INDUCTION AND ANALOGY IN MATHEMATICS
By G. Polya
This book has various aims, closely connected with each other. In the first place, this book intends to serve students and teachers of mathematics in an important but usually neglected way. Yet in a sense the book is also a philosophical essay. It is also a continuation and requires a continuation. I shall touch upon these points, one after the other.
- 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. 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.
We secure our mathematical knowledge by demonstrative reasoning, but we support our conjectures by plausible reasoning. A mathematical proof is demonstrative reasoning, but the inductive evidence of the physicist, the circumstantial evidence of the lawyer, the documentary evidence of the historian, and the statistical evidence of the economist belong to plausible reasoning.
The difference between the two kinds of reasoning is great and manifold. Demonstrative reasoning is safe, beyond controversy, and final. Plausible reasoning is hazardous, controversial, and provisional. Demonstrative reasoning penetrates the sciences just as far as mathematics does, but it is in itself (as mathematics is in itself) incapable of yielding essentially new knowledge about the world around us. Anything new that we learn about the world involves plausible reasoning, which is the only kind of reasoning for which we care in everyday affairs. Demonstrative reasoning has rigid standards, codified and clarified by logic (formal or demonstrative logic), which is the theory of demonstrative reasoning. The standards of plausible reasoning are fluid, and there is no theory of such reasoning that could be compared to demonstrative logic in clarity or would command comparable consensus.
- 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.
This sounds a little paradoxical and I must emphasize a few points to avoid possible misunderstandings.
Mathematics is regarded as a demonstrative science. Yet this is only one of its aspects. Finished mathematics presented in a finished form appears as purely demonstrative, consisting of proofs only. Yet mathematics in the making resembles any other human knowledge in the making. You have to guess a mathematical theorem before you prove it; you have to guess the idea of the proof before you carry through the details. You have to combine observations and follow analogies; you have to try and try again. The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference.
There are two kinds of reasoning, as we said: demonstrative reasoning and plausible reasoning. Let me observe that they do not contradict each other; on the contrary, they complete each other. In strict reasoning the principal thing is to distinguish a proof from a guess, a valid demonstration from an invalid attempt. In plausible reasoning the principal thing is to distinguish a guess from a guess, a more reasonable guess from a less reasonable guess. If you direct your attention to both distinctions, both may become clearer.
A serious student of mathematics, intending to make it his life's work, must learn demonstrative reasoning; it is his profession and the distinctive mark of his science. Yet for real success he must also learn plausible reasoning; this is the kind of reasoning on which his creative work will depend. The general or amateur student should also get a taste of demonstrative reasoning: he may have little opportunity to use it directly, but he should acquire a standard with which he can compare alleged evidence of all sorts aimed at him in modern life. But in all his endeavors he will need plausible reasoning. At any rate, an ambitious student of mathematics, whatever his further interests may be, should try to learn both kinds of reasoning, demonstrative and plausible.
- 1 do not believe that there is a foolproof method to 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.
In what follows, I shall often discuss mathematical discoveries, great and small. I cannot tell the true story how the discovery did happen, because nobody really knows that. Yet I shall try to make up a likely story how the discovery could have happened. I shall try to emphasize the motives underlying the discovery, the plausible inferences that led to it, in short, everything that deserves imitation. Of course, I shall try to impress the reader; this is my duty as teacher and author. Yet I shall be perfectly honest with the reader in the point that really matters: I shall try to impress him only with things which seem genuine and helpful to me.
Each chapter will be followed by examples and comments. The comments deal with points too technical or too subtle for the text of the chapter, or with points somewhat aside of the main line of argument. Some of the exercises give an opportunity to the reader to reconsider details only sketched in the text. Yet the majority of the exercises give an opportunity to the reader to draw plausible conclusions of his own. Before attacking a more difficult problem proposed at the end of a chapter, the reader should carefully read the relevant parts of the chapter and should also glance at the neighboring problems; one or the other may contain a clue. In order to provide (or hide) such clues with the greatest benefit to the instruction of the reader, much care has been expended not only on the contents and the form of the proposed problems, but also on their disposition. In fact, much more time and care went into the arrangement of these problems than an outsider could imagine or would think necessary.
In order to reach a wide circle of readers I tried to illustrate each important point by an example as elementary as possible. Yet in several cases I was obliged to take a not too elementary example to support the point impressively enough. In fact, I felt that I should present also examples of historic interest, examples of real mathematical beauty, and examples illustrating the parallelism of the procedures in other sciences, or in everyday life.
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?" Several passages incorporated in the following text have been suggested by the answers of my students, or my original version has been modified in some other manner by the reaction of my audience.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. In order to be discussed profitably, the question should be put differently. It should be treated differently, too, with less reliance on traditional verbalisms, or on newfangled formalisms, but in closer touch with the practice of scientists. Now, observe that inductive reasoning is a particular case of plausible reasoning. Observe also (what modern writers almost forgot, but some older writers, such as Euler and Laplace, clearly perceived) that the role of inductive evidence in mathematical investigation is similar to its role in physical research. Then you may notice the possibility of obtaining some information about inductive reasoning by observing and comparing examples of plausible reasoning in mathematical matters. And so the door opens to investigating induction inductively.
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.
- This work on Mathematics and Plausible Reasoning, which I have always regarded as a unit, falls naturally into two parts: Induction and Analogy in Mathematics (Vol. 1), and Patterns of Plausible Inference (Vol. II). For the convenience of the student they have been issued as separate volumes. Vol. I is entirely independent of Vol. II, and I think many students will want to go through it carefully before reading Vol.II. It has more of the mathematical "meat" of the work, and it supplies "data" for the inductive investigation of induction in Vol. II. Some readers, who should be fairly sophisticated and experienced in mathematics, will want to go directly to Vol. II, and for these it will be a convenience to have it separately. For ease of reference the chapter numbering is continuous through both volumes. I have not provided an index, since an index would tend to render the terminology more rigid than it is desirable in this kind of work. I believe the table of contents will provide a satisfactory guide to the book.
The present work is a continuation of my earlier book How to Solve It. 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. 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. 4, 6, 8,9,andl0havebeenincorporatedinthefollowingtext. 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 1950, 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. 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 last chapter of Vol. II of the present book, dealing with Invention and Teaching, links the contents more explicitly to the former work of the author and points to a possible sequel.
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. 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.
Copyright © 1996-2018 Alexander Bogomolny