A Formal Cognitive Model of Mathematical Metaphors

Starting from the observation by Lakoff and Núñez (2000) that the process for mathematical discoveries is essentially one of creating metaphors, we show how Information Flow theory (Barwise & Seligman, 1997) can be used to formalise the basic metaphors for arithmetic that ground the basic concepts in the human embodied nature. 1 The Cognition of Mathematics Mathematics is most commonly seen as the uncovering of eternal, absolute truths about the (mostly nonphysical) structure of the universe. Lakatos (1976) and Lakoff and Núñez (2000) argue strongly against the ‘romantic’ (Lakoff and Núñez) or ‘deductivist’ (Lakatos) style in which mathematics is usually presented. Lakatos’s philosophical account of the historical development of mathematical ideas demonstrates how mathematical concepts are not simply ‘unveiled’ but are developed in a process that involves, among other things, proposing formal definitions of mathematical concepts (his main example are polyhedra), developing proofs for properties of those concepts (e.g. Euler’s conjecture that for polyhedra V −E +F = 2, where V , E , F are the number of vertices, edges and faces, respectively) and refining those concepts, e.g. by excluding counterexamples (what Lakatos calls monster barring) or widening the definition to include additional cases. Thus, the concept polyhedron changes in the process of defining its properties, and this casts doubt on whether ployhedra ‘truly exist’. Lakoff and Núñez (2000) describe how mathematical concepts are formed (or, depending on the epistemological view, how mathematical discoveries are made). They claim that the human ability for mathematics is brought about by two main factors: embodiment and the ability to create and use metaphors. They describe how by starting from interactions with the environment we build up (more and more abstract) mathematical concepts by processes of metaphor. They distinguish grounding from linking metaphors (p. 53). In grounding metaphors one domain is embodied and one abstract, e.g. the four grounding metaphors for arithmetic, which we describe below. In linking metaphors, both domains are abstract, which allows the creation of more abstract mathematical concepts, e.g. on the having established the basics of arithmetic with grounding metaphors this knowledge is used to create the concepts points in space and functions (p. 387). ⋆ The research reported here was carried out in the Wheelbarrow project, funded by the EPSRC grant EP/F035594/1. B. Mertsching, M. Hund, and Z. Aziz (Eds.): KI 2009, LNAI 5803, pp. 323–330, 2009. © Springer-Verlag Berlin Heidelberg 2009 324 M. Guhe, A. Smaill, and A. Pease Even if the universal truths view of mathematics were correct, the way in which people construct, evaluate and modify mathematical concepts has received little attention from cognitive science and AI. In particular, it has not yet been described by a computational cognitive model. As part of our research on understanding the cognition of creating mathematical concepts we are building such a model (Guhe, Pease, & Smaill, 2009; Guhe, Smaill, & Pease, 2009), based on the two streams of research outlined above. While there are cognitive models of learning mathematics (e.g. Lebiere 1998; Anderson 2007), there are to our knowledge no models of how humans create mathematics. 2 Structure Mapping, Formal Models and Local Processing Following Gentner (1983; see also Gentner & Markman, 1997, p. 48) we assume that metaphors are similar to analogies. More specifically, there is no absolute distinction between metaphor and analogy; they rather occupy different but overlapping regions of the same space of possible mappings between concepts, with analogies comparatively mapping more abstract relations and metaphors more properties. According to Gentner’s (1983, p. 156) structure mapping theory the main cognitive process of analogy formation is a mapping between the (higher-order) relations of conceptual structures. Although we use this approach for creating computational cognitive models of mathematical discovery (see Guhe, Pease, & Smaill, 2009 for an ACT-R model using the path-mapping approach of Salvucci & Anderson, 2001), in this paper we use a formal model extending the one presented in Guhe, Smaill, and Pease (2009) that specifies the grounding metaphors for arithmetic proposed by Lakoff and Núñez (2000). Formal methods are not commonly used for cognitive modelling, but we consider it a fruitful approach to mathematical metaphors, because, firstly, mathematics is already formalised, and, secondly, if Lakoff and Núñez are correct that metaphors are a general cognitive mechanism that is applied in mathematical thinking, formal methods are an adequately high level of modelling this general purpose mechanism. Furthermore, collecting empirical data on how scientific concepts are created is difficult. This is true for case studies as well as laboratory settings. Case studies (e.g. Nersessian, 2008) are not reproducible (and therefore anecdotal), and they are usually created in retrospect, which means that they will contain many rationalisations instead of an accurate protocol of the thought processes. Laboratory settings in contrast (cf. Schunn & Anderson, 1998) require to limit the participants in their possible responses, and it is unclear whether or how this is different from the unrestricted scientific process. Using the formal method, we take the metaphors used in the discovery process and create a formal cognitive model of the mapping of the high-level relations. The model is cognitive in that it captures the declarative knowledge used by humans, but it does not simulate the accompanying thought processes. For the formalisation we use Information Flow theory (Barwise & Seligman, 1997, see section 4), because it provides means to formalise interrelations (flows of information) between different substructures of a knowledge representation, which we called local contexts in Guhe (2007). A local context contains a subset of the knowledge available to the model that is relevant with respect to a particular focussed element. In cognitive terms, a focussed element is an item (concept, percept) that is currently in the focus of attention. A Formal Cognitive Model of Mathematical Metaphors 325 The ability to create and use suitable local contexts is what sets natural intelligent systems apart from artificial intelligent systems and what allows them to interact efficiently and reliably with their environment. The key benefit of this ability is that only processing a local context (a subset of the available knowledge) drastically reduces the computational complexity and enables the system to interact with the environment under real-time constraints. 3 Lakoff and Núñez’s Four Basic Metaphors of Arithmetic Lakoff and Núñez (2000, chapter 3) propose that humans create arithmetic with four different grounding metaphors that create an abstract conceptual space from embodied experiences, i.e. interactions with the real world. Since many details are required for describing these metaphors adequately, we can only provide the general idea here. Note that the metaphors are not interchangeable. All are used to create the basic concepts of arithmetic. 1. Object Collection. The first metaphor, arithmetic is object collection, describes how by interacting with objects we experience that objects can be grouped and that there are certain regularities when creating collections of objects, e.g. by removing objects from collections, by combining collections, etc. By creating metaphors (analogies), these regularities are mapped into the domain of arithmetic, for example, collections of the same size are mapped to the concept of number and putting two collections together is mapped to the arithmetic operation of addition. 2. Object Construction. Similarly, in the arithmetic is object construction metaphor we experience that we can combine objects to form new objects, for example by using toy building blocks to build towers. Again, the number of objects that are used for the object construction are mapped to number and constructing an object is mapped to addition. 3. Measuring Stick. The measuring stick metaphor captures the regularities of using measuring sticks for the purpose of establishing the size of physical objects, e.g. for constructing buildings. Here numbers correspond to the physical segments on the measuring stick and addition to putting together segments to form longer segments. 4. Motion Along A Path. The motion along a path metaphor, finally, adds concepts to arithmetic that we experience by moving along straight paths. For example, numbers are point locations on paths and addition is moving from point to point.