Eine prägnante, einheitliche Sicht auf die Mathematik und ihre historische Entwicklung. Der Autor richtet sich an Mathematiker, die ein besseres Verständnis für die Zusammenhänge der modernen Mathematik erlangen möchten. Historische Aspekte dienen hier der Motivation und Einheit der Mathematik, ohne dass historische Vorkenntnisse vorausgesetzt werden.
In dem Buch erkundet der preisgekrönte Autor John Stillwell die Konsequenzen, die sich ergeben, wenn man die Unendlichkeit akzeptiert, und diese Konsequenzen sind vielseitig und überraschend. Der Leser benötigt nur wenig über die Schulmathematik hinausgehendes Hintergrundwissen; es reicht die Bereitschaft, sich mit ungewohnten Ideen auseinanderzusetzen. Stillwell führt den Leser sanft in die technischen Details von Mengenlehre und Logik ein, indem jedes Kapitel einem einzigen Gedankengang folgt, der mit einer natürlichen mathematischen Frage beginnt und dann anhand einer Abfolge von historischen Antworten nachvollzogen wird. Auf diese Weise zeigt der Autor, wie jede Antwort ihrerseits zu neuen Fragen führt, aus denen wiederum neue Begriffe und Sätze entstehen. Jedes Kapitel endet mit einem Abschnitt „Historischer Hintergrund“, der das Thema in den größeren Zusammenhang der Mathematik und ihrer Geschichte einordnet.
Stillwell is . . . One of the better current mathematical authors: he writes clearly and engagingly, and makes more of an effort than most to provide historical detail and a sense of how various mathematical ideas tie in with one another. . . . The features we have learned to expect from Stillwell (including, but not limited to, excellent writing) are present in [Elements of Mathematics] as well.--MAA Reviews
A beautiful and relatively elementary account of a part of mathematics where three main fields - algebra, analysis and geometry - meet. The book provides a broad view of these subjects at the level of calculus, without being a calculus book. Its roots are in arithmetic and geometry, the two opposite poles of mathematics, and the source of historic conceptual conflict. The resolution of this conflict, and its role in the development of mathematics, is one of the main stories in the book. Stillwell has chosen an array of exciting and worthwhile topics and elegantly combines mathematical history with mathematics. He covers the main ideas of Euclid, but with 2000 years of extra insights attached. Presupposing only high school algebra, it can be read by any well prepared student entering university. Moreover, this book will be popular with graduate students and researchers in mathematics due to its attractive and unusual treatment of fundamental topics. A set of well-written exercises at the end of each section allows new ideas to be instantly tested and reinforced.
Linking set theory with analysis, this text offers a detailed exploration of the real numbers system. It provides a unique introduction to set theory while thoroughly explaining the fundamental concepts of analysis, filling a gap in standard curricula. The book aims to enhance understanding of both mathematical fields through its comprehensive approach.
This book explores the dual nature of algebra as both abstract and applied mathematics, emphasizing its role in solving concrete geometric problems. It traces algebra's evolution from Euclid's era to the 19th century, highlighting its unifying power across various mathematical disciplines and fostering a deeper appreciation for its principles.
Solutions of equations in integers is the central problem of number theory and is the focus of this book. The amount of material is suitable for a one-semester course. The author has tried to avoid the ad hoc proofs in favor of unifying ideas that work in many situations. There are exercises at the end of almost every section, so that each new idea or proof receives immediate reinforcement.
Exploring the evolution of proof, this book highlights its crucial role in shaping mathematical knowledge from ancient to modern times. It traces the journey from Euclid's geometry to the development of algebra and calculus, illustrating how proof methods have transformed. The discussion extends to number theory, non-Euclidean geometry, and logic, revealing the profound implications of proof on arithmetic and the limitations it imposes on theorem validation. Through historical episodes, it provides a fresh perspective on mathematics' foundational principles and its capacity for innovation.
This book explores four distinct approaches to teaching geometry, highlighting the evolution from Euclid's methods to modern concepts like linear algebra and transformation groups. Each approach is presented in two chapters: one concrete and introductory, the other more abstract, illustrating the richness and diversity of geometric understanding.
Exploring the intersection of set theory and mathematical logic, this book delves into their influence on contemporary mathematics, particularly in number theory and combinatorics. It examines how foundational questions about infinity and the nature of proof have shaped mathematical thought. By tracing the evolution of these ideas, the text provides a comprehensive understanding of their relevance in modern mathematical discussions and developments.