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Originally published in 1901.This volume from the Cornell University Library's print collections was scanned on an APT BookScan and converted to JPG 2000 format by Kirtas Technologies.All titles scanned cover to cover and pages may include marks notations and other marginalia present in the original volume. ... Read more

Customer Reviews (5)

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. ThisDover 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.

A very good Introduction to Number Theory
If you wan a text to introduce you at Number Theory this is your best Philosophical option ... Read more

Product Description
The invention of ideals by Dedekind in the 1870s was well ahead of its time, and proved to be the genesis of what today we would call algebraic number theory. His memoir "Sur la Theorie des Nombres Entiers Algebriques" first appeared in installments in the Bulletin des sciences mathematiques in 1877. This book is a translation of that work by John Stillwell, who adds a detailed introduction giving historical background and who outlines the mathematical obstructions that Dedekind was striving to overcome. Dedekind's memoir offers a candid account of the development of an elegant theory and provides blow by blow comments regarding the many difficulties encountered en route. This book is a must for all number theorists. ... Read more

Customer Reviews (3)

An antidote against too much modernistic algebra
Algebraic number theory is about employing unique factorisation in rings larger than the integers. The classical cases are the quadratic integers and the cyclotomic integers. They came with elaborate theories to deal with the fact that unique factorisation does not always hold. Dedekind generalises and cleans up these theories by developing a general theory of algebraic integers. Kummer's theory of ideal prime factors, which saved unique factorisation in some cases in the cyclotomic integers, is replaced by a beautifully conceptual and streamlined theory of ideals. The power of abstraction has perhaps never been more impressive. Many insights that today are scattered in abstract algebra and linear algebra can be seen here in their original glory, introduced not as soulless axiomatic structures but for their original noble purpose of understanding numbers.

Half the book consists of Stillwell's introduction, which is a brilliant sketch of the history of number theory from Diophantus to Dedekind, of course focusing especially on the prehistory of algebraic number theory.

Emmy Noether called it a must-read
This is Dedekind's famous creation of the theory of (algebraic number) rings and modules, which he presented as an appendix to his edition of Dirichlet's LECTURES ON NUMBER THEORY. In fact it went through several editions, and the translation here is from another article he wrote to make the ideas more accessible. Anyway Noether had her students read every versiom of it. Her watchword was "It is all already in Dedekind", meaning largely this work. And she was right, in a very deep sense the whole modern approach to abstract algebra is in Dedekind, though it took her phenomenal genius to *find* it there.

Dedekind (most of the time) explicitly limits himself to modules of algebraic numbers, but Noether correctly saw that Dedekind already knew (many of) his theorems held for the whole abstract range she would explicate and develop. Benefitting from her, we can even see this generality peeking through in some of his remarks.

Anyone knowing the basic modern ideas of rings and modules can read this with pleasure, both as the origins of abstract algebra with many fine insights to offer, and as a connection to the concrete motives. Of course Dedekind wrote for people who did not know such things. But he assumed they would think very, very hard. He also assumed some arithmetic ideas not widely taught today, but nicely explained in Stillwell's preface. Dedekind is a wonderful writer, well served here by a clear translation. You are apt to fall in love with this book, and want to accompany it with Dirichlet's own LECTURES ON NUMBER THEORY, written up by Dedekind, and also translated to english by Stillwell.

Emmy Noether called it a must-read
This is Dedekind's famous creation of the theory of (algebraic number) rings and modules, as an appendix to his edition of Dirichlet's LECTURES ON NUMBER THEORY. In fact it went through several editions, and Noether insisted that her students read every edition of it. Her watchword was "It is all in Dedekind", meaning largely this work. And she was right, in a very deep sense the whole modern approach to abstract algebra is in Dedekind, though it took her phenomenal genius to *find* it there. Anyone knowing the basic modern ideas of rings and modules can read this with pleasure, both as the origins of abstract algebra with many fine insights to offer, and as a connection to the concrete motives. Of course Dedekind wrote for people who did not know such things. But he assumed they would think very, very hard. He also assumed some arithmetic ideas not widely taught today, but nicely explained in Stillwell's preface. Dedekind is a wonderful writer, well served here by a clear translation. You are apt to fall in love with this book, and want to accompany it with Dirichlet's own LECTURES ON NUMBER THEORY, written up by Dedekind, and also translated to english by Stillwell. ... Read more

Product Description
This volume is a translation of Dirichlet's Vorlesungen über Zahlentheorie which includes nine supplements by Dedekind and an introduction by John Stillwell, who translated the volume.

Lectures on Number Theory is the first of its kind on the subject matter. It covers most of the topics that are standard in a modern first course on number theory, but also includes Dirichlet's famous results on class numbers and primes in arithmetic progressions.

The book is suitable as a textbook, yet it also offers a fascinating historical perspective that links Gauss with modern number theory. The legendary story is told how Dirichlet kept a copy of Gauss's Disquisitiones Arithmeticae with him at all times and how Dirichlet strove to clarify and simplify Gauss's results. Dedekind's footnotes document what material Dirichlet took from Gauss, allowing insight into how Dirichlet transformed the ideas into essentially modern form.

Also shown is how Gauss built on a long tradition in number theory--going back to Diophantus--and how it set the agenda for Dirichlet's work. This important book combines historical perspective with transcendent mathematical insight. The material is still fresh and presented in a very readable fashion.

This volume is one of an informal sequence of works within the History of Mathematics series. Volumes in this subset, "Sources", are classical mathematical works that served as cornerstones for modern mathematical thought. (For another historical translation by Professor Stillwell, see Sources of Hyperbolic Geometry, Volume 10 in the History of Mathematics series.) ... Read more

Customer Reviews (1)

Gauss and then some
Dirichlet is all about quadratic forms. But first there are three preliminary chapters on the tools we will need: unique factorisation, modulo arithmetic, quadratic reciprocity. Then in chapter 4 we get to the quadratic forms, ax^2+2bxy+cy^2. "The whole theory originates in the problem of deciding whether a given number is representable by a given form" (p. 92). (Remember, for example, that Fermat solved the case a=1, b=0, c=1 -- which integers are sums of two squares?) "The number b^2-ac, on which the properties of the form mainly depend, is called the determinant of the form", and two forms are equivalent (represent the same numbers) when one results from the other by applying a variable transformation matrix of determinant 1. And now the problem above reduces to "the two main problems in the theory of equivalence: I. To decide whether two given forms of the same determinant are equivalent. II. To find all substitutions that send one of two equivalent form to the other." (p. 100). We spend the rest of the chapter solving there two problems for any determinant D, and we work out the applications in the cases D=-1,-2,-3,-5 (which includes the theorem of Fermat above). In the cases D=-3,-5 representations are not generally unique (which we secretly think of as the manifestation of the loss of unique factorisation in Z[sqrt(-3)] and Z[sqrt(-5)]) and this goes hand in hand with the fact that the number of equivalence classes (the "class number") of forms in those cases is 2, not 1. Such matters are the motivation for Dirichlet's great contribution: the determination of the class number for any D (chapter 5). Apart from this motivation of measuring "how far quadratic integers deviate from unique prime factorisation", as Stillwell puts it (p. xvii), Dirichlet also assigns his solution of the class number problem great intrinsic beauty: "This problem is the last and most important solved in this book, and is connected with the most beautiful algebraic and analytic investigations of this century" (p. 100).

This is a pleasant book. Dirichlet is a celebrated expositor and quite rightly so. There is also an excellent 10 page introduction by Stillwell and some 70 pages of supplements by Dedekind. The most interesting supplement is certainly Dirichlet's famous L-series proof that there are infinitely many primes in essentially any arithmetic progression. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

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This Book Is In German. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Customer Reviews (1)

Come on . . . it's Dedekind.
Important foundational work by Dedekind in a new translation.It doesn't get better than this. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

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This digital document is an article from Science and Its Times, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 584 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.The histories of science, technology, and mathematics merge with the study of humanities and social science in this interdisciplinary reference work. Essays on people, theories, discoveries, and concepts are combined with overviews, bibliographies of primary documents, and chronological elements to offer students a fascinating way to understand the impact of science on the course of human history and how science affects everyday life. Entries represent people and developments throughout the world, from about 2000 B.C. through the end of the twentieth century. ... Read more

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Chapters: Richard Dedekind, Kurt Tank, Manfred Eigen, Georg Wittig, Nikolaus Hofreiter, Henning Kagermann, Herbert Freundlich, Caesar Rudolf Boettger, Julius Tröger. Source: Wikipedia. Pages: 40. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Julius Wilhelm Richard Dedekind (October 6, 1831 February 12, 1916) was a German mathematician who did important work in abstract algebra (particularly ring theory), algebraic number theory and the foundations of the real numbers. Dedekind was the youngest of four children of Julius Levin Ulrich Dedekind. As an adult, he never employed the names Julius Wilhelm. He was born, lived most of his life, and died in Braunschweig (often called "Brunswick" in English). In 1848, he entered the Collegium Carolinum in Braunschweig, where his father was an administrator, obtaining a solid grounding in mathematics. In 1850, he entered the University of Göttingen. Dedekind studied number theory under Moritz Stern. Gauss was still teaching, although mostly at an elementary level, and Dedekind became his last student. Dedekind received his doctorate in 1852, for a thesis titled Über die Theorie der Eulerschen Integrale ("On the Theory of Eulerian integrals"). This thesis did not reveal the talent evident on almost every page Dedekind later wrote. At that time, the University of Berlin, not Göttingen, was the leading center for mathematical research in Germany. Thus Dedekind went to Berlin for two years of study, where he and Riemann were contemporaries; they were both awarded the habilitation in 1854. Dedekind returned to Göttingen to teach as a Privatdozent, giving courses on probability and geometry. He studied for a while with Dirichlet, and they became close friends. Because of lingering weaknesses in his mathematical knowledge, he studied elliptic and abelian ...More: http://booksllc.net/?id=23475106 ... Read more

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Chapters: Carl Friedrich Gauss, Caroline of Brunswick, Karl Andree, Karl Fiehler, Richard Dedekind, Louis Spohr, Ernest Augustus, Prince of Hanover, Christian Rudolph Wilhelm Wiedemann, Ricarda Huch, Stephanie Storp, Frederick William, Duke of Brunswick-Wolfenbüttel, Duchess Augusta of Brunswick-Wolfenbüttel, Elisabeth Christine of Brunswick-Wolfenbüttel, Heinz Waaske, Axel Freiherr Von Dem Bussche-Streithorst, Bernhard Plockhorst, Karl Lachmann, Charles I, Duke of Brunswick-Wolfenbüttel, Paul Drude, Levin August, Count Von Bennigsen, Otto Grotewohl, Heike Lätzsch, Norbert Schultze, Christiane Kubrick, Johann Christian Martin Bartels, Gerard Krefft, Ernst Sagebiel, Johann Ludwig Christian Gravenhorst, Ernst August Friedrich Klingemann, Emil Selenka, Carl Ludwig Blume, Edward Degener, Alfred Kubel, Johann Karl Wilhelm Illiger, Gertrude of Brunswick, Henning Kagermann, Georg Ferdinand Howaldt, Katrin Kauschke, Michael Green, Wilhelm Nienstädt, Karl Gustav Himly, Johann Leopold Theodor Friedrich Zincken, August Wilhelm, Duke of Brunswick-Bevern, Christian Schwarzer, Ferdinand Albert I, Duke of Brunswick-Lüneburg, Louis Köhler, August Howaldt, Anton Ludwig Ernst Horn, Kurt Reidemeister, Gerd Wedler, Conrad Friedrich Hurlebusch, Richard Andree, Gerhard Schrader, Eberhard Schrader, Emil Fischer, Arend Friedrich August Wiegmann, Heinrich Wolfgang Ludwig Dohrn, Werner Fürbringer, Florian Meyer, Gustav Teichmüller, August Wilhelm Knoch, Jens Pieper, Gustav Von Der Mülbe, Anton August Beck. Source: Wikipedia. Pages: 232. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Johann Carl Friedrich Gauss (pronounced ; German: ·), Latin: ) (30 April 1777 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differe...More: http://booksllc.net/?id=6125 ... Read more

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High Quality Content by WIKIPEDIA articles! Julius Wilhelm Richard Dedekind (October 6, 1831 - February 12, 1916) was a German mathematician who did important work in abstract algebra (particularly ring theory), algebraic number theory and the foundations of the real numbers. While teaching calculus for the first time at the Polytechnic, Dedekind came up with the notion now called a Dedekind cut, now a standard definition of the real numbers. The idea behind a cut is that an irrational number divides the rational numbers into two classes (sets), with all the members of one class (upper) being strictly greater than all the members of the other (lower) class. In 1888, he published a short monograph titled Was sind und was sollen die Zahlen? ("What are numbers and what should they be?"), which included his definition of an infinite set. ... Read more