The origins and structure of quantitative concepts

Abstract

"Number" is the single most influential quantitative dimension in modern human society. It is our preferred dimension for keeping track of almost everything, including distance, weight, time, temperature, and value. How did "number" become psychologically affiliated with all of these different quantitative dimensions? Humans and other animals process a broad range of quantitative information across many psychophysical dimensions and sensory modalities. The fact that adults can rapidly translate one dimension (e.g., loudness) into any other (e.g., handgrip pressure) has been long established by psychophysics research (Stevens, 1975 ). Recent literature has attempted to account for the development of the computational and neural mechanisms that underlie interactions between quantitative dimensions. We review evidence that there are fundamental cognitive and neural relations among different quantitative dimensions (number, size, time, pitch, loudness, and brightness). Then, drawing on theoretical frameworks that explain phenomena from cross-modal perception, we outline some possible conceptualizations for how different quantitative dimensions could come to be related over both ontogenetic and phylogenetic time scales.

Figure 1

Left: Lesion overlap at the intraparietal sulcus of three patients with isolated acalculia. Right: Performance of acalculic patients on subtests of WAIS subtests demonstrates impairment to arithmetic ability, as well as more subtle impairments in perceptual and spatial reasoning, but no impairment on verbal tasks. (Takayama et al., 1994). (Figure reprinted with permission from the American Medical Association.)

Figure 2

Psychophysical functions from Meck & Church (1983). Median probability of a right response as a function of number (left) and duration (right). The methamphetamine condition (black circles) shifted the curves above the baseline (saline—red circles) condition by about 10% for judgments of both number and duration. (Figure reproduced according to the republishing guidelines of the American Psychological Association.)

Figure 3

Top: Neural overlap among combinations of the dimensions of number, size and luminance. Bottom: The distance effect: neural signal differences in the intraparietal sulcus (IPS) between close (c) and far (f) pairs of number stimuli, size stimuli, and luminance stimuli (Pinel et al., 2004). (Figure reprinted with permission from Cell Press).

Figure 4

Figures redrawn from Goldstone et al. (1978). Y-axis is the “average category response” rating, where 1 = “shorter” and 5 = “longer” than a 1-second standard duration. X-axis indicates the duration of the comparison value (in seconds). Higher/lower intensity in the figure legend refers to the order of brightness or loudness value presentation. High-Low pairs induce a bias to rate durations slightly lower, on average, than Low-High pairs. (Figure reproduced with permission from the Psychonomic Society.)

Figure 5

Figure redrawn from Pickens et al. (1994) demonstrating a U-shaped curve (drawn for illustrative purposes) for cross-modal matching of synchronized faces and voices in infants studied longitudinally. Dependent measure on the y-axis is the proportion of total looking time to matching/synchronized face/voice pairs. (Figure reproduced with permission from Elsevier.)

Figure 6

Mean total looking times to displays with a bouncing ball (congruent or incongruent pitch height to object height mapping) or an object decreasing or increasing in sharpness (congruent or incongruent pitch height to sharpness mapping. Measurements taken from Walker et al. (2009). Error bars ±1 SE. (Figure drawn with permission from Walker).