The World of Mathematics: A Four-Volume Set

The World of Mathematics I......Page 1
Introduction......Page 3
TABLE OF CONTENTS I......Page 5
TABLE OF CONTENTS II......Page 762
TABLE OF CONTENTS III......Page 1476
TABLE OF CONTENTS IV......Page 2089
PART I: General Survey......Page 9
COMMENTARY ON: PHILIP E. B. JOURDAIN......Page 10
1. The Nature of Mathematics......Page 12
PART II: Historical and Biographical......Page 81
COMMENTARY ON: The Great Mathematicians......Page 82
1. The Great Mathematicians......Page 83
COMMENTARY ON" The Rhind Papyrus......Page 177
2. The Rhind Papyrus......Page 178
COMMENTARY ON: ARCHIMEDES......Page 187
3. Archimedes......Page 188
COMMENTARY ON: Greek Mathematics......Page 196
4. Greek Mathematics......Page 197
COMMENTARY ON: ROBERT RECORDE......Page 218
5. The Declaration of the Profit of Arithmeticke......Page 220
COMMENTARY ON: KEPLER and LODGE......Page 226
6. Johann Kepler......Page 228
COMMENTARY ON: DESCARTES and AnalyticalGeometry......Page 243
7. The Geometry......Page 247
COMMENTARY ON: ISAAC NEWTON......Page 262
8. Isaac Newton......Page 263
9. Newton, the Man......Page 285
COMMENTARY ON: BISHOP BERKELEY and Infinitesimals......Page 294
10. The Analyst......Page 296
COMMENTARY ON: GAUSS......Page 302
11. The Prince of Mathematicians......Page 303
COMMENTARY ON: CAYLEY and SYLVESTER......Page 348
12. Invariant Twins, Cayley and Sylvester......Page 349
COMMENTARY ON: SRINIV ASA RAMANUJAN......Page 374
13. Srinivasa Ramanujan......Page 376
COMMENTARY ON: BERTRAND RUSSELL......Page 385
14. My Mental Development......Page 389
COMMENTARY ON: ALFRED NORTH WHITEHEAD......Page 403
15. Mathematics as an Element in the History of Thought......Page 410
COMMENTARY ON: Poppy Seeds and Large Numbers......Page 426
PART III: Arithmetic, Numbers andthe Art of Counting......Page 425
1. The Sand Reckoner......Page 428
COMMENTARY ON: The Art of Counting......Page 438
2. Counting......Page 440
3. From Numbers to Numerals and From Numerals to Computation......Page 450
COMMENTARY ON: Idiot Savants......Page 473
4. Calculating Prodigies......Page 475
COMMENTARY ON: Gifted Birds......Page 496
5. The Ability of Birds to "Count"......Page 497
COMMENTARY ON: The Mysteries of Arithmetic......Page 505
6. The Queen of Mathematics......Page 506
COMMENTARY ON (P+PQ)^m/n......Page 527
7. On the Binomial Theorem for Fractional and Negative Exponents......Page 529
COMMENTARY ON: The Number Concept......Page 533
8. Irrational Numbers......Page 536
9. Definition of Number......Page 545
COMMENTARY ON: WILLIAM KINGDON CLIFFORD......Page 554
PART IV: Mathematics of Space and Motion......Page 553
1. The Exactness of Mathematical Laws......Page 556
2. The Postulates of the Science o f Space......Page 560
3. On the Space Theory of Matter......Page 576
COMMENTARY ON: A Famous Problem......Page 578
4. The Seven Bridges of Konigsberg......Page 581
5. Topology......Page 589
COMMENTARY ON: DURER and the Mathematics of Painting......Page 608
6. Durer as a Mathematician......Page 611
7. Projective Geometry......Page 630
COMMENTARY ON: HERMANN VON HELMHOLTZ......Page 650
8. On the Origin and Significance of Geometrical Axioms......Page 655
COMMENTARY ON: Symmetry......Page 677
9. Symmetry......Page 679
The World of Mathematics II......Page 760
PART V: Mathematics and the Physical World......Page 766
COMMENTARY ON: GALILEO GALILEI......Page 767
1. Mathematics of Motion......Page 775
COMMENTARY ON: THE BERNOULLIS......Page 812
2. Kinetic Theory of Gases......Page 815
COMMENTARY ON: A Great Prize, a Long-Suffering Inventor and the First Accurate Clock......Page 819
3. The Longitude......Page 821
COMMENTARY ON: JOHN COUCH ADAMS......Page 861
4. John Couch Adams and the Discovery of Neptune......Page 863
COMMENTARY ON: H. G. J. MOSELEY......Page 881
5. Atomic Numbers......Page 883
COMMENTARY ON: The Small Furniture of Earth......Page 892
6. The Rontgen Rays......Page 895
7. Crystals and the Future of Physics......Page 912
COMMENTARY ON: Queen Dido, Soap Bubbles, and a Blind Mathematician......Page 923
8. What Is Calculus of Variationsand What Are Its Applications?......Page 927
9. The Soap-bubble......Page 932
10. Plateau's Problem......Page 942
COMMENTARY ON: The Periodic Law and Mendeleeff......Page 951
11. Periodic Law of the Chemical Elements......Page 954
12. Mendeleeff......Page 960
COMMENTARY ON: GREGOR MENDEL......Page 973
13. Mathematics of Heredity......Page 978
COMMENTARY ON: J. B. S. HALDANE......Page 991
14. On Being the Right Size......Page 993
15. Mathematics of NaturalSelection......Page 999
COMMENTARY ON: ERWIN SCHRODINGER......Page 1014
16. Heredity and the QuantumTheory......Page 1016
COMMENTARY ON: D'ARCY WENTWORTH THOMPSON......Page 1037
17. On Magnitude......Page 1042
COMMENTARY ON: Uncertainty......Page 1088
18. The Uncertainty Principle......Page 1092
19. Causality and Wave Mechanics......Page 1097
COMMENTARY ON: SIR ARTHUR STANLEY EDDINGTON......Page 1110
20. The Constants of Nature......Page 1115
21. The New Law of Gravitation and the Old Law......Page 1135
COMMENTARY ON: The Theory of Relativity......Page 1146
22. The Theory of Relativity......Page 1148
COMMENTARY ON: The Founder of Psychophysics......Page 1187
1. Gustav Theodor Fechner......Page 1189
COMMENTARY ON: SIR FRANCIS GALTON......Page 1208
2. Classification of Men Accordingto Their Natural Gifts......Page 1214
COMMENTARY ON: THOMAS ROBERT MALTHUS......Page 1230
3. Mathematics of Population and Food......Page 1233
COMMENTARY ON: COURNOT, JEVONS, and the Mathematics of Money......Page 1241
4. Mathematics of Value and Demand......Page 1244
5.  Theory of Political Economy......Page 1258
COMMENTARY ON: A Distinguished Quaker and War......Page 1279
6. Mathematics of War and Foreign Politics......Page 1281
7. Statistics of Deadly Quarrels......Page 1295
COMMENTARY ON: The Social Application of Mathematics......Page 1305
8. The Theory of Economic Behavior......Page 1308
9. Theory of Games......Page 1326
10. Sociology Learns the Language of Mathematics......Page 1335
PART VII: The Laws of Chance......Page 1356
COMMENTARY ON: PIERRE SIMON DE LAPLACE......Page 1357
1. Concerning Probability......Page 1366
2. The Red and the Black......Page 1375
3. The Probability of Induction......Page 1382
COMMENTARY ON: LORD KEYNES......Page 1396
4. The Application of Probability to Conduct......Page 1401
COMMENTARY ON: An Absent-minded Genius and the Laws of Chance......Page 1415
5. Chance......Page 1421
COMMENTARY ON: ERNEST NAGEL and the Laws ofProbability......Page 1436
6. The Meaning of Probability......Page 1439
The World of Mathematics III......Page 1474
PART VIII: Statistics and the Design of Experiments......Page 1480
COMMENTARY ON: An Ingenious Army Captain and on a Generous and Many-sided Man......Page 1481
1. Foundations of Vital Statistics......Page 1486
2. First Life Insurance Tables......Page 1502
COMMENTARY ON: The Law of Large Numbers......Page 1513
3. The Law of Large Numbers......Page 1517
COMMENTARY ON: Statistics and the Lady with a Fine Palate......Page 1521
4. Sampling and Standard Error......Page 1524
5. On the Average and Scatter......Page 1552
6. Mathematics of a Lady Tasting Tea......Page 1577
COMMENTARY ON: The Scientific Aptitude of Mr. George Bernard Shaw......Page 1587
7. The Vice of Gambling and the Virtue of Insurance......Page 1589
PART IX: The Supreme Art of Abstraction: Group Theory......Page 1598
COMMENTARY ON: Certain Important Abstractions......Page 1599
1. The Group Concept......Page 1603
2. The Theory of Groups......Page 1623
COMMENTARY ON: HANS HAHN......Page 1656
PART X:  Mathematics of Infinity......Page 1640
1. Mathematics and the Metaphysicians......Page 1641
2. Infinity......Page 1658
PART XI: Mathematical Truth and the Structure of Mathematics......Page 1678
COMMENTARY ON: The Foundations of Mathematics......Page 1679
1. On the Nature of Mathematical Truth......Page 1684
2. Geometry and Empirical Science......Page 1700
3. The Axiomatic Method......Page 1712
4. Goedel's Proof......Page 1733
5. A Mathematical Science......Page 1761
6. Mathematics and the World......Page 1773
7. Mathematical Postulates and Human Understanding......Page 1788
COMMENTARY ON: An Eristic Controversy......Page 1821
PART XII: The Mathenlatical Way of Thinking......Page 1820
1. The Study That Knows Nothing of Observation......Page 1823
COMMENTARY ON: CHARLES SANDERS PEIRCE......Page 1832
2. The Essence of Mathematics......Page 1838
COMMENTARY ON: ERNST MACH......Page 1849
3. The Economy of Science......Page 1852
COMMENTARY ON: NORMAN ROBERT CAMPBELL......Page 1861
4. Measurement......Page 1862
5. Numerical Laws and the Use of Mathematics in Science......Page 1879
COMMENTARY ON: HERMANN WEYL......Page 1895
6. The Mathematical Way of Thinking......Page 1897
COMMENTARY ON: Symbolic Logic, GEORGE BOOLE and a Horrible Dream......Page 1917
PART XIII: Mathematics and Logic......Page 1916
1. Mathematical Analysis of Logic......Page 1921
2. History of Symbolic Logic......Page 1924
3. Symbolic Notation, Haddocks'Eyes and the Dog-Walking Ordinance......Page 1943
4. Symbolic Logic......Page 1966
PART XIV: The Unreasonableness ofMathematics......Page 1998
COMMENTARY ON: The Importance of Being Absurd......Page 1999
1. Paradox Lost and Paradox Regained......Page 2001
2. The Crisis in Intuition......Page 2021
PART XV: How to Solve It......Page 2042
COMMENTARY ON: The Tears of Mathematics......Page 2043
1. How to Solve It......Page 2045
PART XVI: The Vocabulary of Mathematics......Page 2058
COMMENTARY ON: Double Infinite Rapport and Other Mathematical Jargon......Page 2059
1. New Names for Old......Page 2061
PART XVII: Mathematics as an Art......Page 2076
COMMENTARY ON: JOHN WILLIAM NAVIN SULLIVAN......Page 2077
1. Mathematics as an Art......Page 2080
The World of Mathematics IV......Page 2087
PART XVIII: The Mathematician......Page 2093
COMMENTARY ON: G. H. HARDY......Page 2094
1. A Mathematician's Apology......Page 2097
COMMENTARY ON: The Elusiveness of Invention......Page 2109
2. Mathematical Creation......Page 2111
COMMENTARY ON: The Use of a Top Hatas a Water Bucket......Page 2121
3. The Mathematician......Page 2123
PART XIX: Mathematical Machines:Can a Machine Think?......Page 2135
COMMENTARY ON: Automatic Computers......Page 2136
1. The General and Logical Theory of Automata......Page 2140
2. Can a Machine Think?......Page 2169
3. A Chess-Playing Machine......Page 2194
PART XX: Mathematics in Warfare......Page 2205
COMMENTARY ON: FREDERICK WILLIAMLANCHESTER......Page 2206
1. Mathematics in Warfare......Page 2208
COMMENTARY ON: Operations Research......Page 2228
2. How to Hunt a Submarine......Page 2230
PART XXI: A Mathematical Theory of Art......Page 2251
COMMENTARY ON: GEORGE DAVID BIRKHOFF......Page 2252
1. Mathematics of Aesthetics......Page 2255
PART XXII: Mathematics of the Good......Page 2267
1. A Mathematical Approach to Ethics......Page 2268
PART XXIII: Mathematics in Literature......Page 2279
COMMENTARY ON: The Island of Laputa......Page 2280
1. Cycloid Pudding......Page 2284
COMMENTARY ON: ALDOUS HUXLEY......Page 2291
2. Young Archimedes......Page 2293
COMMENTARY ON: Mr. Fortune......Page 2320
3. Geometry in the South Pacific......Page 2322
COMMENTARY ON: Statistics as a Literary Stimulus......Page 2331
4. Inflexible Logic......Page 2332
5. The Law......Page 2338
PART XXIV: Mathematics and Music......Page 2343
COMMENTARY ON: SIR JAMES JEANS......Page 2344
1. Mathematics of Music......Page 2348
PART XXV: Mathematics as a Culture Clue......Page 2381
COMMENTARY ON: OSWALD SPENGLER......Page 2382
1. Meaning of Numbers......Page 2385
2. The Locus of Mathematical Reality: An Anthropological Footnote......Page 2418
PART XXVI: Amusements, Puzzles, Fancies......Page 2435
COMMENTARY ON: AUGUSTUS DE MORGAN......Page 2436
1. Assorted Paradoxes......Page 2439
COMMENTARY ON: A Romance of Many Dimensions......Page 2453
2. Flatland......Page 2455
COMMENTARY ON: LEWIS CARROLL......Page 2467
3. What the Tortoise Said to Achilles and Other Riddles......Page 2472
COMMENTARY ON: Continuity......Page 2480
4. The Lever of Mahomet......Page 2482
COMMENTARY ON: Games and Puzzles......Page 2484
5. Pastimes of Past and Present Times......Page 2486
6. Arithmetical Restorations......Page 2509
7. The Seven Seven's......Page 2514
COMMENTARY ON: THOMAS JOHN I'ANSON BROMWICH......Page 2519
8. Easy Mathematics and Lawn Tennis......Page 2520
COMMENTARY ON: STEPHEN BUTLER LEACOCK......Page 2525
9. Mathematics for Golfers......Page 2526
10. Common Sense and the Universe......Page 2530
INDEX......Page 2541