Convex hull as a heuristic

Recent studies in the field of numerical cognition quantify the impact of physical properties of an array on its enumeration, demonstrating that enumeration relies on the perception of these properties. This paper marks a shift in reasoning as it changes the focus from demonstrating this effect to explaining it. Interestingly, we were inspired by one of the very first articles in the field, “The power of numerical discrimination” by Stanley Jevons that was published in Nature in 1871. In his report, Jevons attempts to answer the question of how many objects can be perceived in “a single mental beat of attention”. We relate directly to Jevons’s records, putting forward a plausible heuristic mechanism that relies on the physical geometrical properties of the arrays to be enumerated. We use a mathematical theorem and computer simulation to show that the shape of the convex hull, the smallest polygon containing all dots in an array, is a good predictor of numerosity. We show that convex hull downsamples the spatial data, allowing quick and fairly accurate numerical estimation. Moreover, convex hull predictability changes as numerosity grows, corresponding to the psychophysical curve of enumeration shown by Jevons and many others that followed

Putting the world in mind: The case of mental representation of quantity

A reoccurring question in cognitive science concerns the way the world is represented. Cognitive scientists quantify the contribution of a physical attribute to a sensation and try to characterize the underlying mechanism. In numerical cognition, the contribution of physical properties to quantity perception in comparison tasks was widely demonstrated albeit leaving the underlying mechanism unclear. Furthermore, it is unclear whether this contribution is related solely to comparison tasks or to a core, general ability. Here we demonstrate that the shape of the convex hull, the smallest convex polygon containing all objects in an array, plays a role in the transfer function between quantity and its mental representation. We used geometric probability to demonstrate that the shape of the convex hull is correlated with quantity in a way that resembles the behavioral enumeration curve of subitizing and estimation. Then, in two behavioral experiments we manipulated the shape of the convex hull and demonstrated its effect on enumeration. Accordingly, we suggest that humans learn the correlation between convex hull shape and numerosity and use it to enumerate.</p

One tamed at a time: A new approach for controlling continuous magnitudes in numerical comparison tasks

Non-symbolic stimuli (i.e., dot arrays) are commonly used to study numerical cognition. However, in addition to numerosity, non-symbolic stimuli entail continuous magnitudes (e.g., total surface area, convex-hull, etc.) that correlate with numerosity. Several methods for controlling for continuous magnitudes have been suggested, all with the same underlying rationale: disassociating numerosity from continuous magnitudes. However, the different continuous magnitudes do not fully correlate; therefore, it is impossible to disassociate them completely from numerosity. Moreover, relying on a specific continuous magnitude in order to create this disassociation may end up in increasing or decreasing numerosity saliency, pushing subjects to rely on it more or less, respectively. Here, we put forward a taxonomy depicting the relations between the different continuous magnitudes. We use this taxonomy to introduce a new method with a complimentary Matlab toolbox that allows disassociating numerosity from continuous magnitudes and equating the ratio of the continuous magnitudes to the ratio of the numerosity in order to balance the saliency of numerosity and continuous magnitudes. A dot array comparison experiment in the subitizing range showed the utility of this method. Equating different continuous magnitudes yielded different results. Importantly, equating the convex hull ratio to the numerical ratio resulted in similar interference of numerical and continuous magnitudes