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. 
What people are saying  Write a review
User ratings
5 stars 
 
4 stars 
 
3 stars 
 
2 stars 
 
1 star 

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
Review: What Is Mathematics?: An Elementary Approach to Ideas and Methods
User Review  D  Goodreads"The difficulty is that it is easy to prove a problem is easy, but hard to prove that it is hard!" I'm probably not unique in that I made it through a lot of math without ever really understanding ... Read full review
Contents
VI  5 
VII  6 
VIII  7 
IX  9 
X  12 
XI  13 
XII  14 
XIII  15 
CXXX  288 
CXXXI  289 
CXXXII  295 
CXXXIII  297 
CXXXIV  299 
CXXXV  301 
CXXXVI  303 
CXXXVII  305 
XIV  16 
XV  18 
XVI  23 
XVII  25 
XIX  26 
XX  27 
XXI  30 
XXII  31 
XXIII  37 
XXIV  38 
XXV  40 
XXVI  42 
XXVII  46 
XXVIII  48 
XXIX  49 
XXX  52 
XXXI  54 
XXXII  57 
XXXIII  58 
XXXIV  61 
XXXV  63 
XXXVI  66 
XXXVII  68 
XXXVIII  71 
XXXIX  72 
XL  74 
XLI  77 
XLII  79 
XLIII  83 
XLIV  86 
XLV  87 
XLVII  88 
XLVIII  92 
XLIX  98 
L  101 
LI  103 
LII  104 
LIII  108 
LIV  112 
LV  114 
LVI  117 
LVII  120 
LVIII  122 
LIX  125 
LX  127 
LXI  133 
LXII  134 
LXIII  135 
LXIV  137 
LXV  138 
LXVI  140 
LXVIII  142 
LXIX  144 
LXX  145 
LXXI  146 
LXXII  147 
LXXIII  152 
LXXIV  155 
LXXV  158 
LXXVI  161 
LXXVII  162 
LXXVIII  165 
LXXIX  167 
LXXX  168 
LXXXI  170 
LXXXII  172 
LXXXIII  179 
LXXXIV  180 
LXXXV  183 
LXXXVI  185 
LXXXVIII  187 
LXXXIX  188 
XC  190 
XCI  191 
XCIII  193 
XCIV  196 
XCV  198 
XCVI  201 
XCVII  205 
XCVIII  209 
XCIX  212 
C  214 
CI  218 
CII  222 
CIII  223 
CIV  224 
CV  227 
CVI  228 
CVII  230 
CVIII  235 
CIX  236 
CX  241 
CXI  243 
CXII  244 
CXIII  246 
CXIV  248 
CXV  251 
CXVI  255 
CXVII  256 
CXVIII  258 
CXIX  259 
CXX  264 
CXXI  267 
CXXII  269 
CXXIII  272 
CXXIV  273 
CXXV  277 
CXXVI  278 
CXXVII  282 
CXXVIII  283 
CXXIX  286 
CXXXVIII  307 
CXXXIX  309 
CXL  310 
CXLI  312 
CXLII  313 
CXLIII  315 
CXLIV  317 
CXLV  319 
CXLVI  322 
CXLVII  323 
CXLVIII  325 
CXLIX  326 
CL  327 
CLI  329 
CLII  330 
CLIV  332 
CLV  333 
CLVI  336 
CLVII  338 
CLVIII  340 
CLIX  341 
CLX  343 
CLXI  345 
CLXIII  346 
CLXIV  349 
CLXV  351 
CLXVI  352 
CLXVII  353 
CLXVIII  354 
CLXIX  356 
CLXX  358 
CLXXI  359 
CLXXIII  361 
CLXXV  363 
CLXXVI  365 
CLXXVII  366 
CLXXVIII  368 
CLXXIX  370 
CLXXX  372 
CLXXXI  373 
CLXXXII  376 
CLXXXIII  379 
CLXXXIV  381 
CLXXXV  383 
CLXXXVI  384 
CLXXXVII  385 
CLXXXVIII  386 
CLXXXIX  388 
CXC  391 
CXCI  398 
CXCII  399 
CXCIII  401 
CXCIV  404 
CXCV  406 
CXCVI  411 
CXCVII  414 
CXCVIII  416 
CXCIX  418 
CC  421 
CCI  422 
CCII  423 
CCIII  426 
CCV  427 
CCVI  433 
CCVII  436 
CCVIII  439 
CCIX  441 
CCX  442 
CCXI  443 
CCXII  446 
CCXIII  447 
CCXIV  448 
CCXV  451 
CCXVI  453 
CCXVII  454 
CCXVIII  458 
CCXIX  460 
CCXX  462 
CCXXI  464 
CCXXII  465 
CCXXIII  469 
CCXXIV  471 
CCXXV  472 
CCXXVI  477 
CCXXVII  479 
CCXXVIII  482 
CCXXIX  487 
CCXXX  488 
CCXXXI  491 
CCXXXII  493 
CCXXXIII  494 
CCXXXIV  495 
CCXXXV  499 
CCXXXVI  501 
CCXXXVII  505 
CCXXXVIII  507 
CCXXXIX  513 
CCXL  518 
CCXLI  525 
CCXLIII  526 
CCXLIV  532 
CCXLV  533 
CCXLVI  534 
CCXLVII  537 
CCXLVIII  538 
CCXLIX  540 
CCL  542 
549  
553  
559  
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.