The generative basis of natural number concepts

Number concepts must support arithmetic inference. Using this principle, it can be argued that the integer concept of exactly ONE is a necessary part of the psychological foundations of number, as is the notion of the exact equality - that is, perfect substitutability. The inability to support reasoning involving exact equality is a shortcoming in current theories about the development of numerical reasoning. A simple innate basis for the natural number concepts can be proposed that embodies the arithmetic principle, supports exact equality and also enables computational compatibility with real- or rational-valued mental magnitudes.     

Feature integration in natural language concepts

Two experiments measured the joint influence of three key sets of semantic features on the frequency with which artifacts (Experiment 1) or plants and creatures (Experiment 2) were categorized in familiar categories. For artifacts, current function outweighed both originally intended function and current appearance. For biological kinds, appearance and behavior, an inner biological function, and appearance and behavior of offspring all had similarly strong effects on categorization. The data were analyzed to determine whether an independent cue model or an interactive model best accounted for how the effects of the three feature sets combined. Feature integration was found to be additive for artifacts but interactive for biological kinds. In keeping with this, membership in contrasting artifact categories tended to be superadditive, indicating overlapping categories, whereas for biological kinds, it was subadditive, indicating conceptual gaps between categories. It is argued that the results underline a key domain difference between artifact and biological concepts.     

Linear separability in superordinate natural language concepts

Two experiments are reported in which linear separability was investigated in superordinate natural language concept pairs (e.g., toiletry-sewing gear). Representations of the exemplars of semantically related concept pairs were derived in two to five dimensions using multidimensional scaling (MDS) of similarities based on possession of the concept features. Next, category membership, obtained from an exemplar generation study (in Experiment 1) and from a forced-choice classification task (in Experiment 2) was predicted from the coordinates of the MDS representation using log linear analysis. The results showed that all natural kind concept pairs were perfectly linearly separable, whereas artifact concept pairs showed several violations. Clear linear separability of natural language concept pairs is in line with independent cue models. The violations in the artifact pairs, however, yield clear evidence against the independent cue models.     

Exemplars and prototypes in natural language concepts: A typicality-based evaluation

Are natural language categories represented by instances of the category or by a summary representation? We used an exemplar model and a prototype model, both derived within the framework of the generalized context model (Nosofsky, 1984, 1986), to predict typicality ratings for 12 superordinate natural language concepts. The models were fitted to typicality ratings averaged across participants and to the typicality judgments of individual participants. Both analyses yielded results in favor of the exemplar model. These results suggest that higher-level natural language concepts are represented by their subordinate members, rather than by a summary representation.     

Development of natural language concepts: IV. The relationship between semantic organization and learning

In Experiment 1, 30 subjects at each of three age levels were shown pictures of unfamiliar “animals”. They were told two semantic attributes for each animal (e.g., kind and strong) and were asked to learn these sets so that, when shown a picture, the appropriate attributes could be recalled. The results indicated that the dimensional structure of the attributes had a strong effect on new learning for young children. Incongruent pairs of attributes (e.g., kind and ugly) were difficult to learn compared to congruent (e.g., kind and beautiful) or unrelated (e.g., kind and tall) sets. These results indicate: (a) Attribute structure is a factor in new learning; (b) 6-year-olds tend to organize the verbal labels for attributes into bipolar dimensions, rather than into independent clusters of attributes. The latter findings require reexamination of the meaning of previous word-association data which had been interpreted as indicating that 6-year-olds tend not to organize meanings bipolarly. Experiment 2 showed that the disruptive effects of incongruence on new learning disappears by college age.     

Cardinal limits: Evidence from language awareness and bilingualism for developing concepts of number

The study presented here investigated children's acquisition of cardinality in terms of a framework that isolates two cognitive processing components previously shown to be involved in children's metalinguistic development. These components are called analysis of knowledge and control of processing. In Study 1, children from 3 to 5 years of age were asked to solve three problems that required an understanding of cardinality. The problems were designed to place different demands on these processing components and examine their involvement in specific problems. In Study 2, bilingual children were given two of the problems to compare their performance to a new group of monolinguals. The results from both studies contribute to our knowledge of the development of cardinality by assessing children's ability across several tasks, by classifying those tasks in terms of their reliance on distinct processing components, and by providing a means for relating children's development of number concepts to their development of language. The results also contribute to our knowledge of the cognitive abilities of bilingual children. The implications of these results are that aspects of symbolic development may be a broadly based process that extends beyond domain-specific boundaries.     

Iconic and non-iconic stages in number development: the role of language

Is language the key to number? This article argues that the human language faculty provides the cognitive equipment that enables humans to develop a systematic number concept. Importantly, the number concept is based on non-iconic representations that involve relations between relations: relations between numbers are linked with relations between objects. In contrast to this, language-independent numerosity concepts provide only iconic representations. The pattern of forming relations between relations lies at the heart of our language faculty, suggesting that it is language that enables humans to make the step from these iconic representations, which we share with other species, to a generalized concept of number.     

How to learn the natural numbers: inductive inference and the acquisition of number concepts

Theories of number concepts often suppose that the natural numbers are acquired as children learn to count and as they draw an induction based on their interpretation of the first few count words. In a bold critique of this general approach, Rips, Asmuth, Bloomfield [Rips, L., Asmuth, J. & Bloomfield, A. (2006). Giving the boot to the bootstrap: How not to learn the natural numbers. Cognition, 101, B51-B60.] argue that such an inductive inference is consistent with a representational system that clearly does not express the natural numbers and that possession of the natural numbers requires further principles that make the inductive inference superfluous. We argue that their critique is unsuccessful. Provided that children have access to a suitable initial system of representation, the sort of inductive inference that Rips et al. call into question can in fact facilitate the acquisition of larger integer concepts without the addition of any further principles.     

The numbers 1 to 5 in the development of children's number concepts

The structure of preschool children's number concepts was investigated. It was hypothesized that preschoolers have number concepts based on the numbers 1 to 5 rather than on the decade structure. In Experiment 1, children were taught how to solve problems using the number 5 or the number 10. Children given the former solved almost all the problems correctly. In Experiment 2, children were asked to resolve numbers into 5's and x's or to find supplements to 10. It was significantly easier and faster for children to resolve numbers than to find supplements. In Experiment 3, children were given both addition and subtraction problems. Performance in addition and subtraction was comparable. Strategies suggested that children understood numbers to 5 as a privileged anchor. These results are discussed in terms of a new model of a representational system of numbers.     

Comparing natural and abstract categories: A case study from computer science

In order to address the issue of whether abstract and concrete objects are categorized similarly, this paper looks at how computer scientists categorize the common concepts of their field. Using the methodology of Rosch, Mervis, Gray, Johnson, and Boyes-Braem (1976) and of Rosch and Mervis (1975) to identify the categorizations used by computer scientists, this study found that the categorization phenomena reported for concrete objects are quite general, operating over a wider range than was previously thought.     

Religious authority and the transmission of abstract god concepts

According to the Standard Model account of religion, religious concepts tend to conform to “minimally counterintuitive” schemas. Laypeople may, to varying degrees, verbally endorse the abstract doctrines taught by professional theologians. But, outside the Sunday school exam room, the implicit representations that tend to guide people’s everyday thinking, feeling, and behavior are about minimally counterintuitive entities. According to the Standard Model, these implicit representations are the essential thing to be explained by the cognitive science of religion (CSR). It is argued here that this theoretical orientation of mainstream CSR misses a whole dimension of religiosity—the acceptance of certain religious authorities, that is, the acceptance of other people’s superior expertise. Average believers (especially in doctrinal traditions) tend to accept the authority of religious experts who espouse highly counterintuitive ideas that they (the laypeople) understand in a distorted form, if at all. These highly counterintuitive ideas are culturally successful because laypeople see them as being justified by people they have reason to regard as epistemic authorities. The tendency for people to endorse (without fully understanding) highly counterintuitive religious ideas espoused by intellectuals may explain parallels in the development of separate traditions (e.g., Judaism and Hinduism), as religious philosophers follow parallel lines of reasoning.     

An investigation of the nature of abstract concepts

Eight abstract concepts were submitted to a procedure designed to test the logical nature of their definition. The hypothesis that they would show a polymorphous prototype structure similar to that found for concrete categories (Hampton, 1979; Rosch & Mervis, 1975) was confirmed for five of the concepts. Reasons for the lack of fit of the prototype model to the remaining concepts and implications for the generality of existing theories of semantic memory are discussed.     

Content differences for abstract and concrete concepts

Concept properties are an integral part of theories of conceptual representation and processing. To date, little is known about conceptual properties of abstract concepts, such as idea. This experiment systematically compared the content of 18 abstract and 18 concrete concepts, using a feature generation task. Thirty-one participants listed characteristics of the concepts (i.e., item properties) or their relevant context (i.e., context properties). Abstract concepts had significantly fewer intrinsic item properties and more properties expressing subjective experiences than concrete concepts. Situation components generated for abstract and concrete concepts differed in kind, but not in number. Abstract concepts were predominantly related to social aspects of situations. Properties were significantly less specific for abstract than for concrete concepts. Thus, abstractness emerged as a function of several, both qualitative and quantitative, factors.     

Categorization of novel stimuli in well-known natural concepts: a case study

In this study, we investigated to what extent exemplar-based and prototype predictors can be applied to predicting categorization in natural language concepts. Participants categorized novel tropical foods into two well-known natural language concepts:fruits and vegetables. The results indicate that both the prototype predictors and the exemplar predictors contribute significantly in accounting for the categorization choices but that the contribution of the prototype predictor comes from just a limited number of features.     

Japanese children's numerical competencies: age- and schooling-related influences on the development of number concepts and addition skills

Using a cutoff design (J. Bisanz, F. J. Morrison, & M. Dunn, 1995) to separate school-related influences from those that are age related, the study investigated the development of number concepts and addition skills in Japanese children. Three groups of kindergarten and 1st grade children who differed in age and/or school experiences completed tasks on their numerical competencies 1 and 6 months after school entrance. Children's use of addition strategies, rather than their solution accuracy, changed primarily as a function of schooling, not age. Children's Base 10 number concepts improved with the amount of schooling, as well as with other social and age-related factors. Results suggest that schooling is an important determinant in developing Japanese-speaking children's numerical competencies, which were not explained solely by their language characteristics or by age-related factors.