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Benacerraf's Dilemma Revisited
Author(s) -
Hale Bob,
Wright Crispin
Publication year - 2002
Publication title -
european journal of philosophy
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.42
H-Index - 36
eISSN - 1468-0378
pISSN - 0966-8373
DOI - 10.1111/1468-0378.00151
Subject(s) - wright , dilemma , metaphysics , the arts , library science , art history , philosophy , history , computer science , art , epistemology , visual arts
One of the most influential articles 1 in the last half century of philosophy of mathematics begins by suggesting that accounts of mathematical truth have been motivated by two quite distinct concerns: (1) the concern for having a homogeneous semantical theory in which the semantics for the statements of mathematics parallel the semantics for the rest of the language (2) the concern that the account of mathematical truth mesh with a reasonable epistemology [403] Observing that the two concerns are liable to pull strongly in opposite directions, he proposes two conditions which an acceptable account of mathematical truth should satisfy. One—the semantic constraint—has it that: any theory of mathematical truth [should] be in conformity with a general theory of truth … which certifies that the property of sentences that the account calls 'truth' is indeed truth [408] The other—the epistemological constraint—is that: a satisfactory account of mathematical truth … must fit into an overall account of knowledge in a way that makes it intelligible how we have the mathematical knowledge that we have. An acceptable semantics for mathematics must fit an acceptable epistemology [409] 2 Whilst some further explanation—of what Benacerraf takes to be required for an acceptable epistemology—is obviously needed before the exact force of the epistemological constraint can be clear, its general drift and point is evident. The precise import of the semantic constraint, however, is less immediately apparent. It is clear that Benacerraf takes it to require, at a minimum, giving a uniform semantic account of our language as a whole, including mathematical language. But sometimes he seems to suggest a stronger requirement. Observing that his examples [405]: (1) There are at least three large cities older than New York (2) There are at least three perfect numbers greater than 17 share, on the face of it, the same 'logicogrammatical' form, 2 which he represents as: (3) There are at least three FGs that bear R to a he writes: … if we are to meet this requirement [the semantic constraint], we shouldn't be satisfied with an account that fails to treat (1) and (2) in parallel fashion, on the model of (3). There may well be differences, but I expect these to emerge at the level of the analysis of the reference of the singular terms and predicates.[408] Taken on its own, this suggests a very exacting reading of the semantic constraint, under which it can …