The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

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 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.1
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 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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