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This monograph considers several well-known mathematical theorems
and asks the question, "Why prove it again?" while examining
alternative proofs. It explores the different rationales
mathematicians may have for pursuing and presenting new proofs of
previously established results, as well as how they judge whether
two proofs of a given result are different. While a number of books
have examined alternative proofs of individual theorems, this is
the first that presents comparative case studies of other methods
for a variety of different theorems. The author begins by laying
out the criteria for distinguishing among proofs and enumerates
reasons why new proofs have, for so long, played a prominent role
in mathematical practice. He then outlines various purposes that
alternative proofs may serve. Each chapter that follows provides a
detailed case study of alternative proofs for particular theorems,
including the Pythagorean Theorem, the Fundamental Theorem of
Arithmetic, Desargues' Theorem, the Prime Number Theorem, and the
proof of the irreducibility of cyclotomic polynomials. Why Prove It
Again? will appeal to a broad range of readers, including
historians and philosophers of mathematics, students, and
practicing mathematicians. Additionally, teachers will find it to
be a useful source of alternative methods of presenting material to
their students.
Kurt Gödel (1906-1978) was the most outstanding logician of the twentieth century. This second volume of a comprehensive edition of Gödel's works collects the remainder of his published work, covering the period 1938-1974. (Volume I included all of his publications from 1929-1936). Each article or closely related group of articles is preceded by an introductory note that elucidates it and places it in historical context. The aim is to make the full body of Gödel's work as accessible and useful to as wide an audience as possible, without in any way sacrificing the requirements of historical and scientific accuracy.
This monograph considers several well-known mathematical theorems
and asks the question, "Why prove it again?" while examining
alternative proofs. It explores the different rationales
mathematicians may have for pursuing and presenting new proofs of
previously established results, as well as how they judge whether
two proofs of a given result are different. While a number of books
have examined alternative proofs of individual theorems, this is
the first that presents comparative case studies of other methods
for a variety of different theorems. The author begins by laying
out the criteria for distinguishing among proofs and enumerates
reasons why new proofs have, for so long, played a prominent role
in mathematical practice. He then outlines various purposes that
alternative proofs may serve. Each chapter that follows provides a
detailed case study of alternative proofs for particular theorems,
including the Pythagorean Theorem, the Fundamental Theorem of
Arithmetic, Desargues' Theorem, the Prime Number Theorem, and the
proof of the irreducibility of cyclotomic polynomials. Why Prove It
Again? will appeal to a broad range of readers, including
historians and philosophers of mathematics, students, and
practicing mathematicians. Additionally, teachers will find it to
be a useful source of alternative methods of presenting material to
their students.
Kurt Gödel was the most outstanding logician of the 20th century and a giant in the field. This book is part of a five volume set that makes available all of Gödels writings. The first three volumes, already published, consist of the papers and essays of Gödel. The final two volumes of the set deal with Gödel's correspondence with his contemporary mathematicians, this fifth volume consists of material from correspondents from H-Z.
Kurt Gödel was the most outstanding logician of the 20th century and a giant in the field. This book is part of a five volume set that makes available all of Gödels writings. The first three volumes, already published, consist of the papers and essays of Gödel. The final two volumes of the set deal with Gödel's correspondence with his contemporary mathematicians, this fourth volume consists of material from correspondents from A-G.
Kurt Goedel (1906 - 1978) was the most outstanding logician of the
twentieth century, famous for his hallmark works on the
completeness of logic, the incompleteness of number theory, and the
consistency of the axiom of choice and the continuum hypothesis. He
is also noted for his work on constructivity, the decision problem,
and the foundations of computability theory, as well as for the
strong individuality of his writings on the philosophy of
mathematics. He is less well known for his discovery of unusual
cosmological models for Einstein's equations, in theory permitting
time travel into the past. The Collected Works is a landmark
resource that draws together a lifetime of creative thought and
accomplishment. The first two volumes were devoted to Goedel's
publications in full (both in original and translation), and the
third volume featured a wide selection of unpublished articles and
lecture texts found in Goedel's Nachlass. These long-awaited final
two volumes contain Goedel's correspondence of logical,
philosophical, and scientific interest. Volume IV covers A to G,
with H to Z in volume V; in addition, Volume V contains a full
inventory of Goedel's Nachlass. All volumes include introductory
notes that provide extensive explanatory and historical commentary
on each body of work, English translations of material originally
written in German (some transcribed from the Gabelsberger
shorthand), and a complete bibliography of all works cited. Kurt
Goedel: Collected Works is designed to be useful and accessible to
as wide an audience as possible without sacrificing scientific or
historical accuracy. The only comprehensive edition of Goedel's
work available, it will be an essential part of the working library
of professionals and students in logic, mathematics, philosophy,
history of science, and computer science and all others who wish to
be acquainted with one of the great minds of the twentieth century.
Kurt Gödel is regarded as one of the most outstanding logician of the twentieth century, famous for his work on logic and number theory. This third volume of a comprehensive edition of Godel's works comprises a selection of previously unpublished manuscripts and lectures. It includes introductory notes that provide extensive explanations and historical commentary on each of the papers. This book is accessible to a wide audience without sacrificing historical or scientific accuracy and will be an essential part of the working library of both professionals and students.
This scarce antiquarian book is a selection from Kessinger
Publishing's Legacy Reprint Series. 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 to protecting,
preserving, and promoting the world's literature. Kessinger
Publishing is the place to find hundreds of thousands of rare and
hard-to-find books with something of interest for everyone!
Kurt Gödel (1906-1978) was the most outstanding logician of the twentieth century, famous for his hallmark works on the completeness of logic, the incompleteness of number theory and stronger systems, and the consistency of the axiom of choice and the continuum hypothesis. He is also noted for his work on constructivity, the decision problem, the foundations of computation theory, unusual cosmological models, and for the strong individuality of his writings on the philosophy of mathematics. The Collected Works is a landmark resource that draws together a lifetime of creative accomplishment. The first two volumes were devoted to Gödel's publications in full (both in the original and translation). This third volume features a wide selection of unpublished articles and lecture texts found in Gödel's Nachlass, documents that enlarge considerably our appreciation of his scientific and philosophical thought and add a great deal to our understanding of his motivations. Continuing the format of the earlier volumes, the present volume includes introductory notes that provide extensive explanatory and historical commentary on each of the papers, English translations of material originally written in German (some transcribed from Gabelsberger shorthand), and a complete bibliography. A succeeding volume is to contain a comprehensive selection of Gödel's scientific correspondence and a complete inventory of his Nachlass. The books are designed to be accessible and useful to as wide an audience as possible without sacrificing scientific or historical accuracy. The only complete edition available in English, it will be an essential part of the working library of professionals and students in logic, mathematics, philosophy, history of science, and computer science.
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