# Mathematical Reasoning : Analogies, Metaphors, ...

How we reason with mathematical ideas continues to be a fascinating and challenging topic of research--particularly with the rapid and diverse developments in the field of cognitive science that have taken place in recent years. Because it draws on multiple disciplines, including psychology, philosophy, computer science, linguistics, and anthropology, cognitive science provides rich scope for addressing issues that are at the core of mathematical learning. Drawing upon the interdisciplinary nature of cognitive science, this book presents a broadened perspective on mathematics and mathematical reasoning. It represents a move away from the traditional notion of reasoning as "abstract" and "disembodied", to the contemporary view that it is "embodied" and "imaginative." From this perspective, mathematical reasoning involves reasoning with structures that emerge from our bodily experiences as we interact with the environment; these structures extend beyond finitary propositional representations. Mathematical reasoning is imaginative in the sense that it utilizes a number of powerful, illuminating devices that structure these concrete experiences and transform them into models for abstract thought. These "thinking tools"--analogy, metaphor, metonymy, and imagery--play an important role in mathematical reasoning, as the chapters in this book demonstrate, yet their potential for enhancing learning in the domain has received little recognition. This book is an attempt to fill this void. Drawing upon backgrounds in mathematics education, educational psychology, philosophy, linguistics, and cognitive science, the chapter authors provide a rich and comprehensive analysis of mathematical reasoning. New and exciting perspectives are presented on the nature of mathematics (e.g., "mind-based mathematics"), on the array of powerful cognitive tools for reasoning (e.g., "analogy and metaphor"), and on the different ways these tools can facilitate mathematical reasoning. Examples are drawn from the reasoning of the preschool child to that of the adult learner.

## Mathematical Reasoning : Analogies, Metaphors, ...

Analogical cognition, which embraces all cognitive processesinvolved in discovering, constructing and using analogies, is broaderthan analogical reasoning (Hofstadter 2001; Hofstadter and Sander2013). Understanding these processes is an important objective ofcurrent cognitive science research, and an objective that generatesmany questions. How do humans identify analogies? Do nonhuman animalsuse analogies in ways similar to humans? How do analogies andmetaphors influence concept formation?

Many people take the concept of model-theoretic isomorphism to set the standard for thinking about similarity and its role inanalogical reasoning. They propose formal criteria forevaluating analogies, based on overall structural or syntacticalsimilarity. Let us refer to theories oriented around such criteria asstructuralist.

Analogy plays a significant role in problem solving, as well as decision making, argumentation, perception, generalization, memory, creativity, invention, prediction, emotion, explanation, conceptualization and communication. It lies behind basic tasks such as the identification of places, objects and people, for example, in face perception and facial recognition systems. It has been argued that analogy is "the core of cognition".[3] Specific analogical language comprises exemplification, comparisons, metaphors, similes, allegories, and parables, but not metonymy. Phrases like and so on, and the like, as if, and the very word like also rely on an analogical understanding by the receiver of a message including them. Analogy is important not only in ordinary language and common sense (where proverbs and idioms give many examples of its application) but also in science, philosophy, law and the humanities. The concepts of association, comparison, correspondence, mathematical and morphological homology, homomorphism, iconicity, isomorphism, metaphor, resemblance, and similarity are closely related to analogy. In cognitive linguistics, the notion of conceptual metaphor may be equivalent to that of analogy. Analogy is also a basis for any comparative arguments as well as experiments whose results are transmitted to objects that have been not under examination (e.g., experiments on rats when results are applied to humans).

Steven Phillips and William H. Wilson[35][36] use category theory to mathematically demonstrate how the analogical reasoning in the human mind, that is free of the spurious inferences that plague conventional artificial intelligence models, (called systematicity), could arise naturally from the use of relationships between the internal arrows that keep the internal structures of the categories rather than the mere relationships between the objects (called "representational states"). Thus, the mind may use analogies between domains whose internal structures fit according with a natural transformation and reject those that do not.

Paper #1 identifies analogical reasoning as a way of thinking in the context of advanced mathematics. There has been critique of the use of analogies for the purpose of students learning new content because students may fail to appropriately recognize the analogical connections developed by instructors. I counter that students can productively reason by analogy to understand new mathematics when provided with settings to develop this way of thinking. In this paper, I use examples from the work of mathematicians to argue for the important role of analogy for the purpose of mathematical discovery. I then provide an illustration of an undergraduate student engaged in similar productive analogical reasoning as they develop analogs between structures in group and ring theory. Through this process, the student showed increasing awareness of how and why they were engaging with such reasoning. This observation evidences the potential for students to reason by analogy for mathematical discovery. 041b061a72