Numbers: Their Tales, Types, and Treasures.
Chapter 2: Numbers and Psychology
2.1.CORE KNOWLEDGE OF THE WORLD
How would life be in a primitive foraging society of hunters and gatherers? Can we think of circumstances that would obviate the need for counting, numbers, and arithmetic? Today, it is almost impossible to find people who haven't been influenced by modern civilization, but there are still smaller groups living in remote regions of the Amazon jungle that apparently haven't invented counting. If we lived in such a society, where cultural needs would not force us to learn how to count, how would that affect our knowledge about numbers?
As we have seen, the number concept is based on some fundamental knowledge of the world. The objects in our environment have permanence and individuality and are encountered either alone or in pairs or in larger groups. Evolution has probably ingrained some of this rudimentary knowledge in the neuronal structure of our brain, to the extent that it is helpful for surviving. Thus, it seems natural to assume that some apprehension of number is already hardwired in the brain of a newborn.
The field of science that is concerned with these questions is called mathematical cognition. It has been established in recent years as a new domain of cognitive science. Its subject is to investigate how the human brain does mathematics. An impressive amount of research has been devoted to, among other things, the following questions: What are the neuronal structures in the brain that represent numbers? Do animals have a sense of number? Is there inherited knowledge about numbers? How do we obtain knowledge about numbers? What mathematical faculties are obtained through culture and learning, and what is innate? Is the understanding of numbers and arithmetic connected with the ability to speak a language, or does it have preverbal roots? When do we learn to count, and which steps do we master in that process?
With these questions we also enter the domain of “genetic epistemology.” Epistemology (from Greek episteme, meaning “knowledge”) investigates the nature and scope of knowledge and was long considered the exclusive domain of philosophy. Jean Piaget (1896–1980), perhaps the most influential developmental psychologist of the twentieth century, argued that epistemology should also take into account the findings of cognitive science about the psychological and sociological origins of knowledge. He developed the field of genetic epistemology to investigate how knowledge is achieved through cognitive processes taking place in every individual.
In the mid-twentieth century, Piaget held the opinion that we are born without any knowledge, the brain being an empty page, fully ignorant of its environment but endowed with some fundamental mechanisms for learning. Sensual input would trigger processes of mental organization and adaptation in our brain, creating internal representations of aspects of the external world. Further refinements and adjustments of these mental concepts are constructed by the human brain in a continuous effort to harmonize internal representations with sensual impressions.
Concerning the abilities of newborns, we have a quite different view today. There is now sufficient evidence that we are already born with inherited neuronal structures that originated in evolutionary processes. These “core-knowledge systems” represent some basic knowledge of the external world and help us to interpret sensual inputs and guide our acquisition of further abilities. Research in mathematical cognition has identified two basic mental representations of the number concept: an exact representation for small numbers up to three or four and an approximate number sense for larger quantities. Here, we describe these neuronal foundations of number very briefly. Those seeking more detail are referred to the excellent book The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene (1965–), one of the pioneers in the field of cognitive neuroscience.