Looking at nothing but seeing something: The resilient numerical representation of empty sets.

Numbers are mentally represented as magnitudes along a continuum, as evidenced by distance and end effects in comparison tasks. Empty sets, representing the absence rather than presence of quantity, pose a challenge to this framework. Previous research has shown that empty sets can exhibit numerical processing signatures, yet the conditions enabling this representation remain unclear. The present studies examined whether perceptual homogeneity is necessary for empty sets to be perceived as numerical entities. In Experiment 1 (N = 72), participants performed nonsymbolic magnitude comparisons under conditions where stimuli backgrounds were homogeneous or heterogeneous. In Experiment 2 (N = 72), homogeneity was varied through the presence or absence of frames surrounding the presented stimuli. Contrary to our hypothesis that perceptual heterogeneity would disrupt numerical processing, distance and end effects for comparisons to empty sets emerged across all conditions in both experiments, including heterogeneous presentations. However, effect sizes varied systematically across conditions, with perceptual features (background patterns, frame presence, frame location) modulating processing efficiency without eliminating the core numerical representation. These findings demonstrate that empty sets are robustly processed as numerical entities across diverse perceptual contexts. This robustness may reflect the intrinsic congruence between empty sets' numerical value (zero) and perceptual properties (absence of elements), rendering them less sensitive to external perceptual manipulations than non-empty numerosities.