Beyond the number domain

Abstract

In a world without numbers, we would be unable to build a skyscraper, hold a national election, plan a wedding or pay for a chicken at the market. The numerical symbols used in all these behaviors build on the approximate number system (ANS) which represents the number of discrete objects or events as a continuous mental magnitude. Here, we first discuss evidence that the ANS bears a set of behavioral and brain signatures that are universally displayed across animal species, human cultures and development. We then turn to the question of whether the ANS constitutes a specialized cognitive and neural domain - a question central to understanding how this system works, the nature of its evolutionary and developmental trajectory and its physical instantiation in the brain.

Figure 1

Numerical judgments in monkeys and humans follow Weber’s Law. (A) In a recent study from our lab, monkeys and adults were trained to discriminate stimuli based on their best estimate of numerical value. Adults were specifically instructed to avoid verbally counting and to respond as rapidly as possible. For both groups, accuracy (B) and response times (C) were modulated by the numerical ratio between the stimuli. In addition, accuracy performance is well-fit by a mathematical model of performance that adheres to Weber’s law. Reprinted with permission from [19].

Figure 2

Brain damage reveals distinct functional localization of numerical and nonnumerical semantic judgments. (A) Patients with temporal lobe atrophy can show impairments in categorization and recognition tasks involving animals, objects, or letters but be unimpaired on tasks requiring numerical computation, recognition, and counting. (B) By contrast, patients with right parietal lesions can be impaired at tasks involving mathematical operations such as addition and subtraction as well as number comparison and bisection but be unimpaired in categorization and recognition of non-numerical stimuli. Drawn with permission from [23, 24].

Figure 3

Infants can discriminate multiple quantities. Six-month-old infants show similar sensitivity to discriminations to time, surface area, and number. During looking-time studies testing discrimination of each of these dimensions, infants look longer at novel values when they are in a 1:2 ratio to the familiar stimuli but they do not look longer to novel stimuli presented in a 2:3 ratio to the familiar stimuli. These findings suggest that infants can discriminate time, surface area, and number at a 1:2 ratio but not at a 2:3 ratio. But, while at 6 months of age babies require a 2-fold change in number, time, or surface area to detect a change, by 9 or 10 months of age a 2:3 ratio is sufficient for discrimination. Drawn with permission from [20, 65, 66].

Figure 4

Activation of parietal cortex by magnitude processing. Neuroimaging studies have reported activity in parietal cortex, including regions along the IPS, in response to stimuli from a variety of magnitudes including number, size, time, brightness, and ordinal position judgments. Each point on the cortical template represents peak parietal coordinates from studies of: Number: [–, –78, 80, 82, 97, 98]; Time: [85, 87, 88, 90]; Brightness: [77, 78, 90]; Size: [75, 78, 80, 81]; Ordinal memory: [83]; Ordinal position: [82]. Maps generated using Caret software (http://brainmap.wustl.edu/caret; [99]).