[Part II] Approximate Number System: The Emergence of a Modern Theory of Numerical Perception

Abstract

How the theory of approximate quantity perception developed, from early models to research on humans and other primates.

Introduction

The first part of this review examined research on subitizing - the remarkable human ability to determine the number of small sets of objects almost instantaneously and with high accuracy. The pioneering work of Kaufman et al., followed by the theory proposed by Trick and Pylyshyn, demonstrated that this process is not merely an exceptionally rapid form of counting but instead relies on mechanisms of visual attention and object individuation. However, this explanation applies only to relatively small numerosities, typically no more than three or four objects.

This naturally raises a fundamental question: How are humans able to rapidly estimate much larger quantities when sequential counting becomes too slow and simultaneous individuation of every object is no longer possible? Although it is impossible to instantly count several dozen items, most people can easily distinguish a set of twenty objects from a set of forty and can reliably estimate the relative size of different groups. Moreover, similar abilities have been observed not only in adults but also in infants and in numerous animal species that have never learned symbolic arithmetic.

These findings led researchers to propose that, in addition to mechanisms responsible for the precise identification of small sets, the brain possesses a separate system specialized for the approximate estimation of larger numerosities. During the late twentieth and early twenty-first centuries, this idea evolved into what is now known as the Approximate Number System (ANS), one of the central theoretical frameworks in contemporary cognitive psychology and the neuroscience of numerical cognition.

This article reviews the historical development of the Approximate Number System, its experimental foundations, the principal theoretical models, and current perspectives on the mechanisms underlying approximate numerical perception. Particular attention is given to studies demonstrating the relationship between ANS and Weber's Law, research on numerical cognition in infants and non-human animals, and current debates concerning the relationship between ANS and subitizing.

The Emergence of the Modern Concept (2004)

Although experimental evidence for approximate quantity estimation accumulated throughout the second half of the twentieth century, the modern concept of the Approximate Number System did not fully emerge until the beginning of the twenty-first century. A major milestone was the influential review by Feigenson, Dehaene, and Spelke, Core Systems of Number (2004), which integrated findings from studies of infants, animals, and adult participants into a unified theoretical framework.

For example:

A defining characteristic of ANS is its conformity to Weber's Law. The accuracy of numerical discrimination depends not on the absolute difference between two quantities but on their ratio. For instance, distinguishing between sets of 10 and 20 objects is considerably easier than distinguishing between 90 and 100, despite the identical absolute difference of ten objects.

Current Perspectives

Despite decades of research, the relationship between subitizing and the Approximate Number System remains an open question. Three principal viewpoints can be identified in the contemporary literature.

Two Independent Systems

The most widely accepted position holds that subitizing and ANS represent distinct cognitive mechanisms. Subitizing is associated with attentional processes and object individuation (object files), whereas ANS is viewed as a system for representing approximate numerical magnitude (mental magnitude) that obeys Weber's Law. This interpretation is supported by researchers including Dehaene, Feigenson, Piazza, Nieder, and many others.

A Unified Numerical System

According to an alternative hypothesis, numerical perception relies on a single underlying mechanism. Differences between subitizing and ANS arise not from distinct cognitive systems but from a gradual decline in precision as the number of objects increases. Within this framework, subitizing represents the high-precision operating range of a general numerical processing system.

The Hybrid Model

In recent years, an intermediate view has gained increasing attention. According to this model, numerical perception consists of several successive stages:

Rather than competing mechanisms, subitizing and ANS are interpreted as different levels within a unified hierarchy of numerical processing.

Studies in Non-Human Primates

Important evidence for the evolutionary origins of numerical cognition has come from studies of non-human primates. In 1985, Tetsuro Matsuzawa demonstrated that the chimpanzee Ai was capable of associating Arabic numerals with corresponding quantities of objects.

Later, Tomonaga and Matsuzawa (2002) directly compared humans and chimpanzees in tasks involving the enumeration of briefly presented visual arrays. Their results showed that chimpanzees, like humans, exhibit a transition from rapid processing of small sets to slower processing of larger numerosities, suggesting that these mechanisms have deep evolutionary roots.

Current Directions of Research

Today, research is gradually shifting away from the question of how many numerical processing systems exist toward more fundamental issues.

• How does the visual system individuate objects before numerical estimation begins?
• How do attention, working memory, and the Approximate Number System interact?
• Do subitizing and ANS share common neural mechanisms?
• Is the boundary between subitizing and approximate estimation fixed, or does it depend on task demands and viewing conditions?
• Can principles of approximate processing be extended beyond numerosity to other aspects of visual perception, such as spatial relationships, symmetry, or structural regularities?

Among these questions, the last has attracted increasing attention. While the existence of an Approximate Number System for numerical quantities is supported by extensive experimental evidence, whether comparable mechanisms operate in the perception of visual structure remains an open question.

Conclusion

The development of the Approximate Number System has significantly expanded our understanding of numerical cognition by demonstrating that the brain can rapidly estimate not only small but also large quantities without sequential counting.

Despite considerable progress, many important questions remain unresolved. Researchers continue to debate the relationship between subitizing and ANS, the roles of attention and working memory, and whether these processes reflect independent cognitive systems or different stages within a common processing architecture.

Today, the Approximate Number System is regarded as one of the central models in cognitive psychology and the neuroscience of numerical cognition. Ongoing research continues to refine our understanding of its mechanisms and its relationship to broader processes of visual perception and attention.