Primarily, this book addresses beginning graduate students expecting to become mathematicians or mathematically oriented computer scientists. Accordingly, the presentation is conditioned in content as well as in form by the assumption that the reader has already made an internal commitment to mathematics and is seeking not only mathematical information but also active involvement with mathematical pursuits. The general aim of what follows is to present basic mathematical concepts and techniques in familiar contexts in such a way as to illuminate the nature of mathematics as an art. Thus, the selection and organization of the material is based on considerations regarding the philosophical significance of various mathematical notions and results, their interdependence and their accessi bility. In other words, this text concentrates on displaying coherent mathe matical material meriting exceptionally careful and expansive contemplation. It should not be regarded as a reference for the most frequently used results or methods of elementary mathematics. The exposition is meant to be reasonably self-contained and to embody a growth pattern of mathematical ideas (for which no historical validity is claimed, of course). In order to avoid burying the essentials under routine technicalities, a style has been adopted that relies on the reader's active involvement somewhat more than is customary in texts for beginners. The exercises proposed at the end of each chapter are frequently extensions of the chapter content, rather than mere illustrations. They are designed to be manageable in a straightforward fashion within the framework provided by the text.weiterlesen