Mathematical reasoning with higher-order anti-unifcation

Mathematical reasoning with higher-order anti-unifcation Markus Guhe, Alison Pease, Alan Smaill (m.guhe j a.pease j a.smaill@ed.ac.uk) University of Edinburgh, School of Informatics, Informatics Forum, 10 Crichton Street, Edinburgh E H 8 9 A B, Scotland Martin Schmidt, Helmar Gust, Kai-Uwe K uhnberger, Ulf Krumnack (martisch j hgust j kkuehnbe j krumnack@uni-osnabrueck.de) University of Osnabr¨uck, Institute of Cognitive Science, Albrechtstr. 28, 49076 Osnabr¨uck, Germany Abstract We show how heuristic-driven theory projection (H D T P, a method based on higher-order anti-unification) can be used to model analogical reasoning in mathematics. More precisely, H D T P provides the framework for a model of the inductive analogy-making process involved in establishing the fundamen- tal concepts of arithmetic. This process is a crucial component for being able to generalise from the concrete experiences that humans have due to their embodied and embedded nature. Such generalisations are a cornerstone of the ability to create an ab- stract domain like arithmetic. In addition to generalisations, H D T P can also transfer concepts from one domain into an- other, which is, for example, needed to introduce the concept Z E R O into arithmetic. The approach presented here is closely related to the theories of Information Flow and Institutions. The latter in particular provides a compelling way to integrate concept blending into the HDTP approach. Keywords: mathematical cognition; mathematical reasoning; analogy; anti-unification; concept blending Mathematical reasoning as a cognitive process Although mathematics is usually presented in terms of axioms, concise proofs, theorems and so on, the actual cognitive pro- cess of mathematical reasoning is very different. For example, when a mathematician changes a definition this affects the proofs that use it, but such changes are not discussed in mathe- matical papers. Additionally, mathematics, at least partly, does not consist of discovering eternal, Platonic ideals but in cre- ating mathematical concepts. For example, Lakatos’s (1976) account of the history of Euler’s conjecture illuminates how the concept P O LY H E D R O N can differ and how its definition depends on the current circumstances and needs of the mathe- matician. Put differently, if the Platonic ideal P O LY H E D R O N does exist, it is not clear how it can be identified by mathemat- ical means – what cognitive processes mathematicians can use to find the correct definition. Thus, mathematical concepts are not necessarily the same as the ideals. Lakoff and N´u n ˜ ez (2000) describe how our embodied, situ- ated experience is the basis on which abstract mathematical concepts are developed by a process of metaphorical abstrac- tion and transfer. In chapter 3, they describe how basic arith- metic is created from four everyday experiences, which are the source domains of the metaphors. In this way, arithmetic is grounded in situated cognition. To motivate that these four domains in particular are source domains, Lakoff and N´u n ˜ ez analyse linguistic expressions used in the target domain, arith- metic, which they trace back to these four domains. For exam- ple, we use the terms add and take away in arithmetic. Lakoff and N´u n ˜ ez argue that these terms were originally used for talking about collections of objects, such as physically placing an object into a container, e.g. adding an onion to the soup, or physically removing a substance or an object from a container, e.g. take a book out of the box. Analogical reasoning is a central component of the pro- cess transforming knowledge of this kind into mathematical concepts. For present purposes we assume that metaphor and analogy are essentially the same cognitive process (Gentner, Bowdle, Wolff, & Boronat, 2001), and we have demonstrated how structure mapping (Gentner, 1983; Gentner & Markman, 1997) – a basic method to compute analogical relations – can account for the overall cognitive process (Guhe, Pease, & Smaill, 2009). In this paper, we describe a formal cognitive model of this process. This has a twofold motivation: firstly, we want to spec- ify the cognitive processes that mathematicians use, to better understand how mathematical discovery works; secondly, we want to use the model to improve automated theorem provers by incorporating cognitive mechanisms. In Guhe, Smaill, and Pease (2009a, 2009b) we presented formal representations of the four grounding metaphors (the 4Gs) and suggested how Information Flow theory (Barwise & Seligman, 1997) may be used to model the analogies involved. The 4Gs are: (1) arith- metic is object collection, (2) arithmetic is object construction, (3) measuring stick and (4) arithmetic is motion along a path. Here, we present a proof-of-concept of how performing anti-unification (Plotkin, 1970) on such representations can account for aspects of the analogical reasoning involved in the 4Gs. This inductive kind of reasoning provides us with a procedural version of the otherwise static Information Flow models and enables us to computationally determine the re- lationships between the domains (classifications in the case of Information Flow). More precisely, we will use Heuristic- Driven Theory Projection (H D T P; Schwering, Krumnack, K¨uhnberger, & Gust, 2009), a general framework for making analogies. H D T P provides us with the means to generalise over two of Lakoff and N´u n ˜ ez’s domains to establish a basis for arithmetic as well as the means to generalise over one of the domains as source domain and arithmetic as the target domain to add concepts to arithmetic that are only present in one of the grounding domains. We will also outline how this conception of mathematical reasoning is linked to Goguen’s (2006) notion of concept blending (which is based on notions by G¨ardenfors, 2000 and Fauconnier & Turner, 2002), a fur- ther cognitive process for creating mathematical concepts.