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Step 4 of 7~8 min read~37 min left
Benacerraf's Dilemma
Examine the two constraints that define the central problem of philosophy of mathematics.
“Benacerraf's dilemma: any adequate philosophy of mathematics must satisfy two constraints simultaneously. The semantic constraint: the semantics of mathematical statements should be continuous with the semantics of ordinary statements, '7 is prime' should work like 'the cat is black'; both refer to objects and predicate properties. The epistemological constraint: our account of mathematical knowledge must fit with a credible account of human knowledge in general, we are physical beings who gain knowledge through causal interaction with our environment. The problem: if mathematical objects are abstract and causally inert (satisfying the semantic constraint), it is mysterious how we can know about them. If they are physical or we deny that mathematical statements refer to objects (satisfying the epistemological constraint), the semantics of mathematics becomes systematically misleading. — Benacerraf, 'Mathematical Truth' (1973); SEP 'Philosophy of Mathematics'; IEP 'Philosophy of Mathematics'”