The Mysteries of Richard Dedekind

Abstract

Everybody loves a mystery. And I have three. Pending their resolutions, any mathematical portrait of Richard Dedekind remains unfinished. For the mysteries are mathematical veils across the face of Dedekind’s work. The first of the three involves a “proof” of something most set theorists do not prove and a conclusion most people would not allow. What set theorists normally do not prove is the axiom of infinity, the assertion that there exists an infinite set. What Dedekind offers us seems to be a mysterious “proof” of exactly that.¹ Its logical heart is Dedekind’s assertion that the Gedankenwelt, his own thoughtworld, constitutes such an infinite collection. It is also Dedekind’s view that from any infinite collection one can garner the unique series of natural numbers.² Can we conclude, therefore, that the one and only natural numbers, the numbers so painstakingly defined in Was Sind and Was Sollen die Zahlen? are to be located within the world of his — Dedekind’s — thought? Is this what Dedekind wants us to believe? The second mystery revolves around Dedekind’s famous essay on continuity, Stetigkeit and irrationale Zahlen. We uncover the mystery by asking, “How can Dedekind claim to have captured — in his definition of ‘real number’ — the essence of the continuum and, at the very same time, describe for us a continuum which we cannot see to satisfy that definition?” We suppose that, in the “Continuity” essay, Dedekind first unveils his definition of the continuum, the one formulated in terms of Dedekind cuts. We do not often remark upon the fact that, before setting out that definition, Dedekind attempts to motivate it by assuming the existence and examining the character of another, seemingly distinct continuum. This structure Dedekind calls ‘the straight line.’ Although Dedekind, in presenting reals in terms of cuts, claims to have captured the essence of the continuum, he makes no effort to prove a uniqueness result. He does not even assay the prospect that the two continua of his own article might be related in some fully satisfactory, mathematical way. Later — in a letter (Dedekind 1932, p. 478) — Dedekind will insist, in effect, that the two structures, the straight line and the collection of Dedekind reals, cannot be proved isomorphic. How many Dedekindian continua are there — one or two? That is the second mystery. On the face of it, the third mystery seems to be one of historical classification rather than mathematical individuation. Dedekind is generally acknowledged to be the father of classical set-theoretic algebra and, by every measure logical and mathematical, was no constructivist. Yet, Dedekind avers — often and throughout his writings — that mathematical entities are not mind-independent, Platonistic abstracta, but are literally geistig or mental. More troubling still, Dedekind seems to think them not sempiternal but brought into being by discrete mental episodes, acts he calls ‘freie Schöpfung’ — free creation. This is romantic, even revolutionary, talk and just the idiom constructivists such as Brouwer chose to herald the anticlassical revolt. “Free mathematics! A free creation of the human mind!” is a constructivist’s — and not a classical mathematician’s — rallying cry.

For it is in light of the idea of a creative [schöpferische] reason that we so guide the empirical employment of our reason as to secure its greatest possible extension.(Kant 1965, p. 551)

My heartfelt thanks to colleagues Chip Bolyard, Anil Gupta and Luise Prior McCarty for their comments and suggestions regarding the content of this essay.