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Über dieses Buch

The main intention of this book is to describe and develop the conceptual, structural and abstract thinking of mathematics. Specific mathematical structures are used to illustrate the conceptual approach; providing a deeper insight into mutual relationships and abstract common features. These ideas are carefully motivated, explained and illustrated by examples so that many of the more technical proofs can be omitted. The book can therefore be used:

· simply as an overview of the panorama of mathematical structures and the relations between them, to be supplemented by more detailed texts whenever you want to acquire a working knowledge of some structure

· by itself as a first introduction to abstract mathematics

· together with existing textbooks, to put their results into a more general perspective

· to gain a new and hopefully deeper perspective after having studied such textbooks

Mathematical Concepts has a broader scope and is less detailed than standard mathematical textbooks so that the reader can readily grasp the essential concepts and ideas for individual needs. It will be suitable for advanced mathematicians, postgraduate students and for scientists from other fields with some background in formal reasoning.

Inhaltsverzeichnis

Chapter 1. Overview and Perspective

Abstract
This book is concerned with the conceptual structure of mathematics. In this introductory chapter, we discuss distinctions, relations, abstractions, and generation.
Jürgen Jost

Chapter 2. Foundations

Abstract
A set is a collection of distinct or distinguishable objects, its elements. But how can these elements be distinguished? Possibly, by certain specific intrinsic properties that they possess in contrast to others. Better, by specific relations that they have with other elements. In this chapter, we introduce the fundamental relations, operations and structures that will appear in this book.
Jürgen Jost

Chapter 3. Relations

Abstract
Relations take place between elements, and in order to conceptualize this, we can either take the elements or the relations as primary. Alternatively, we can try to combine these two dual approaches. In this chapter, we discuss this issue with tools from graph theory. We also analyze symmetry groups of graphs and their representations.
Jürgen Jost

Chapter 4. Spaces

Abstract
In the history of science, there have been different conceptualizations of the relation between physical objects or bodies and space. In this chapter, we discuss the general concept of a topological space.
Jürgen Jost

Chapter 5. What Is Space?

Abstract
How can the notion of space be conceptualized and formalized? What is the space that we are living in? In this chapter, we discuss the concepts of a manifold (including an introduction to Riemannian geometry), a simplicial complex and a scheme as possible answers.
Jürgen Jost

Chapter 6. Spaces of Relations

Abstract
Generalizing the concept of a graph introduced earlier, here we consider a framework of relations as being constitutive for a space. This chapter will also provide an introduction to the basic concepts of algebraic topology, homology, cohomology, and homotopy.
Jürgen Jost

Chapter 7. Structures

Abstract
A mathematical structure consists of a set of generators, a (possibly empty) set of relations between them (constraints), and a set of operations with which the other elements of the structure are generated from the generators.
Jürgen Jost

Chapter 8. Categories

Abstract
This chapter will introduce the abstract and universal constructions of category theory.
Jürgen Jost

Chapter 9. Topoi

Abstract
In this chapter, we describe and analyze the concept of a topos that provides a general framework for the mathematical structures of geometry and logic A topos is a category of structured families of sets where the type of structure characterizes the topos in question.
Jürgen Jost

Chapter 10. A Review of Examples

Abstract
In this chapter, we systematically describe the simplest examples of the structures that are discussed in the main text.
Jürgen Jost

Backmatter

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