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[ G.Polya ] Modern heuristic endeavours to understand the process of solving problems, especially the mental operations typically useful in this process. It has various sources of information none of which should be neglected. A serious study of heuristic should take into account both the logical and psychological background, it should not neglect what should older writers Pappus , Descartes , Leibnitz , and Bolzano have to say about the subject, but it should least neglect unbiased experience. Experience in solving problems and experience in watching other people solving problems must be the basis on which heuristic is built. In this study, we should not neglect any sort of problem, and should find out common features in the way of handling all sorts of problems; we should aim at general features, independent of the subject matter of the problem. The study of heuristic has "practical" aims; a better understanding of the mental operations typically useful in solving problems could exert some good influence of teaching, especially on the teaching of mathematics. The present book is a first attempt toward the realisation of this program. We are going to discuss how the various articles of this dictionary fit into the program. 1. Our list is, in fact, a list of mental operations typically useful in solving problems; the questions and suggestions listed hint at such operations. Some of these operations are described again in the Second Part, and some of them are more thoroughly discussed and illustrate in the he First Part. For additional information about particular questions and suggestions of the list, the reader should refer to those fifteen articles of the Dictionary whose titles are the first sentences of the fifteen paragraphs of the list: WHAT IS THE UNKNOWN? IS IT POSSIBLE TO SATISFY THE CONDITION ? IT IS POSSIBLE TO SATISFY THE CONDITION? DRAW A FIGURE ... CAN YOU USE THE RESULT? The reader, wishing information about a particular item of the list, should look at the first word of the paragraph in which the item is contained and then look up the article in the Dictionary that has those first words as title. For instance, the suggestion " Go back to definitions" is contained in the paragraph of the list whose first sentence is : COULD YOU RESTATE THE PROBLEM? Under this title, the reader finds a cross-reference to DEFINITION in which article the suggestion in question is explained and illustrated. 2. The process of solving problems is a complex process that has several different aspects. The twelve principal articles of this Dictionary study certain of these aspects at some length; we are going to mention their titles in what follows. When we are working intensively, we feel keenly the progress of work; we are elated when our progress is rapid, we are depressed when it is slow. What is essential to PROGRESS AND ACHIEVEMENT in solving problems? The article discussing this question is often quoted in other parts of the Dictionary and should be read fairly early. Trying to solve a problem, we consider different aspects of it in turn, we roll it over and over incessantly in our mind; VARIATION OF THE PROBLEM is essential to our work. We may vary the problem by DECOMPOSING AND RECOMBINING its elements, or by going back to the DEFINITION of certain of its terms, or we may use the great resources of GENERALISATION , SPECIALISATION and ANALOGY . Variation of the problem may lead us to AUXILIARY ELEMENTS or to the discovery of a more accessible AUXILIARY PROBLEM . We have to distinguish carefully between two kinds of problems, PROBLEMS TO FIND, PROBLEMS TO PROVE. Our list is specially adapted to "problems to find." We have to revise it and change some of its question and suggestions in order to apply it also to "problems to prove." In all sorts of problems, but especially in mathematical problems which are not too simple, suitable NOTATION and geometrical FIGURES are a great and often indispensable help. 3. The process of solving problems has many aspects but some of them are not considered at all in this book and others only very briefly. It is justified, I think to exclude from a first short exposition points which could appear too subtle, or too technical, or too controversial. Provisional, merely plausible HEURISTIC REASONING is important in discovering the solution, but you should not teach it for a proof; you must guess, but also EXAMINE YOUR GUESS . The nature of heuristic arguments is discussed in SIGNS OF PROGRESS but the discussion could go further. The consideration of certain logical patterns is important in our subject but it appeared advisable not to introduce any technical article. There are only two articles predominantly devoted to psychological aspect, on DETERMINATION,HOPE,SUCCESS , and on SUBCONSCIOUS WORK . There is incidental remark on animal psychology; see WORKING BACKWARDS It is emphasised that all sorts of problems, especially PRACTICAL PROBLEMS , and even PUZZLES , are within the scope of heuristic. It is also emphasised that infallible RULES OF DISCOVERY are beyond the scope of serious re-search. Heuristic discusses human behaviour in the face of problems; this has been fashion, presumably, since the beginning of human society, and the quintessence of such ancient discussions seems to be preserved in the WISDOM OF PROVERBS . 4. A few articles on particular questions are included and some articles on more general aspects an expanded because they could be, or parts of them could be, of special interest to students or teachers. There are articles discussing methodical questions often important in elementary mathematics, as PAPPUS , WORKING BACKWARDS (already quoted under 3), REDUCTIO AD ABSURDUM AND INDIRECT PROOF INDUCTION AND MATHEMATICAL INDUCTION , SETTING UP EQUATIONS , TEST BY DIMENSION , and WHY PROOFS A few articles address themselves more particularly to teachers as ROUTINE PROBLEMS and DIAGNOSIS , and others to students somewhat more ambitious than the average, as THE INTELLIGENT PROBLEM SOLVER , THE INTELLIGENT READER , and THE FUTURE MATHEMATICIAN . It may be mentioned here that the dialogues between the teacher and his students, given in sections 8,10,18,19,20 and in various articles of the Dictionary ay serve as models not only to the teacher who tries to guide his class but also to the problem-solver who works by himself. To describe thinking as "mental discourse," as a sort of conversation of the thinker with himself, is not in appropriate. The dialogues in question show the progress of the solution; the problem-solver, talking with himself may progress along a similar line. 5. We are not going to exhaust the remaining titles; just a few groups will be mentioned. Some articles contain remarks on the history of our subject, on DESCARTES , LEIBNITZ , BOLZANO , on HEURISTIC , on TERMS,OLD AND NEW , and on PAPPUS (this last one has been quoted already under 4). A few articles explain technical terms: CONDITION , COROLLARY , LEMMA Some articles contain only cross-references (they are marked with daggers [+] in the Table of Contents). 6. Heuristic aims at generality, at the study of procedures which are independent of the subject-matter and apply to all sorts of problems. The present exposition, however quotes almost exclusively elementary mathematical problems as examples. It should not be overlooked that this is a restriction but it is hoped that this restriction does not impair seriously the trend of our study. In fact, elementary mathematical problems present all the desirable variety, and the study of their solution is particularly accessible and interesting. Moreover, non-mathematical problems although seldom quoted as examples are never completely forgotten. More advanced mathematical problems are never directly quoted but constitute the real background of the present exposition. The expert mathematician who has some interest for this sort of study can easily add examples from his own experience to elucidate the point illustrated by elementary examples here. 7. The write of this book wishes to acknowledge indebtedness and express his gratitude to a few modern authors, not quoted in the article on HEURISTIC they are the physicist and the philosopher Ernst mach, the mathematician Jacques Hadamard, the psychologists William James and Wolfgang Kohler''. He wishes also to quote the psychologist K. Duncker and the mathematician F. Krauss whose work (published after his own research was fairly advanced, and partly published) shows certain parallel remarks.