truth (1973)
In the conversation
- Question What is Benacerraf’s dilemma, in words a school leaver can follow?
- Question If numbers aren’t physical, how could we ever know anything about them?
- Debate Mackie’s argument from queerness proves too much: by its standards arithmetic is queer too
- Debate Gödel’s “perception” of sets explains nothing about how we actually know mathematics
- Reading group Reading group: Benacerraf’s “What Numbers Could Not Be” and “Mathematical Truth”
In the Coordinates
The questions of philosophy where Paul Benacerraf takes a place, from The Coordinates of Thought.
- 1.1.4 Abstract Objects Does the number seven exist?
- 6.9 Philosophy of Mathematics Are numbers discovered or invented?