Essays on the Theory of Numbers (Dover Books on Mathematics)
by Mathematics, Richard Dedekind
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Book Summary Information
Author: Mathematics, Richard DedekindEdition: Paperback
Audio: English (Unknown); English (Original Language); English (Published)
Published: 1963-01-01
ISBN: 0486210103
Number of pages: 115
Publisher: Dover Publications
Book Reviews of Essays on the Theory of Numbers (Dover Books on Mathematics)
Book Review: Will Appeal to Students of Mathematics and Philosophy
Summary: 4 Stars
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 I had some difficulty with The Nature and Meaning of Numbers as some of Dedekind's terminology is outdated and unfamiliar.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.
Summary of Essays on the Theory of Numbers (Dover Books on Mathematics)
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.
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