1967 edition; 4th printing; original copyright 1940; copyright renewed 1967;This book lures the reader into the realm of higher mathematics where the human mind must transcend "common sense" and intuition.  This includes the "Googolplex --- the largest definite number that anyone has yet bothered to conceive of, pi, famous puzzles, paradoxes, "laws" of chance, rubber-sheet geometry and more.

Mathematics and the Imagination

Humor & Entertainment

James R. Newman

Edward Kasner

Puzzles & Games

Fireside

A small library of the literature of mathematics from A'h-mose the Scribe to Albert Einstein, presented with commentaries and notes by the author.

The World of Mathematics

Science & Math

Mathematics

Simon&Schuster

The objective of this anthology is to introduce intelligent laypersons to the scope, power, significance and beauty of mathematics. It comprises a four-volume set with the selection of articles arranged within topic groupings.

Touchstone

In 1931 Kurt Gödel published his fundamental paper, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems." This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. Gödel received public recognition of his work in 1951 when he was awarded the first Albert Einstein Award for achievement in the natural sciences—perhaps the highest award of its kind in the United States. The award committee described his work in mathematical logic as "one of the greatest contributions to the sciences in recent times." However, few mathematicians of the time were equipped to understand the young scholar's complex proof. Ernest Nagel and James Newman provide a readable and accessible explanation to both scholars and non-specialists of the main ideas and broad implications of Gödel's discovery. It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject. New York University Press is proud to publish this special edition of one of its bestselling books. With a new introduction by Douglas R. Hofstadter, this book will appeal students, scholars, and professionals in the fields of mathematics, computer science, logic and philosophy, and science.

Godel's Proof

Ernest Nagel

NYU Press

From the Introduction. In 1931 there appeared in a German scientific periodical a relatively short paper with the forbidding title "Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme" ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems"). Its author was Kurt Gödel, then a young mathematician of 25 at the University of Vienna and since 1938 a permanent member of the Institute for Advanced Study at Princeton. The paper is a milestone in the history of logic and mathematics. When Harvard University awarded Gödel an honorary degree in 1952, the citation described the work as one of the most important advances in logic in modern times. At the time of its appearance, however, neither the title of Gödel's paper nor its content was intelligible to most mathematicians. The Principia Mathematica mentioned in the title is the monumental three-volume treatise by Alfred North Whitehead and Bertrand Russell on mathematical logic and the foundations of mathematics; and familiarity with that work is not a prerequisite to successful research in most branches of mathematics. Moreover, Gödel's paper deals with a set of questions that has never attracted more than a comparatively small group of students. The reasoning of the proof was so novel at the time of its publication that only those intimately conversant with the technical literature of a highly specialized field could follow the argument with ready comprehension. Nevertheless, the conclusions Gödel established are now widely recognized as being revolutionary in their broad philosophical import. It is the aim of the present essay to make the substance of Gödel's findings and the general character of his proof accessible to the non-specialist. Gödel's famous paper attacked a central problem in the foundations of mathematics. It will be helpful to give a brief preliminary account of the context in which the problem occurs. Everyone who has been exposed to elementary geometry will doubtless recall that it is taught as a deductive discipline. It is not presented as an experimental science whose theorems are to be accepted because they are in agreement with observation. This notion, that a proposition may be established as the conclusion of an explicit logical proof, goes back to the ancient Greeks, who discovered what is known as the "axiomatic method" and used it to develop geometry in a systematic fashion. The axiomatic method consists in accepting without proof certain propositions as axioms or postulates (e.g., the axiom that through two points just one straight line can be drawn), and then deriving from the axioms all other propositions of the system as theorems. The axioms constitute the ''foundations" of the system; the theorems are the "superstructure," and are obtained from the axioms with the exclusive help of principles of logic.

CreateSpace Independent Publishing Platform

The World of mathematics: A small library of the literature of mathematics from Aʻh-mosé the scribe to Albert Einstein

MICROSOFT PRESS,U.S.

See the best price to sell, buy, or rent books by James R. Newman. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Books by James R. Newman

See the best price to sell, buy, or rent books by James R. Newman. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Mathematics and the Imagination

The World of Mathematics

The World of Mathematics

Godel's Proof

Godel's Proof

The World of mathematics: A small library of the literature of mathematics from Aʻh-mosé the scribe to Albert Einstein