Arabic Numerals and the Problem of Mathematical Sophistication
This chapter begins with a critique of views on which the reference of a complex Arabic numeral is fixed by a sophisticated mathematical description. The chapter proposes instead that that reference is fixed by a convention that assigns to the numeral “1” the number one and that assigns to the numeral for n+1 the number greater by one than the number assigned to the numeral for n. This view, which presupposes that the sequence of Arabic numerals is learned independently of any principle of referential interpretation, evades the objections to sophisticated descriptivist theories. Toward the end of the chapter, a view of the referents of the numerals is defended according to which these are the finite cardinality properties. This view is argued to be the only one compatible with a number of principles constituting the intuitive conception of number and presupposed in the referential intentions of users of the numerals.