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