Dedekind Richard (1831–1916), German mathematician, one of the most important figures in the mathematical analysis of foundational questions that took place in the late nineteenth century. Philosophically, three things are interesting about Dedekind’s work: (1) the insistence that the fundamental numerical systems of mathematics must be developed independently of spatiotemporal or geometrical notions; (2) the insistence that the numbers systems rely on certain mental capacities fundamental to thought, in particular on the capacity of the mind to ‘create’; and (3) the recognition that this ‘creation’ is ‘creation’ according to certain key properties, properties that careful mathematical analysis reveals as essential to the subject matter. (1) is a concern Dedekind shared with Bolzano, Cantor, Frege, and Hilbert; (2) sets Dedekind apart from Frege; and (3) represents a distinctive shift toward the later axiomatic position of Hilbert and somewhat away from the concern with the individual nature of the central abstract mathematical objects which is a central concern of Frege. Much of Dedekind’s position is sketched in the Habilitationsrede of 1854, the procedure there being applied in outline to the extension of the positive whole numbers to the integers, and then to the rational field. However, the two works best known to philosophers are the monographs on irrational numbers (Stetigkeit und irrationale Zahlen, 1872) and on natural numbers (Was sind und was sollen die Zahlen?, 1888), both of which pursue the procedure advocated in 1854. In both we find an ‘analysis’ designed to uncover the essential properties involved, followed by a ‘synthesis’ designed to show that there can be such systems, this then followed by a ‘creation’ of objects possessing the properties and nothing more. In the 1872 work, Dedekind suggests that the essence of continuity in the reals is that whenever the line is divided into two halves by a cut, i.e., into two subsets A1 and A2 such that if p 1 A1 and q 1 A2, then p ‹ q and, if p 1 A1 and q ‹ p, then q 1 A1, and if p 1 A2 and q ( p, then q 1 A2 as well, then there is real number r which ‘produces’ this cut, i.e., such that A1 % {p; p ‹ r}, and A2 % {p: r m p}. The task is then to characterize the real numbers so that this is indeed true of them. Dedekind shows that, whereas the rationals themselves do not have this property, the collection of all cuts in the rationals does. Dedekind then ‘defines’ the irrationals through this observation, not directly as the cuts in the rationals themselves, as was done later, but rather through the ‘creation’ of ‘new (irrational) numbers’ to correspond to those rational cuts not hitherto ‘produced’ by a number. The 1888 work starts from the notion of a ‘mapping’ of one object onto another, which for Dedekind is necessary for all exact thought. Dedekind then develops the notion of a one-toone into mapping, which is then used to characterize infinity (‘Dedekind infinity’). Using the fundamental notion of a chain, Dedekind characterizes the notion of a ‘simply infinite system,’ thus one that is isomorphic to the natural number sequence. Thus, he succeeds in the goal set out in the 1854 lecture: isolating precisely the characteristic properties of the natural number system. But do simply infinite systems, in particular the natural number system, exist? Dedekind now argues: Any infinite system must contain a simply infinite system (Theorem 72). Correspondingly, Dedekind sets out to prove that there are infinite systems (Theorem 66), for which he uses an infamous argument (reminiscent of Bolzano’s from thirty years earlier) involving ‘my thought-world,’ etc. It is generally agreed that the argument does not work, although it is important to remember Dedekind’s wish to demonstrate that since the numbers are to be free creations of the human mind, his proofs should rely only on the properties of the mental. The specific act of ‘creation,’ however, comes in when Dedekind, starting from any simply infinite system, abstracts from the ‘particular properties’ of this, claiming that what results is the simply infinite system of the natural numbers.
See also CANTOR, CONTINUUM PROBLEM , PHILOSOPHY OF MATHEMATIC. M.H.