Introduction: Structures in Mathematics
One: Structures in the Images of Mathematics
1 Structures in Algebra: Changing Images
1.1 Jordan and Hölder: Two Versions of a Theorem
1.2 Heinrich Weber:Lehrbuch der Algebra
1.3 Bartel L. van der Waerden:Moderne Algebra
1.4 Other Textbooks of Algebra in the 1920s
2 Richard Dedekind: Numbers and Ideals
2.1 Lectures on Galois Theory
2.1 Algebraic Number Theory
2.2.1 Ideal Prime Numbers
2.2.2 Theory of Ideals: The First Version (1871)
2.2.3 Later Versions
2.2.4 The Last Version
2.2.5 Additional Contexts
2.3 Ideals andDualgruppen
2.4 Dedekind and the Structural Image of Algebra
3 David Hilbert: Algebra and Axiomatics
3.1 Algebraic Invariants
3.2 Algebraic Number Theory
3.2 Hilbert’s Axiomatic Approach
3.4 Hilbert and the Structural Image of Algebra
3.5 Postulational Analysis in the USA
4 Concrete and Abstract: Numbers, Polynomials, Rings
4.1 Kurt Hensel: Theory of p-adic Numbers
4.2 Ernst Steinitz:Algebraische Theorie der Körper
4.3 Alfred Loewy:Lehrbuch der Algebra
4.4 Abraham Fraenkel: Axioms forp-adicSystems
4.5 Abraham Fraenkel: Abstract Theory of Rings
4.6 Ideals and Abstract Rings after Fraenkel
4.7 Polynomials and their Decompositions
5 Emmy Noether: Ideals and Structures
5.1 Early Works
5,2 Idealtheorie in Ringbereichen
5.3 Abstrakter Aufbau der Idealtheorie
5.4 Later Works
5.5 Emmy Noether and the Structural Image of Algebra
Two: Structures in the Body of Mathematics
6 Oystein Ore: Algebraic Structures
6.1 Decomposition Theorems and Algebraic Structures
6.2 Non-Commutative Polynomials and Algebraic Structure
6.3 Structures and Lattices
6.4 Structures in Action
6.5 Universal Algebra, Model Theory, Boolean Algebras
6.6 Ore’s Structures and the Structural Image of Algebra
7 Nicolas Bourbaki: Theory of Structures
7.1 The Myth
7.2 Structures and Mathematics
7.3 Structures and the Body of Mathematics
7.3.1 Set Theory
7.3.2 Algebra
7.3.3 General Topology
7.3.4 Commutative Algebra
7.4 Structures and the Structural Image of Mathematics
8 Category Theory: Early Stages
8.1 Category Theory: Basic Concepts
8.2 Category Theory: A Theory of Structures
8.3 Category Theory: Early Works
8.4 Category Theory: Some Contributions
8.5 Category Theory and Bourbaki
9 Categories and Images of Mathematics
9.1 Categories and the Structural Image of Mathematics
9.2 Categories and the Essence of Mathematics
9.3 What is Algebra and what has it been in History?
Author Index
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