A guide to the practical art of plausible reasoning, this book has relevance in every field of intellectual activity. Professor Polya, a world-famous mathematician from Stanford University, uses mathematics to show how hunches and guesses play an important part in even the most rigorously deductive science. He explains how solutions to problems can be guessed at; good guessing is often more important than rigorous deduction in finding correct solutions. Vol. I, on Induction and Analogy in Mathematics, covers a wide variety of mathematical problems, revealing the trains of thought that lead to solutions, pointing out false bypaths, discussing techniques of searching for proofs. Problems and examples challenge curiosity, judgment, and power of invention.
Author(s): Polya G.
Publisher: Princeton
Year: 1954
Language: English
Pages: 296
Cover
Title Page
Preface
Hints to the Reader
Table of Contents
Chapter I Induction
1. Experience and Belief
2. Suggestive Contacs
3. Supporting Contacts
4. The Inductive Attitude
Chapter II Generalization Specialization, Analogy
1. Generalization, Specialization, Analogy, and Induction
2. Generalization
3. Specialization
4. Analogy
5. Generalizations, Specializations, and Analogy
6. Discovery by Analogy
7. Analogy and Induction
Examples and Comments on Chapter II
First Part
Second Part
Chapter III Induction in Solid Geometry
1. Polyhedra
2. First Supporting Contacts
3. More Supporting Contacts
4. A Severe Test
5. Verifications and Verifications
6. A Very different case
7. Analogy
8. The Partition of Space
9. Modifying the Problem
10. Generalization, Specialization and Analogy
11. An Analogous Problem
12. An Array of Analogous Problems
13. Many Problems may be easier than just one
14. A Conjecture
15. Prediction and Verification
16. Again and Better
17. Induction Suggests deduction; the particular case suggestions the general proof
18. More Conjectures
Examples and Comments on Chapter III
Chapter IV Induction in the Theory of Numbers
1. Right triangles in integers
2. Sum of Squares
3. On the Sum of four odd squares
4. Examining the Example
5. Tabulating the Observations
6. What is the rule
7. On the nature of inductive discovery
8. On the nature of inductive evidence
Examples and Comments on Chapter IV
Chapter V Miscellneous Examples of Induction
1. Expansions
2. Approximations
3. Limits
4. Trying to Disprove it
5. Trying to Prove it
6. The role of inductive phase
Examples and Comments on Chapter V
Chapter VI A More General Statement
1. Euler
2. Eulers memoir
3. Transition to a more general viewpoint
4. Schematic outline of Eulers Memoir
Examples and Comments on Chapter VI
Chapter VII Mathematical Induction
1. The Inductive Phase
2. The Demonstrative Phase
3. Examining Transitions
4. The Technique of Mathematical Induction
Examples and Comments on Chapter VII
Chapter VIII Maxima and Minima
1. Patterns
2. Example
3. The Pattern of the tangent level line
4. Examples
5. The Pattern of Partial Variation
6. The Theorem of the Arithmetic and Geometric Means and its first Consequences
Examples and Comments on Chapter VIII
First Part
Second Part
Chapter IX Physical Mathematics
1. Optical Interpretation
2. Mechanicl Interpretation
3. Reinterpretation
4. Jean Bernoullis discovery of the Brachistochrone
5. Archimedes Discovery of Integral Calculus
Examples and Comments on Chapter IX
Chapter X The Isoperimetric Problem
1. Descartes inductive reasons
2. Latent Reasons
3. Physical Reasons
4. Lord Rayleighs Inductive reasons
5. Deriving Consequences
6. Verifying Consequences
7. Very Close
8. Three forms of the Isoperimetric Theorem
9. Applications and Questions
Examples and Comments on Chapter X
First Part
Second Part
Chapter XI Further Kinds of Plausible Reasons
1. Conjecture and Conjectures
2. Judging by a Related Case
3. Judging by the General Case
4. Prefering the Simpler Case
5. Background
6. Inexhaustible
7. Usual Heuristic Assumptions
Examples and Comments on Chapter XI
Final Remark
Solutions
Chapter I
Chapter II
Chapter III
Chapter IV
Chapter V
Chapter VI
Chapter VII
Chapter VIII
Chapter IX
Chapter X
Chapter XI
Bibliography
