Number systems are present in most human languages and cultures, and a world without access to numbers seems unimaginable and foreign to adults in many linguistic and cultural communities. From nursery rhymes to baking recipes, from calendars to engineering plans, from sports rules to financial transactions, numbers are present in every aspect of human activity. Number is deeply embedded in human culture in ways that have transformed societies through the ages. Number is also, fundamentally, a product of the human mind and a powerful domain for discovering the mechanisms that govern how the human mind works.
Many things about number come easily to us. It is generally agreed that a rudimentary ability to perceive and represent number is present in human infants (and many kinds of animals). Furthermore, young children across different cultures and languages seem capable of gaining access to the system of natural numbers before they are exposed to formal schooling, simply by interacting with people in their environment. Nevertheless, other aspects of number knowledge come less easily to humans and seem to require considerable cultural support. Even though children learn number word meanings before they begin formal education, they initially lack full appreciation of their logic. Furthermore, the ability to engage in exact mathematical operations (“Ambition, Distraction, Uglification, and Derision,” in Lewis Carroll’s famously mocking words) can seem daunting to many older children and even adults. For cultures without number words (or other equivalent symbolic tools), precise numerical calculations appear severely limited or downright impossible.
From the perspective of modern cognitive science, the complex nature of number knowledge touches on several foundational issues in the study of human cognition: How do people represent number and quantity? How much of this representational machinery is unique to humans? How does the ability to reason about number develop from infancy to early childhood, to adulthood and throughout the lifespan? How does number make contact with the language system? Are there universal aspects of the representation of number? Can our brain architecture explain what makes numbers sometimes so easy and sometimes so hard? Attempts to answer these questions have generated some of the most elegant, influential and broad-ranging work in cognitive science. Beyond its intellectual appeal, the effort to understand how number is coded in the human mind has immediate relevance for a host of societal and policy issues, including how we approach mathematics education, or treat math-specific learning disabilities.
These considerations led the present author and the leadership of the Society for Language Development to organize a symposium on the topic of number representation on November 6, 2014, at Boston University. The invited speakers were Elizabeth S. Spelke, Elizabeth M. Brannon and Jessica F. Cantlon. The goal of the symposium was to bring some of the broad findings, theorizing and debates in the number domain to the attention of the language acquisition community, and to foster discussion of the origins and development of number representations and their relation to language. The current special issue of Language Learning and Development contains papers that the three invited speakers and their collaborators later wrote on the same theme.
In her article, Elizabeth S. Spelke addresses the origins of natural numbers. She reviews the proposal (first made by Randy Gallistel and Rochel Gelman) that natural numbers are a product of cognitive evolution and the more recent alternative proposal (put forth by Susan Carey) according to which numbers are a product of human cultural history. Spelke’s own proposal is that natural language is critically involved in the discovery of natural numbers. Spelke assumes that humans possess an innate system that delivers approximate numerical magnitudes (also shared with other animals), as well as innate systems that deliver representations for individual objects and object kinds. To arrive at the kind of exact representation underlying natural numbers, Spelke argues, children use linguistic resources to combine outputs from these innate core systems. The proposal seeks to explain how numbers can be both innate and learned, and how certain aspects of our number knowledge (including the outputs of the approximate number system) seem to come naturally to us and others (arguably, those linked to exact representations of natural numbers) require additional effort or resources.
The other two papers in this volume further explore how the systems underlying approximate and exact (symbolic) numerical representations and computations overlap and affect each other. Emily Szkudlarek and Elizabeth M. Brannon provide an overview of the properties of the approximate number system using evidence from human infants and non-human primates. They then present different types of behavioral and neural evidence suggesting that there is a relation between the acuity of the approximate number system system and symbolic math abilities. One issue with this claim is that the relation between the two systems does not consistently surface in empirical studies. The authors propose an explanation for the mixed state of the evidence and offer a range of mechanistic characterizations of the relation between the two systems that can be more thoroughly tested in future work.
Alyssa J. Kersey and Jessica F. Cantlon introduce several strands of evidence to argue that the exact number concepts available to humans build on inborn, primitive representational and logical abilities. Some of this evidence comes from the development of number concepts and number language (including the counting routine) in infancy and childhood. Other evidence comes from recent neuroimaging studies showing continuity in the neural substrates of numerical cognition in humans and other animals, and studies of numerical development showing commonalities in patterns of brain activation during symbolic and non-symbolic tasks. Despite these links between symbolic and non-symbolic computational mechanisms, the authors also argue for uniquely human aspects of numerical reasoning based on comparisons between human and non-human primate neuroimaging research.
Together, the papers in this special issue provide a far-ranging overview of the state of the art in number cognition, number development and the relation between number and language. Even though many unanswered questions remain, the current volume raises theoretical hypotheses and avenues for empirical work that promise to engage the field for years to come.