Like most cognitive scientists, I take concepts to be mental symbols. Mental symbols are not all concepts, as there are also sensory representations, motor representations, and perceptual representations. From the perspective of cognitive science, a theory of concepts must specify their format, what computations they enter into, what determines their content, and how they differ from other types of mental symbols. In a recent book (Carey, in press) I present case studies of the acquisition of several important domains of conceptual representations, arguing that the details of the acquisition process adjudicate among rival theories of concepts within cognitive science. The case studies bear on the existence and nature of innate concepts, and on the existence and nature of discontinuities in development. Human conceptual development involves the construction of representational resources that go beyond those from which they are built in theoretically interesting ways. As Fodor (1975Fodor ( , 1980 has forcefully argued, characterizing these discontinuities and explaining how they are possible is a formidable challenge to cognitive science. Meeting this challenge informs our theories of the human conceptual system.Here I illustrate the lessons I draw from these case studies by touching on one of them: accounting for the origin of concepts of natural number. Explaining the human capacity for representing natural numbers has been a project in philosophy for centuries (e.g., Mill, 1874) and in psychology since its emergence as a scientific discipline during the past century (e.g., Piaget, 1952). As natural number is the backbone of all of arithmetic, an understanding how representations of natural number arise provides a good start on a theory of the human capacity for mathematics.Accounting for the origin of any conceptual requires specifying the innate building blocks from which the representations are built, and specifying the learning mechanisms that accomplish the feat. Two distinct research programs should be, but often are not, distinguished. In one, the logical program, the building blocks are conceived of as logically necessary prerequisites for the capacity in question. In the case of natural number representations these might include the capacity for carrying out recursive computations, the capacity to represent sets, and various logical capacities, such as those captured in second order predicate calculus. Sometimes arguments within the logical program seek to specify some necessary computational ability that other animals lack (e.g., see Hauser, Chomsky & Fitch's, 2002, proposal that non-humans lack the capacity for recursion) that might explain why only humans have the target conceptual ability. A full account of the building blocks for some representational capacity within the logical program must include all of the necessary ones. A second research program, the ontogenetic program, conceives of the building blocks as specific representational systems out of which the target representational capacity is ...