Richard Dedekind (1831–1916) published in 1888 a paper entitled Was sind und was sollen die Zahlen? variously translated as What are numbers and what should they be? or, as Beman did, The Nature of Meaning of Numbers. Dedekind added a second preface to the second edition in 1893. The second edition was translated into English by Wooster Woodruff Beman in 1901.

Much of Dedekind’s terminology and notation as translated by Beman differs from the modern standards, so I offer these notes, using current terminology and notation, to make his work more understandable.

The pdf file is here.

Even today, the importance of rigorously establishing the nature and properties of real numbers is underappreciated. In a book I cite in a previous post, Theory and Application of Infinite Series, the author, Knopp, takes great pains to emphasize the properties of real numbers. A reviewer on Amazon.com complained about this, but this is the very attitude that Knopp seeks to dispel.