The purpose of this book is to introduce the basic ideas of mathematical proof to students embarking on university mathematics. The emphasis is on helping the reader in understanding and constructing proofs and writing clear mathematics. This is achieved by exploring set theory, combinatorics and number theory, topics which include many fundamental ideas which are part of the tool kit of any mathematician. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic (...) methods of proof, and includes some of the classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. Over 250 problems include questions to interest and challenge the most able student as well as plenty of routine exercises to help familiarize the reader with the basic ideas. (shrink)
Popper and Eccles present two different notions of Interactionism. Popper's arguments arise out of the traditional philosophical debate, whereas Eccles' arguments arise out of a mixture of neurophysiology and personal belief. Popper's three-world ontology is the philosophical foundation of both their positions. However, it is precisely against the background of the three Worlds that the considerable differences between their positions are apparent. Despite these defects, Interactionism is a productive notion since it does not place the Self beyond experimental (...) investigation. Indeed, both Popper and Eccles have made implicit suggestions as to experimental procedure which might contribute to the investigation of the Self. (shrink)
Physics and metaphysics (with a letter from K.R. Popper) -- Relation between cause and effect (with a letter from V. Tonini) -- Chance or finality? (with two letters from L. Pauling and two from J. Eccles) -- The individual and metaphysics : life existentalism and the mystery of life (with a letter from H. Von Balthasar) -- The hidden roots of pain (with a letter from P. Pavan).
Mainstream cognitive neuroscience typically ignores the role of quantum physical effects in the neural processes underlying cogni¬tion and consciousness. However, many unsolved problems remain, suggesting the need to consider new approaches. We propose that quantum theory, especially through an ontological interpretation due to Bohm and Hiley, provides a fruitful framework for addressing the neural correlates of cognition and consciousness. In particular, the ontological interpretation suggests that a novel type of 'active information', connected with a novel type of 'quantum potential energy', (...) plays a key role in quantum physical processes. After introducing the ontological interpretation we illustrate its value for cognitive neuroscience by discussing it in the light of a proposal by Beck and Eccles about how quantum tunneling could play a role in controlling the frequency of synaptic exocytosis. In this proposal, quantum tunneling would enable the 'self' to control its brain without violating the energy conservation law. We argue that the ontological interpretation provides a sharper picture of what actually could be taking place in quantum tunneling in general and in synaptic exocytosis in particular. Based on the notions of active information and quantum potential energy, we propose a coherent way of understanding how mental processes (understood as involving non-classical physical processes) can act on traditional, classically describable neural processes without violating the energy conservation law. (shrink)