The Logical Structure of Mathematics: An Overview of Mathematical Concepts
暫譯: 數學的邏輯結構：數學概念概論
Serovajsky, Simon
商品描述
The Logical Structure of Mathematics is devoted to the presentation of the logical structure of mathematics from its foundations to its main standard areas and sections. It is intended to give an idea of mathematics as a uniform, integral system of knowledge, where different areas and concepts are intertwined with multilateral connections.
The book is conceived as a brief mathematical encyclopedia but presents the necessary information about the main sections of mathematics and the most important mathematical concepts not in alphabetical order, but in their logical sequence. All the concepts and statements considered are illustrated by numerous examples. Due to the breadth of the material covered and the emphasis on identifying interdisciplinary connections, the book can help mathematicians expand the scope of their scientific interests and draw new ideas from related mathematical areas. Due to the thorough explanation of the meaning of newly introduced concepts, the presence of rich illustrative material, and the absence of cumbersome technical calculations, the book can help young researchers and students navigate the boundless ocean of mathematical ideas.
Features
- Brief introduction to the main sections of mathematics from the most fundamental ones such as algebra, topology, and analysis to more specialized ones such as differential equations, probability theory, calculus of variations, and differential and algebraic geometry
- Comprehensive treatment of the basic concepts of mathematics in their logical interconnection
- Significant portion of the material is included in appendices to parts and chapters, which allows the book to be used by readers with different levels of preparation and interests
- Includes numerous illustrations and examples
- Requires minimal mathematical prerequisites
- Extensive subject index makes the book easy to use as a reference book
商品描述(中文翻譯)
數學的邏輯結構旨在介紹數學從基礎到主要標準領域與分支的邏輯結構。本書希望呈現數學作為一套統一且完整的知識體系，其中不同領域與概念透過多方面的關聯彼此交織。
本書以簡明數學百科全書為構想，但對數學主要分支與最重要數學概念的介紹，並非按照字母順序排列，而是依其邏輯順序編排。書中討論的所有概念與命題，皆以大量範例加以說明。由於涵蓋的內容廣泛，並著重於辨識不同學科之間的關聯，本書能協助數學家拓展科學興趣的範圍，並從相關數學領域汲取新的構想。另一方面，由於本書詳盡解釋新引入概念的意義，提供豐富的圖示與說明材料，且避免繁複的技術性計算，因此也能幫助年輕研究人員與學生在浩瀚的數學思想海洋中找到方向。
特色
- 簡要介紹數學的主要分支，涵蓋代數、拓撲與分析等最基礎的領域，以及微分方程、機率論、變分法、微分幾何與代數幾何等較專門的領域
- 全面介紹數學的基本概念，並闡明其邏輯上的相互關聯
- 大量內容收錄於各篇章與各章的附錄中，使不同準備程度與興趣的讀者都能使用本書
- 包含大量圖示與範例
- 僅需具備最低限度的數學先備知識
- 完整的主題索引使本書便於作為參考書使用
作者簡介
Simon Serovajsky. Professor of Mathematics of al-Farabi Kazakh National University, Almaty, Kazakhstan.
Research Interests. Optimal Control Theory, Partial Differential Equations, Nonlinear Functional Analysis, Mathematical and Computational Modeling, History and Philosophy of Mathematics, Foundations of Mathematics. Scientific Achievements. Definition of the extended operator derivative. Extended differentiability of the solution for the nonlinear infinite dimensional systems with respect to parameter without its Gateaux differentiability. Necessary conditions of optimality for nonlinear infinite dimensional control systems. Definition of the weakened approximate solution of optimal control problems. Definition of the sequential model of systems. Sequential extension of optimal control problems. Mathematical modeling of problems in medicine and epidemiology.
General Books:
1. Counterexamples in the Optimal Control Theory. - Brill Academic Press. Netherlands, Utrecht, Boston, 2004. - 176 p.; Berlin, Boston, De Gruyter, 2011 (reprint).
2. Optimization and Differentiation. - Chapman and Hall / CRC Press. Taylor and Francis group, London, 2017. - 517 p.
3. Sequential models of the Mathematical Physics. - Boca Raton, London, New York, CRC Press, Taylor and Francis Group, 2019.- 266 p.
4. History of Mathematics. Evolutions of mathematical ideas. Part 1. Number Theory. Geometry. Topology. - Moscow, URSS, 2019. - 224 p.; Moscow, URSS, 2021 (reprint); Moscow, URSS, 2022 (reprint).
5. History of Mathematics. Evolutions of mathematical ideas. Part 2. Algebra. Analysis. Differential Equations. Extremum Theory. - Moscow, URSS, 2019. - 368 p.; Moscow, URSS, 2021 (reprint); Moscow, URSS, 2022 (reprint).
6. History of Mathematics. Evolutions of mathematical ideas. Part 3. Computational Mathematics. Probability Theory. Computer Science. Mathematical Logic. - Moscow, URSS, 2019. - 240 p.; Moscow, URSS, 2021 (reprint); Moscow, URSS, 2022 (reprint).
7. Architecture of Mathematics. - Boca Raton, London, New York, Chapman and Hall/CRC Press. - 2020. - 394 p.
8. Mathematical Modelling. - Boca Raton, London, New York, CRC Press. Taylor and Francis Group, 2022. - 441 p.
9. Optimization. 100 Examples. - New York, Chapman and Hall/CRC, 2024. - 556 p.
作者簡介(中文翻譯)
Simon Serovajsky。哈薩克斯坦阿拉木圖，al-Farabi Kazakh National University 數學教授。
研究興趣。最佳控制理論、偏微分方程、非線性泛函分析、數學與計算建模、數學史與數學哲學、數學基礎。
學術成就。定義延伸算子導數。在不具備 Gâteaux 可微性的情況下，研究非線性無限維系統解對參數的延伸可微性。建立非線性無限維控制系統的最優性必要條件。定義最優控制問題的弱化近似解。定義系統的序列模型。最優控制問題的序列延伸。醫學與流行病學問題的數學建模。
一般著作：
1. 《Counterexamples in the Optimal Control Theory》。Brill Academic Press，荷蘭烏特勒支、波士頓，2004 年，176 頁；Berlin、Boston，De Gruyter，2011 年（再版）。
2. 《Optimization and Differentiation》。Chapman and Hall / CRC Press，Taylor and Francis 集團，倫敦，2017 年，517 頁。
3. 《Sequential Models of the Mathematical Physics》。Boca Raton、倫敦、紐約，CRC Press，Taylor and Francis 集團，2019 年，266 頁。
4. 《History of Mathematics. Evolutions of Mathematical Ideas. Part 1. Number Theory. Geometry. Topology》。莫斯科，URSS，2019 年，224 頁；莫斯科，URSS，2021 年（再版）；莫斯科，URSS，2022 年（再版）。
5. 《History of Mathematics. Evolutions of Mathematical Ideas. Part 2. Algebra. Analysis. Differential Equations. Extremum Theory》。莫斯科，URSS，2019 年，368 頁；莫斯科，URSS，2021 年（再版）；莫斯科，URSS，2022 年（再版）。
6. 《History of Mathematics. Evolutions of Mathematical Ideas. Part 3. Computational Mathematics. Probability Theory. Computer Science. Mathematical Logic》。莫斯科，URSS，2019 年，240 頁；莫斯科，URSS，2021 年（再版）；莫斯科，URSS，2022 年（再版）。
7. 《Architecture of Mathematics》。Boca Raton、倫敦、紐約，Chapman and Hall/CRC Press，2020 年，394 頁。
8. 《Mathematical Modelling》。Boca Raton、倫敦、紐約，CRC Press，Taylor and Francis 集團，2022 年，441 頁。
9. 《Optimization. 100 Examples》。紐約，Chapman and Hall/CRC，2024 年，556 頁。
