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What Is Mathematics? : An Elementary Approach to Ideas and Methods - Richard Courant

What Is Mathematics?

An Elementary Approach to Ideas and Methods

Paperback Published: 18th July 1996
ISBN: 9780195105193
Number Of Pages: 592

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For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. This new edition of Richard Courant's and Herbert Robbins's classic work seeks to address this problem. Its goal is to put the meaning back into mathematics.
Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics?, Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts. Brought up to date with a new chapter by Ian Stewart, What is Mathematics?, Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved.
Formal mathematics is like spelling and grammar--a matter of the correct application of local rules. Meaningful mathematics is like journalism--it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature--it brings a story to life before your eyes and involves you in it, intellectually and emotionally. What is Mathematics is like a fine piece of literature--it opens a window onto the world of mathematics for anyone interested to view.

Still a book that all prospective mathematics teachers should read and experience. A rare book that has retained its "freshness" and readability for more than 50 years....Very readable. * Stephen Krulik, Temple University * It is a work of high perfection, whether judged by aesthetic, pedagogical or scientific standards. It is astonishing to what extent What is Mathematics? has succeeded in making clear by means of the simplest examples all the fundamental ideas and methods which we mathematicians consider the life blood of our science. * Herman Weyl * This is not a book in philosophy; but there are probably few philosophers who can not gain instruction and clarification from it. It succeeds brilliantly in conveying the intellectual excitement of mathematical inquiry and in communicating the essential ideas and methods."Journal of Philosophy This book is a work of art. * Marston Morse * It contains an excellent selection of material for students who have no desire to develop mathematical skills but who may be willing to look briefly into this field of intellectual activity....For the inquiring student who wishes to know what real mathematics is about, or for the trained engineer or physicist who has some interest in the justification of procedures he uses, it should prove a source of great pleasure and satisfaction. * Journal of Applied Physics * A work of extraordinary perfection. * Mathematical Reviews * Without doubt, the work will have great influence. It should be in the hands of everyone, professional or otherwise, who is interested in scientific thinking. * The New York Times * A lucid representation of the fundamental concepts and methods of the whole field of mathematics. It is an easily understandable introduction for the layman and helps to give the mathematical student a general view of the basic principles and methods. * Albert Einstein * A great book. * Ludwig Otto, Paul Quinn College * Can...be read with great profit by anyone desiring general mathematical literacy. * Mathematics Abstracts *

Preface to Second Edition
Preface to Revised Editions
How to Use the Book
What is Mathematics?
The Natural Numbersp. 1
Supplement to Chapter I. the Theory of Numbersp. 21
The Number System of Mathematicsp. 52
Supplement to Chapter II. The Algebra of Setsp. 108
Geometrical Constructions. The Algebra of Number Fieldsp. 117
Projective Geometry. Axiomatics. Non-Euclidean Geometriesp. 165
Topologyp. 235
Functions and Limitsp. 272
Supplement to Chapter VIp. 322
Maxima and Minimap. 329
The Calculusp. 398
Supplement to Chapter VIIIp. 462
Recent Developmentsp. 487
Appendix: Supplementary Remarks, Problems, and Exercisesp. 525
Suggestions for Further Readingp. 549
Suggestions for Additional Readingp. 553
Indexp. 559
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780195105193
ISBN-10: 0195105192
Series: Oxford Paperbacks
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 592
Published: 18th July 1996
Publisher: Oxford University Press
Country of Publication: GB
Dimensions (cm): 22.86 x 15.24  x 2.54
Weight (kg): 0.82
Edition Number: 2
Edition Type: Revised