Core systems of number

What representations underlie the ability to think and reason about number? Whereas certain numerical concepts, such as the real numbers, are only ever represented by a subset of human adults, other numerical abilities are widespread and can be observed in adults, infants and other animal species. We review recent behavioral and neuropsychological evidence that these ontogenetically and phylogenetically shared abilities rest on two core systems for representing number. Performance signatures common across development and across species implicate one system for representing large, approximate numerical magnitudes, and a second system for the precise representation of small numbers of individual objects. These systems account for our basic numerical intuitions, and serve as the foundation for the more sophisticated numerical concepts that are uniquely human.     

核心知识系统及其对相关研究的启示

核心知识系统出现于人类个体发展和种系发展的早期,在人类复杂认知能力的发生发展中起着建构模块的作用。该文以表征物体和数量关系的两类核心知识系统的性质及二者在儿童的“数”概念形成和成人的数学思维中所起的作用为例,综述了以灵长类动物、婴儿、儿童和成人为被试的研究证据,并进而提出了核心知识系统给相关研究带来的若干启示。     

Neural signatures of number processing in human infants: evidence for two core systems underlying numerical cognition

Behavioral research suggests two cognitive systems are at the foundations of numerical thinking: one for representing 1-3 objects in parallel and one for representing and comparing large, approximate numerical magnitudes. We tested for dissociable neural signatures of these systems in preverbal infants, by recording event-related potentials (ERPs) as 6-7.5 month-old infants (n = 32) viewed dot arrays containing either small (1-3) or large (8-32) sets of objects in a number alternation paradigm. If small and large numbers are represented by the same neural system, then the brain response to the arrays should scale with ratio for both number ranges, a behavioral and brain signature of the approximate numerical magnitude system obtained in animals and in human adults. Contrary to this prediction, a mid-latency positivity (P500) over parietal scalp sites was modulated by the ratio between successive large, but not small, numbers. Conversely, an earlier peaking positivity (P400) over occipital-temporal sites was modulated by the absolute cardinal value of small, but not large, numbers. These results provide evidence for two early developing systems of non-verbal numerical cognition: one that responds to small quantities as individual objects and a second that responds to large quantities as approximate numerical values. These brain signatures are functionally similar to those observed in previous studies of non-symbolic number with adults, suggesting that this dissociation may persist over vast differences in experience and formal training in mathematics.     

Calibrating the mental number line

Human adults are thought to possess two dissociable systems to represent numbers: an approximate quantity system akin to a mental number line, and a verbal system capable of representing numbers exactly. Here, we study the interface between these two systems using an estimation task. Observers were asked to estimate the approximate numerosity of dot arrays. We show that, in the absence of calibration, estimates are largely inaccurate: responses increase monotonically with numerosity, but underestimate the actual numerosity. However, insertion of a few inducer trials, in which participants are explicitly (and sometimes misleadingly) told that a given display contains 30 dots, is sufficient to calibrate their estimates on the whole range of stimuli. Based on these empirical results, we develop a model of the mapping between the numerical symbols and the representations of numerosity on the number line.     

Two core systems of numerical representation in infants

Two nonverbal representation systems, the analog magnitude system (AMS) and the object tracking system (OTS), have been proposed to explain how humans and nonhuman animals represent numerosities. There has long been debate about which of the two systems is responsible for representing small numerosities (<4). This review focuses on findings with human infants to inform that debate. We argue that the empirical data cannot all be explained by a single system, and in particular, infants’ failures to compare small and large numerosities – the boundary effect – undermines the claim that the AMS can account for infants’ numerical abilities in their entirety. We propose that although the two systems coexist throughout the lifespan, competition between the systems is primarily a developmental phenomenon. Potential factors that drive the engagement of each system in infancy, such as stimulus features and task demands, are discussed, and directions for future research are suggested.     

Sensorimotor foundations of speech perception in infancy

The perceptual system for speech is highly organized from early infancy. This organization bootstraps young human learners’ ability to acquire their native speech and language from speech input. Here, we review behavioral and neuroimaging evidence that perceptual systems beyond the auditory modality are also specialized for speech in infancy, and that motor and sensorimotor systems can influence speech perception even in infants too young to produce speech-like vocalizations. These investigations complement existing literature on infant vocal development and on the interplay between speech perception and production systems in adults. We conclude that a multimodal speech and language network is present before speech-like vocalizations emerge.     

Multiple spatially overlapping sets can be enumerated in parallel

A system for nonverbally representing the approximate number of items in visual and auditory arrays has been documented in multiple species, including humans. Although many aspects of this approximate number system are well characterized, fundamental questions remain unanswered: how does attention select which items in a scene to enumerate, and how many enumerations can be computed simultaneously? Here we show that when presented an array containing different numbers of spatially overlapping dots of many colors, human adults can select and enumerate items on the basis of shared color and can enumerate approximately three color subsets from a single glance. This three-set limit converges with previously observed three-item limits of parallel attention and visual short-term memory. This suggests that participants can select a subset of items from a complex array as a single individual set, which then serves as the input to the approximate number system.     

Spontaneous analog number representations in 3‐year‐old children

When enumerating small sets of elements nonverbally, human infants often show a set‐size limitation whereby they are unable to represent sets larger than three elements. This finding has been interpreted as evidence that infants spontaneously represent small numbers with an object‐file system instead of an analog magnitude system ( Feigenson, Dehaene & Spelke, 2004 ). In contrast, non‐human animals and adult humans have been shown to rely on analog magnitudes for representing both small and large numbers ( Brannon & Terrace, 1998 ; Cantlon & Brannon, 2007 ; Cordes, Gelman, Gallistel & Whalen, 2001). Here we demonstrate that, like adults and non‐human animals, children as young as 3 years of age spontaneously employ analog magnitude representations to enumerate both small and large sets. Moreover, we show that children spontaneously attend to numerical value in lieu of cumulative surface area. These findings provide evidence of young children’s greater sensitivity to number relative to other quantities and demonstrate continuity in the process they spontaneously recruit to judge small and large values.     

Experiments on the emergence of human communication

Children learn language from their parents and then use the acquired system throughout the rest of their life with little change. At least that is commonly assumed. But a recent paper by Galantucci adds to the growing evidence that adults (and children) are able to create and negotiate complex communication systems from scratch and relatively quickly, without a prior model. This raises questions of what cognitive mechanisms are implied in this joint construction of communication systems, and what the implications are for the origins of human language.     

Measurement of Human Performance in Complex Task Environments

This article outlines a model of human performance representing an alterna- tive to studying human interaction in complex systems solely as a function of traditional performance measures (i.e., speed- and accuracy-based measures taken from the system itself). The model described was developed to focus on pilot/operator strategies as they are formulated in the context of the task en- vironment. This model facilitated the development of human-system interac- tion (HSI) measures which allow the quantitative assessment of operator strategies, in terms of the functions affected and the sequence of actions taken in response to external challenges. In conjunction with the more tradi- tional pilot performance measures, HSI measures can be used to investigate how strategic responses impact overt performance.     

Semantic representation of Kinship systems

Four representational systems are examined with respect to their adequacy for representing reasoning processes within kinship systems; the systems are associative, semantic feature, logical predicate, and an “algebraic” one based on set mappings. The associative and semantic feature systems are shown to be inadequate, and the logical predicate system is shown to be somewhat clumsy and unintuitive. The algebraic is proposed as the correct representation of kinship terms. It is suggested that certain other conceptual domains can also be best represented by an algebraic system.     

Foundational numerical capacities and the origins of dyscalculia

One important cause of very low attainment in arithmetic (dyscalculia) seems to be a core deficit in an inherited foundational capacity for numbers. According to one set of hypotheses, arithmetic ability is built on an inherited system responsible for representing approximate numerosity. One account holds that this is supported by a system for representing exactly a small number (less than or equal to four4) of individual objects. In these approaches, the core deficit in dyscalculia lies in either of these systems. An alternative proposal holds that the deficit lies in an inherited system for sets of objects and operations on them (numerosity coding) on which arithmetic is built. I argue that a deficit in numerosity coding, not in the approximate number system or the small number system, is responsible for dyscalculia. Nevertheless, critical tests should involve both longitudinal studies and intervention, and these have yet to be carried out.     

Do Older Adults and Those Recovered from Low Back Injury Share Common Muscle Activation Adaptations?

Theoretical models suggest trunk muscle activation compensates for spinal systems impairments. The purpose of this study was to determine if two populations (older adults and those recovered from a lower back injury (rLBI)) with spinal system impairments have similar muscle activation patterns to each other, but differ from controls. Trunk electromyograms collected from 12 older adults, 16 rLBI, and 19 controls during two dynamic tasks showed that older adults and rLBI had higher activation amplitudes, sustained temporal and more synergistic activation relative to controls. However, differences found between older adults and rLBI suggest that spinal system impairments differed between groups or that recent pain (rLBI) uniquely influenced muscle activation. This sheds light on our understanding of the relationship between spinal system impairments and muscle activation.     

Discrimination of small quantities by fish (redtail splitfin, Xenotoca eiseni)

Discrimination of quantity has been argued to rely on two non-verbal representational systems: an object file system (OFS) for representing small values (≤3–4) and an analog magnitude system (AMS) for representing large magnitudes (>4). Infants’ ability to discriminate 1 versus 2, 1 versus 3, 2 versus 3, but not 1 versus 4 or 2 versus 4 seems to prove the independence of such systems. Here, we show that redtail splitfin fish (Xenotoca eiseni) performed relative quantity estimations preferring to approach the location previously occupied by the larger in number between two groups of conspecifics (no longer visible at test) in sets of 1 versus 2 and 2 versus 3 items, but failed at 3 versus 4 items, thus showing the same set-size limit as infants for discrimination of small quantities. However, when tested with quantities that spanned the boundary of the two systems, that is, 1 versus 4 and 2 versus 4, fish succeeded. These results thus point to either the use of continuous physical variables and/or the use of the AMS also for small numerousness in fish in this task.     

One, two, three, four, or is there something more? Numerical discrimination in day-old domestic chicks

Human adults master sophisticated, abstract numerical calculations that are mostly based on symbolic language and thus inimitably human. Humans may nonetheless share a subset of non-verbal numerical skills, available soon after birth and considered the evolutionary foundation of more complex numerical reasoning, with other animals. These skills are thought to be based on the two systems: the object file system which processes small values (<3) and the analogue magnitude system which processes large magnitudes (>4). Infants’ ability to discriminate 1 vs. 2, 1 vs. 3, 2 vs. 3, but not 1 vs. 4, seems to indicate that the two systems are independent, implying that the conception of a continuous number processing system is based on precursors that appear to be interrupted at or about the number four. The findings from the study being presented here indicating that chicks are able to make a series of discriminations regarding that borderline number (1 vs. 4, 1 vs. 5, 2 vs. 4) support the hypothesis that there is continuity in the number system which processes both small and large numerousness.     

Age-Related Differences in Lay Conceptions of Well-Being and Experienced Well-Being

Conceptions of well-being are defined as a system of beliefs concerning the nature and experience of well-being and can be described generally by the degree to which four dimensions, representing (1) the experience of pleasure, (2) avoidance of negative experience, (3) selfdevelopment, and (4) contribution to others, are emphasized. A first main objective of the current study was to investigate age-related differences in younger and older adults’ conceptions of well-being. A second main objective was to address whether conceptions of well-being are differentially associated with experienced well-being in younger and older adults. Results indicated several age-related differences in conceptions of well-being, with younger adults reporting more emphasis on the experience of pleasure and self-development, older adults reporting more emphasis on avoidance of negative experience, and younger and older adults reporting similar levels of emphasis on contribution to others. Results further indicated several age-related differences in associations between the experience of pleasure and avoidance of negative experience dimensions and well-being, with these two dimensions being more strongly and positively associated with well-being in older adults. Self-development and contribution to others were found to be positively associated with well-being regardless of age.     

How writing system and age influence spatial representations of actions: a developmental, cross-linguistic study

Recently, researchers reported a bias for placing agents predominantly on the left side of pictures. Both hemispheric specialization and cultural preferences have been hypothesized to be the origin of this bias. To evaluate these hypotheses, we conducted a study with participants exposed to different reading and writing systems: Germans, who use a left-to-right system, and Israelis, who use a right-to-left system. In addition, we manipulated the degree of exposure to the writing systems by testing preschoolers and adults. Participants heard agent-first or recipient-first sentences and were asked to draw the content of the sentences or to arrange transparencies of protagonists and objects such that their arrangement depicted the sentences. Although preschool-age children in both countries showed no directional bias, adults manifested a bias that was consistent with the writing system of their language. These results support the cultural hypothesis regarding the origin of spatial-representational biases.     

Dissociable neural systems support retrieval of how and why action knowledge

In everyday discourse, people typically represent actions in one of two ways: how they are performed or why they are performed. In the present study, we determined the neural systems that support these natural modes of representing actions. Participants underwent functional magnetic resonance imaging while identifying how and why people perform various familiar actions. Identifying how actions are performed produced activity in premotor areas that support the execution of actions and in higher-order visual areas that support the perception of action-related objects; this finding supports an embodied view of action knowledge. However, identifying why actions are performed preferentially engaged areas of the brain associated with representing and reasoning about mental states; these areas were right temporoparietal junction, precuneus, dorsomedial prefrontal cortex, and posterior superior temporal sulcus. Our results suggest that why action knowledge is not sufficiently constituted by information in motor and visual systems, but requires a system for representing states of mind, which do not have reliable motor correlates or visual appearance.     

Olfactory acuity and cognitive function converge in older adulthood: support for the common cause hypothesis

Visual and auditory thresholds and cognitive variables have shown converging losses in old age, which might exist because standard cognitive tests rely on these modalities for assessment. The present study investigated the common cause hypothesis in another sensory modality. Structural equation modeling tested the fit of a model representing the common cause hypothesis for olfactory acuity and cognitive function data from 98 nondemented older adults and 103 younger adults. The model fit better the data from the older adults, consistent with the common cause hypothesis. In addition, unique influences of age beyond those shared between cognitive and sensory variables were also demonstrated. The results suggest that olfactory functioning may be a valid indicator of the integrity of the aging brain in older adults.     

Consumers Need Information: Supplementing Teleosemantics with an Input Condition

The success of a piece of behaviour is often explained by its being caused by a true representation (similarly, failure falsity). In some simple organisms, success is just survival and reproduction. Scientists explain why a piece of behaviour helped the organism to survive and reproduce by adverting to the behaviour's having been caused by a true representation. That usage should, if possible, be vindicated by an adequate naturalistic theory of content. Teleosemantics cannot do so, when it is applied to simple representing systems (Godfrey-Smith 1996). Here it is argued that the teleosemantic approach to content should therefore be modified, not abandoned, at least for simple representing systems. The new 'infotel-semantics' adds an input condition to the output condition offered by teleosemantics, recognising that it is constitutive of content in a simple representing system that the tokening of a representation should correlate probabilistically with the obtaining of its specific evolutionary success condition.