Category theory and the foundations of mathematics: Philosophical excavations

The aim of this paper is to clarify the role of category theory in the foundations of mathematics. There is a good deal of confusion surrounding this issue. A standard philosophical strategy in the face of a situation of this kind is to draw various distinctions and in this way show that the confusion rests on divergent conceptions of what the foundations of mathematics ought to be. This is the strategy adopted in the present paper. It is divided into 5 sections. We first show that already in the set theoretical framework, there are different dimensions to the expression foundations of. We then explore these dimensions more thoroughly. After a very short discussion of the links between these dimensions, we move to some of the arguments presented for and against category theory in the foundational landscape. We end up on a more speculative note by examining the relationships between category theory and set theory.Various versions of this paper have been read by many people, many of whom have made crucial comments. Needless to say, I am entirely responsible for the claims made in this paper. I would particularly like to thank, in alphabetical order, Mario Bunge, Marta Bunge, Michael Hallett, Andrew Irvine, Saunders Mac Lane, Collin McLarty, Peneloppe Maddy and Mihaly Makkai. Part of the work was done while the author was a visiting fellow at REHSEIS in Paris and at the Center for Philosophy of Science in Pittsburgh. I would like to thank everyone for his or her help and support. I gratefully acknowledge the financial support received from the SSHRC of Canada while this work was done.     

Socially desirable responding and the factorial stability of the NEO PI-R

The goal of the present investigation is to compare the factor structure of the revised NEO Personality Inventory (NEO PI-R; P. T. Costa & R. R. McCrae, 1992) in samples of respondents differentially motivated to respond in a socially desirable manner. In the French sample, the authors compared the NEO PI-R structure of job applicants (N=12,631) subgrouped by relative degree of socially desirable responding with that of a normative sample (N=801). In the Belgian sample, the authors compared the NEO PI-R structure in naturally occurring groups, including job applicants (N=3,028), individuals receiving career counseling (N=221), and a normative sample (N=549). Results indicate that the NEO PI-R factor structure remained stable across all samples.     

Laboratory experimentation and social field experimentation: An ambiguous distinction

Critical and epistemological reflections on the use of experimental methods in social psychology are often made to distinguish between two types of settings where these methods are used: The laboratory and the social field. But it is by no means certain that such a division has, from the start and at its own level, real significance: for one thing, because in social psychology, laboratory work is probably never merely laboratory work; and for another, because an analysis of what happens in practice would show that much experimental research work carried out in the social field merely transfers into that setting tools, concepts and a theoretical intention derived from the laboratory. If that is so, it follows that it is not at its own level that the laboratory-social field distinction has a heuristic and epistemological significance, nor is that the real location of the breaking-point between the methodologies. Perhaps the real ‘dividing’ line is one which separates experimental work which, whether in the laboratory or in the social field, concentrates on social interactions which are miniaturized or able to be so, from work which, in the social field, aims to concentrcte on processes which cannot be simulated or miniaturized without becoming distorted and seeks to reach social systems which are relatively complete and essentially intact.     

Dimensionality of the Rosenberg Self-Esteem Scale and its relationships with the Three-and the Five-factor personality models

We investigated the dimensionality of the French version of the Rosenberg Self-Esteem Scale (RSES; Rosenberg, 1965) using confirmatory factor analysis. We tested models of 1 or 2 factors. Results suggest the RSES is a 1-dimensional scale with 3 highly correlated items. Comparison with the Revised NEO-Personality Inventory (NEO-PI-R; Costa, McCrae, & Rolland, 1998) demonstrated that Neuroticism correlated strongly and Extraversion and Conscientiousness moderately with the RSES. Depression accounted for 47% of the variance of the RSES. Other NEO-PI-R facets were also moderately related with self-esteem.     

Developing motor planning over ages

Few studies have explored the development of response selection processes in children in the case of object manipulation. In the current research, we studied the end-state comfort effect, the tendency to ensure a comfortable position at the end rather than at the beginning of simple object manipulation tasks. We used two versions of the unimanual bar transport task. In Experiment 1, only 10-year-olds reached the same level of sensitivity to end-state comfort as adults, and 8-year-olds were less efficient than 6-year-olds. In each age group, children’s sensitivity did not increase during a session: i.e., either clearly showed the sensitivity or showed no sensitivity at all. Experiment 2 replicated these results when the bar was replaced by a pencil and when the task did not require much precision. However, when the task required more precision, 8-year-olds increased their level of sensitivity to the end-state comfort effect, whereas this was not the case for younger children. These results describe the development of advanced planning processes from 4 to 10 years of age as well as the positive effect of task constraints on the end-state comfort effect for 8-year-olds.     

The development of analogy making in children: Cognitive load and executive functions

The aim of the current study was to investigate the performance of 6-, 8-, and 14-year-olds on an analogy-making task involving analogies in which there are competing perceptual and relational matches. We hypothesized that the selection of the common relational structure requires the inhibition of other salient features, in particular, perceptual matches. Using an A:B::C:D paradigm, we showed that children’s performance in analogy-making tasks depends crucially on the nature of the distractors. Children chose more perceptual distractors having a common feature with C compared with A or B (Experiment 1). In addition, they were also influenced by unstructured random textures. When measuring reaction times instead of accurate responses, only the 8-year-olds’ reaction times were significantly influenced by perceptual distractors. The 6-year-olds seemed to select the first match they noticed, and the 14-year-olds were not influenced (or much less influenced) by featural distractors. These results are compatible with an analogy-making account based on varying limitations in executive functioning at different ages.     

Scoring the DSM-IV personality disorders using the Five-Factor Model: development and validation of normative scores for North American, French, and Dutch-Flemish samples

Five-Factor Model (FFM) personality disorder (PD) counts have demonstrated significant convergent and discriminant validity with DSM-IV PD symptoms. However, these FFM PD counts are of limited clinical use without normative data because it is difficult to determine what a specific score means with regard to the relative level of elevation. The current study presents data from three large normative samples that can be used as norms for the FFM PD counts in the respective countries: United States (N = 1,000), France (N = 801), and Belgium-Netherlands (N = 549). The present study also examines the performance, with regard to diagnostic efficiency, of statistically-defined cut-offs at 1.5 standard deviations above the mean (T > or = 65) versus previously identified cut-offs using receiver-operator characteristics (ROC) analyses. These cut-offs are tested in three clinical samples-one from each of the aforementioned countries. In general, the T > or = 65 cut-offs performed similarly to those identified using ROC analyses and manifested properties relevant to a screening instrument. These normative data allow FFM data to be used in a flexible and comprehensive manner, which may include scoring this type of personality data in order to screen for DSM-IV PD constructs.