Xu, Fei
Fei
Xu and Elizabeth Spelke
What is distinctive about our number representations
that allows humans, and only humans, to develop a notion of discrete infinity,
systems of measurement, and formal mathematics? In this talk, we propose that
the human talent for mathematics depends on the coordination of two core systems
of representation. Because each of the systems is shared by other primates,
uniquely human talents are not explained by the core systems alone. What is
special about human mathematical cognition stems from our ability to combine
these systems, and that ability in turn depends on human language. We first
review evidence that human infants and other primates are able to discriminate
small numbers of objects and to take account of simple additions and
subtractions of objects. These abilities, we propose, depend on a system for
representing objects: a system that makes explicit the notion of a discrete
individual but not the notion of a set. Next, we present evidence that human
infants, like many other animals, are able to discriminate large numerosities
when other correlated variables are controlled, provided that the difference
ratio between the numerosities is large. This ability, we propose, depends on a
system for representing collections: a system that makes explicit the notion of
a set but not the notion of an individual. Finally, we consider the process by
which children learn the meanings of the number words and the counting routine.
Drawing on existing and new evidence, we propose that children learn to count by
conjoining their representations of objects and sets so as to form new
representations of sets of discrete individuals. These combinations, we argue,
are unique to humans and constitute the first step toward the development of
mathematics.
Xu, Fei
Department of Psychology
Northeastern University
Boston, MA 02115, USA