Nicolas Bourbaki and the concept of mathematical structure

In the present article two possible meanings of the term mathematical structure are discussed: a formal and a nonformal one. It is claimed that contemporary mathematics is structural only in the nonformal sense of the term. Bourbaki's definition of structure is presented as one among several attempts to elucidate the meaning of that nonformal idea by developing a formal theory which allegedly accounts for it. It is shown that Bourbaki's concept of structure was, from a mathematical point of view, a superfluous undertaking. This is done by analyzing the role played by the concept, in the first place, within Bourbaki's own mathematical output. Likewise, the interaction between Bourbaki's work and the first stages of category theory is analyzed, on the basis of both published texts and personal documents.Several persons have read and criticized earlier drafts of this paper. I thank them all for forcing me to formulate my ideas more simply and convincingly. Special thanks are due to Professors Pierre Cartier (Bures-sur-Yvette), Giorgio Israel (Rome), and Andrée C. Ehresmann (Amiens) for stimulating and illuminating conversations.     

The content and structure of laypeople's concept of pleasure

Five studies were conducted to map the content and structure of laypeople's conceptions of pleasure. Instances of the pleasure concept collected in Study 1 consisted predominantly of objects, events or persons described as sources of pleasure. Content analysis suggested that the pleasure category, like emotional response categories, might be formed at an implicit level where various pleasure antecedents are grouped based on common phenomenological qualities of the affective experience. Studies 2 and 3a showed that the pleasure category possesses a graded structure and fuzzy boundaries. Results further revealed that, either when explicitly presented with labels (Study 3b) or left to their own implicit categories during a sorting task (Study 4), laypeople represented pleasure as a hierarchical concept in which differentiated pleasure types (i.e., intellectual, emotional, social and physical) were subsumed under a higher level unitary form of pleasure. In this structure, unitary and differentiated pleasures shared a set of common affective qualities but were also distinguishable by unique and distinctive affective characteristics (Study 5). Ties to prior theories of pleasure and implications for decision making and behavioural research are discussed.     

Application of the concept of simple structure to Alexander's data

A battery of 20 tests originally analyzed by Alexander (1) was reworked according to the principle of simple structure. His results were sustained in general. Both analyses yielded five factors in the first-order domain. Of these, three factors in the re-analysis (v,X andF) have almost exactly the same loadings as the corresponding factors in the original work, and were interpreted in the same way. The loading pattern of a fourth factor,Z, left uninterpreted in the original study, happened to be more clear in the re-analysis, and an interpretation was attempted. It appears to be a factor of perceptual synthesis, and seems to play an important role in intellectual processes. A fifth factor, not present in Alexander's results, appeared in the new analysis: the reasoning factor, involved in inductive and deductive thinking. All four cognitive factors are related to a general factor that can be thought of as representing abstraction and education of relations and correlates, these processes being, therefore, the essential feature underlying intellectual behavior, at least in that sector surveyed by the tests of the present battery.The analysis of the data was done in the Psychometric Laboratory of The University of Chicago, under a fellowship from The University of Madrid (Junta de Relaciones Culturales). The author wishes to express his gratitude to Professor L. L. Thurstone, whose advice as a scientist and kindness as a friend, have been the principal stimulus for this work.     

Classifying types of concept and conceptual structure: Some taxonomies

Abstract

The study of concepts can advance further by clarifying the meanings of basic terms such as “prototype” and by constructing a large-scale primary taxonomy of concept types. This paper presents the latter, based on concept structure, two secondary concept taxonomies and one of conceptual structures. The primary taxonomy organises much data and several previous taxonomies into a single framework. It suggests that many concept types exist, and that type determines how a concept is learned, is used and how it develops. As well as enumerating and interrelating some major types, the taxonomy predicts that many individuals' concepts will consist of mixes of information types. The present study suggests that the field will progress faster by ceasing to look for the single way anything is done. A better strategy is to enumerate concept types, ways each can be learned and developmental patterns, and then see what variables affect these and how common each is.     

The generalized polymorphous concept account of graded structure in abstract categories

Abstract categories present with graded structure. The extent to which feature commonality between exemplars and category provides a satisfying account of this graded structure varies from one abstract category to the other (Hampton, 1981). We investigate whether the incorporation of features that exemplars share with external categories yields an improved account of abstract categories' graded structures. In doing so, we follow the suggestion that abstract categories are relational in nature (Goldstone, 1996; Wiemer-Hastings & Xu, 2005). The generalized polymorphous concept model, which incorporates both types of features, is found to improve the account of typicality and category membership in three of seven studied abstract categories. These three categories are found to be the most abstract, suggesting that it is appropriate to think of abstract categories as varying along a continuum of abstractness/interrelatedness rather than as a distinct type of category altogether.     

Connectionist representation,semantic compositionality,and the instability of concept structure

Summary In the present paper connectionist approaches to the problem of internal representation and the nature of concepts are discussed. In the first part the concept of representation that underlies connectionist modeling is made explicit. It is argued that the connectionist view of representation relies on a correlational theory of semantic content- i.e., the covariation between internal and external states is taken as the basis for ascribing meaning to internal states. The problems and virtues of such a correlational approach to internal representation are addressed. The second part of the paper is concerned with whether connectionism is capable of accounting for the apparent productivity and systematicity of language and thought. There is an evaluation of the recent arguments of Fodor and Pylyshyn, who claim that systematicity can only be explained if one conceives of mental representations as structured symbols composed of context-free constituents. There is a review of empirical evidence that strongly suggests that concepts are not fixed memory structures and that the meaning of constituent symbols varies, depending on the context in which they are embedded. On the basis of this review it is concluded that the meaning of a complex expression is not computed from the context-free meanings of the constituents, and that strong compositionality, as endorsed by Fodor and Pylyshyn (1988), seems implausible as a process theory for the comprehension of complex concepts. Instead, the hypothesis is endorsed that constraint satisfaction in distributed connectionist networks may allow for an alternative account of weak compositionality compatible with the context sensitivity of meaning. In the final section, it is argued that neither mere implementation of a language of thought in connectionist networks nor radical elimination of symbol systems seems to be a fruitful research strategy, but that it might be more useful to discuss how connectionist systems can develop the capacity to use external symbol systems like language or logic without instantiating symbol systems themselves.     

Sequence and structure in the concept development of children with emotional and behavioral difficulties

We investigated the order of acquisition of concepts and cognitive processes in children with emotional and behavioral difficulties and the extent to which these children show similarity to ordinary children in sequence and structure of development. Similarity in developmental sequence was strongly supported by scalogram analyses and has clear implications for organizing teaching, but similarity in structure was less clear. Cognitive processes however, and the nature of the children's reasoning were clarified by the method of exploration used. This was particularly fruitful for providing insight into the difficulties children may have. The findings thus contribute to understanding with a view toward furthering concept development in children with emotional and behavioral difficulties.     

Features of graded category structure: Generalizing the family resemblance and polymorphous concept models

Many real-world categories contain graded structure: certain category members are rated as more typical or representative of the category than others. Research has shown that this graded structure can be well predicted by the degree of commonality across the feature sets of category members. We demonstrate that two prominent feature-based models of graded structure, the family resemblance (Rosch & Mervis, 1975) and polymorphous concept models (Hampton, 1979), can be generalized via the contrast model (Tversky, 1977) to include both common and distinctive feature information, and apply the models to the prediction of typicality in 11 semantic categories. The results indicate that both types of feature information play a role in the prediction of typicality, with common features weighted more heavily for within-category predictions, and distinctive features weighted more heavily for contrast-category predictions. The same pattern of results was found in additional analyses employing rated goodness and exemplar generation frequency. It is suggested that these findings provide insight into the processes underlying category formation and representation.     

Age and gender invariance of self‐concept factor structure: An investigation of a newly developed Chinese self‐concept instrument

A new instrument, the Chinese Adolescent Self‐Esteem Scales (CASES), was developed to measure the self‐concepts of the young people in Hong Kong in seven aspects: social, academic, appearance, moral, family, physical/sport, and general self‐esteem. LISREL procedures were utilized to test the extent of factorial invariance for age and gender based on the responses to CASES of 551 Hong Kong adolescents. It was found that CASES possesses the necessary invariance properties for between‐group measurement in terms of the number and pattern of the underlying factors, item factor loadings, and inter‐factor relations, but not in terms of item uniqueness. Implications of these findings are discussed in terms of the use of CASES and of empirical support for the equivalence of self‐concept factor structure for age and gender groups for both Western and non‐Western adolescents.     

Flow and structure of time experience – concept,empirical validation and implications for psychopathology

We present a conceptual framework on the experience of time and provide a coherent basis on which to base further inquiries into qualitative approaches concerning time experience. We propose two Time-Layers (Macro-Layer and Micro-Layer) and two Time-Formats (Flow and Structure) forming four Time-Domains. Micro-Flow and Micro-Structure represent the implicit phenomenal basis, from which the explicit experiences of Macro-Flow and Macro-Structure emerge. Complementary to this theoretical proposal, we present empirical results from qualitative content analysis obtained from 25 healthy participants. The data essentially corroborate the theoretical proposal. With respect to Flow, the phenomenally accessible time experience appeared as a continuous passage reaching from the past through the present into the future. With respect to Structure, the individual present was embedded in the individual biography, emerging from past experiences and comprising individual plans and goals. New or changing plans and goals were being integrated into the existing present, thus forming a new present. The future appeared as changeable within the present, by means from the past, and therefore as a space of potential opportunities. Exemplarily, we discuss these results in relation to previous empirical findings on deviant experiences of time in Autism Spectrum Disorder (ASD) that is presumably characterized by a breakdown of Flow and concomitant compensatory repetition resulting in an overly structured time. Finally, we speculate about possible implications of these findings both for psychopathological and neuroscientific research.     

Beard's concept of neurasthenia and Freud's concept of the actual neuroses

Beard's concept of neurasthenia and Freud's concept of the actual neuroses are described and evaluated critically. Despite differences in the content of their theories, especially in the role of sexual factors and sexual mechanisms, there are important formal and logical similarities. Both Beard and Freud incorrectly identified the causes of the neuroses being studied; both used inadequate methods of assessing the strengths of the causal factors; and the central theoretical concepts of both were equally vacuous. Freud adopted the same method of study as did Beard, but does not seem to have been influenced directly by him.     

The concept of intelligence

Rosch's theory of concepts, applied to the concept of intelligence, suggests that one's intelligence is just the degree to which one resembles a prototypically intelligent person. Because no single characteristics defines the prototype, there can be no adequate process-based definition of intelligence. In principle, a combination of many empirically derived measures into a single index—as in a Binet test—would be appropriate. In practice, many of the relevant characteristics are simply impossible to measure.     

The concept of validity

This article advances a simple conception of test validity: A test is valid for measuring an attribute if (a) the attribute exists and (b) variations in the attribute causally produce variation in the measurement outcomes. This conception is shown to diverge from current validity theory in several respects. In particular, the emphasis in the proposed conception is on ontology, reference, and causality, whereas current validity theory focuses on epistemology, meaning, and correlation. It is argued that the proposed conception is not only simpler but also theoretically superior to the position taken in the existing literature. Further, it has clear theoretical and practical implications for validation research. Most important, validation research must not be directed at the relation between the measured attribute and other attributes but at the processes that convey the effect of the measured attribute on the test scores.     

Children's concept of word

The development of the concept of word as a unit of speech is a metalinguistic insight that usually corresponds to the acquisition of literacy skills. Further, as a fundamental aspect of language structure, it converges as well with skills involved in bilingualism. A framework that describes the structure of metalinguistic awareness is applied to this task in order to relate the development of word concept to literacy and bilingualism. Children between 4 and 6 years old who are prereaders or early readers, and bilingual or monolingual, are given three tests of word concept. The results show that concept of word emerges gradually and is evident to some extent even in the 4-year-olds. Further, literacy and bilingualism are shown to promote specific aspects of its developmentThe research reported in this paper was unded by grant No. A2559 from the Natural Sciences and Engineering Research Council of Canada. I am indebted to Sandra Owston, who assisted in conducting the study.