Essays on the Theory of Numbers

By Paul Coggle and Heiner Schenke

Two most important essays by the famous German mathematician: one provides an arithmetic, rigorous foundation for the irrational numbers, thereby a rigorous meaning of continuity in analysis. The other is an attempt to give logical basis for transfinite numbers and properties of the natural numbers.

• Format: Paperback
• Language: English
• ISBN: 0486210103
• ISBN13: 9780486210100
• Release Date: January 1963
• Publisher: Dover Publications
• Length: 128 Pages
• Weight: 0.35 lbs.
• Dimensions: 0.3" x 5.4" x 8.0"

Customer Reviews

Foundations of the reals, foundations of the integers

The first essay (27 pages) is Dedekind's excellent exposition of his "Dedekind cuts" definition of the real numbers. This constructs the reals from the rationals and proves that there are no gaps left. The only downside, Dedekind anticipates, is that the principles on which this proof is built are so "common-place" that "the majority of my readers will be very much disappointed" (p. 11) that there is nothing more to it. The...

Will Appeal to Students of Mathematics and Philosophy

Richard Dedekind (1831-1916) is recognized as one of the great pioneers in the logical and philosophical analysis of the foundations of mathematics. Dedekind completed his doctoral studies under Gauss, was a friend of Cantor and Riemann, and worked under Dirichlet. This inexpensive, 115-page book, Essays on the Theory of Numbers, contains two essays: his brief, famous essay Continuity and Irrational Numbers and his longer...

An interesting pair of historical essays

Richard Dedekind is one of the fathers of modern mathematical proofs. Reading his work will give you a glimpse into the early stages of this development. Indeed, his essay on Continuity and Irrational Numbers was, in part, written because Dedekind was trying to provide some rigor to what was not yet a rigorous science. The first essay is a classic. It is his description of a means of defining a number in a given space, which...

Accessible genius

This is not a book of "number theory" in the usual sense. It is a book combining two essays by Dedekind: "Continuity and irrational numbers" is Dedekind's way of defining the real numbers from rational numbers; and "The nature and meaning of numbers" where Dedekind offers a precise explication of the natural numbers (using what are now called the Peano axioms, since Peano made so much of them after reading Dedekind). They...