When you think about it, it seems obvious: The only mathematical ideas that human beings can have are ideas that the human brain allows. We know a lot about what human ideas are like from research in Cognitive Science. Most ideas are unconscious, and that is no less true of the mathematical ones. Abstract ideas, for the most part, arise via conceptual metaphor-a mechanism for projecting embodied (that is, sensory-motor) reasoning to abstract reasoning. This book argues that conceptual metaphor plays a central, defining role in mathematical ideas within the cognitive unconscious-from arithmetic and algebra to sets and logic to infinity in all of its forms: transfinite numbers, points at infinity, infinitesimals, and so on. Even the real numbers, the imaginary numbers, trigonometry, and calculus are based on metaphorical ideas coming out of the way we function in the everyday physical world.This book is about mathematical ideas, about what mathematics means-and why. The authors believe that understanding the metaphors implicit in mathematics will make mathematics make more sense. Moreover, understanding mathematical ideas and how they arise from our bodies and brains will make it clear that the brain's mathematics is mathematics, the only mathematics we know or can know.
George Lakoff is Professor of Linguistics at the University of California, Berkeley, and the co-author, with Mark Johnson, of
Metaphors We Live By and
Philosophy in the Flesh. He has also written
Women, Fire, and Dangerous Things,
Moral Politics, and, with Mark Turner,
More Than Cool Reason: A Field Guide to Poetic Metaphor. He was one of the founders of the generative-semantics movement in linguistics in the 1960s, a founder of the field of cognitive linguistics in the 1970s, and one of the developers of the neural theory of language in the 1980s and '90s.
Rafael Núñez, currently at the Department of Psychology of the University of Freiburg, is a research associate of the University of California, Berkeley. He has worked for more than a decade on the foundations of embodied cognition, with special research into the nature and origin of mathematical concepts. He has published in several languages in a variety of areas, and has taught in leading academic institutions in Europe, the United States, and South America. He is the co-editor of Reclaiming Cognition: The Primacy of Action, Intention, and Emotion.
