I should add that for many of the stories told the final form resulted from a sort of informal psychological experiment. I discussed the subject with several different classes, interrupting my exposition frequently with such questions as: "Well, what would you do in such a situation? When a biologist attempts to investigate some general problem, let us say, of genetics, it is very important that he should choose some particular species of plants or animals that lends itself well to an experimental study of his problem.
When a chemist intends to investigate some general problem about, let us say, the velocity of chemical reactions, it is very important that he should choose some particular substances on which experiments relevant to his problem can be conveniently made. The choice of appropriate experimental material is of great importance in the inductive investigation of any problem.
It seems to me that mathematics is, in several respects, the most appropriate experimental material for the study of inductive reasoning. This study involves psychological experiments of a sort: you have to experience how your confidence in a conjecture is swayed by various kinds of evidence. Thanks to their inherent simplicity and clarity, mathematical subjects lend themselves to this sort of psychological experiment much better than subjects in any other field. On the following pages the reader may find ample opportunity to convince himself of this.
It is more philosophical, I think, to consider the more general idea of plausible reasoning instead of the particular case of inductive reasoning. It seems to me that the examples collected in this book lead up to a definite and fairly satisfactory aspect of plausible reasoning. Yet I do not wish to force my views upon the reader. In fact, I do not even state them in Vol.
I I want the examples to speak for themselves. The first four chapters of Vol. II, however, are devoted to a more explicit general discussion of plausible reasoning. There I state formally the patterns of plausible inference suggested by the foregoing examples, try to systematize these patterns, and survey some of their relations to each other and to the idea of probability.
I do not know whether the contents of these four chapters deserve to be called philosophy. If this is philosophy, it is certainly a pretty low-brow kind of philosophy, more concerned with understanding concrete examples and the concrete behavior of people than with expounding generalities.
I know still less, of course, how the final judgment on my views will turn out. Yet I feel pretty confident that my examples can be useful to any reasonably unprejudiced student of induction or of plausible reasoning, who wishes to form his views in close touch with the observable facts. The present work is a continuation of my earlier book How to Solve It.
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The reader interested in the subject should read both, but the order does not matter much. The present text is so arranged that it can be read independently of the former work. In fact, there are only few direct references in the present book to the former and they can be disregarded in a first reading.
Yet there are indirect references to the former book on almost every page, and in almost every sentence on some pages.
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In fact, the present work provides numerous exercises and some more advanced illustrations to the former which, in view of its size and its elementary character, had no space for them. The present book is also related to a collection of problems in Analysis by G. Szeg6 and the author see Bibliography. The problems in that collection are carefully arranged in series so that they support each other mutually, provide cues to each other, cover a certain subject-matter jointly, and give the reader an opportunity to practice various moves important in problem-solving.
In the treatment of problems the present book follows the method of presentation initiated by that former work, and this link is not unimportant. Two chapters in Vol. II of the present book deal with the theory of probability. The first of these chapters is somewhat related to an elementary exposition of the calculus of probability written by the author several years ago see the Bibliography. The underlying views on probability and the starting points are the same, but otherwise there is little contact.
Some of the views offered in this book have been expressed before in my papers quoted in the Bibliography.
Extensive passages of papers no. Acknowledgment and my best thanks are due to the editors of the American Mathematical Monthly, Etudes de Philosophie des Sciences en Hommage a Ferdinand Gonseth, and Proceedings of the International Congress of Mathematicians , who kindly gave permission to reprint these passages.
Most parts of this book have been presented in my lectures, some parts several times. In some parts and in some respects, I preserved the tone of oral presentation.
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I do not think that such a tone is advisable in printed presentation of mathematics in general, but in the present case it may be appropriate, or at least excusable. The efficient use of plausible reasoning plays an essential role in problem solving. The present book tries to illustrate this role by many examples, but there remain other aspects of problem-solving that need similar illustration. Many points touched upon here need further work. My views on plausible reasoning should be confronted with the views of other authors, the historical examples should be more thoroughly explored, the views on invention and teaching should be investigated as far as possible with the methods of experimental psychology, and so on.gekarevi.cf
INDUCTION AND ANALOGY IN MATHEMATICS
Several such tasks remain, but some of them may be thankless. The present book is not a textbook. Yet I hope that in time it will influence the usual presentation of the textbooks and the choice of their problems. The task of rewriting the textbooks of the more usual subjects along these lines need not be thankless. Up Contact Front page Contents Books.
What is what? Strictly speaking, all our knowledge outside mathematics and demonstrative logic which is, in fact, a branch of mathematics consists of conjectures. There are, of course, conjectures and conjectures. There are highly respectable and reliable conjectures as those expressed in certain general laws of physical science.
Mathematics and Plausible Reasoning, Volume 1 – Induction and Analogy in Mathematics
There are other conjectures, neither reliable nor respectable, some of which may make you angry when you read them in a newspaper. And in between there are all sorts of conjectures, hunches, and guesses. Another point concerning the two kinds of reasoning deserves our attention. Everyone knows that mathematics offers an excellent opportunity to learn demonstrative reasoning, but I contend also that there is no subject in the usual curricula of the schools that affords a comparable opportunity to learn plausible reasoning.
I address myself to all interested students of mathematics of all grades and I say: Certainly, let us learn proving, but also let us learn guessing. At any rate, I if there is such a method, I do not know it, and quite certainly I do not pretend to offer it on the following pages. The efficient use of plausible reasoning is a practical skill and it is learned, as any other practical skill, by imitation and practice. I shall try to do my best for the reader who is anxious to learn plausible reasoning, but what I can offer are only examples for imitation and opportunity for practice.
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In short, I tried to use all my experience in research and teaching to give an appropriate opportunity to the reader for intelligent imitation and for doing things by himself. The examples of plausible reasoning collected in this book may be put to another use: they may throw some light upon a much agitated philosophical problem: the problem of induction. The crucial question is: Are there rules for induction? Some philosophers say Yes, most scientists think No.
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