Beliefs about the nature of numbers

Nearly all psychologists think that cardinality is the basis of number knowledge. When they test infants’ sensitivity to number, they look for evidence that the infants grasp the cardinality of groups of physical objects. And when they test older children’s understanding of the meaning of number words, they look for evidence that the children can, for example, "Give [the experimenter] three pencils" or can "Point to the picture of four balloons." But when people think about the positive integers, do they single them out by means of the numbers’ cardinality, by means of the ordinal relations that hold among them, or in some other way? This chapter reviews recent research in cognitive psychology that compares people’s judgments about the integers’ cardinal and ordinal properties. It also presents new experimental evidence suggesting that, at least for adults, the integers’ cardinality is less central than their number-theoretic and arithmetic properties.