Peripheral vision has different functional priorities for mammals than foveal vision. One of its roles is to monitor the environment while central vision is focused on the current task. Becoming distracted too easily would be counterproductive in this perspective, so the brain should react to behaviourally relevant changes. Gist processing is good for this purpose, and it is therefore not surprising that evidence from both functional brain imaging and behavioural research suggests a tendency to generalize and blend information in the (...) periphery. This may be caused by the balance of perceptual influence in the periphery between bottom-up and top-down processing channels. Here, we investigated this interaction behaviourally using a peripheral numerosity discrimination task with top-down and bottom-up manipulations. Participants compared numerosity between the left and right peripheries of a screen. Each periphery was divided into a centre and a surrounding area, only one of which was a task relevant target region. Our top-down task modulation was the instruction which area to attend – centre or surround. We varied the signal strength by altering the stimuli durations i.e., the amount of information presented/processed. We found that numerosity perceived in target regions was affected by contextual information in neighbouring areas. This effect appeared as soon as stimulus duration allowed the task to be reliably performed and persisted even at the longest duration. We compared the pattern of results with an ideal-observer model and found a qualitative difference in the way centre and surround areas interacted perceptually in the periphery. When participants reported on the central area, the irrelevant surround would affect the response as a weighted combination – consistent with the idea of a receptive field focused in the target area to which irrelevant surround stimulation leaks in. When participants report on surround, we can best describe the response with a model in which occasionally the attention switches from task relevant surround to task irrelevant centre – consistent with a selection model of two competing streams of information. Overall our results show that the influence of spatial context in the periphery is mandatory but task dependent. (shrink)

The common view of the transition between subitizing and numerosity estimation regimes is that there is a hard bound on the subitizing range, and beyond this range, people estimate. However, this view does not adequately address the behavioral signatures of enumeration under conditions of attentional load or in the immediate post-subitizing range. The possibility that there might exist a numerosity range where both processes of subitizing and estimation operate in conjunction has so far been ignored. Here, we investigate (...) this new proposal, that people strategically combine the processes of subitizing and estimation to maximize accuracy and precision, given time or attentional constraints. We present a process-level account of how subitizing and estimation can be combined through strategic deployment of attention to maximize the precision of perceived numerosity given time constraints. We then describe a computational model of this account and apply it in two experimental simulations to demonstrate how it can explain key findings in prior enumeration research. While recent modeling work has argued that the behavioral signatures of enumeration can best be explained through a single numerosity system with a single form of representation, we argue that our model demonstrates how the traditional two-systems view of numerical representation accounts for behavioral data through coordination with a unified attentional mechanism, rather than a unified representation. (shrink)

It has been claimed that empirical work in psychology requires the attribution of representational content to perceptual states: that is, the attribution of veridicality conditions to those states. This is a claim that can only be evaluated by the examination of actual empirical research. In this paper I argue that talk of ‘representation’ in at least one area of research in the psychology of perception can be reinterpreted so as to avoid the attribution of veridicality conditions. This area is (...) the study of human capacities to perceive the relative numerosities of collections of objects. (shrink)

Leibovich et al. opened up an important discussion on the nature and origins of numerosity perception. The authors rightly point out that non-numerical features of stimuli influence this ability. Despite these biases, there is evidence that from birth, humans perceive and represent numerosities, and not just non-numerical quantitative features such as item size, density, and convex hull.

Numerosity perception is a key ability for human and non-human species, probably mediated by dedicated brain mechanisms. Electrophysiological studies revealed the existence of both early and mid-latency components of the Electrophysiological (EEG) signal sensitive to numerosity changes. However, it is still unknown whether these components respond to physical or perceived variation in numerical attributes. We here tackled this point by recording electrophysiological signal while participants performed a numerosity adaptation task, a robust psychophysical method yielding changes in (...) perceived numerosity judgments despite physical numerosity invariance. Behavioral measures confirmed that the test stimulus was consistently underestimated when presented after a high numerous adaptor while perceived as veridical when presented after a neutral adaptor. Congruently, EEG results revealed a potential at around 200 ms (P2p) which was reduced when the test stimulus was presented after the high numerous adaptor. This result was much prominent over the left posterior cluster of electrodes and correlated significantly with the amount of adaptation. No earlier modulations were retrievable when changes in numerosity were illusory while both early and mid-latency modulations occurred for physical changes. Taken together, our results reveal that mid-latency P2p mainly reflects perceived changes in numerical attributes, while earlier components are likely to be bounded to the physical characteristics of the stimuli. These results suggest that short-term plastic mechanisms induced by numerosity adaptation may involve a relatively late processing stage of the visual hierarchy likely engaging cortical areas beyond the primary visual cortex. Furthermore, these results also indicate mid-latency electrophysiological correlates as a signature of the internal representation of numerical information. (shrink)

Clarke and Beck assume that approximate number system representations should be assigned referents from our scientific ontology. However, many representations, both in perception and cognition, do not straightforwardly refer to such entities. If we reject Clarke and Beck's assumption, many possible contents for ANS representations besides number are compatible with the evidence Clarke and Beck cite.

Perceptual judgments result from a dynamic process, but little is known about the dynamics of number‐line estimation. A recent study proposed a computational model that combined a model of trial‐to‐trial changes with a model for the internal scaling of discrete numbers. Here, we tested a surprising prediction of the model—a situation in which children's estimates of numerosity would be better than those of adults. Consistent with the model simulations, task contexts led to a clear developmental reversal: children made more (...) adult‐like, linear estimates when to‐be‐estimated numbers were descending over trials (i.e., backward condition), whereas adults became more like children with logarithmic estimates when numbers were ascending (i.e., forward condition). In addition, adults’ estimates were subject to inter‐trial differences regardless of stimulus order. In contrast, children were not able to use the trial‐to‐trial dynamics unless stimuli varied systematically, indicating the limited cognitive capacity for dynamic updates. Together, the model adequately predicts both developmental and trial‐to‐trial changes in number‐line tasks. (shrink)

There is ongoing debate about whether number is perceived directly. Clarke and Beck suggest that what plagues this debate is a lack of shared understanding about what it means to perceive number in the first place. I agree. I argue that the perception of number is held to a different standard than, say, the perception of objecthood; considering this, I explore what it might mean for the number system to represent rational numbers.

One of the most prominent social influences on human decision making is conformity, which is even more prominent when the perceptual information is ambiguous. The Bayes optimal solution to this problem entails weighting the relative reliability of cognitive information and perceptual signals in constructing the percept from self‐sourced/endogenous and social sources, respectively. The current study investigated whether humans integrate the statistics (i.e., mean and variance) of endogenous perceptual and social information in a Bayes optimal way while estimating numerosities. Our results (...) demonstrated adjustment of initial estimations toward group means only when group estimations were more reliable (or “certain”), compared to participants’ endogenous metric uncertainty. Our results support Bayes optimal social conformity while also pointing to an implicit form of metacognition. (shrink)

Less salient, lower contrast disks appear to be more numerous than more salient, higher contrast disks when intermingled in equal numbers into the same display, but they are equal in perceived numerosity when segregated into different displays. Comparative judgements indicate that the apparent numerosity of the lower contrast disks is unaffected by being intermingled with high contrast disks, whereas the high contrast disks are reduced in numerosity by being intermingled with the low contrast ones. Here, we report (...) that this illusion also occurs for absolute judgements of the numerosities of displays of from 20 to 80 disks. A model based on luminance-difference contrast normalization explains the illusory loss of high-contrast items along with veridical perception of the low-contrast ones. The model correctly predicts that perceived numerosity is linearly related to the square-root of the number of disks, with the extent of the illusion depending on an attentionally-weighted function of contrast and assimilation. (shrink)

: Fodor argues that perceptual representations are a subset of iconic representations, which are distinguished from symbolic/discursive representations. Iconic representations are nonconceptual and they do not support the abilities afforded by concepts. Iconic representations, for example, cannot support object individuation. If someone thinks that perception or some of its parts has imagistic NCC, they face the following dilemma. Either they will have to accept that this NCC does not allow for object individuation, but it represents instead conglomerations of properties (...) and at some stage of visual processing it must interface with cognition and its conceptual capacities for the visual objects to be individuated. Or, they will have to hold that the imagistic, NCC of perception, allows for object individuation. I opt for the second thesis because I think there is strong empirical evidence that objects are individuated during early vision. I also think that early vision individuates objects by means of, what I had previously called nonconceptual perceptual demonstrative reference. I argue, first, why Fodor’s view that iconic NCC does not enable object individuation is false. I also argue, contra Fodor, that early vision allows the perception of the cardinality of sets of objects. Keywords : Early Vision; Analog Representations; Object Individuation; Arithmetic Cognition; Cardinality of Sets L'individuazione di oggetti mediante il contenuto iconico: come è rappresentata la numerosità nella rappresentazione iconica? Riassunto: Per Fodor le rappresentazioni percettive sono un sottoinsieme delle rappresentazioni iconiche, distinte dalle rappresentazioni simbolico/discorsive. Le rappresentazioni iconiche sono nonconcettuali e non supportano le abilità richieste dai concetti. Le rappresentazioni iconiche, per esempio, non supportano l'individuazione di oggetti. Se si pensa che la percezione o qualche sua parte abbia un contenuto nonconcettuale come immagine, si ci si imbatte nel seguente dilemma. O si accetta che il NCC non permetta di individuare oggetti, ma che rappresenti conglomerati di proprietà e che si interfacci con la cognizione e le sue capacità concettuali, per individuare gli oggetti visivi. O si dice che l'immagine, il NCC, della percezione consenta di individuare oggetti. Propendo per la seconda tesi, poiché penso che ci sia solida evidenza empirica per cui gli oggetti vengono individuati durante le prime fasi della visione, in cui l’individuazione avviene mediante quanto definito come riferimento dimostrativo del contenuto nonconcettuale percettivo. Chiarirò le ragioni per cui la concezione di Fodor, per cui il contenuto nonconcettuale iconico non supporta l’individuazione di oggetti, è falsa. Inoltre, contra Fodor, sosterrò che le prime fasi della visione permettono la percezione della cardinalità di insiemi di oggetti. Parole chiave: Prime fasi della visione; Rappresentazioni analogiche; Individuazione di oggetti; Cognizione aritmetica; Cardinalità degli insiemi. (shrink)

Clarke and Beck's defense of the theoretical construct “approximate number system” is flawed in serious ways – from biological misconceptions to mathematical naïveté. The authors misunderstand behavioral/psychological technical concepts, such as numerosity and quantical cognition, which they disdain as “exotic.” Additionally, their characterization of rational numbers is blind to the essential role of symbolic reference in the emergence of number.

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. (shrink)

Over the past 50 years, philosophers and psychologists have perennially argued for the existence of analog mental representations of one type or another. This study critically reviews a number of these arguments as they pertain to three different types of mental representation: perceptual representations, imagery representations, and numerosity representations. Along the way, careful consideration is given to the meaning of “analog” presupposed by these arguments for analog mental representation, and to open avenues for future research.

Weber-Fechner Law states that the perceived intensity is proportional to the logarithm of the stimulus. Recent experiments suggest that this law also holds true for perception of numerosity. Therefore, the use of a logarithmic scale for the quantification of the perceived intensity may also depend on how the cognitive apparatus processes information. If Weber-Fechner law is the result of natural selection, then the logarithmic scale should be better, in some sense, than other biologically feasible scales. We consider the (...) minimization of the relative error as the target of natural selection and we provide a formal proof that the logarithmic scale minimizes the maximal relative error. (shrink)

People’s awareness of tactile stimuli has been investigated in far less detail than their awareness of stimuli in other sensory modalities. In an attempt to fill this gap, we provide an overview of studies that are pertinent to the topic of tactile consciousness. We discuss the results of research that has investigated phenomena such as “change blindness”, phantom limb sensations, and numerosity judgments in tactile perception, together with the results obtained from the study of patients affected by deficits (...) that can adversely affect tactile perception such as neglect, extinction, and numbsense. The similarities as well as some of the important differences that have emerged when visual and tactile conscious information processing have been compared using similar experimental procedures are highlighted. We suggest that conscious information processing in the tactile modality cannot be separated completely from the more general processing of spatial information in the brain. Finally, the importance of considering tactile consciousness within the larger framework of multisensory information processing is also discussed. (shrink)

What constitutes our number concept? What makes it possible for us to employ numbers the way we do; which mental faculties contribute to our grasp of numbers? What do we share with other species, and what is specific to humans? How does our language faculty come into the picture? This 2003 book addresses these questions and discusses the relationship between numerical thinking and the human language faculty, providing psychological, linguistic and philosophical perspectives on number, its evolution and its development in (...) children. Heike Wiese argues that language as a human faculty plays a crucial role in the emergence of systematic numerical thinking. She characterises number sequences as powerful and highly flexible mental tools that are unique to humans and shows that it is language that enables us to go beyond the perception of numerosity and to develop such mental tools. (shrink)

Although many researchers theorize that primitive numerosity processing abilities may lay the foundation for whole number concepts, other classes of numbers, like fractions, are sometimes assumed to be inaccessible to primitive architectures. This research presents evidence that the automatic processing of nonsymbolic magnitudes affects processing of symbolic fractions. Participants completed modified Stroop tasks in which they selected the larger of two symbolic fractions while the ratios of the fonts in which the fractions were printed and the overall sizes of (...) the compared fractions were manipulated as irrelevant dimensions. Participants were slower and less accurate when nonsymbolic dimensions of printed fractions were incongruent with the symbolic comparison decision. Results indicated a robust basic sensitivity to nonsymbolic ratios that exceeds prior conceptions about the accessibility of fraction values. Results also indicated a congruity effect for overall fraction size, contrary to findings of prior research. These findings have implications for extending theory about the nature of human number sense and mathematical cognition more generally. (shrink)

To understand the number sense, we need to understand its function. We argue that numerosity estimation is fundamental not only for perception, but also preparation and control of action. We outline experiments that link numerosity estimation with action, pointing to a generalized numerosity system that serves both perception and action preparation.

Leibovich et al. propose that non-symbolic numerosity abilities develop from the processing of more basic, continuous magnitudes such as size, area, and density. Here I review similar arguments arising in the visual perception field and further propose that the evolvement of discrete representations from continuous stimulus properties may be a fundamental characteristic of cognitive development.

In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system represents numbers or numerosities, and why the ANS represents rational numbers.

Epistemic consequentialism has been challenged on the grounds that it is overly demanding. According to the Epistemic Junk Problem, this view implies that we are often required to believe junk propositions such as ‘the Great Bear Lake is the largest lake entirely in Canada’ and long disjunctions of things we already believe. According to the Numerosity Problem, this view implies that we are frequently required to have an enormous number of beliefs. This paper puts forward a novel version of (...) epistemic consequentialism which avoids these twin demandingness problems. The key is to recognise, first, that the final epistemic value of a true belief depends at least in part on the duration for which it is retained by the agent and, second, that our cognitive makeup places important constraints on which beliefs are retained and for how long. (shrink)

In describing numerosity as “a kind of ersatz number,” Clarke and Beck fail to consider a familiar and compelling definition of numerosity, which conceptualizes numerosity as the cognitive counterpart of the mathematical concept of cardinality; numerosity is the magnitude, whereas number is a scale through which numerosity/cardinality is measured. We argue that these distinctions should be considered.

Leibovich et al. challenge the prevailing view that non-symbolic number sense is innate, that detection of numerosity is distinct from detection of continuous magnitude. In the present commentary, the authors' viewpoint is discussed in light of the integrative theory of numerical development along with implications for understanding mathematics disabilities.

Early modern empiricists thought that the nature of perceptual experience is given by citing the object presented to the mind in that experience. Hallucination and illusion suggest that this requires untenable mind-dependent objects. Current orthodoxy replaces the appeal to direct objects with the claim that perceptual experience is characterized instead by its representational content. This paper argues that the move to content is problematic, and reclaims the early modern empiricist insight as perfectly consistent, even in cases of illusion, with the (...) realist contention that these direct objects of perception are the persisting mind-independent physical objects we all know and love. (shrink)

The ability to represent, discriminate, and perform arithmetic operations on discrete quantities (numerosities) has been documented in a variety of species of different taxonomic groups, both vertebrates and invertebrates. We do not know, however, to what extent similarity in behavioral data corresponds to basic similarity in underlying neural mechanisms. Here, we review evidence for magnitude representation, both discrete (countable) and continuous, following the sensory input path from primary sensory systems to associative pallial territories in the vertebrate brains. We also speculate (...) on possible underlying mechanisms in invertebrate brains and on the role played by modeling with artificial neural networks. This may provide a general overview on the nervous system involvement in approximating quantity in different animal species, and a general theoretical framework to future comparative studies on the neurobiology of number cognition. (shrink)

In the Garden of Eden, we had unmediated contact with the world. We were directly acquainted with objects in the world and with their properties. Objects were simply presented to us without causal mediation, and properties were revealed to us in their true intrinsic glory.