The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview
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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.
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.
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.


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