The Logical Structure of Mathematics

An Overview of Mathematical Concepts
1. Auflage 2026
ISBN: 978-1-041-01522-2
Verlag: Taylor & Francis Ltd

Buch, Englisch, 552 Seiten, Format (B × H): 178 mm x 254 mm

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

Zielgruppe

Undergraduate Advanced and Undergraduate Core

Autoren/Hrsg.

Serovajsky, Simon

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Zahlentheorie
• Mathematik | Informatik Mathematik Mathematik Allgemein Mathematische Logik
• Mathematik | Informatik Mathematik Geometrie
• Mathematik | Informatik Mathematik Mathematik Allgemein Geschichte der Mathematik
• Mathematik | Informatik Mathematik Mathematik Allgemein Philosophie der Mathematik

Inhaltsverzeichnis

Part 1: Languages  Part 2: Sets  Part 3: Numbers  Part 4: Objects  4A. Ordered Objects  4B. Algebraic Objects  4C. Topological Objects  Part 5: Theories  5A. Applications of Linear Algebra and Geometry  5B. Topological Algebra and Functional Analysis  5C. Calculus  5D. Algebraic Geometry and Algebraic Topology  Part 6: Categories

Über Autor(innen)

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.