b. Brunswick, Germany, 1831; d. Brunswick, Germany, 1916
Algebraic number theory; algebraic curves; set theory;
foundations of mathematics
Dedekind spent most of his life as a professor at the Technische Hochschule in Brunswick, Germany (his and GAUSS’S [VI.26] home town), having spent the years 1858-62 at the Polytechnikum in Zürich (which later became known as ETH). He obtained his mathematical education at Göttingen, being Gauss’s last Ph.D. student and subsequently a pupil of DIRICHLET [VI.36] and RIEMANN [VI.49]. Dedekind was a retiring man with, as KLEIN [VI.57] said, a “contemplative nature;” he remained a bachelor, living with his mother and sister. Nevertheless, he had an impact upon a select group of contemporaries (especially CANTOR [VI.54], FROBENIUS [VI.58], and Heinrich Weber) through his rich correspondence.
A key figure in the emergence of modern set-theoretic mathematics, and particularly the notion of a mathematical structure, Dedekind is best known for his work on the foundations of the REAL NUMBER SYSTEM [I.3 §1.4]. His main contribution, however, was in algebraic number theory. Indeed, he shaped modern number theory as we know it, presenting it as a theory of ideals in rings of integers (see ALGEBRAIC NUMBERS [IV.1 §§4-7]). This was first made public in 1871, within Supplement X to his edition of Dirichlet’s Vorlesungen über Zahlentheorie, where he established unique decomposition of ideals into prime ideals for all rings of algebraic integers. In the process, he formulated the concepts of field, ring, ideal, and module (see [I.3 §2.2] and [III.81]), always within the particular context of the complex numbers. It was also in the context of algebra (Galois theory) and number theory that Dedekind started systematic work with quotient structures, isomorphisms, homomorphisms, and automorphisms.
In subsequent editions of Dirichlet’s Vorlesungen (1879 and 1894) Dedekind went on refining his presentation of ideal theory, making it more purely set-theoretic. In 1882, together with Weber, he offered a theory of ideals in fields of algebraic functions, which made it possible to give a rigorous treatment of Riemann’s results on algebraic curves up to the RIEMANNROCH THEOREM [V.31]. This work paved the way for modern algebraic geometry.
Intimately linked with Dedekind’s work in algebra and number theory were his reflections on the foundations of the real number system. In 1858 (published 1872) he formulated a definition of the real numbers using what are now known as “Dedekind cuts” in the set of rational numbers. During the 1870s (published 1888) he elaborated a purely set-theoretic definition of the natural numbers as “simply infinite” sets, which led him to crystallize the DEDEKIND-PEANO AXIOMS [III.67]. In this work, as in his more advanced research, sets, structures, and mappings form the essential building blocks, the very foundations of pure mathematics. In the light of (now superseded) conceptions of logic, this led Dedekind to the view that “arithmetic (algebra, analysis) is only a part of logic.” From a modern viewpoint, his contributions show that SET THEORY [IV.22] is a sufficient foundation for classical mathematics. Thus he contributed as much as anybody else to the set-theoretic reformulation of modern mathematics.
Corry, L. 2004. Modern Algebra and the Rise of Mathematical Structures, second revised edn. Basel: Birkhäuser.
Ewald, W., ed. 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, two volumes. Oxford: Oxford University Press.
Ferreirós, J. 1999. Labyrinth of Thought. A History of Set Theory and Its Role in Modern Mathematics. Basel: Birkhäuser.