Abstract:
There is a puzzle about the occurrence
of number words in natural language that goes back to Frege. Number
words like "two" and "four" can occur as singular terms, but also as
adjectives or determiners. These two syntactic occurrences correspond
to two quite different semantic functions, and these two semantic
functions are related to two rather different philosophical visions of
arithmetic. The puzzle is how number words can do both of these things.
I will argue that this puzzle has never been properly solved, but that
a solution is forthcoming once we take a close look at number words as
determiners. The solution I will propose will overcome what I take to
be the biggest obstacle in a defense of logicism. Logicism will,
however, not be understood as the claim that math is reducible to
logic, but rather as a defense of the grand philosophical vision of
mathematics that such a reduction was supposed to establish.