What are the meaning and the purpose of numbers? Continuity and irrational numbers. Edited by Stefan Müller-Stach (Q2361501)

`Es steht alles schon bei Dedekind' (Everything can be found already in the writings of Dedekind) is a famous sentence attributed to Emmy Noether as witnessed by van der Waerden. However, Dedekind and his works are still not very well known (and read) today although (or because?) he anticipated many of the important ideas of `modern' mathematics. Hence, a book on Dedekind's mathematical breakthroughs seems to be more than welcome in these days. The book centres around the two famous books \textit{Was sind und was sollen die Zahlen?} (What are numbers and what should they be?) [JFM 20.0049.05; JFM 58.0042.09] and \textit{Stetigkeit und irrationale Zahlen} (Continuity and irrational numbers) [JFM 58.0042.09] which are reprinted here in their last (German) editions; the 10th edition in case of the first, the 7th edition in case of the latter book. The first chapter is concerned with a historic introduction by the editor Stefan Müller-Stach. Besides a short biography this chapter includes important remarks concerning Dedekind's mathematical environment (including his contemporary colleagues), the correspondence, his Habilitationsvortrag of 1854, and remarks on Dedekind's stile and influence. In the second chapter, Dedekind's investigations concerning the `Zahlbegriff' (concept of number) are thoroughly discussed. Dedekind cuts are found here as are the axioms of natural numbers and the famous recursion theorem. Then, the reprints of the two books by Dedekind follow in Chapter 3. The fourth chapter gives explanations of the two texts in modern language resulting in two readable summaries. Chapter 5 gives an overview on the history of the reception of Dedekind's works. The last Chapter 6 is devoted to the `Wirkungsgeschichte', i.e.\ the history of the influence of Dedekind's works. Here, recursive functions and computability, the further development of set theory, and the further development of the concept of number and arithmetic are discussed. The book ends with two appendices; one a list of Dedekind's publications, the other the famous letter to Keferstein of 27th February 1890. A list of references and an index conclude the book. The present book is a marvel in the history of modern mathematics. The editor being an expert mathematician has succeeded in presenting a readable historical survey around the two famous books by Dedekind. The book is written in German.