A mathematical theory of sign-mediated concept formation Yair Neuman a, * , Ophir Nave b a Office for Interdisciplinary Research, Ben-Gurion University of the Negev, 84105 Beer-Sheva, Israel b Department of Mathematics, Ben-Gurion University of the Negev, 84105 Beer-Sheva, Israel Abstract Cognitive theories of concept formation have usually ignored the fact that, as a natural part of their cognitive devel- opment, people form concepts through the linguistic mediation of signs. In this paper, we propose a mathematical theory of sign-mediated concept formation (SMCF). This theory relies heavily on ideas from category theory, specifically the ideas of pushout and pullback. The theory addresses some of the major problems in modeling concept formation, including fam- ily resemblance and contextuality, and has potential applications for a wide variety of tasks involving concept computation. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Concept formation; Category theory; Pushout; Pullback; Semiotics; Interdisciplinary research 1. Introduction Human intelligence involves the unique capacity to generate concepts through the mediation of linguistic signs. Thinking is not a solitary activity; we are born into a world inhabited by others, who mediate the con- struction of our conceptual system through language. For example, a mother interacting with her child may point to a Chihuahua, a Great Dane and a German Shepherd and name them all ‘‘Dog”. Through this inter- action, the mother mediates the emergence of the dog-concept, which, roughly speaking, is the mental object corresponding to the linguistic sign ‘‘Dog” [1]. 1 Although non-human organisms and even other biological systems such as the immune system clearly use concepts, they do not construct them through linguistic mediation and therefore their concepts seem to be qualitatively different [2]. For example, their responses indicate that non-human organisms have fewer con- cepts than human beings and that these concepts are limited to concrete behavioral contexts; for the predator the world is divided into prey and non-prey, and other differentiations are secondary. Deducing the existence of a concept from a response might be mistakenly perceived as the expression of an anachronistic behaviorist perspective. However, this deduction resonates with the interactionist perspective as 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.12.001 * Corresponding author. E-mail address: yneuman@bgu.ac.il (Y. Neuman). 1 For clarity we denote the linguistic sign by an initial capital letter (e.g., Dog) and the corresponding concept by an asterisk (e.g., Dog * ). Available online at www.sciencedirect.com Applied Mathematics and Computation 201 (2008) 72–81 www.elsevier.com/locate/amc