A Cognitive Model of Discovering Commutativity - eScholarship
Markus Guhe, Alison Pease, Alan Smail · Proceedings of the Annual Meeting of the Cognitive Science Society · 2009
A Cognitive Model of Discovering Commutativity Markus Guhe ([email protected]) Alison Pease ([email protected]) Alan Smaill ([email protected]) School of Informatics, Informatics Forum, 10 Crichton Street Edinburgh EH8 9AB, UK The Role of Metaphors Abstract In this paper, we focus on the approach by Lakoff and N´u n ˜ ez (2000). They propose that the human embodied mind brings mathematics into being. That is, human mathematics is grounded in the bodily experience of a physical world, and mathematical entities inherit properties of objects in the world, such as being consistent or stable over time. Via exploration of the physical world we build up mini-domains, which we then map to abstract mathematical domains, allowing us to make inferences in the abstract world by transferring knowledge about the physical world. The main process enabling humans to make this transfer is the ability to use metaphors. Metaphors and analogies in mathematics have so far been mainly documented by educators (for example, English, 1997; Sfard, 1996). Despite the importance of metaphors and analo- gies for discovering new concepts in mathematics, historians and philosophers of mathematics, and mathematicians them- selves have tended to be silent on the matter, with notable exceptions such as Lakatos (1976, p 9; who recommends em- bedding a conjecture in a distant body of knowledge, eg a conjecture about solids in the theory of rubber sheets), Polya (1954, p 15–22; who describes and analyses Euler’s appli- cation of rules for finite equations to infinite equations) and Weil (see Krieger, 2003; who discusses a number of fruitful historical mathematical analogie.s) Mathematics is often seen as the uncovering of eternal truths that exist independently of the human mind. However, even if this epistemological view is correct, the mathematics that humans can know can only be the result of cognitive processes. We investigate this ability of the human mind to make math- ematical discoveries. More precisely, we present a cognitive model of how the ability to use metaphors and analogies plays a key role in such discoveries. As a proof of concept we present an AC T- R model that uses path-mapping and that is capable of discovering the commutativity property of addition. Keywords: analogy; metaphor; mathematics; scientific discov- ery; cognitive modelling. Mathematical Discoveries The Cognition of Mathematics The way in which people construct, evaluate and modify math- ematical concepts has received relatively little attention from cognitive science. Likewise, automated mathematical theory formation has so far put little emphasis on cognitively plausi- ble mechanisms. In the Wheelbarrow project, we are, therefore, working towards a cognitive theory of mathematical thought in order to substantiate the existing theories and to improve automated theory formation systems. We build on two streams of research: embodied conceptual- isation, which analyses mathematical ideas from a cognitive perspective (Lakoff & N´u n ˜ ez, 2000), and societal conceptuali- sation based on Lakatos’s (1976) philosophical account of the evolution of mathematical ideas. Both argue strongly against the ‘romantic’ (Lakoff and N´u n ˜ ez) or ‘deductivist’ (Lakatos) style in which mathematics is presented as an ever-increasing set of universal, absolute, certain truths which exist indepen- dently of humans. Our main interest is how mathematical concepts are formed and modified by the embodied and situated human mind. For instance, Euclid formulated geometric axioms to describe the physical world – the foundations of Euclidean geometry. Eu- clidean geometry was later modified by rejecting the parallel postulate (one of the axioms), and non-Euclidean geometries were formed, along with new sets of concepts. On a less cele- brated but equally remarkable level children are able to formu- late and modify mathematical rules about their environment such as transitivity or the commutativity in arithmetic. Lakoff and N´u n ˜ ez’s theory of embodied mathematics and Lakatos’s philosophy of mathematics suggest explanation of how this may work. Metaphors and Detecting New Scientific Concepts Lakoff and N´u n ˜ ez (2000) argue that our ordinary conceptual system is fundamentally metaphorical in nature: metaphor makes abstract thought possible, and the development of thought is the process of developing better metaphors. They characterise metaphors as a ‘grounded, inference-preserving cross domain mapping’ (p 6), thus enabling us to use the infer- ential structure of one domain to reason about another. They catalogue a large number of mathematical metaphors, thus suggesting how highly abstract mathematical ideas may be discovered and understood, and how they can be traced back to human embodiment. Lakoff and N´u n ˜ ez show how conceptual metaphors are re- vealed by everyday language, eg the expression adding onions to soup, suggests that add can mean physically placing objects in a container. They place great emphasis on the type of do- mains in a metaphor and distinguish two types of metaphor: grounding metaphors, in which one domain is embodied and the other abstract, and linking metaphors, in which both do- mains are abstract. Many linking metaphors in mathematics conceptualise some domain of mathematics in terms of arith-