A study of the cognitive science of mathematical ideas.

Traditional cognitive science is Cartesian in the sense that it takes as fundamental the distinction between the mental and the physical, the mind and the world. This leads to the claim that cognition is representational and best explained using models derived from AI and computational theory. The authors depart radically from this model.

Originating from a Dagstuhl seminar, the collection of papers presented in this book constitutes on the one hand a representative state-of-the-art survey of embodied artificial intelligence, and on the other hand the papers identify the important research trends and directions in the field. Following an introductory overview, the 23 papers are organized into topical sections on - philosophical and conceptual issues - information, dynamics, and morphology - principles of embodiment for real-world applications - developmental approaches - artificial evolution and self-reconfiguration

What are human beings like? How is knowledge possible? What is truth? Where do moral values come from? Questions like these have stood at the center of Western philosophy for centuries. In addressing them, philosophers have made certain fundamental assumptions-that we can know our own minds by introspection, that most of our thinking about the world is literal, and that reason is disembodied and universal-that are now called into question by well-established results of cognitive science. It has been shown empirically that:Most thought is unconscious. We have no direct conscious access to the mechanisms of thought and language. Our ideas go by too quickly and at too deep a level for us to observe them in any simple way.Abstract concepts are mostly metaphorical. Much of the subject matter of philosopy, such as the nature of time, morality, causation, the mind, and the self, relies heavily on basic metaphors derived from bodily experience. What is literal in our reasoning about such concepts is minimal and conceptually impoverished. All the richness comes from metaphor. For instance, we have two mutually incompatible metaphors for time, both of which represent it as movement through space: in one it is a flow past us and in the other a spatial dimension we move along.Mind is embodied. Thought requires a body-not in the trivial sense that you need a physical brain to think with, but in the profound sense that the very structure of our thoughts comes from the nature of the body. Nearly all of our unconscious metaphors are based on common bodily experiences.Most of the central themes of the Western philosophical tradition are called into question by these findings. The Cartesian person, with a mind wholly separate from the body, does not exist. The Kantian person, capable of moral action according to the dictates of a universal reason, does not exist. The phenomenological person, capable of knowing his or her mind entirely through introspection alone, does not exist. The utilitarian person, the Chomskian person, the poststructuralist person, the computational person, and the person defined by analytic philosopy all do not exist.Then what does?Lakoff and Johnson show that a philosopy responsible to the science of mind offers radically new and detailed understandings of what a person is. After first describing the philosophical stance that must follow from taking cognitive science seriously, they re-examine the basic concepts of the mind, time, causation, morality, and the self: then they rethink a host of philosophical traditions, from the classical Greeks through Kantian morality through modern analytic philosopy. They reveal the metaphorical structure underlying each mode of thought and show how the metaphysics of each theory flows from its metaphors. Finally, they take on two major issues of twentieth-century philosopy: how we conceive rationality, and how we conceive language.

A landmark book in the debate over free will that makes the case for compatibilism. In this landmark 1984 work on free will, Daniel Dennett makes a case for compatibilism. His aim, as he writes in the preface to this new edition, was a cleanup job, “saving everything that mattered about the everyday concept of free will, while jettisoning the impediments.” In Elbow Room, Dennett argues that the varieties of free will worth wanting—those that underwrite moral and artistic responsibility—are not threatened by advances in science but distinguished, explained, and justified in detail. Dennett tackles the question of free will in a highly original and witty manner, drawing on the theories and concepts of fields that range from physics and evolutionary biology to engineering, automata theory, and artificial intelligence. He shows how the classical formulations of the problem in philosophy depend on misuses of imagination, and he disentangles the philosophical problems of real interest from the “family of anxieties” in which they are often enmeshed—imaginary agents and bogeymen, including the Peremptory Puppeteer, the Nefarious Neurosurgeon, and the Cosmic Child Whose Dolls We Are. Putting sociobiology in its rightful place, he concludes that we can have free will and science too. He explores reason, control and self-control, the meaning of “can” and “could have done otherwise,” responsibility and punishment, and why we would want free will in the first place. A fresh reading of Dennett's book shows how much it can still contribute to current discussions of free will. This edition includes as its afterword Dennett's 2012 Erasmus Prize essay.

Collection of the most interesting recent writings on the philosophy of mathematics written by highly respected researchers from philosophy, mathematics, physics, and chemistry Interdisciplinary book that will be useful in several fields—with a cross-disciplinary subject area, and contributions from researchers of various disciplines

How Humans Learn to Think Mathematically describes the development of mathematical thinking from the young child to the sophisticated adult. Professor David Tall reveals the reasons why mathematical concepts that make sense in one context may become problematic in another. For example, a child's experience of whole number arithmetic successively affects subsequent understanding of fractions, negative numbers, algebra, and the introduction of definitions and proof. Tall's explanations for these developments are accessible to a general audience while encouraging specialists to relate their areas of expertise to the full range of mathematical thinking. The book offers a comprehensive framework for understanding mathematical growth, from practical beginnings through theoretical developments, to the continuing evolution of mathematical thinking at the highest level.