What is Mathematics?: An Elementary Approach to Ideas and MethodsFor 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 FourColor 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 grammara matter of the correct application of local rules. Meaningful mathematics is like journalismit tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literatureit 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 literatureit opens a window onto the world of mathematics for anyone interested to view. 
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Review: What Is Mathematics?: An Elementary Approach to Ideas and Methods
User Review  Dan  GoodreadsVery well done. About half of this can be appreciated by the bright high school senior; the rest might require a couple of college math courses. Read full review
Review: What Is Mathematics?: An Elementary Approach to Ideas and Methods
User Review  Ron Banister  GoodreadsOne of my favorite books on mathematics. Also, one of Einstein's favorites. Read full review
Contents
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Common terms and phrases
algebraic altitude triangle angle axioms calculus circle complex numbers concept conic consider construction continuous function coordinates corresponding crossratio decimal defined definition denote differential digits dimension distance domain doubling the cube elements ellipse equal equation Euclidean geometry Euler's Euler's formula example Exercise expression fact Fermat Figure finite number formula function geometry given Hence hyperbola infinite integers intersection interval intuitive inverse inverse function irrational numbers Jordan curve theorem Leibniz length limit mathematical induction mathematicians minimum multiplication nested intervals obtain parallel plane polygon polynomial positive integer positive number prime problem projective projective geometry proof properties prove quadratic residue quantity radius rational numbers real numbers root segment sequence side simple solution square Steiner Steiner's problem straight line surface symbol tangent tends to infinity theorem theory tion topological transformation variable vertices zero
Popular passages
Page 1  A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician. If this description were accurate, mathematics could not attract any intelligent person. It would be a game with definitions, rules, and syllogisms, without motive or goal.
Page 1  True, the element of constructive invention, of directing and motivating intuition, is apt to elude a simple philosophical formulation ; but it remains the core of any mathematical achievement, even in the most abstract fields.