Mathematics and Plausible Reasoning, both volumes combined

Volume I: Induction and Analogy in Mathematics

Preface v 
Hints to the Reader xi 
@=18
I. Induction 3 
   1. Experience and belief  3
   2. Suggestive contact  4
   3. Supporting contacts 5
   4. The inductive attitude  7
   Examples and Comments on I 8
 
 
II. Generalization, Specialization, Analogy 12  
   1. Generalization, specialization, analogy, and induction  12
   2. Generalization 12
   3. Specialization 13
   4. Analogy 13
   5. Generalization, specialization, and analogy 15
   6. Discovery by analogy  17
   7. Analogy and induction  21
   Examples and Comments on II 22
 
 
III. Induction in Solid Geometry 35  
   1. Polyhedra 35
   2. First supporting contacts 38
   3. More supporting contacts 38
   4. A severe test 39
   5. Verifications and verification  40
   6. A very different case 41
   7. Analogy 42
   8. The partition of space  43
   9. Modifying the problem 44
   10. Generalization, specialization, analogy 44
   11. An analogous problem 45
   12. An array of analogous problems 46
   13. Many problems may be easier than just one  47
   14. A conjecture 47
   15. Prediction and verification 49
   16. Again and better 49
   17. Induction suggests deduction, the particular case suggests the general proof 50
   18. More conjectures  51
   Examples and Comments on III 52
 
 
IV. Induction in the Theory of Numbers 59 
   1. Right triangles in integers 59
   2. Sums of squares 62
   3. On the sum of four odd squares 63
   4. Examining an example 64
   5. Tabulating the observations 65
   6. What is the rule? 65
   7. On the nature of inductive discovery 68
   8. On the nature of inductive evidence  68
   Examples and Comments on IV 70
 
 
V. Miscellaneous Examples of Induction 76 
   1. Expansions 76
   2. Approximations 77
   3. Limits 79
   4. Trying to disprove it 80
   5. Trying to prove it 81
   6. The role of the inductive phase  83
   Examples and Con1ments on V 84
 
 
VI. A More General Statement 90 
   1. Euler  90
   2. Euler\'s memoir 90
   3. Transition to a more general viewpoint  99
   4. Schematic outline of Euler\'s memoir  99
   Examples and Comments on VI 100
 
 
VII. Mathematical Induction 108 
   1. The inductive phase 108
   2. The demonstrative phase 110
   3. Examining transitions 110
   4. The technique of mathematical induction  111
   Examples and Comments on VII 116
 
 
VIII.  Maxima and Minima 121 
   1. Patterns 121
   2. Example 122
   3. The pattern of the tangent level line 123
   4. Examples 126
   5. The pattern of partial variation 128
   6. The theorem of the arithmetic and geometric means and its first consequences  130
   Examples and Comments on VIII 131
 
 
IX. Physical Mathematics 142 
   1. Optical interpretation 142
   2. Mechanical interpretation 146
   3. Reinterpretation 149
   4. Jean Bernoulli\'s discovery of the brachistochrone 152
   5. Archimedes\' discovery of the integral calculus  155
   Examples and Comments on IX 158
 
 
X. The Isoperimetric Problem 168 
   1. Descartes\' inductive reasons 168
   2. Latent reasons 169
   3. Physical reasons 170
   4. Lord Rayleigh\'s inductive reasons 170
   5. Deriving consequences 171
   6. Verifying consequences 174
   7  Very close 177
   8. Three forms of the Isoperimetric Theorem 179
   9. Applications and questions  180
   Examples and Comments on X 181
 
 
XI. Further Kinds of Plausible Reasons 190 
   1. Conjectures and conjectures 190
   2. Judging by a related case  190
   3. Judging by the general case 192
   4. Preferring the simpler conjecture 194
   5. Background 196
   6. Inexhaustible 198
   7. Usual heuristic assumptions  199
   Examples and Comments on XI 200
 
Final Remark 210 
Solutions to Problems 213 
   Chapter I  213
   Chapter II 214
   Chapter III 222
   Chapter IV 227
   Chapter V 232
   Chapter VI 236
   Chapter VII 240
   Chapter VIII 244
   Chapter IX 258
   Chapter X 266
   Chapter XI 273
Bibliography 279 



Volume II: Patterns of Plausible Inference

Contents ix 
Preface v 
Hints to the Reader vii 
@=12
XII. Some Conspicuous Patterns 3 
   1. Verification of a consequence 3
   2. Successive verification of several consequences 5
   3. Verification of an improbable consequence 7
   4. Inference from analogy 9
   5. Deepening the analogy  10
   6. Shaded analogical inference  12
   Examples and Comments on  XII 12
 
 
XIII. Further Patterns and First Links  18
   1. Examining a consequence 18
   2. Examining a possible ground  19
   3. Examining a conflicting conjecture 20
   4. Logical term  20
   5. Logical links between patterns of plausible inference 23
   6. Shaded inference 23
   7. A table 25
   8. Combination of simple pattern  26
   9. On inference from analogy 27
   10. Qualified inference 28
   11. On successive verifications 30
   12. The influence of rival conjecture  31
   13. On judicial proof  32
   Examples and Comments on  XIII 37
 
 
XIV. Chance, the Ever-present Rival Conjecture  55
   1. Random mass phenomena 55
   2. The concept of probability  57
   3. Using the bag and the balls 60
   4. The calculus of probability. Statistical hypotheses 64
   5. Straightforward prediction of frequencies  65
   6. Explanation of phenomena 70
   7. Judging statistical hypothese  74
   8. Choosing between statistical hypotheses 78
   9. Judging nonstatistical conjectures 84
   10. Judging mathematical conjectures  95
   Examples and Comments on  XIV 98
 
 
XV. The Calculus of Probability and the Logic of Plausible Reasoning 109  
   1. Rules of plausible reasoning 109
   2. An aspect of demonstrative reasoning 111
   3. A corresponding aspect of plausible reasoning  113
   4. An aspect of the calculus of probability. Difficulties 116
   5. An aspect of the calculus of probability. An attempt 118
   6. Examining a consequence 119
   7. Examining a possible ground 122
   8. Examining a conflicting conjecture 123
   9. Examining several consequences in succession 124
   10. On circumstantial evidence  126
   Examples and Comments on  XV 128
 

XVI. Plausible Reasoning in Invention and Instruction 142 
   1. Object of the present chapter 142
   2. The story of a little discovery 142
   3. The process of solution 145
   4. Deus ex machina 146
   5. Heuristic justification 147
   6. The story of another discovery 148
   7. Some typical indications 152
   8. Induction in invention 153
   9. A few words to the teacher 157
   Examples and Comments on Chapter XVI, 1-13 160

 
Solutions to Problems 171  
   Chapter XII 171
   Chapter XIII 174
   Chapter XIV 178
   Chapter XV 185
   Chapter XVI 186
Bibliography 189