Essays on the Theory of Numbers

by Wooster Woodruff Beman and Richard Dedekind

Book Overview

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 second essay develops basic set theory and uses it in particular to build a foundation for the integers in terms of the successor function and induction. It's a plain definition-theorem-proof account, just as may be found in any foundations of mathematics book of today. There is none of the enthusiasm of the first essay. Dedekind even says that working out this theory was a "wearisome labor" (p. 41, referring to building the theory of infinite sets from the definition that a set is infinite if it can be put in one-to-one correspondence with a subset of itself---of course, Dedekind doesn't make this remark out of dislike of mathematics, but for the sake of alerting others to the brilliance of his work, which incidentally seems to have been one of his key motivations throughout).

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 paper The Nature and Meaning of Numbers. This Dover edition (1963) is an unabridged and unaltered copy of the 1901 authorized English translation by mathematician W. W. Beman. I particularly enjoyed his famous essay on the Dedekind cut and irrational numbers. Dedekind writes clearly and carefully and this first paper should appeal to all students of mathematics. The intent of the longer essay was to provide a logical basis for finite and infinite numbers as well as demonstrating the logical validity of mathematical induction. I had some difficulty with The Nature and Meaning of Numbers as some of Dedekind's terminology is outdated and unfamiliar. Some statements can be reformulated easily to modern terminology. For example, simply substitute set for system and proper set for proper system. Dedekind uses the term transformation for function (or mapping). Inverse transformations and identical transformations are the same as inverse functions and identical mappings. A system may be compounded from other systems (same concept as union of sets). The community of systems A, B, and C is the same as intersection of sets A, B, and C. While admitting that a null system has some value, Dedekind deliberately avoided using the concept of a null set in these essays. I did not at first recognize that similar or distinct transformations were equivalent to one-to-one mappings. I had difficulty with the Dedekind's use of the term chain when discussing the transformation of a system S into itself. Dedekind was not successful in imposing his terminology on later mathematicians. Nonetheless, Dedekind's essays had considerable influence on mathematics, not only for their content, but for their clarity of expression. Minor points: This 1901 translation often employs an unusual positioning of the verb 'is': If R, S are similar systems, then is every part of S also similar to a part of R. Also, while I encountered a few typos, none were particularly troublesome.

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 has since been referred to as a "Dedekind cut." His descriptions and proofs are exceptionally clear and straightforward. The second essay is a discussion of how a number system is constructed and its characteristics. It, too, shows Dedekind to possess a excellent ability to explain the ideas very clearly and simply.There are two difficulties with the book, which I found serious enough to warrant only four stars. First, the terminology is rather antiquated, so that the descriptions are clear only once you are able to translate Dedekind's phrases; for instance, "a system S is compounded from the systems A and B" would today be written "the set S is the union of sets A and B." Second, there are a fairly large number of typos in the book, given its importance and the rigorousness of the work; for example, in the proof in paragraph 42 of The Meaning of Numbers, the phrase (not in Dedekind's shorthand) "the transformation of A is contained in B" should read "the transformation of A is contained in A." Most typos are as minor as this, but annoying in the unnecessary effort needed to bull ones way through them. A couple errors are more significant. I blame the translator and proofreader, not Dedekind.All in all, the book is well worth the price and the effort to understand it.

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 are essays in logic, or foundations of mathematics, or philosophy, as you like. And they are brilliant, readable, works of genius.Probably the main value of the book is as an introduction to Dedekind's way of thinking about mathematics: his clarity, precision, and way of cutting to the bare core of a subject. You can find the same genius in Dedekind's THEORY OF ALGEBRAIC INTEGERS (available in a fine English translation by John Stillwell) but of course that is a more advanced text. The same style of thought works powerfully in all of Dedekind's mathematics. But most of it is very hard stuff. Here you see it in easily accessible form, suitable for even a smart high school student willing to think hard.