15 CRISPIN WRIGHTI am grateful to Michael Dummett for his searching comments on my paper.
Since he seems to have found so much of it unclear, I shall here try to clarify
some of the essentials of the view I was defending, and to respond to his
Response to Dummett
I. FOUR NEO-FREGEAN CLAIMSLoosely expressed, the neo-Fregean thesis about arithmetic is that a knowledge of its fundamental laws (essentially, the Dedekind--Peano axioms)--
and hence of the existence of a range of objects which satisfy them--may
be based a priori on the explanatory principle, N=. More specifically, the
thesis involves four ingredient claims:
|i. ||that the vocabulary of higher-order logic plus the cardinality operator, 'Nx: . . . x . . .', provides a sufficient definitional basis for a statement
of the basic laws of arithmetic;|
|ii. ||that when they are so stated, N= provides for a derivation of those
laws within higher-order logic;|
|iii. ||that someone who understood a higher-order language to which the
cardinality operator was to be added would learn, on being told that N= is
analytic of that operator, all that it is necessary to know in order to construe any of the new statements that would then be formulable.|
|iv. ||Finally and crucially, that N= may be laid down without significant
epistemological obligation: that it may simply be stipulated as an explanation of the meaning of statements of numerical identity, and that--
beyond the issue of the satisfaction of the truth-conditions it thereby lays
down for such statements--no competent demand arises for an independent assurance that there are objects whose conditions of identity are as it
Questia, a part of Gale, Cengage Learning. www.questia.com
Book title: The Philosophy of Mathematics Today.
Contributors: Matthias Schirn - Editor.
Publisher: Clarendon Press.
Place of publication: Oxford.
Publication year: 1998.
Page number: 389.
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