First published in 1958, John von Neumann's classic work "The Computer and the Brain" explored the analogies between computing machines and the living human brain. Von Neumann showed that the brain operates both digitally and analogically, but also has its own unique statistical language. And more than fifty years after its inception the "von Neumann architecture" - an organizational framework for computer design - still lies at the heart of today's machines. In his foreword to this new edition, Ray Kurzweil, a futurist famous for his own musings on the relationship between technology and consciousness, places von Neumann's work in a historical context and shows how it remains relevant today.

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations.

From the Back Cover
Es gibt einige Bücher, die die naturwissenschaftliche Welt verändert haben: John von Neumanns Buch über die Quantenmechanik gehört dazu! Mit dieser richtungsweisenden Studie legte er den Grundstein für seine späteren, weltberühmten Arbeiten in den USA. Das Buch ist nicht nur von historischem Interesse, sondern kann immer noch als elementare Einführung in die Grundbegiffe der Quantenmechanik mit Gewinn studiert werden. Seine besondere Stärke ist die Verbindung zwischen physikalischer Idee und mathematischer Exaktheit. Ein Gewinn für jeden Leser aus den Bereichen Mathematik und Physik - schon ab dem 4. Semester.

From the reviews: "The theory is systematically developed by the axiomatic method that has, since von Neumann, dominated the general approach to linear functional analysis and that achieves here a high degree of lucidity and clarity...The book contains about 350 well placed and instructive problems, which cover a considerable part of the subject. All in all this is an excellent work, of equally high value for both student and teacher." --ZENTRALBLATT FUR MATHEMATIK

From the Preface: "This book was written for the active reader. The first part consists of problems, frequently preceded by definitions and motivation, and sometimes followed by corollaries and historical remarks...The second part, a very short one, consists of hints...The third part, the longest, consists of solutions: proofs, answers, or contructions, depending on the nature of the problem...This is not an introduction to Hilbert space theory. Some knowledge of that subject is a prerequisite: at the very least, a study of the elements of Hilbert space theory should proceed concurrently with the reading of this book."

From the Reviews: "...He (the author) uses the language and notation of ordinary informal mathematics to state the basic set-theoretic facts which a beginning student of advanced mathematics needs to know...Because of the informal method of presentation, the book is eminently suited for use as a textbook or for self-study. The reader should derive from this volume a maximum of understanding of the theorems of set theory and of their basic importance in the study of mathematics." - "Philosophy and Phenomenological Research".

First published in 1961, this collection of playful and provocative stories by the eminent physicist is returned to print with an additional story and a new introduction. Annotation copyrighted by Book News, Inc., Portland, OR

新的科学突破将在于物质与信息边缘的地方。对于麦克斯韦妖这个假想的科学怪物的研究证实在朝向这个方向努力。逻辑计算能够不产生熵吗？计算上的不可逆和物理上的不可逆究竟有什么关系？信息和物质究竟以怎样的方式纠缠在一起？这本书是一个论文集，对物理与信息科学的边界进行探讨。

'The book is outstanding and admirable in many respects ...is necessary reading for all kinds of readers from undergraduate students to top authorities in the field' - "Journal of Symbolic Logic". Written by two experts in the field, this is the only comprehensive and unified treatment of the central ideas and applications of Kolmogorov complexity. The book presents a thorough treatment of the subject with a wide range of illustrative applications. Such applications include the randomness of finite objects or infinite sequences, Martin-Loef tests for randomness, information theory, computational learning theory, the complexity of algorithms, and the thermodynamics of computing. It will be ideal for advanced undergraduate students, graduate students, and researchers in computer science, mathematics, cognitive sciences, philosophy, artificial intelligence, statistics, and physics. The book is self-contained in that it contains the basic requirements from mathematics and computer science. Included are also numerous problem sets, comments, source references, and hints to solutions of problems. New topics in this edition include Omega numbers, Kolmogorov-Loveland randomness, universal learning, communication complexity, Kolmogorov's random graphs, time-limited universal distribution, Shannon information and others.