The Theory of Plane Area at the Crossroads: Philosophical, Historical, and Logical Perspectives
Giovannini, Eduardo N., Haeusler, Edward Hermann, Lassalle Casanave, Abel
- 出版商: Springer
- 出版日期: 2025-01-04
- 售價: $4,690
- 貴賓價: 9.5 折 $4,456
- 語言: 英文
- 頁數: 146
- 裝訂: Hardcover - also called cloth, retail trade, or trade
- ISBN: 3031709152
- ISBN-13: 9783031709159
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商品描述
This book explores a cluster of philosophical, historical, and logical problems concerning the foundations of the theory of plane area in elementary geometry. The motivation of this study is a notable geometrical proposition known as De Zolt's postulate, which asserts that a polygon cannot be equal in area to a proper polygonal part. The book is the first systematic investigation of the philosophical and foundational significance of this proposition, which can also be described as the "fundamental theorem" of the theory of plane area.
This volume provides a comparative study of Euclid's development of the theory of area in the Elements and its modern reinterpretation in Hilbert's classical monograph Foundations of Geometry. It connects the historical reflections on De Zolt's postulate with the nineteenth-century program of providing a purely geometrical foundation for Euclidean geometry, uncovering a rich array of intertwined conceptual problems. It also shifts the perspective and provides a logical analysis of this geometrical postulate within an original development of the abstract theory of magnitudes, called compatible magnitudes. Finally, it extends the previous formal treatment of De Zolt's postulate to the case of three-dimensional geometry by producing a type system for polyhedral geometrical mereology. The innovative combination of philosophical, historical, and logical perspectives results in a novel discussion of a fascinating problem at the crossroads of (late) nineteenth-century geometry. This volume will interest readers in the fields of history and philosophy of mathematics, logic, and formal philosophy.
商品描述(中文翻譯)
本書探討了一系列與平面面積理論基礎相關的哲學、歷史和邏輯問題。本研究的動機源自一個著名的幾何命題，稱為 De Zolt 的公設，該公設主張一個多邊形不能與其適當的多邊形部分在面積上相等。本書是對這一命題的哲學和基礎意義的首次系統性研究，這一命題也可以被描述為平面面積理論的「基本定理」。
本卷提供了對歐幾里得在《幾何原本》中面積理論發展的比較研究，以及其在希爾伯特的經典專著《幾何的基礎》中現代的重新詮釋。它將對 De Zolt 公設的歷史反思與十九世紀提供歐幾里得幾何純粹幾何基礎的計畫相連結，揭示出一系列錯綜複雜的概念問題。它還轉變了視角，並在一個名為相容量的抽象量理論的原創發展中，對這一幾何公設進行了邏輯分析。最後，它將對 De Zolt 公設的先前形式處理擴展到三維幾何的情況，通過為多面體幾何的部分論建立一個類型系統。哲學、歷史和邏輯視角的創新結合，導致了對一個迷人問題的新討論，該問題位於十九世紀晚期幾何的交叉點上。本卷將吸引數學史與哲學、邏輯及形式哲學領域的讀者。
作者簡介
Eduardo N. Giovannini is Associate Researcher at the National Scientific and Technological Research Council (CONICET, Argentina) and Assistant Professor at the National University of Litoral (Argentina). He also has held research positions at the Department of Philosophy of the University of Vienna and the University of California at Berkeley (USA). His research areas include the history and philosophy of mathematics, especially in the nineteenth and early twentieth centuries; the history of modern axiomatic geometry, with a particular emphasis on David Hilbert's metatheoretical investigations, and the history and philosophy of modern formal logic, particularly model theory. He has published extensively on Hilbert's foundational work in geometry, the history of the modern axiomatic method, the geometrical roots of model theory. He is the author of David Hilbert y los fundamentos de la geometría (College Publications, 2015).
Edward Hermann Haeusler is Associate Professor of the Department of Informatics at the Pontifical Catholic University of Rio de Janeiro (PUC-Rio, Brazil). His research areas are proof theory, logic, and theory of computation. Besides papers and book chapters, he is the author of Category Theory for Computer Science (Sagra Luzzato and UFRGS), co-editor of Celebration of Dag Prawitz's Work (Springer-Verlag, 2013), co-editor of A Question is More Illuminating than an Answer. A Festschrift for Paulo A. S. Veloso (College Publications, 2021), and co-editor of Why is this a Proof ? Festschrift for Luiz Carlos Pereira (College Publications, 2015). He is also co-editor of the special issue volume 1, number 18 of the Logic Journal of IGPL (Oxford University Press, 2009) and of the special issue of Computación y Sistemas 21(3) (2017).
Abel Lassalle-Casanave is Full Professor of the Department of Philosophy at the Federal University of Bahia (UFBA, Brazil) and Researcher at the National Council of Research and Technological Development (CNPq, Brazil). His area of research is philosophy of formal sciences, with emphasis in verbal, symbolic and diagrammatic proofs, modern philosophy of mathematics and Hilbertian formalism. Beside papers and chapters of books, he is the author of Por construção de conceitos: em torno da filosofia kantiana da matemática (Editora PUC-Rio, 2019), editor of Symbolic Knowledge from Leibniz to Husserl (College Publications, 2012), and co-editor of Visualização nas ciências formais (College Publications, 2012, with Frank Sautter) and of El árbol de los números: cognición, lógica y práctica matemática(Editorial Universidad de Sevilla, 2016, with José Ferreirós).
作者簡介(中文翻譯)
Eduardo N. Giovannini 是阿根廷國家科學與技術研究委員會（CONICET）的副研究員，以及阿根廷利托拉爾國立大學的助理教授。他曾在維也納大學哲學系和美國加州大學伯克利分校擔任研究職位。他的研究領域包括數學的歷史與哲學，特別是十九世紀和二十世紀初期；現代公理幾何的歷史，特別強調大衛·希爾伯特的元理論研究，以及現代形式邏輯的歷史與哲學，特別是模型理論。他在希爾伯特的幾何基礎工作、現代公理方法的歷史以及模型理論的幾何根源方面發表了大量著作。他是《David Hilbert y los fundamentos de la geometría》（College Publications, 2015）的作者。
Edward Hermann Haeusler 是巴西里約熱內盧天主教大學（PUC-Rio）資訊學系的副教授。他的研究領域包括證明理論、邏輯和計算理論。除了論文和書籍章節外，他還是《Category Theory for Computer Science》（Sagra Luzzato 和 UFRGS）的作者，《Celebration of Dag Prawitz's Work》（Springer-Verlag, 2013）的共同編輯，《A Question is More Illuminating than an Answer. A Festschrift for Paulo A. S. Veloso》（College Publications, 2021）的共同編輯，以及《Why is this a Proof? Festschrift for Luiz Carlos Pereira》（College Publications, 2015）的共同編輯。他還是《Logic Journal of IGPL》（Oxford University Press, 2009）第1卷第18期的特刊共同編輯，以及《Computación y Sistemas》21(3)（2017）的特刊共同編輯。
Abel Lassalle-Casanave 是巴西巴伊亞聯邦大學（UFBA）哲學系的正教授，以及巴西國家研究與技術發展委員會（CNPq）的研究員。他的研究領域是形式科學的哲學，重點在於口頭、符號和圖示證明、現代數學哲學和希爾伯特形式主義。除了論文和書籍章節外，他是《Por construção de conceitos: em torno da filosofia kantiana da matemática》（Editora PUC-Rio, 2019）的作者，《Symbolic Knowledge from Leibniz to Husserl》（College Publications, 2012）的編輯，以及《Visualização nas ciências formais》（College Publications, 2012，與 Frank Sautter 共同編輯）和《El árbol de los números: cognición, lógica y práctica matemática》（Editorial Universidad de Sevilla, 2016，與 José Ferreirós 共同編輯）的共同編輯。
