Frege on Cardinality

Abstract

Frege invented a theory of quantification in order to provide a means to test validity of proofs in mathematics. A striking innovation in this theory was to construe quantifiers as second-level which applied to first-level concepts. A few years later he introduced his theory of arithmetic in Die Grundlagen der Arithmetik. In Grundlagen, Frege launched an extensive critique of views of others concerning arithmetic. After finding these views unsatisfactory, he concluded that the content of a statement of number is an assertion about a ([i], ?46). Although he presented his view as an improvement on earlier attempts, failure of other views served mainly as independent support for an already motivated theory. This central thesis of Grundlagen follows from earlier work on logic, for attributions of are special cases of Frege's theory of quantification. According to this result, attributions of apply higher-level cardinality concepts to lower-level concepts. However, Frege did not conclude that cardinal numbers are kinds of quantifiers. In Grundlagen he proceeded to argue that cardinal numbers are objects, not concepts. He finally defined numbers as certain objects, extensions of those concepts. The close connection between quantification and has not received attention it deserves. This is not surprising, for it is in apparent conflict with thesis that numbers are objects. Frege's ontology encompassed two ontological categories, function and object.' These ontological categories were mutually exclusive; no function was an object. And were a type of function, namely those which took a single argument, yielding a truth-value. So no concept could be an object. Frege introduced quantifiers as certain second-level concepts, with no