Modern algebra and the rise of mathematical structures
type: Item
instance of: scholarly article
title: Modern algebra and the rise of mathematical structures
zbMATH Open document ID: 0858.01022
author: Leo Corry
published in: Science Networks. Historical Studies
publication date: 1996-03-20T00:00:00Z
review text: The concept of mathematical structures is often used to describe the development of mathematics in the 20th century. But, as the author points out, the idea of a mathematical structure is an idea whose nature, meaning and role are not clearly understood and have seldom been systematically discussed. The author tries to answer some of the questions which arise in connection with the emergence of mathematical structures. To do this, he examines the problem in a much broader context and makes the distinction between the body of knowledge and the images of knowledge. According to his definition the former includes statements that are answers to questions related to the subject matter of the given discipline, theories, facts, methods and open problems, and the latter includes the claims which express knowledge about the discipline qua discipline. Both, body and images are organically interconnected and the author compares their relationship with that between a text and its context. The book is divided in two parts and the headings of the chapters are the following: Introduction: Structures in mathematics. Part One: Structures in the images of mathematics. 1. Structures in algebra: changing images; 2. Richard Dedekind: Numbers and ideals; 3. David Hilbert: Algebra and axiomatics; 4. Concrete and abstract: Numbers, polynomials, rings; 5. Emmy Noether: Ideals and structures; Part Two: Structures in the body of mathematics. 6. Oystein Ore: Algebraic structures; 7. Nicolas Bourbaki: Theory of structures; 8. Category theory: early stages; 9. Categories and images of mathematics. As seen from the contents the author chose the ideal theory to exemplify his ideas. In the first part of the book he describes the rise of the ideal theory from its beginnings in the work of R. Dedekind up to those of Emmy Noether. Furthermore, he explains what he understands under the structural approach to algebra. Therefore he compares the presentation of algebraic knowledge in the textbooks at the …
reviewed by: Karl Heinz Schlote
Mathematics Subject Classification ID: 01A60
Mathematics Subject Classification ID: 01-02
Mathematics Subject Classification ID: 00A30
Mathematics Subject Classification ID: 00A35
Mathematics Subject Classification ID: 03B30
zbMATH DE Number: 858494
zbMATH Keywords: mathematical structure
zbMATH Keywords: modern algebra
zbMATH Keywords: category theory
zbMATH Keywords: algebra
zbMATH Keywords: ideals
zbMATH Keywords: algebraic structures
zbMATH Keywords: Richard Dedekind
zbMATH Keywords: axiomatics
zbMATH Keywords: polynomials
zbMATH Keywords: rings
zbMATH Keywords: Emmy Noether
zbMATH Keywords: Nicolas Bourbaki
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 858494