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The nineteenth century saw the paradoxes and obscurities of
eighteenth-century calculus gradually replaced by the exact
theorems and statements of rigorous analysis. It became clear that
all analysis could be deduced from the properties of the real
numbers. But what are the real numbers and why do they have the
properties we claim they do? In this charming and influential book,
Richard Dedekind (1831-1916), Professor at the Technische
Hochschule in Braunschweig, showed how to resolve this problem
starting from elementary ideas. His method of constructing the
reals from the rationals (the Dedekind cut) remains central to this
day and was generalised by Conway in his construction of the
'surreal numbers'. This reissue of Dedekind's 1888 classic is of
the 'second, unaltered' 1893 edition.
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 instalments in the 'Bulletin
des sciences mathematiques' in 1877. This is a translation of that
work by John Stillwell, who also adds a detailed introduction that
gives the historical background as well as outlining the
mathematical obstructions that Dedekind was striving to overcome.
Dedekind's memoir gives a candid account of his development of an
elegant theory as well as providing blow-by-blow comments as he
wrestled with the many difficulties encountered en route. A must
for all number theorists.
Great mathematicians write for the future and Georg Friedrich
Bernhard Riemann (1826-66) was one of the greatest mathematicians
of all time. Edited by Heinrich Martin Weber, with assistance from
Richard Dedekind, this edition of his collected works in German
first appeared in 1876. Riemann's interests ranged from pure
mathematics to mathematical physics. He wrote a short paper on
number theory which provided the key to the prime number theorem,
and his zeta hypothesis has given mathematicians the most famous of
today's unsolved problems. Moreover, his famous 1854 lecture 'On
the hypotheses which underlie geometry' set in motion studies which
culminated in Einstein's general theory of relativity. Even
Riemann's over-optimistic use of the Dirichlet principle to prove
the conformal mapping theorem turned out to be immensely fruitful.
The alert reader will further profit from finding here the seeds of
modern distribution theory, algebraic topology and measure theory.
1. VORSTELLUNG DES ZAHLENGEBIETES Wir konnen jede ganze Zahl
bildlich oder geometrisch darstellen. Nehmen wir zum Beispiel eine
Linie von beliebiger Lange an, und auf derselben einen Punkt o. So
konnen wir die Zahl eins so darstellen, indem wir eine beliebige
konstante Lange auf dieser vom Nullpunkt aus nach rechts auftragen.
Dieses Stuck reprasen- tirt uns also die Zahl eins. Wollen wir die
Zahl 2 geometrisch darstellen, so wissen wir, dass 2 = 1 + 1 ist.
Wir haben also nur die Einheit zweimal vom Nullpunkt aus
aufzutragen, oder von 1 aus noch einmal und erhalten das
geometrische Bild der Zahl 2 . Urn das Bild der Zahl 3 zu erhalten,
konnen wir unsere Langeneinheit dreimal vom Nullpunkt aus
auftragen. Ebenso k- nen wir 4,5,6,7,8 ... bis bildlich darstellen.
Wollen wir hingegen eine gebrochene Zahl geometrisch darstellen,
zum Beispiel t, so waren wir dies mit unsern Langeneinheiten 7 3 3
nicht imstande, denn 4 = 14 ' und 4 ist eine Grosse, die kleiner
ist als 1. Wir mussen daher unsere Lange in noch klei- nere Theile
eintheilen und zwar in Viertel. Dann sind wir erst 7 imstande, 4
geometrisch darzustellen.
Zur Rechtfertigung dieser Edition von Dedekinds beruhmtem elften
Supplement zu Dirichlets "Vorlesungen uber Zahlentheorie" kann ich
keine besseren Worte finden als die von Dedekind selbst am Schluss
seines Vorworts zur zweiten Auflage dieser "V orlesungen" (1871):
"Endlich habe ich mich bemuht, uberall, wo es mir moglich war, auf
die Quellen zu verweisen, um den Leser zum Studium der Original
werke zu veranlassen und in ihm ein Bild von den Fortschritten der
Wissenschaft zu erwecken, deren ebenso tiefe wie erhabene
Wahrheiten einen Schatz bilden, welcher die unvergangliche Frucht
eines wahrhaft edelen Wett kampfes der europaischen Volker ist. "
Das elfte Supplement, das zuerst in der dritten Auflage erschien,
war eine Neufassung eines bedeutenden Abschnittes ( 159-170) des
zehnten Supplementes der zweiten Auflage. Uber diesen Abschnitt
schreibt Dedekind im Vorwort zur zweiten Auflage: "Endlich habe ich
in dieses Supplement eine allgemeine Theorie der Ideale
aufgenommen, um auf den Hauptgegenstand des ganzen Buches von einem
hoheren Standpunkte aus ein neues Licht zu werfen; hierbei habe ich
mich freilich auf die Darstellung der Grundlagen beschranken
mussen, doch hoffe ich, dass das Streben nach charakteristischen
Grund begriffen, welches in anderen Teilen der Mathematik mit so
schonen Erfolgen gekront ist, mir nicht ganz missgluckt sein moge.
" Schon vor Dedekind hatte Kronecker eine Idealtheorie der
algebraischen Zahlkorper entwickelt, aber die Dedekindsche Theorie
ist unabhangig von der Kroneckerschen entstanden."
The nineteenth century saw the paradoxes and obscurities of
eighteenth-century calculus gradually replaced by the exact
theorems and statements of rigorous analysis. It became clear that
all analysis could be deduced from the properties of the real
numbers. But what are the real numbers and why do they have the
properties we claim they do? In this charming and influential book,
Richard Dedekind (1831 1916), Professor at the Technische
Hochschule in Braunschweig, showed how to resolve this problem
starting from elementary ideas. His method of constructing the
reals from the rationals (the Dedekind cut) remains central to this
day and was generalised by Conway in his construction of the
'surreal numbers'. This reissue of Dedekind's 1888 classic is of
the 'second, unaltered' 1893 edition.
This Is A New Release Of The Original 1901 Edition.
This Is A New Release Of The Original 1901 Edition.
This book is the first English translation of the classic long
paper Theorie der algebraischen Functionen einer Veranderlichen
(Theory of algebraic functions of one variable), published by
Dedekind and Weber in 1882. The translation has been enriched by a
Translator's Introduction that provides historical background, and
extensive commentary embedded in the translation itself. The
translation, introduction, and commentary provide the first easy
access to this important paper for a wide mathematical audience:
students, historians of mathematics, and professional
mathematicians. Why is the Dedekind-Weber paper important? In the
1850s, Riemann initiated a revolution in algebraic geometry by
interpreting algebraic curves as surfaces covering the sphere. He
obtained deep and striking results in pure algebra by intuitive
arguments about surfaces and their topology. However, Riemann's
arguments were not rigorous, and they remained in limbo until 1882,
when Dedekind and Weber put them on a sound foundation. The key to
this breakthrough was to develop the theory of algebraic functions
in analogy with Dedekind's theory of algebraic numbers, where the
concept of ideal plays a central role. By introducing such concepts
into the theory of algebraic curves, Dedekind and Weber paved the
way for modern algebraic geometry.
I. Continuity and Irrational Numbers. II. The Nature and Meaning of
Numbers.
I. Continuity and Irrational Numbers. II. The Nature and Meaning of
Numbers.
I. Continuity and Irrational Numbers. II. The Nature and Meaning of
Numbers.
This scarce antiquarian book is a facsimile reprint of the
original. Due to its age, it may contain imperfections such as
marks, notations, marginalia and flawed pages. Because we believe
this work is culturally important, we have made it available as
part of our commitment for protecting, preserving, and promoting
the world's literature in affordable, high quality, modern editions
that are true to the original work.
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