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The number zero holds a special status among numbers, indispensable for developing a comprehensive number theory. 1,2,3,4 Despite its importance in mathematics, the neuronal foundation of zero in the human brain is unknown. We conducted single-neuron recordings in neurosurgical patients 5,6,7 while they made judgments involving nonsymbolic number representations (dot numerosity), including the empty set, and symbolic numbers (Arabic numerals), including numeral zero. Neurons showed responsiveness to either the empty set or numeral zero, but not both. Neuronal activity to zero in both nonsymbolic and symbolic formats exhibited a numerical distance effect, indicating that zero representations are integrated together with countable numerosities and positive integers at the low end of the number line. 8,9 A boundary in neuronal coding existed between the nonsymbolic empty set and small numerosities, correlating with the relative difficulty in discriminating numerosity zero behaviorally. Conversely, no such boundary was found for symbolic zero activity, suggesting that symbolic representations integrate zero with other numerals along the number line, reconciling its outlier role. The status of zero as a special nonsymbolic numerical quantity is reflected in the activity of neurons in the human brain, which seems to serve as a scaffold for more advanced representations of zero as a symbolic number. • Single-neuron recordings in the medial temporal lobe of patients • Neurons signal both nonsymbolic and symbolic zero • Neuronal representations of zero integrated along the number line • Nonsymbolic empty sets distinctly encoded from other small numbers Kutter et al. identify and characterize single-neuron activity in the human brain correlated with the representation of both nonsymbolic (dot numerosities) and symbolic (Arabic numerals) number zero. [ABSTRACT FROM AUTHOR]

While domestic horses can be trained to choose the highest numerosity, spontaneous choice tests led to contrasting results, which in part might have depended upon the adopted experimental protocols. To bring further evidence, we carried out two experiments based on binary choice tests (1:4, 1:2, 2:3, 4:6 apple halves). We tested both horses used for schooling (N = 25) and semi-wild bred horses (N = 10), using two different arenas. In both experiments, horses did not choose the highest quantity, irrespective of test difficulty. Horses preferentially chose the stimulus located at their left, while no other tested variable affected their choice. The consistency between the results suggests that the lack of discrimination did not depend on test protocols. The strong preference for a given side and high within-individual repeatability indicate that lateralization may affect the results of spontaneous tests. [ABSTRACT FROM AUTHOR]

Despite their young age, zebrafish larvae have a well-developed visual system and can distinguish between different visual stimuli. First, we investigated if the first visual surroundings the larvae experience during the first days after hatching shape their habitat preference. Indeed, these animals seem to "imprint" on the first surroundings they see and select visual stimuli accordingly at 7 days post fertilization (dpf). In particular, if zebrafish larvae experience a bar background just after hatching, they later on prefer bars over white stimuli, and vice versa. We then used this acquired preference for bars to investigate innate numerical abilities. We wanted to specifically test if the zebrafish larvae show real numerical abilities or if they rely on a lower-level mechanism—i.e. spatial frequency—to discriminate between two different numerosities. When we matched the spatial frequency in stimuli with different numbers of bars, the larvae reliably selected the higher numerosity. A previous study has ruled out that 7 dpf zebrafish larvae use convex hull, cumulative surface area and density to choose between two numerosities. Therefore, our results indicate that zebrafish larvae rely on real numerical abilities rather than other cues, including spatial frequency, when spontaneously comparing two sets with different numbers of bars. [ABSTRACT FROM AUTHOR]

Our everyday perceptual experiences are grounded in the integration of information within and across our senses. Due to this direct behavioural relevance, cross‐modal integration retains a certain degree of contextual flexibility, even to social relevance. However, how social relevance modulates cross‐modal integration remains unclear. To investigate possible mechanisms, Experiment 1 tested the principles of audio‐visual integration for numerosity estimation by deriving a Bayesian optimal observer model with perceptual prior from empirical data to explain perceptual biases. Such perceptual priors may shift towards locations of high salience in the stimulus space. Our results showed that the tendency to over‐ or underestimate numerosity, expressed in the frequency and strength of fission and fusion illusions, depended on the actual event numerosity. Experiment 2 replicated the effects of social relevance on multisensory integration from Scheller & Sui, 2022 JEP:HPP, using a lower number of events, thereby favouring the opposite illusion through enhanced influences of the prior. In line with the idea that the self acts like a prior, the more frequently observed illusion (more malleable to prior influences) was modulated by self‐relevance. Our findings suggest that the self can influence perception by acting like a prior in cue integration, biasing perceptual estimates towards areas of high self‐relevance. [ABSTRACT FROM AUTHOR]

Vittorio Benussi (1878–1927) is known for numerous studies on optical illusions, visual and haptic perception, spatial and time perception. In Padova, he had a brilliant student who carefully worked on the topic of how people estimate numerosity, Silvia De Marchi (1897–1936). Her writings have never been translated into English before. Here we comment on her work and life, characterized also by the challenges faced by women in academia. The studies on perception of numerosity from her thesis were published as an article in 1929. We provide a translation from Italian, a redrawing of its 23 illustrations and of the graphs. It shows an original experimental approach and an anticipation of what later became known as magnitude estimation. [ABSTRACT FROM AUTHOR]

Whether estimating the size of a crowd or rating a restaurant on a five-star scale, humans frequently navigate between subjective sensory experiences and shared formal systems. Here we ask how people manage this in the case of estimating number. We present participants with arrays of dots and ask them to report how many dots there are. Our results produce two novel findings. First, people's estimates are best fit by a bilinear function in log space, rather than the traditional power law described in previous literature. Second, we find that people's estimates do not have a stable coefficient of variation at higher magnitudes, and that the likely cause of this is a "drift" in people's estimate calibration over many trials which has not previously been identified. Building on these results, we present a model of the mapping function from subjective numerosity to symbolic number that relies primarily on a constrained set of previous estimates and familiar numerosities, rather than the robust internal scale used in existing models. Our model is able to generate an accurate mapping with limited data and reproduce notable aspects of estimation seen in our experimental results. This suggests that human number estimation, and perhaps other domains in which we must navigate between subjective representations and formal systems, is governed by a relatively simple decision process that primarily seeks to maintain consistency with previous estimates. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Introduction: Associating multimodal information is essential for human cognitive abilities including mathematical skills. Multimodal learning has also attracted attention in the field of machine learning, and it has been suggested that the acquisition of better latent representation plays an important role in enhancing task performance. This study aimed to explore the impact of multimodal learning on representation, and to understand the relationship between multimodal representation and the development of mathematical skills. Methods: We employed a multimodal deep neural network as the computational model for multimodal associations in the brain. We compared the representations of numerical information, that is, handwritten digits and images containing a variable number of geometric figures learned through single- and multimodal methods. Next, we evaluated whether these representations were beneficial for downstream arithmetic tasks. Results: Multimodal training produced better latent representation in terms of clustering quality, which is consistent with previous findings on multimodal learning in deep neural networks. Moreover, the representations learned using multimodal information exhibited superior performance in arithmetic tasks. Discussion: Our novel findings experimentally demonstrate that changes in acquired latent representations through multimodal association learning are directly related to cognitive functions, including mathematical skills. This supports the possibility that multimodal learning using deep neural network models may offer novel insights into higher cognitive functions. [ABSTRACT FROM AUTHOR]

Everyday experiences suggest that a container, such as a box of cereal, can convey pertinent information about the nature and quantity of its content. This study investigated how well people can judge large quantities of objects in a container through haptic perception. Stimuli consisted of plastic drinking straws cut to "small" (1.5 cm) or "big" (4.5 cm) pieces contained in plastic food containers. Participants performed both a magnitude estimation of the number of objects and a direct estimation of the proportion of the container perceived to be filled with objects. Overall, participants demonstrated considerable accuracy for both tasks and irrespective of the size of the content. Post-experiment interviews revealed three potential strategies. Participants either focused on the container's contents, the excess space in the container, or the perceived weight of the container (content). [ABSTRACT FROM AUTHOR]

In the literature on numerical cognition, the presence of the capacity to distinguish between numerosities by attending to the number of items, rather than continuous properties of stimuli that correlate with it, is commonly taken as sufficient indication of numerical abilities in cognitive agents. However, this literature does not take into account that there are non-numerical methods of assessing numerosity, which opens up the possibility that cognitive agents lacking numerical abilities may still be able to represent numerosity. In this paper, I distinguish between numerical and non-numerical methods of assessing numerosity and show that the most common models of the internal mechanisms of the so-called number sense rely on non-numerical methods, despite the claims of their proponents to the contrary. I conclude that, even if it is established that agents attend to numerosity, rather than continuous properties of stimuli correlated with it, an answer to the question of the existence of the number sense is still pending the investigation of a further issue, namely, whether the mechanisms the brain uses to assess numerosity qualify as numerical or non-numerical.

Many sensory brain areas are organized as topographic maps where neural response preferences change gradually across the cortical surface. Within association cortices, 7-Tesla fMRI and neural model-based analyses have also revealed many topographic maps for quantities like numerosity and event timing, often in similar locations. Numerical and temporal quantity estimations also show behavioral similarities and even interactions. For example, the duration of high-numerosity displays is perceived as longer than that of low-numerosity displays. Such interactions are often ascribed to a generalized magnitude system with shared neural responses across quantities. Anterior quantity responses are more closely linked to behavior. Here, we investigate whether common quantity representations hierarchically emerge by asking whether numerosity and timing maps become increasingly closely related in their overlap, response preferences, and topography. While the earliest quantity maps do not overlap, more superior maps overlap increasingly. In these overlapping areas, some intraparietal maps have consistently correlated numerosity and timing preferences, and some maps have consistent angles between the topographic progressions of numerosity and timing preferences. However, neither of these relationships increases hierarchically like the amount of overlap does. Therefore, responses to different quantities are initially derived separately, then progressively brought together, without generally becoming a common representation. Bringing together distinct responses to different quantities may underlie behavioral interactions and allow shared access to comparison and action planning systems.

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Everyday experiences suggest that a container, such as a box of chocolate sprinkles, can convey pertinent information about the nature of its content. Despite the familiarity of the experience, we do not know whether people can perceive the number of objects in the container from touch alone and how accurately they can do so. In three experiments, participants handled containers holding between one and five objects and verbally estimated their number. Containers were small cardboard jewelry boxes, and objects were round beads of varying diameter and weight. Any useful visual and auditory cues were precluded. Experiment 1 demonstrated very accurate performance, provided the objects were of sufficient weight. Experiment 2 demonstrated that withholding information about the possible number of objects inside the container does not affect accuracy at a group level but does produce occasional overestimations at an individual level. Experiment 3 demonstrated that removing the weight cue leads to systematic underestimations but does not eliminate people's ability to distinguish between different numbers of objects in the container. This study contributes to a growing picture that container haptics is surprisingly capable. [ABSTRACT FROM AUTHOR]

There are many situations in which we interact with collections of objects, from a crowd of people to a bowl of blackberries. There is an experience of the quantity of these items, although not a precise number, and we have this impression quickly and effortlessly. It can be described as an expressive property of the whole. In the literature, the study of this sense of numerosity has a long history, which is reviewed here with examples. I argue that numerosity is a direct perceptual experience, and that all experiences of numerosity, not only estimations, are affected by perceptual organisation. [ABSTRACT FROM AUTHOR]

Development of the cardinality principle, an understanding that the last number-word recited in counting a collection of items specifies the number of items in that collection, is a critical milestone in developing a concept of number. Researchers in early number development have endeavored to theorize its development. Here we critique two widely respected hypotheses that explain cardinality-principle development as building on young learners' ability to subitize small numbers. These hypotheses consider subitizing to be the basis of cardinality-principle development. We argue that there is a qualitative and significant difference between subitizing and the cardinality principle and that the explanation provided are insufficient to account for a change of this magnitude. We then propose a conjecture intended to better explain this change. The conjecture describes counting as the medium for a series of reflective abstractions leading to the cardinality principle. [ABSTRACT FROM AUTHOR]

Both symbolic (digits) and non-symbolic (dots) numerals are spatially coded, with relatively small numbers being responded faster with a left key and large numbers being responded faster with a right key (spatial–numerical association of response codes [SNARC]). The idea of format independent SNARC seems to support the existence of a common system for symbolic and non-symbolic numerical representations, although evidence in the field is still mixed. The aim of the present study is to investigate whether symbolic and non-symbolic numerals interact in the SNARC effect when both information is simultaneously displayed. To do so, participants were presented with dice-like patterns, with digits being used instead of dots. In two separate magnitude classification tasks, participants had to respond either to the number of digits presented on the screen or to their numerical size. In the non-symbolic task, they had to judge whether the digits on the screen were more or less than three, irrespective of the numerical value of the digits. In the symbolic task, participants had to judge whether the digits on the screen were numerically smaller or larger than three, irrespective of the number of digits being present. The results show a consistent SNARC effect in the symbolic task and no effect in the non-symbolic one. Furthermore, congruency between symbolic and non-symbolic numerals did not modulate the response patterns, thus supporting the idea of independent representations and questioning some propositions of current theoretical accounts. [ABSTRACT FROM AUTHOR]

Ponzo is a familiar name in psychology because of the illusion that takes his name. He had a long and productive career in Italy, and some of his work was translated for international journals already in his lifetime. However, few of these papers are available in English. We provide a commentary that considers how his name came to be associated with an illusion he did not discover. We explain the content of several papers, some of which are often cited in a wrong context in the literature (i.e., papers on touch mentioned in relation to the Ponzo illusion). More importantly, we discuss his contribution to the study of perceived numerosity, and provide a full translation of his important 1928 paper, including a redrawing of its 28 illustrations. [ABSTRACT FROM AUTHOR]

On a now orthodox view, humans and many other animals areendowed with a “number sense”, or approximate number system(ANS), that represents number. Recently, this orthodox view hasbeen subject to numerous critiques, with critics maintaining eitherthat numerical content is absent altogether, or else that someprimitive analog of number (‘numerosity’) is represented as opposedto number itself. We distinguish three arguments for these claims –the arguments from congruency, confounds, and imprecision – andshow that none succeed. We then highlight positive reasons forthinking that the ANS genuinely represents numbers. The upshot isthat proponents of the orthodox view should not feel troubled byrecent critiques of their position.

This study investigated the perception of numbers in humans in 3D stimuli. Recent research has shown that number processing relies on "number sense" for small values, in line with Weber's law. While previous studies have reported 3D numerosity overestimation mainly in higher numerical values, our experiment examined whether this phenomenon occurs at lower numerical values. We also explored whether the Weber ratio follows Weber's law when comparing 2D and 3D stimuli in terms of the number of elements. Observers were presented with pairs of stimuli on a monitor and were asked to identify the stimulus with a larger number of elements. Using the constant method, we calculated the point of subjective equality (PSE), just noticeable difference (JND), and Weber ratios from the collected data. As a result, it was confirmed that the phenomenon of over-estimation of 3D numerical values occurs even when the numerical values are small. Additionally, we observed that the Weber fraction adhered to Weber's law within the measured range. These findings contribute to the existing body of research, supporting the existence of distinct mechanisms for perceiving numerosity and density. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Can you reduce uncertainty by thinking? Intuition suggests that this happens through the elusive process of attention: if we expend mental effort, we can increase the reliability of our sensory data. Models based on "rational inattention" formalize this idea in terms of a trade-off between the costs and benefits of attention. This paper surveys the origin of these models in economics, their connection to rate-distortion theory, and some of their recent applications to psychology and neuroscience. We also report new data from a numerosity judgment task in which we manipulate performance incentives. Consistent with rational inattention, people are able to improve performance on this task when incentivized, in part by increasing the reliability of their sensory data. [ABSTRACT FROM AUTHOR]

If you are stuck in a traffic jam, the more numerous the queuing cars are, the longer you expect to wait. Time and numerosity are stimulus dimensions often associated in the same percept and whose interaction can lead to misjudgements. At brain level it is unclear to which extent time and numerosity recruit same/different neural populations and how their perceptual integration leads to changes in these populations' responses. Here we used high-spatial-resolution functional magnetic resonance imaging with neural model-based analyses to investigate how the topographic representations of numerosity and time change when these dimensions are varied together on the same visual stimulus in a congruent (the more numerous the items, the longer the display time) or incongruent manner. Compared to baseline conditions, where only one dimension was changed at a time, the variation of both stimulus dimensions led to changes in neural population responses that became more sensitive either to the two features or to one of them. Magnitude integration led also to degradation of topographies and shifts in response preferences. These changes were more pronounced in the comparison between parietal and frontal maps. Our results while pointing to partially distinct representations of time and numerosity show a common neural response to magnitude integration. [ABSTRACT FROM AUTHOR]

Visual numerosity is represented automatically and rapidly, but much remains unknown about the computations underlying this perceptual experience. For example, it is unclear whether numerosity is represented with an opponent channel or multichannel coding system. Within an opponent channel system, all numerical values are represented via the relative activity of two pools of neurons (i.e., one pool with a preference for small numerical values and one pool with a preference for large numerical values). However, within a multichannel coding system, all numerical values are represented directly, with separate pools of neurons for each (discriminable) numerical value. Using an adaptation paradigm, we assessed whether the visual perception of number is better characterized by an opponent channel or multichannel system. Critically, these systems make distinct predictions regarding the pattern of aftereffects exhibited when an observer is adapted to an intermediate numerical value. Opponent channel coding predicts no aftereffects because both pools of neurons adapt equally. By contrast, multichannel coding predicts repulsive aftereffects, wherein numerical values smaller than the adapter are underestimated and those larger than the adapter are overestimated. Consistent with multichannel coding, visual adaptation to an intermediate value (50 dots) yielded repulsive aftereffects, such that participants underestimated stimuli ranging from 10-50 dots, but overestimated stimuli ranging from 50-250 dots. These findings provide novel evidence that the visual perception of number is supported by a multichannel, not opponent channel, coding system, and raise important questions regarding the contributions of different cortical regions, such as the ventral and lateral intraparietal areas, to the representation of number. [ABSTRACT FROM AUTHOR]

In four experiments we investigated the Solitaire illusion. In this illusion, most observers see as more numerous a set of dots that forms a single central group, compared to dots on the outside of that group. We confirmed and extended the effect to configurations with much higher numerosity than the original and of various colours. Contrary to prediction, separating the two groups, so that they are presented side by side, reduced but did not abolish or reverse the illusion. In this illusion, therefore, neither total size of the region (area), not average distance of the elements has the expected effect. In Experiments 3 and 4 we eliminated the regularity of the pattern, by sampling 50% (Exp 3) or only a 10% (Exp 4) of the elements. These produces quasi‐random configurations. For these configurations the bias for the inner groups was still present, and it was only eliminated when the groups were shown as separate. However, the effect never reversed (no bias for the outer group, despite its larger area). We conclude that the Solitaire illusion is evidence of a strong bias in favour of centrally located elements, a bias that can overcome other factors. [ABSTRACT FROM AUTHOR]

In this study, we investigated the influence of negative emotions on numerosity estimation and whether this influence changes with aging during adulthood. Young and older adults were asked to estimate and compare the numerosity of collections of elements (cars or dots) with a two-digit number. Collections of elements were preceded by emotionally neutral (e.g., mushrooms) or emotionally negative (e.g., a corpse) pictures. Stimuli were easier (i.e., small-ratio) or harder (i.e., large-ratio) items. Young and older participants obtained similar numerosity estimation performance. Interestingly, participants were less accurate under negative emotions than under neutral emotions when they estimated numerosity of collections of abstract elements (i.e., dots). In contrast, participants improved their performance under negative emotions while estimating collections of non-abstract, daily-life elements (i.e., cars). These findings have important implications for furthering our understanding of the role of negative emotions in numerosity estimation and age-related differences therein. [ABSTRACT FROM AUTHOR]

In the literature on numerical cognition, the presence of the capacity to distinguish between numerosities by attending to the number of items, rather than continuous properties of stimuli that correlate with it, is commonly taken as sufficient indication of numerical abilities in cognitive agents. However, this literature does not take into account that there are non-numerical methods of assessing numerosity, which opens up the possibility that cognitive agents lacking numerical abilities may still be able to represent numerosity. In this paper, I distinguish between numerical and non-numerical methods of assessing numerosity and show that the most common models of the internal mechanisms of the so-called number sense rely on non-numerical methods, despite the claims of their proponents to the contrary. I conclude that, even if it is established that agents attend to numerosity, rather than continuous properties of stimuli correlated with it, an answer to the question of the existence of the number sense is still pending the investigation of a further issue, namely, whether the mechanisms the brain uses to assess numerosity qualify as numerical or nonnumerical. [ABSTRACT FROM AUTHOR]

IntroductionWhen items are connected together, they tend to be perceived as an integrated whole rather than as individual dots, causing a strong underestimation of the numerosity of the ensemble. Previous evidence on grouping-induced biases of numerosity has shown a dependency on autistic-like personality traits in neurotypical adults, with a weaker tendency for grouping into meaningful segmented objects in individuals with strong autistic traits. Here we asked whether this result would generalize to the autistic population.MethodsTwenty-two adults with a diagnosis of Autism Spectrum Disorder (ASD) and 22 matched neurotypical controls judged the numerosity of clouds of dot-pairs connected by thin lines.ResultsResults showed no significant group difference in discrimination precision, suggesting that both groups were equally capable performing the task. However, while connecting pairs of dots at moderate numerosities caused large changes in apparent numerosity in the neurotypical controls, particularly those with low autistic-like traits, it had little effect in the group of autistic participants, suggesting significant differences in numerosity estimation between autistic and neurotypical perception. Consistent with earlier studies, the magnitude of the effect covaried strongly with AQ-defined autistic traits in the neurotypical range, reinforcing the idea that autistic traits predict the strength of grouping.DiscussionThese results provide strong support for the theories of autistic perception that highlight dissimilarities in global vs. local processing, and open the door to study grouping mechanisms indirectly, by asking participants to report on the apparent numerosity rather than on the grouping organization per se.

Magnitude information is often correlated in the external world, providing complementary information about the environment. As if to reflect this relationship, the perceptions of different magnitudes (e.g. time and numerosity) are known to influence one another. Recent studies suggest that such magnitude interaction is similar to cue integration, such as multisensory integration. Here, we tested whether human observers could integrate the magnitudes of two quantities with distinct physical units (i.e. time and numerosity) as abstract magnitude information. The participants compared the magnitudes of two visual stimuli based on time, numerosity, or both. Consistent with the predictions of the maximum-likelihood estimation model, the participants integrated time and numerosity in a near-optimal manner; the weight of each dimension was proportional to their relative reliability, and the integrated estimate was more reliable than either the time or numerosity estimate. Furthermore, the integration approached a statistical optimum as the temporal discrepancy of the acquisition of each piece of information became smaller. These results suggest that magnitude interaction arises through a similar computational mechanism to cue integration. They are also consistent with the idea that different magnitudes are processed by a generalized magnitude system.

A significant amount of research on numerosity demonstrates that product perceptions are often influenced by the scale on which numerical attribute information is presented. However, fewer studies have examined how self-control is influenced by the numerosity of cost information (e.g. price, nutritional content) in situations that may violate a personal goal. The present research demonstrates that, in such situations, the numerosity of cost information has a stronger influence on self-control when consumers are highly focused on restraint. Because restrained consumers regulate their behavior by anticipating the negative emotions from violating their goals, they experience a narrowing of attention during self-control decisions that makes them more reliant on numerosity as a cue for judgment. The results of eight experiments demonstrate that consumers who are primed or predisposed to be high in restraint display less self-control when cost information is presented on a contracted scale with small numbers compared to an expanded scale with large numbers. When consumers are less focused on restraint, numerosity has less of an effect on self-control because unrestrained consumers do not experience an analogous narrowing of attention. [ABSTRACT FROM AUTHOR]

Consumers typically infer greater quantity from larger numbers. For instance, a 500 gram box of chocolates appears heavier than a.5 kilogram box. By expressing quantities in alternative units or attribute dimensions, one can represent an otherwise identical quantity in a more versus less discretized manner (e.g. a box containing 25 chocolates vs. a box weighing 500 grams). Seven experimental studies show that a difference between more discretized quantities (e.g. 25 vs. 50 chocolates) appears larger relative to a difference between less discretized quantities (e.g. 500 grams vs. 1,000 grams), above and beyond effects of number magnitude. More discretized quantity expressions enhance the consumers' ability to discriminate between choice options and can also nudge consumers to more favorable choices. Because more discretized quantities are more evaluable, expressing a quantity in terms of a collection of elements particularly helps individuals who are less numerically proficient. By identifying how discretization functions as a novel antecedent of evaluability and by distinguishing two different conceptualizations of numerosity (i.e. symbolic and perceptual numerosity), this article draws important connections between the numerical cognition literature and General Evaluability Theory. [ABSTRACT FROM AUTHOR]

Modern society frequently requires that we express our sub-jective senses in objective, shared formal systems; this en-tails mapping multiple internal variables onto a common scale.Here we ask whether we accomplish this feat in the case ofestimating number by learning a single mapping between ex-plicit numbers and one integrated subjective estimate of nu-merosity, or if we separately map different perceptual featuresonto numbers. We present people with arrays of dots and askthem to report how many dots there are; we rely on the sys-tematic under/overestimation of number at higher quantities toestimate error in the mapping function. By comparing how thiserror changes over time, as the mapping fluctuates for differentvisual cues to numerosity, we can evaluate whether these cuesshare a single mapping, or are mapped onto number individu-ally. We find that area, size, and density all share a commonmapping, indicating that people obtain a unified subjective es-timate of numerosity before mapping it onto the formal numberline.

Numerosity is informative for living organisms. It cantransmit, among many things, amount of food available,heading direction of the troop, which group could win aterritorial dispute, the decision of were to build a beehive.Given its ecological importance, we test the hypothesisthat numerosity captures visual selection. In five exper-iments we confirmed that an irrelevant visual stimulusthat was numerically large slowed down participants indetecting a task-relevant visual target (Exp. 1 and 2). Thiscapture was not driven by sensory variables that could cor-relate with numerosity: cumulative area (Exp. 3) and ele-ment size (Exp. 4). We also confirmed that the underlyingnumerosity representations were analogue, not set-based(Exp. 5). In a crowded visual scene numerosity is a rele-vant cue for visual selection, but represented only in ap-proximate/coarse fashion.

Visual numerosity is represented automatically and rapidly, but much remains unknown about the computations underlying this perceptual experience. For example, it is unclear whether numerosity is represented with an opponent channel or multichannel coding system. Within an opponent channel system, all numerical values are represented via the relative activity of two pools of neurons (i.e., one pool with a preference for small numerical values and one pool with a preference for large numerical values). However, within a multichannel coding system, all numerical values are represented directly, with separate pools of neurons for each (discriminable) numerical value. Using an adaptation paradigm, we assessed whether the visual perception of number is better characterized by an opponent channel or multichannel system. Critically, these systems make distinct predictions regarding the pattern of aftereffects exhibited when an observer is adapted to an intermediate numerical value. Opponent channel coding predicts no aftereffects because both pools of neurons adapt equally. By contrast, multichannel coding predicts repulsive aftereffects, wherein numerical values smaller than the adapter are underestimated and those larger than the adapter are overestimated. Consistent with multichannel coding, visual adaptation to an intermediate value (50 dots) yielded repulsive aftereffects, such that participants underestimated stimuli ranging from 10–50 dots, but overestimated stimuli ranging from 50–250 dots. These findings provide novel evidence that the visual perception of number is supported by a multichannel, not opponent channel, coding system, and raise important questions regarding the contributions of different cortical regions, such as the ventral and lateral intraparietal areas, to the representation of number.

In the present study, we investigated the effects of auditory numerosity and magnitude (loudness) on visual numerosity processing. Participants compared numerosities of two sequential dot arrays. The second dot array was paired with a tone array that was independent of visual comparison. The numerosity (One-tone vs. Multiple-tone) and the non-numerical magnitude of tones (loudness) were manipulated in Experiments 1 and 2, respectively. In Experiment 1, participants' inverse efficiency score (IES), that is, the quotient between response time and accuracy, was significantly smaller in the One-tone and Multiple-tone conditions than that in the No-tone condition, and linear trend analyses showed that the IES decreased with the number of tones. In Experiment 2, the IES in the Loud-tone condition was significantly smaller than that in the No-tone condition, and the IES decreased as the loudness of the tones increased. In Experiment 3, both auditory numerosity and magnitude were manipulated. For soft tones, the IES was smaller in the Multiple-tone condition than in the One-tone condition, whereas no significant difference was found between two conditions in loud tones. In sum, these findings suggest that the visual numerical representation can be spontaneously affected by the numerosity and non-numerical magnitude of stimuli from another modality. [ABSTRACT FROM AUTHOR]

• The single-trial ERP-behavioral correlation approach was applied to show how the ERP was associated with behavior. Accumulated functional magnetic resonance imaging (fMRI) and electroencephalography evidence indicate that numerosity is first processed in the occipito-parietal cortex. fMRI evidence also indicates right-lateralized processing of numerosity, but there is no consistent evidence from event-related potential (ERP) studies. This study investigated the ERP of numerosity processing in the left, right, and bilateral visual fields. The single-trial ERP-behavioral correlation was applied to show how the ERP was associated with behavioral responses. The results showed a significant early behavioral-ERP correlation on the right N1 component when stimuli were presented in the left visual field rather than in the right visual field. The behavioral ERP correlation was termed BN1. There was bilateral BN1 based on the reaction time or error rate, but the right BN1 was larger than that the left BN1 when the stimulus was present in the bilateral visual field. Therefore, this study provided a new neural marker for individual differences in processing numerosity and suggested that processing numerosity was supported by the right occipito-parietal cortex. [ABSTRACT FROM AUTHOR]

There are many situations in which we interact with collections of objects, from a crowd of people to a bowl of blackberries. There is an experience of the quantity of these items, although not a precise number, and we have this impression quickly and effortlessly. It can be described as an expressive property of the whole. In the literature, the study of this sense of numerosity has a long history, which is reviewed here with examples. I argue that numerosity is a direct perceptual experience, and that all experiences of numerosity, not only estimations, are affected by perceptual organisation.

Most animals, from humans to invertebrates, possess an ability to estimate numbers. This evolutionary advantage facilitates animals’ choice of environments with more food sources, more conspecifics to increase mating success, and/or reduced predation risk among others. However, how the brain processes numerical information remains largely unknown. There are currently two lines of research interested in how numerosity of visual objects is perceived and analyzed in the brain. The first argues that numerosity is an advanced cognitive ability processed in high-order brain areas, while the second proposes that “numbers” are attributes of the visual scene and thus numerosity is processed in the visual sensory system. Recent evidence points to a sensory involvement in estimating magnitudes. In this Perspective, we highlight this evidence in two highly evolutionary distant species: humans and flies. We also discuss the advantages of studying numerical processing in fruit flies in order to dissect the neural circuits involved in and required for numerical processing. Based on experimental manipulation and the fly connectome, we propose a plausible neural network for number sense in invertebrates.

In the study of consciousness, qualia, the individual subjective experience, is neglected. It remains impenetrable because the objective perspective used for scientific investigations misses its subjective nature. In 1974, Thomas Nagel suggested that studying qualia requires an "objective phenomenology method" whose goal would be to describe the subjective character of experiences in an independent manner. We introduce a corresponding theoretical and experimental framework based on the "idiosyncrasy principle." Accordingly, subjectivity depends on the idiosyncratic composition of physical properties to qualia. This allows conceptualizing an idiosyncratic transfer function between the world and its representation. The main challenge in delineating such a transfer function is to come up with an objective measure for another person's perspective. Numerosity, as opposed to other perceived contents of the physical world, allows reporting subjective experience in an objective manner. On the basis of this unique attribute of numerosity, we suggest a tentative neurocognitive research plan aimed at delineating such idiosyncratic transfer functions, permitting one person to adopt the perspective of another and linking qualia to its mechanism. [ABSTRACT FROM AUTHOR]

The "mental number line" (MNL) is a form of spatial numeric representation that associates small and large numbers with the left and right spaces, respectively. This spatio-numeric organization can be found in adult humans and has been related to cultural factors such as writing and reading habits. Yet, both human newborns and birds order numbers consistently with an MNL, thus raising the question of whether culture is a main explanation for MNL. Here, we explored the numeric sense of honey bees and show that after being trained to associate numbers with a sucrose reward, they order numbers not previously experienced from left to right according to their magnitude. Importantly, the location of a number on that scale varies with the reference number previously trained and does not depend on low-level cues present on numeric stimuli. We provide a series of neural explanations for this effect based on the extensive knowledge accumulated on the neural underpinnings of visual processing in honey bees and conclude that the MNL is a form of numeric representation that is evolutionarily conserved across nervous systems endowed with a sense of number, irrespective of their neural complexity. [ABSTRACT FROM AUTHOR]

In 1939, the influential psychophysicist S. S. Stevens proposed definitional distinctions between the terms 'number,' 'numerosity,' and 'numerousness.' Although the definitions he proposed were adopted by syeveral psychophysicists and experimental psychologists in the 1940s and 1950s, they were almost forgotten in the subsequent decades, making room for what has been described as a "terminological chaos" in the field of numerical cognition. In this paper, I review Stevens's distinctions to help bring order to this alleged chaos and to shed light on two closely related questions: whether it is adequate to speak of a number sense and how philosophers can make sense of the claim by cognitive scientists that numbers are perceptual entities. Moreover, I offer further support to Stevens's distinction between numerosity and numerousness by showing that they are relational properties that emerge through different relationships between agents and their environments. The final conclusion is that, by adopting Stevens's distinctions, numbers do not need to be seen as perceptual entities, since the so-called number sense is better described as a sense of numerousness. [ABSTRACT FROM AUTHOR]

There is ample evidence from literature and clinical practice indicating mathematical difficulties in individuals with ADHD, even when there is no concomitant diagnosis of developmental dyscalculia. What factors underlie these difficulties is still an open question. Research on dyscalculia and neurotypical development suggests visual perception of numerosity (the number sense) as a building block for math learning. Participants with lower numerosity estimation thresholds (higher precision) are often those with higher math capabilities. Strangely, the role of numerosity perception in math skills in ADHD has been neglected, leaving open the question whether math difficulties in ADHD also originate from a deficitary visual number sense. In the current study we psychophysically measured numerosity thresholds and accuracy in a sample of children/adolescents with ADHD, but not concomitant dyscalculia (N = 20, 8–16 years). Math abilities were also measured by tasks indexing different mathematical competences. Numerosity performance and math scores were then compared to those obtained from an age-matched control group (N = 20). Bayesian statistics indicated no difference between ADHD and controls on numerosity perception, despite many of the symbolic math tasks being impaired in participants with ADHD. Moreover, the math deficits showed by the group with ADHD remained substantial even when numerosity thresholds were statistically regressed out. Overall, these results indicate that math difficulties in ADHD are unlikely to originate from an impaired visual number sense. [ABSTRACT FROM AUTHOR]

Animals' negotiations with the environment frequently involve quantitative assessments. However, it is largely unknown how different nervous systems can deal with information about magnitude and perform operations on it. Here we review some of the literature on this topic and discuss a few issues worthy of debate that can guide future research directions. First, we present experimental evidence suggesting that, in addition to the cortical (pallial) brain regions that are widely acknowledged to play a role in magnitude estimation, subcortical (more generally, subpallial) brain regions also play an important role. Second, we discuss interactions between different domains of magnitude and put forward a hypothesis to account for the directionality of associations between discrete and continuous magnitude. Finally, we suggest how the distinction between the concepts of number and discrete quantity should foster more attention to the role of sensory areas and circuits in assessing discrete quantities. [ABSTRACT FROM AUTHOR]

Background: Despite the interest and potential of multi touch devices, there are limited published studies researching their effectiveness and usability specifically with children with Down syndrome, one of the most common groups of children with an intellectual disability. This is particularly true for mathematical learning, an area in which many experience particular difficulty. Objectives: The study set out to evaluate a bespoke digital game in which children learned to select which was "more," a foundational skill for understanding magnitude. Methods: A mixed methods approach was adopted with eight single case studies of children aged 9–14 years. Probes (untaught examples) were introduced to examine progress and the impact on performance in non‐digital contexts. Results: Five pupils improved their performance on the digital games and this was sustained at the time of the delayed post‐test. Four pupils showed improved performance in non‐taught, non‐digital contexts for both taught and untaught ratios. Disaggregated data reveals the variability in performance, with peak performances occurring at different points of the intervention. The introduction of a two‐player version improved performance for five pupils through promoting sustained attention and strategic responses to winning. One child performed at chance level on the digital game throughout but made gains in non‐digital settings. Conclusions: Bespoke learning tools have good potential to promote attention to numerosity. However, progress in digital contexts does not automatically transfer to non‐digital contexts. The study reveals the individual nature of the learning affordances of different pedagogic tools and the place of bespoke games within teachers' repertoires. Lay Description: Research suggests that it is important to tailor intervention to the particular strengths and needs of children with Down syndrome but there are few bespoke games in the field of mathematics.This study focuses on the ability to discriminate between quantities, a foundation for mathematical learning.Children's discrimination of "which is more" improved during the period of intervention and at follow‐up. They did not however automatically transfer these skills to other contexts. Many attended best when playing the game with another person. Additionally, one pupil performed better on non‐digital activities.The study emphasizes the importance of practitioners looking carefully at each pupil's performance in a range of digital and nondigital contexts, to identify which best meets their current learning needs. [ABSTRACT FROM AUTHOR]

Observers can quickly estimate the quantity of sets of visual elements. Many aspects of this ability have been studied and the underlying system has been called the Approximate Number Sense (Dehaene, 2011). Specific visual properties, such as size and clustering of the elements, can bias an estimate. For intermediate numerical quantities at low density (above five, but before texturization), human performance is predicted by a model based on the region of influence of elements (occupancy model: Allïk & Tuulmets, 1991). For random 2D configurations we computed ten indices based on graph theory, and we compared them with the occupancy model: independence number, domination, connected components, local clustering coefficient, global clustering coefficient, random walk, eigenvector centrality, maximum clique, total degree of connectivity, and total edge length. We made comparisons across a range of parameters, and we varied the size of the region of influence around each element. The analysis of the pattern of correlations suggests two main groups of graph-based measures. The first group is sensitive to the presence of local clustering of elements, the second seems more sensitive to density and the way information spreads in graphs. Empirical work on perception of numerosity may benefit from comparing, or controlling for, these properties. [ABSTRACT FROM AUTHOR]