Modern Algebra and the Rise of Mathematical Structures
Originally published as volume 17 in the series: Science Networks Hist. Studies
2nd revised ed. 2004, 451 p.
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The notion of a mathematical structure is among the most pervasive ones in twentieth-century mathematics. Modern Algebra and the Rise of Mathematical Structures describes two stages in the historical development of this notion: first, it traces its rise in the context of algebra from the mid-nineteenth century to its consolidation by 1930, and then it considers several attempts to formulate elaborate theories after 1930 aimed at elucidating, from a purely mathematical perspective, the precise meaning of this idea.
Part one dicusses the process whereby the aims and scope of the discipline of algebra were deeply transformed, turning it into that branch of mathematics dealing with a new kind of mathematical entities: the "algebraic structures". The transition from the classical, nineteenth-century, image of the discipline to the thear of ideals, from Richard Dedekind to Emmy Noether, and culminating with the publication in 1930 of Bartel L. van der Waerden's Moderne Algebra. Following its enormous success in algebra, the structural approach has been widely adopted in other mathematical domains since 1930s. But what is a mathematical structure and what is the place of this notion within the whole fabric of mathematics? Part Two describes the historical roots, the early stages and the interconnections between three attempts to address these questions from a purely formal, mathematical perspective: Oystein Ore's lattice-theoretical theory of structures, Nicolas Bourbaki's theory of structures, and the theory of categories and functors.
Content Level »Research
Keywords »Algebra - History of Mathematics - YellowSale2006
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 ofp-adicNumbers.- 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.- 5,2Idealtheorie in Ringbereichen.- 5.3Abstrakter 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 ofStructures.- 7.1 The Myth.- 7.2 Structures and Mathematics.- 7.3Structuresand the Body of Mathematics.- 7.3.1 Set Theory.- 7.3.2 Algebra.- 7.3.3 General Topology.- 7.3.4 Commutative Algebra.- 7.4Structuresand 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.