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.     

How to Assess a Model's Testability and Identifiability

Formal definitions are given of the following intuitive concepts: (a) A model is quantitatively testable if its predictions are highly precise and narrow. (b) A model is identifiable if the values of its parameters can be ascertained from empirical observations. (c) A model is redundant if the values of some parameters can be deduced from others or if the values of some observables can be deduced from others. Various rules of thumb for nonredundant models are examined. The Counting Rule states that a model is quantitatively testable if and only if it has fewer parameters than observables. This rule can be safely applied only to identifiable models. If a model is unidentifiable, one must apply a generalization of the Counting Rule known as the Jacobian Rule. This rule states that a model is quantitatively testable if and only if the maximum rank (i.e., the number of linearly independent columns) of its Jacobian matrix (i.e., the matrix of partial derivatives of the function that maps parameter values to the predicted values of observables) is smaller than the number of observables. The Identifiability Rule states that a model is identifiable if and only if the maximum rank of its Jacobian matrix equals the number of parameters. The conclusions provided by these rules are only presumptive. To reach definitive conclusions, additional analyses must be performed. To illustrate the foregoing, the quantitative testability and identifiability of linear models and of discrete-state models are analyzed. Copyright 2000 Academic Press.     

QUAID: A questionnaire evaluation aid for survey methodologists

QUAID (question-understanding aid) is a software tool that assists survey methodologists, social scientists, and designers of questionnaires in improving the wording, syntax, and semantics of questions. The tool identifies potential problems that respondents might have in comprehending the meaning of questions on questionnaires. These problems can be scrutinized by researchers when they revise questions to improve question comprehension and, thereby, enhance the reliability and validity of answers. QUAID was designed to identify nine classes of problems, but only five of these problems are addressed in this article: unfamiliar technical term, vague or imprecise relative term, vague or ambiguous noun phrase, complex syntax, and working memory overload. We compared the output of QUAID with ratings of language experts who evaluated a corpus of questions on the five classes of problems. The corpus consisted of 505 questions on 11 surveys developed by the U.S. Census Bureau. Analyses of hit rates, false alarm rates,d′ scores, recall scores, and precision scores revealed that QUAID was able to identify these five problems with questions, although improvements in QUAID’s performance are anticipated in future research and development.     

An Integral Psychology Response to Helminiak's (2001) “Treating Spiritual Issues in Secular Psychotherapy”

This article is a response to D. A. Helminiak's (2001) “Treating Spiritual Issues in Secular Psychotherapy” from the perspective of K. Wilber's integral psychology. The article consists of 3 sections: (a) a selective summary of integral psychology, including the perennial philosophy, 10 levels of development, lines of development, temporary states, types of orientations, the self, and the 4 quadrants; (b) various conceptual issues; and (c) usefulness to mental health practitioners (MHPs). Although D. A. Helminiak's endeavor to formulate a clear and functional definition of spirituality is commended, his claim of novelty is shown to be unwarranted. It is argued that K. Wilber's integral model is more comprehensive, clear, coherent, and helpful to MHPs.     

The sounds of dominance.

A study was conducted to determine the effects of vocal cues on judgements of dominance in an interpersonal influence context. Physical measures of human vocal cues and participant ratings of dominance were obtained from videotapes of actors delivering short influence messages. After controlling for linguistic and visual content of messages, results indicated that mean amplitude and amplitude standard deviation were positively associated with dominance judgments, whereas speech rate was negatively associated with dominance judgments. An unexpected interaction revealed that mean fundamental frequency (F₀) was positively associated with dominance judgments for male speakers but not significantly associated with dominance judgments for female speakers. F₀ standard deviation was not significantly associated with dominance judgments. Results support the conclusion that dominance judgments are inferred from multiple sources of information and that some vocal markers of dominance are more influential than others.     

Structural duality of 1/3⟨111⟩ twin-boundary disconnections

We investigate the structure of 1/3?111? disconnections at Σ3 {111} twin boundaries in gold. Our high-resolution transmission electron microscopy (HREM) observations and atomistic simulations show that the core relaxation of this defect is dramatically affected by reversing the sign of the Burgers vector, as oriented with respect to the twin boundary. In particular, we find two distinct, relaxed structures: one with a localized core and the other with a dissociated core composed of a stacking fault terminated by a Shockley partial dislocation. An analysis of the specific pathways available for the defect to relax and of the elastic interactions of the components of the dissociated dislocation for these cases explains the structural difference.