Visual sense of number vs. sense of magnitude in humans and machines

Abstract

Numerosity perception is thought to be foundational to mathematical learning, but its computational bases are strongly debated. Some investigators argue that humans are endowed with a specialized system supporting numerical representations; others argue that visual numerosity is estimated using continuous magnitudes, such as density or area, which usually co-vary with number. Here we reconcile these contrasting perspectives by testing deep neural networks on the same numerosity comparison task that was administered to human participants, using a stimulus space that allows the precise measurement of the contribution of non-numerical features. Our model accurately simulates the psychophysics of numerosity perception and the associated developmental changes: discrimination is driven by numerosity, but non-numerical features also have a significant impact, especially early during development. Representational similarity analysis further highlights that both numerosity and continuous magnitudes are spontaneously encoded in deep networks even when no task has to be carried out, suggesting that numerosity is a major, salient property of our visual environment.

Visual number sense in untrained deep neural networks

“…Previously, it was suggested that the reported number sense might be observed because of the correlation between the numerosity and other continuous magnitudes such as total area, because it is hardly possible to design visual stimuli in which numerosity is completely independent of other nonnumerical visual magnitudes (e.g., if the total area is fixed, then the single dot size must decrease as numerosity increases) (35). However, this scenario was refuted in a number of commentaries (36)(37)(38)(39)(40) and after further studies (41,42), from the fact that the observed number sense does not depend on various visual features in each controlled test. To address this issue carefully, we devised a new numerosity comparison task that could be successfully performed, only when the network makes decisions based on estimation of abstract numerosity.…”

Numerosities and Other Magnitudes in the Brains: A Comparative View

“…All models for numerical cognition come roughly to the same results, namely, the ability of artificial neural networks to simulate human and non-human animals' behaviors in numerical judgments, i.e., the discrimination of one numerosity from another (Rapp et al, 2020), comparison tasks for relationships of sameness (Nasr et al, 2019), or inequality (Testolin et al, 2020a). However, their relevant implications and proposals regarding the implementation of numerical ability rely on their architectures and main functional properties.…”

“…These types of neural networks have been also usefully employed in order to disentangle the spiny issue of nonnumerical perceptual features that covary with numerosity (Testolin et al, 2020a). Using the stimulus space designed in another study (DeWind et al, 2015), they have been able to statistically estimate the contribute of each non-numerical feature to numerical comparison tasks.…”

“…Concerning this tight relationship of numerosity with other systems of magnitude, some authors have advanced the hypothesis that continuous magnitudes which naturally covary with numbers-such as area, perimeter, and density-might be the leading information processed in tasks on numerosity (Leibovich et al, 2017). Several studies try to control for these variables, at least once at a time so as to tease apart their impact on numerosity tasks from effects of ANS (Rugani et al, 2011(Rugani et al, , 2013bDeWind et al, 2015;Testolin et al, 2020a). Animals for example should be aware of the direction of the correlation between numerosity and the alternative continuous variable, in order to rely on the second one to complete those tasks (Ferrigno and Cantlon, 2017).…”

Poor numerical performance of guppies tested in a Skinner box

“…Conversely, if guppies succeeded only with the easier ratio (0.6), we would conclude that the conditioning chamber allows to study learning but does not provide increased learning efficiency in the context of numerical discrimination. Besides manipulating numerosity, we also manipulated other visual features to investigate the impact of non-numerical magnitudes in discrimination performance (see 25 ). In the "incongruent" condition, cumulative area was equated in the two stimuli, implying that individual item size was incongruent with number (i.e., stimuli with larger numbers had smaller items).…”