Stetigkeit und irrationale Zahlen

Braunschweig: Friedrich Vieweg, 1872. First edition. Hardcover. A LANDMARK WORK ON THE FOUNDATIONS OF MATHEMATICS.

First edition, very rare in commerce, of Dedekind's great work on the foundations of mathematics. "This short work [Stetigkeit und irrationale Zahlen] marks a significant epoch in the movement known as the arithmetization of analysis, that is, the replacement of intuitive geometric notions by concepts described in precise words" (Landmark Writings, p. 553). "This article, whose central idea was worked out by Dedekind while he was teaching in Zürich in 1858, presents a rigorous arithmetical foundation for the theory of real numbers ... Despite Dedekind's assertion in the introductory paragraphs of Continuity and irrational numbers that he originally did not publish his theory because he did not regard it as being very fruitful, it laid the foundations for much of modern-day real analysis and point-set topology" (Ewald, pp. 765-6). No copies listed on ABPC/RBH.

"In 1858, Dedekind had noted the lack of a truly scientific foundation of arithmetic in the course of his Zürich lectures on the elements of differential calculus. On 24 October, Dedekind succeeded in producing a purely arithmetic definition of the essence of continuity and, in connection with it, an exact formulation of the concept of the irrational number. Fourteen years later, he published the result of his considerations, Stetigkeit und irrationale Zahlen (Brunswick, 1872, and later editions), and explained the real numbers as "cuts" in the realm of rational numbers. He arrived at concepts of outstanding significance for the analysis of number through the theory of order. The property of the real numbers, conceived by him as an ordered continuum, with the conceptual aid of the cut that goes along with this, permitted tracing back the real numbers to the rational numbers: Any rational number a produces a resolution of the system R of all rational numbers into two classes A1, A2, in such a way that each number a1 of the class A2 is smaller than each number a2 of the second class A2. (Today, the term "set" is used instead of "system.") The number a is either the largest number of the class A1 or the smallest number of the class A2. A division of the system R into the two classes A1, A2, whereby each number a1 in A1 is smaller than each number a2 in A2 is called a "cut" (A1, A2) by Dedekind. In addition, an infinite number of cuts exist that are not produced by rational numbers. The discontinuity or incompleteness of the region R consists in this property. Dedekind wrote, "Now, in each case when there is a cut (A1, A2) which is not produced by any rational number, then we create a new, irrational number α, which we regard as completely defined by this cut; we will say that this numberα corresponds to this cut, or that it produces this cut" (Stetigkeit, § 4).

"Occasionally Dedekind has been called a "modern Eudoxus" because an impressive similarity has been pointed out between Dedekind's theory of the irrational number and the definition of proportionality in Eudoxus' theory of proportions (Euclid, Elements, bk. V, def. 5). Nevertheless, Oskar Becker correctly showed that the Dedekind cut theory and Eudoxus' theory of proportions do not coincide: Dedekind's postulate of existence for all cuts and the real numbers that produce them cannot be found in Eudoxus or in Euclid. With respect to this, Dedekind said that the Euclidean principles alone-without inclusion of the principle of continuity, which they do not contain-are incapable of establishing a complete theory of real numbers as the proportions of the quantities. On the other hand, however, by means of his theory of irrational numbers, the perfect model of a continuous region would be created, which for just that reason would be capable of characterizing any proportion by a certain individual number contained in it (letter to Rudolph Lipschitz, 6 October 1876).

"With his publication of 1872, Dedekind had become one of the leading representatives of a new epoch in basic research, along with Weierstrass and Georg Cantor. This was the continuation of work by Cauchy, Gauss, and Bolzano in systematically eliminating the lack of clarity in basic concepts by methods of demonstration on a higher level of rigor. Dedekind's and Weierstrass' definition of the basic arithmetic concepts, as well as Georg Cantor's theory of sets, introduced the modern development, which stands "completely under the sign of number," as David Hilbert expressed it.

"Dedekind entered the University of Göttingen in 1850; he studied mathematics and physics, attending Gauss's lectures on the method of least squares and on advanced geodesy. One of his friends was a fellow mathematics student, five years older than he, Bernhard Riemann. In 1852 Dedekind took his doctorate; the dissertation, written under the supervision of Gauss, was on the theory of Eulerian integrals. Both Riemann and Dedekind qualified as university lecturers in 1854 ... In 1855, P.G. Lejeune-Dirichlet left Berlin to succeed to Gauss' professorship in Göttingen ... From 1858 to 1862 he taught at the Polytechnic in Zürich; it was during this time that he developed his ideas on the foundations of real analysis. In 1862 he was appointed to a professorship at the Polytechnic in his native city of Brunswick; he remained there until his death" (Ewald, pp. 753-4).

Honeyman 840. Ewald (ed.), From Kant to Hilbert, 1996. Landmark Writings in Western Mathematics 1640-1940, I. Grattan-Guinness (ed.), Chapter 43. Parkinson, Breakthroughs, p. 415. Stedall, Mathematics Emerging: A Sourcebook 1540-1900, 2008.

8vo (202 x 130 mm), pp. 31, [1]. Contemporary cloth-backed marbled boards (light browning throughout). Preserved in a folding clamshell case.