Was sind und was sollen die Zahlen?

Was sind und was sollen die Zahlen? (What Are Numbers and What Should They Be?) is Richard Dedekind's 1888 treatise on the foundations of arithmetic and number theory. Through a series of logical propositions and proofs, Dedekind constructs the natural numbers from first principles using set theory. The work proceeds methodically through definitions of basic mathematical concepts like chains, mappings, and infinite systems. Dedekind introduces his now-famous "cuts" as a way to define real numbers and establishes key properties of number systems. The text develops its mathematical framework without relying on geometric intuition or physical analogies, marking a shift toward pure logical foundations in mathematics. Its influence extends beyond number theory to set theory and abstract algebra. This groundbreaking work stands as a cornerstone of modern mathematics, presenting a rigorous treatment of concepts that had previously been taken as intuitive. Its emphasis on precise definitions and logical structure helped establish the abstract approach that characterizes contemporary mathematical thinking.

Most readers note this is a challenging mathematical text that requires careful study. Several mathematicians and students on Goodreads mention that Dedekind's construction of the real numbers influenced their understanding of mathematical foundations. Readers appreciate: - Clear logical progression - Rigorous development of number systems - Historical importance in mathematics Common criticisms: - Dense writing style - Complex notation that takes time to understand - Limited availability of quality English translations Ratings: Goodreads: 4.3/5 (31 ratings) No Amazon ratings found From reader reviews: "Made me rethink everything I thought I knew about numbers" - Mathematics student on Goodreads "The formalism is difficult but rewarding once you work through it" - Math professor review "Needed to read some sections 3-4 times to grasp the concepts" - Graduate student comment Several readers recommend Dover's English translation but suggest reading secondary sources alongside it.

Principia Mathematica by Alfred North Whitehead, Bertrand Russell. This work builds a logical foundation for mathematics from first principles through set theory and number theory.

Grundlagen der Arithmetik by Gottlob Frege. The text establishes arithmetic through logical concepts and explores the nature of numbers as logical objects.

Introduction to Mathematical Philosophy by Bertrand Russell. The book examines mathematical concepts through philosophical analysis with focus on number theory and set theory foundations.

The Foundations of Arithmetic by Edmund Husserl. This work investigates the philosophical basis of arithmetic and number theory through phenomenological methods.

Grundlagen der Geometrie by David Hilbert. The text presents an axiomatic foundation for geometry that parallels Dedekind's treatment of numbers.

🔢 Dedekind wrote this groundbreaking work (translated as "What Are Numbers and What Should They Be?") in 1888, presenting the first precise definition of natural numbers using set theory. 📚 The book introduced what are now known as "Dedekind cuts," a method for constructing real numbers from rational numbers that remains fundamental to modern mathematics. 🎓 Despite its revolutionary content, Dedekind had to publish the book at his own expense because no publisher would accept it, believing it wouldn't sell enough copies. ✍️ The work was written in an unusually accessible style for a mathematics text of its era, as Dedekind wanted it to be understandable to his former student Emmy Noether. 🌟 The concepts presented in the book influenced many prominent mathematicians, including Georg Cantor in his development of set theory, and Bertrand Russell in his work on mathematical logic.