The origins of abstract concepts: Natural number - Elizabeth Spelke
On 19 January 2016, Harvard cognitive psychologist Elizabeth Spelke gave lectures and seminars at the departments of psychology at Uppsala University and Karolinska Institute, focusing on the origins of knowledge of number. She also gave a lecture at YBC gymnasium.
Liz Spelke is an internationally recognized cognitive scientist, particularly well-known for her research on infants and young children, and in recent years she has had an important role in the debate on cognitive differences between men and women. She defends the position that there is no scientific evidence of any significant disparity in the intellectual facilities of males and females. Liz is the author of over 200 hundred scientific papers on how children learn about objects, numbers and how they interact with others.
The natural numbers may be our simplest system of abstract concepts: two axioms, together with some logic, suffice to generate all of them. Natural number concepts also are extremely useful: it is hard to think of any product of our culture, from measurement to money to mathematics, that does not depend on them.
Moreover, natural number concepts have been richly studied over the last century, beginning with the pioneering research of Piaget and accelerating with every successive decade. Nevertheless, basic questions concerning the origins and nature of these concepts continue to be debated. Is the system of natural number part of our innate endowment, or do we construct it as children? Are these concepts unique to humans or shared by other animals? Are they universal across all human cultures, or accessible only to human groups whose ancestors discovered them and wove them into the groups' contemporary language and cultural practices?
Today, theories of the development of natural number tend to cluster around two sets of answers to these questions. On one view, the system of natural number is innate in humans, shared by other animals, and universal across cultures. On a second view, the system is learned by children, unique to humans, and variable across cultures. Here I argue for a third view. Spelke suggests that natural number concepts emerge over the course of human development and are unique to humans: in this sense, they are not innate. Nevertheless, natural number concepts also are universal across humans and depend on three innate, early emerging cognitive systems.
Spelke's talk focused on these three systems. First, humans and other animals have a core system of number: an innate sense of approximate numerical magnitudes. With this system, we can compare sets of objects or events on the basis of their relative numerosity, we can match numbers of visible objects to approximately equivalent numbers of sounds, and we can add, subtract, multiply and divide numbers with approximate precision.
Second, humans and other animals have a core system of naive physics, by which we can track up to three objects in parallel and follow their interactions. This system serves to represent objects whether they are visible or hidden, to establish relations of one-to-one correspondence between two small sets of objects, and to make sense of events in which a single object joins or is removed from such a set.
Third, humans have as a species the unique capacity to learn a productively combinatorial natural language. As children learn their native language, they master the meanings of the smallest number words by combining information from the first two systems. Then they learn to combine number words so as to formulate expressions that designate larger numbers. Because the combinatorial rules of natural language are productive, they can serve to generate the infinite series of natural numbers.
In her talk, I Spelke discussed research that bears on this view and its two more popular rivals. She also considered future research directions that might serve to distinguish between them definitively.