Mathematics and Plausible Reasoning, Volume 1

Name: Mathematics and Plausible Reasoning, Volume 1 | Induction and Analogy in Mathematics
Brand: Princeton University Press
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Induction and Analogy in Mathematics

George Polya(Autor*in)

Princeton University Press

1. Auflage

Erschienen am 1. September 2020

296 Seiten

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Inhalt

Cover Page
Half-title page
Title page
Copyright page
Preface
Hints to the Reader
Contents
Chapter I: Induction
1. Experience and belief
2. Suggestive contacts
3. Supporting contacts
4. The inductive attitude
Examples and Comments on Chapter I.
Chapter II: Generalization, Specialization, Analogy
1. Generalization, Specialization, Analogy, and Induction
2. Generalization
3. Specialization
4. Analogy
5. Generalization, Specialization, and Analogy
6. Discovery by analogy
7. Analogy and induction
Examples and Comments on Chapter II
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, 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 suggests 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. Sums of squares
3. On the sum of four odd squares
4. Examining an 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: Miscellaneous Examples of Induction
1. Expansions
2. Approximations
3. Limits
4. Trying to disprove it
5. Trying to prove it
6. The role of the inductive phase
Examples and Comments on Chapter V
Chapter VI: A More General Statement
1. Euler
2. Euler's memoir
3. Transition to a more general viewpoint
4. Schematic outline of Euler's memoir
Examples and Comments
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. Examples
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
Chapter IX: Physical Mathematics
1. Optical interpretation
2. Mechanical interpretation
3. Reinterpretation
4. Jean Bernoulli's discovery of the Brachistochrone
5. Archimedes' discovery of the 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 Rayleigh's 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
Chapter XI: Further Kinds of Plausible Reasons
1. Conjectures and conjectures
2. Judging by a related case
3. Judging by the general case
4. Preferring the simpler conjecture
5. Background
6. Inexhaustible
7. Usual heuristic assumptions
Examples and Comments on Chapter XI
Final Remark
Solutions to Problems
Bibliography

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