The Code of Mathematics: Proof and Truth

Müller-Stach, Stefan

  • 出版商: Springer
  • 出版日期: 2024-09-04
  • 售價: $2,340
  • 貴賓價: 9.5$2,223
  • 語言: 英文
  • 頁數: 238
  • 裝訂: Quality Paper - also called trade paper
  • ISBN: 3662694824
  • ISBN-13: 9783662694824
  • 海外代購書籍(需單獨結帳)

相關主題

商品描述

Inspired by recent developments in dependent type theory and infinity categories, this book presents a history of ideas around the topics of truth, proof, equality and equivalence. Besides selected ideas of Platon, Aristoteles, Leibniz, Kant, Frege and others, the results of Gödel and Tarski on incompleteness, undecidability and truth in deductive systems and their semantic models are covered. The main focus of this textbook is on dependent type theory and its recent variant homotopy type theory. Such theories contain identity types, which give a new understanding of equality, symmetry, equivalence and isomorphism in a conceptual way. The interaction of type theory and infinity category theory yields a new paradigm for a structural view on mathematics. This supports the tendencies towards formalising mathematics with the help of proof assistants.

This book was first published in German. The translation was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content.

商品描述(中文翻譯)

受到最近在依賴類型理論和無窮類別方面的發展啟發,本書呈現了圍繞真理、證明、平等和等價等主題的思想歷史。除了選取的柏拉圖、亞里士多德、萊布尼茨、康德、弗雷格等人的思想外,還涵蓋了哥德爾和塔爾斯基在不完全性、不可決定性以及在演繹系統及其語義模型中的真理方面的成果。本教科書的主要焦點是依賴類型理論及其最近的變體——同倫類型理論。這些理論包含身份類型,從概念上提供了對平等、對稱、等價和同構的新理解。類型理論與無窮類別理論的互動為數學的結構觀提供了一種新範式。這支持了利用證明助手形式化數學的趨勢。

本書首次以德文出版。翻譯是借助人工智慧完成的,隨後進行了主要針對內容的人為修訂。

作者簡介

Stefan Müller-Stach is Vice President for Research and Early Career Academics at Johannes Gutenberg University Mainz and professor for number theory. His main research focus is on algebraic and arithmetic geometry, mathematical physics and history of science.

作者簡介(中文翻譯)

Stefan Müller-Stach 是美因茨約翰·古騰堡大學的研究與早期職業學者副校長,也是數論教授。他的主要研究重點是代數幾何、算術幾何、數學物理以及科學史。