"Zeroing" in on mathematics in the monkey brain

Sometimes nothing is simply nothing: Automatic processing of empty sets

Previous work using the numerical comparison task has shown that an empty set, the nonsymbolic manifestation of zero, can be represented as the smallest quantity of the numerical magnitude system. In this study, we examined whether an empty set can be represented as such under conditions of automatic processing in which deliberate processing of stimuli magnitudes is not required by the task. In Experiment 1, participants performed physical and numerical comparisons of empty sets (i.e., empty frames) and of other numerosities presented as framed arrays of 1 to 9 dots. The physical sizes of the frames varied within pairs. Both tasks revealed a size congruity effect (SCE) for comparisons of non-empty sets. In contrast, comparisons to empty sets produced an inverted SCE in the physical comparison task, whereas no SCE was found for comparisons to empty sets in the numerical comparison task. In Experiment 2, participants performed an area comparison task using the same stimuli as Experiment 1 to examine the effect of visual cues on the automatic processing of empty sets. The results replicated the findings of the physical comparison task in Experiment 1. Taken together, our findings indicate that empty sets are not perceived as "zero," but rather as "nothing," when processed automatically. Hence, the perceptual dominance of empty sets seems to play a more important role under conditions of automatic processing, making it harder to abstract the numerical meaning of zero from empty sets.

On the indicators for perceiving empty sets as zero

The question whether human beings process empty sets as zero has received little research attention. In this study, we used the distance and end effects as indicators for treating empty sets as a numerical entity that represents an absence of quantity. In a series of experiments, participants performed a magnitude comparison task. They were presented with empty sets and other numerosities from 1 to 9, presented as dot arrays. We manipulated task instructions relevant to the target (i.e., "choose the target that contains more/less dots" in Experiment 1) or the given numerical range mentioned in the instructions (i.e., 0-9 or 1-9 in Experiment 2) to create conditions in which an empty set would be perceived as the smallest value of the experimental numerical range. The results revealed distance effects for comparisons to empty sets, irrespective of task instructions. In Experiment 3, we manipulated the response mode. Two groups of participants responded to target location, one group with a key-press and the other vocally, while the third group responded vocally to target color. The results revealed distance effects for comparisons to empty sets only when responding to target location, regardless of the response mode, indicating that spatial features should be primed in order to perceive an empty set as a numerical entity. These findings show that perceiving an empty set as nothing or as zero depends on the context in which it is presented.

The discrimination of magnitude: A review and theoretical analysis

In a discrimination based on magnitude, the same stimulus is presented at two different magnitudes and an outcome, such as food, is signalled by one magnitude but not the other. The review presented in the first part of the article shows that, in general, such a discrimination is acquired more readily when the outcome is signalled by the larger rather than the smaller of the two magnitudes. This asymmetry is observed with magnitudes based on sound, odour, temporal duration, quantity, and physical length. The second part of the article, explores the implications of this pattern of results for the theory of discrimination learning presented by Pearce (1994). The asymmetry found with discriminations based on magnitude contradicts predictions derived from the original version of the theory, but it can be explained by a modified version. The asymmetry also has important implications for understanding how animals represent magnitudes.