BookMark180001：《Logical Foundations of Mathematics and Computational Complexity: A Gentle Introduction》

https://www.amazon.cn/gp/product/3319001183/ref=ox_sc_sfl_title_19?ie=UTF8&psc=1&smid=A1AJ19PSB66TGU

 

The two main themes of this book, logic and complexity, are both essential for understanding the main problems about the foundations of mathematics. Logical Foundations of Mathematics and Computational Complexity covers a broad spectrum of results in logic and set theory that are relevant to the foundations, as well as the results in computational complexity and the interdisciplinary area of proof complexity. The author presents his ideas on how these areas are connected, what are the most fundamental problems and how they should be approached. In particular, he argues that complexity is as important for foundations as are the more traditional concepts of computability and provability.

Emphasis is on explaining the essence of concepts and the ideas of proofs, rather than presenting precise formal statements and full proofs. Each section starts with concepts and results easily explained, and gradually proceeds to more difficult ones. The notes after each section present some formal definitions, theorems and proofs.

Logical Foundations of Mathematics and Computational Complexity is aimed at graduate students of all fields of mathematics who are interested in logic, complexity and foundations. It will also be of interest for both physicists and philosophers who are curious to learn the basics of logic and complexity theory.

 

“This monograph by the outstanding Czech logician Pavel Pudlák provides a broad but also deep survey of work in logic and computer science relevant to foundational issues, interpreted in a wide sense. … This is a fine overview of logic and complexity theory that can be confidently recommended to anybody who would like to orient themselves in an increasingly intricate and difficult field.” (Alasdair Urquhart, Philosophia Mathematica, Vol. 23 (3), October, 2015)

“For the non-expert it offers indeed a ‘gentle introduction’ to logic that is well selected and excellently explained. And for the logician it certainly offers some of the best introductions to those topics outside their area of direct expertise. … it contains plenty of informal explanations, intuition and motivation. … It is truly a gift to the logic and wider communities … . This book is very enjoyable to read and I wish it all success.” (Olaf Beyersdorff, Mathematical Reviews, August, 2014)

“It spans the historical, logical, and at times philosophical underpinnings of the theory of computational complexity. Students of mathematics seeking a transition to higher mathematics will find it helpful, as will mathematicians with expertise in other areas. … an excellent choice for a first text in studying complexity, or as a clarifying adjunct to any assigned text in this area. … a compact guide for graduate students with a need for or interest in computational complexity and its foundations.” (Tom Schulte, MAA Reviews, July, 2014)
“This book, exactly as indicated by its title, deals with the main philosophical, historical, logical and mathematical aspects … in a quite approachable and attractive way. … the prospective readers of this book are mathematicians with an interest in the foundations, philosophers with a good background in mathematics, and also philosophically minded scientists. Due to the author’s nice style, the book will be a very good choice for the first text in studying this subject.” (Branislav Boričić, zbMATH, Vol. 1270, 2013)

 

本书的两个主题，逻辑和复杂性，对理解数学基础的主要问题都是必不可少的。数学和计算复杂性的逻辑基础覆盖了与基础相关的逻辑和集合理论的广泛结果，以及计算复杂性和证据复杂性的跨学科领域的结果。作者提出了他关于这些领域如何相互联系的想法，最根本的问题是什么，以及如何应对它们。特别地，他认为复杂性对于基础是同样重要的，因为更传统的可计算性和可证性概念。

重点是解释概念的本质和证明的想法，而不是提出精确的正式声明和完整的证明。每个部分从概念和结果开始容易解释，并逐渐进行到更困难的。每个部分后面的注释提供了一些正式的定义，定理和证明。

数学和计算复杂性的逻辑基础面向对逻辑，复杂性和基础感兴趣的所有数学领域的研究生。这也将是物理学家和哲学家的兴趣，他们很好奇学习逻辑和复杂性理论的基础。

 

“这本由杰出的捷克逻辑学家PavelPudlák撰写的专着提供了广泛而深入的逻辑和计算机科学方面的研究，与基础问题相关，在广义上进行解释。 ...这是对逻辑和复杂性理论的精细概述，可以自信地推荐给任何想要将自己定位在日益复杂和困难的领域的人。“（Alasdair Urquhart，Philosophia Mathematica，Vol。23（3），October， ）

“对于非专家，它提供了一个”温柔的介绍“的逻辑，被选择和优秀的解释。对于逻辑学家来说，它肯定会提供一些对他们直接专业领域以外的主题的最好的介绍。 ...它包含大量的非正式的解释，直觉和动机。 ...它是真正的礼物给逻辑和更广泛的社区...。这本书是非常愉快的阅读，我希望它的一切成功。“（Olaf Beyersdorff，数学评论，2014年8月）

“它跨越了计算复杂性理论的历史，逻辑和有时的哲学基础。数学的学生寻求过渡到高等数学将会发现有用的，数学家在其他领域的专业知识。 ...对于学习复杂性的第一个文本，或作为这个领域任何指定文本的澄清附件的一个很好的选择。 ...对于需要或对计算复杂性及其基础感兴趣的研究生的紧凑指南“（Tom Schulte，MAA评论，2014年7月）
“这本书，正如它的标题所指出的，以一种相当平易近人和有吸引力的方式处理主要的哲学，历史，逻辑和数学方面。 ...这本书的潜在读者是对基金会感兴趣的数学家，有良好数学背景的哲学家，也有哲学思想的科学家。由于作者的风格，这本书将是学习这个主题的第一篇文章的一个很好的选择。“（BranislavBoričić，zbMATH，Vol。1270，2013）

 

《数学的逻辑基础和计算复杂性——温柔的介绍》  https://site.douban.com/145723/widget/notes/18386132/note/606247927/

为什么想要读这本书  https://book.douban.com/review/8150344/

北京大学 逻辑学 王彦晶  http://www.phil.pku.edu.cn/personal/wangyj/Teaching/Delta16.html