Polya Vol 1 Induction & Analogy 11001

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CONTENTS Preface v Hints to the reader X1 Chapter I. Induction 3 1. Experience and belief. 2. Suggestive contacts. 3. Supporting contacts. 4. The inductive attitude Examples and Comments on Chapter I, 1-14. [12. Yes and No. 13. Experience and behavior. 14. The logician, the mathematician, the physicist, and the engineer.] Chapter II. Generalization, Specialization, Analogy 12 1. Generalization, specialization, analogy, and induction. 2. Generalization. 3. Specialization. 4. Analogy. 5. Generaliz
  CONTENTSPreface vHints to the reader X1Chapter I. Induction 31. Experience and belief. 2. Suggestive contacts. 3. Support-ing contacts. 4. The inductive attitudeExamples and Comments on Chapter I, 1-14. [12. Yes and No. 13. Experience and behavior. 14. The logician, themathematician, the physicist, and the engineer.]Chapter II. Generalization, Specialization, Analogy 121. Generalization, specialization, analogy, and induction.2. Generalization. 3. Specialization. 4. Analogy. 5. Generali-zation, specialization, and analogy. 6. Discovery by analogy.7. Analogy and inductionExamples and Comments on Chapter II, 1-46; [First Part,1-20; Second Part, 21-46]. [1. The right generalization. 5. Anextreme special case. 7. A leading special case. 10. A represen-tative special case. 1 1. An analogous case. 18. Great analogies.19. Clarified analogies. 20. Quotations. 21. The conjecture E.44. An objection and a first approach to a proof. 45. A secondapproach to a proof. 46. Dangers of analogy.]Chapter III. Induction in Solid Geometry 351. Polyhedra. 2. First supporting contacts. 3. More support-ing 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. 1 1 . 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. Againand better. 17. Induction suggests deduction, the particular casesuggests the general proof. 18. More conjecturesExamples and Comments on Chapter III, 1-41. [21.Induction: adaptation of the mind, adaptation of the language.31. Descartes' work on polyhedra. 36. Supplementary solidangles, supplementary spherical polygons.]…..  INDUCTIONIt will seem not a little paradoxical to ascribe a greatimportance to observations even in that part of the mathematicalsciences which is usually called Pure Mathematics, since thecurrent opinion is that observations are restricted to physicalobjects that make impression on the senses. As we must refer thenumbers to the pure intellect alone, we can hardly understand howobservations and quasi-experiments can be of use in investigatingthe nature of the numbers. Yet, in fact, as I shall show here withvery good reasons, the properties of the numbers known todayhave been mostly discovered by observation, and discovered longbefore their truth has been confirmed by rigid demonstrations.There are even many properties of the numbers with which we arewell acquainted, but which we are not yet able to prove; onlyobservations have led us to their knowledge. Hence we see that inthe theory of numbers, which is still very imperfect, we can placeour highest hopes in observations; they will lead us continually tonew properties which we shall endeavor to prove afterwards. Thekind of knowledge which is supported only by observations andis not yet proved must be carefully distinguished from the truth; it isgained by induction, as we usually say. Yet we have seen casesin which mere induction led to error. Therefore, we should takegreat care not to accept as true such properties of the numberswhich we have discovered by observation and which aresupported by induction alone. Indeed, we should use such a  discovery as an opportunity to investigate more exactly theproperties discovered and to prove or disprove them; in both caseswe may learn something useful. — EULER 1i. Experience and belief. Experience modifies human beliefs. Welearn from experience or, rather, we ought to learn fromexperience. To make the best possible use of experience is one of the great human tasks and to work for this task is the proper vocation of scientists.A scientist deserving this name endeavors to extract the mostcorrectbelief from a given experience and to gather the most appropriateexperiencein order to establish the correct belief regarding a given question.The1 Euler, Opera Omnia, ser. 1, vol. 2, p. 459, Specimen de usuobservationum in mathesipura.3
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