ABSTRACT
Reasoning by analogy is widely accepted to be a core component of human cognition. Its contribution to cognition has been defined in varying ways. A recent definition of analogical reasoning introduced the concept of “mental leaps.” Holyoak and Thagard (1995) refer to the act of forming an analogy as seeing one thing as if it were another, an activity that requires a kind of mental leap between domains. In chapter 1 of this volume, Lyn English defines analogy as “the ability to reason with relational patterns.” She notes that analogy involves the detection of patterns, and the identification of the recurrence of patterns in the face of variation in their elements. I will argue that both relational patterns and mental leaps are essential to mathematical activity. So is a third aspect of analogical reasoning, the creative aspect. Analogical thinking has been at the core of some of the greatest discoveries in science. A striking example of this is the analogy that led to Kekule's (1865) theory about the molecular structure of benzene (see Holyoak & Thagard, 1995). In a dream, Kekule had a visual image of a snake biting its own tail. This gave him the idea that the carbon atoms in benzene could be arranged in a ring. The similarity between the snake and the carbon atoms was at a purely structural/relational level—the level of circular arrangement. Recognizing creative abstract similarities such as these is also an important part of mathematical thinking. Put simply, analogy is part of how mathematicians think. Mathematicians themselves recognized this a long time ago (e.g., Polya, 1957).
