ABSTRACT
The purpose of this chapter is to examine what Piaget and other theorists mean by ‘the concept of number’, to consider the implications of their accounts, and to suggest an alternative description of knowing about number. My main criticism of Piaget's account is that it is ‘essentialist’, which is to say that it characterizes knowing about number in terms of the possession of certain essential principles that constitute the necessary and sufficient conditions for the possession of the concept of number; specifically, Piaget claims mat the understanding of one—one correspondence is necessary and sufficient to indicate the number concept while the understanding of class inclusion and seriation are individually necessary and jointly sufficient conditions. Thus Piaget assumes that all the uses of number have a common factor, or essence, such that once someone possesses the essential principles of number they ‘have’ the number concept for all situations. An essentialist view supposes that meaning can be caught in an intensional definition, and thus assumes that the applications of number have a high common factor and are not merely related to one another. I will argue here that, in fact, the uses of number may be similar, or related, but that there are important differences between them which cannot be captured by a common factor.
