Ebbinghaus illusion changes numerosity perception independent of density perception

It has been posited that number and space are processed with shared mechanisms. We examined whether the stimulus configuration that leads to the Ebbinghaus size illusion would change numerosity perception. We first prepared visual stimuli with which perceived densities were equated for given dot number and area size (Experiment 1). In the subsequent experiments using the prepared stimuli, two sets of dots were surrounded by two sets of inducers (one consisted of small circles and the other of large circles). Participants reported which area contained the larger number of dots (Experiment 2) and which area where the dots appeared was larger (Experiment 3). The results showed that dots surrounded by the small (large) inducers appeared more (less) numerous and occupied a larger (smaller) spatial extent. These results suggest that visual processes of numerosity interact with those of size perception independent of visual processes for perceived density.

Exploring the classical and numerical Delboeuf illusion: the impact of transcranial alternating current stimulation on magnitude processing

Understanding cognitive and neural mechanisms underlying quantity processing is crucial for unraveling human cognition. The existence of a single magnitude system, encompassing non-symbolic number estimation alongside other magnitudes like time and space, is still highly debated since clear evidence is limited. Recent research examined whether spatial biases also influence numerosity judgments, using visual illusions like the Delboeuf illusion. While findings support a generalized magnitude system, direct comparisons of spatial and numerical Delboeuf illusions are missing. This study explored whether perceptual biases similarly affect different magnitude processing and whether transcranial alternating current stimulation (tACS) modulates these processes. Participants underwent three tACS conditions (seven Hz, 18 Hz, placebo) while performing tasks involving the classic and numerical Delboeuf illusions. We hypothesized that theta-frequency tACS (seven Hz) would enhance visual integration and illusion strength, while beta tACS (18 Hz) would reduce it by promoting visual segregation. Results indicated higher discrimination accuracy in area-based tasks than numerical judgments. Nonetheless, a significant correlation between performances in spatial and numerical illusions supported the existence of a shared mechanism for magnitude processing. Contrary to expectations, seven Hz tACS reduced the perceptual illusion's strength. No significant interaction emerged between tACS frequency and discrimination abilities. These findings deepen our understanding of the cognitive processes involved in magnitude perception, potentially supporting the hypothesis of a generalized magnitude system. They also highlight the potential and limitations of non-invasive brain stimulation techniques, such as tACS, in modulating perceptual processes, offering insights into the neural underpinnings of quantity perception.

Investigating acoustic numerosity illusions in professional musicians

Various studies have reported an association between musical expertise and enhanced visuospatial and mathematical abilities. A recent work tested the susceptibility of musicians and nonmusicians to the Solitaire numerosity illusion finding that also perceptual biases underlying numerical estimation are influenced by long-term music training. However, the potential link between musical expertise and different perceptual mechanisms of quantitative estimation may be either limited to the visual modality or universal (i.e., modality independent). We addressed this question by developing an acoustic version of the Solitaire illusion. Professional musicians and nonmusicians listened to audio file recordings of piano and trombone notes and were required to estimate the number of piano notes. The stimuli were arranged to form test trials, with piano and trombone notes arranged in a way to form the Solitaire pattern, and control trials, with randomly located notes to assess their quantitative abilities in the acoustic modality. In the control trials, musicians were more accurate in numerical estimation than nonmusicians. In the presence of illusory patterns, nonmusicians differed from musicians in the esteem of regularly arranged vs. randomly arranged notes. This suggests that the association between long-term musical training and different perceptual mechanisms underlying numerical estimation may not be confined to the visual modality. However, neither musicians nor nonmusicians seemed to be susceptible to the acoustic version of the Solitaire illusion, suggesting that the emergence of this illusion may be stimulus and task-dependent.

Implied occlusion and subset underestimation contribute to the weak-outnumber-strong numerosity illusion

Four experimental studies are reported using a total of 712 participants to investigate the basis of a recently reported numerosity illusion called "weak-outnumber-strong" (WOS). In the weak-outnumber-strong illusion, when equal numbers of white and gray dots (e.g., 50 of each) are intermixed against a darker gray background, the gray dots seem much more numerous than the white. Two principles seem to be supported by these new results: 1) Subsets of mixtures are generally underestimated; thus, in mixtures of red and green dots, both sets are underestimated (using a matching task) just as the white dots are in the weak-outnumber-strong illusion, but 2) the gray dots seem to be filled in as if partially occluded by the brighter white dots. This second principle is supported by manipulations of depth perception both by pictorial cues (partial occlusion) and by binocular cues (stereopsis), such that the illusion is abolished when the gray dots are depicted as closer than the white dots, but remains strong when they are depicted as lying behind the white dots. Finally, an online investigation of a prior false-floor hypothesis concerning the effect suggests that manipulations of relative contrast may affect the segmentation process, which produces the visual bias known as subset underestimation.

The influence of visual illusion perception on numerosity estimation could be evolutionarily conserved: exploring the numerical Delboeuf illusion in humans (Homo sapiens) and fish (Poecilia reticulata)

Discriminating between different quantities is an essential ability in daily life that has been demonstrated in a variety of non-human vertebrates. Nonetheless, what drives the estimation of numerosity is not fully understood, as numerosity intrinsically covaries with several other physical characteristics. There is wide debate as to whether the numerical and spatial abilities of vertebrates are processed by a single magnitude system or two different cognitive systems. Adopting a novel approach, we aimed to investigate this issue by assessing the interaction between area size and numerosity, which has never been conceptualized with consideration for subjective experience in non-human animals. We examined whether the same perceptual biases underlying one of the best-known size illusions, the Delboeuf illusion, can be also identified in numerical estimation tasks. We instructed or trained human participants and guppies, small teleost fish, to select a target numerosity (larger or smaller) of squares between two sets that actually differed in their numerosity. Subjects were also presented with illusory trials in which the same numerosity was presented in two different contexts, against a large and a small background, resembling the Delboeuf illusion. In these trials, both humans and fish demonstrated numerical biases in agreement with the perception of the classical version of the Delboeuf illusion, with the array perceived as larger appearing more numerous. Thus, our results support the hypothesis of a single magnitude system, as perceptual biases that influence spatial decisions seem to affect numerosity judgements in the same way.

Nonsymbolic numerosity in sets with illusory-contours exploits a context-sensitive, but contrast-insensitive, visual boundary formation process

The visual mechanisms underlying approximate numerical representation are still intensely debated because numerosity information is often confounded with continuous sensory cues (e.g., texture density, area, convex hull). However, numerosity is underestimated when a few items are connected by illusory contours (ICs) lines without changing other physical cues, suggesting in turn that numerosity processing may rely on discrete visual input. Yet, in these previous works, ICs were generated by black-on-gray inducers producing an illusory brightness enhancement, which could represent a further continuous sensory confound. To rule out this possibility, we tested participants in a numerical discrimination task in which we manipulated the alignment of 0, 2, or 4 pairs of open/closed inducers and their contrast polarity. In Experiment 1, aligned open inducers had only one polarity (all black or all white) generating ICs lines brighter or darker than the gray background. In Experiment 2, open inducers had always opposite contrast polarity (one black and one white inducer) generating ICs without strong brightness enhancement. In Experiment 3, reverse-contrast inducers were aligned but closed with a line preventing ICs completion. Results showed that underestimation triggered by ICs lines was independent of inducer contrast polarity in both Experiment 1 and Experiment 2, whereas no underestimation was found in Experiment 3. Taken together, these results suggest that mere brightness enhancement is not the primary cause of the numerosity underestimation induced by ICs lines. Rather, a boundary formation mechanism insensitive to contrast polarity may drive the effect, providing further support to the idea that numerosity processing exploits discrete inputs.

Perceived number is not abstract

To support the claim that the approximate number system (ANS) represents rational numbers, Clarke and Beck (C&B) argue that number perception is abstract and characterized by a second-order character. However, converging evidence from visual illusions and psychophysics suggests that perceived number is not abstract, but rather, is perceptually interdependent with other magnitudes. Moreover, number, as a concept, is second-order, but number, as a percept, is not.

Number is not just an illusion: Discrete numerosity is encoded independently from perceived size

While seminal theories suggest that nonsymbolic visual numerosity is mainly extracted from segmented items, more recent views advocate that numerosity cannot be processed independently of nonnumeric continuous features confounded with the numerical set (i.e., such as the density, the convex hull, etc.). To disentangle these accounts, here we employed two different visual illusions presented in isolation or in a merged condition (e.g., combining the effects of the two illusions). In particular, in a number comparison task, we concurrently manipulated both the perceived object segmentation by connecting items with Kanizsa-like illusory lines, and the perceived convex-hull/density of the set by embedding the stimuli in a Ponzo illusion context, keeping constant other low-level features. In Experiment 1, the two illusions were manipulated in a compatible direction (i.e., both triggering numerical underestimation), whereas in Experiment 2 they were manipulated in an incompatible direction (i.e., with the Ponzo illusion triggering numerical overestimation and the Kanizsa illusion numerical underestimation). Results from psychometric functions showed that, in the merged condition, the biases of each illusion summated (i.e., largest underestimation as compared with the conditions in which illusions were presented in isolation) in Experiment 1, while they averaged and competed against each other in Experiment 2. These findings suggest that discrete nonsymbolic numerosity can be extracted independently from continuous magnitudes. They also point to the need of more comprehensive theoretical views accounting for the operations by which both discrete elements and continuous variables are computed and integrated by the visual system.

Anisotropy of perceived numerosity: Evidence for a horizontal-vertical numerosity illusion

Many studies have investigated whether numerical and spatial abilities share similar cognitive systems. A novel approach to this issue consists of investigating whether the same perceptual biases underlying size illusions can be identified in numerical estimation tasks. In this study, we required adult participants to estimate the number of white dots in arrays made of white and black dots displayed in such a way as to generate horizontal-vertical illusions with inverted T and L configurations. In agreement with previous literature, we found that participants tended to underestimate the target numbers. However, in the presence of the illusory patterns, participants were less inclined to underestimate the number of vertically aligned white dots. This reflects the perceptual biases underlying horizontal-vertical illusions. In addition, we identified an enhanced illusory effect when participants observed vertically aligned white dots in the T shape compared to the L shape, a result that resembles the length bisection bias reported in the spatial domain. Overall, we found the first evidence that numerical estimation differs as a function of the vertical or horizontal displacement of the stimuli. In addition, the involvement of the same perceptual biases observed in spatial tasks supports the idea that spatial and numerical abilities share similar cognitive processes.