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15
Response to Dummett
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
principal criticisms.
I. FOUR NEO-FREGEAN CLAIMS
Loosely expressed, the neo-Fregean thesis about arithmetic is that a knowl-
edge 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 oper-
ator, '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 con-
strue any of the new statements that would then be formulable. 1
iv. Finally and crucially, that N= may be laid down without significant
epistemological obligation: that it may simply be stipulated as an ex-
planation 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 indepen-
dent assurance that there are objects whose conditions of identity are as it
stipulates.
____________________
1 Or anyway, any which corresponded to something finite-arithmetical--there is no need
for a neo-Fregean about arithmetic to make larger claims about larger cardinals.

-389-

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Publication Information: 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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