出版社: Princeton University Press
副标题: A New Aspect of Mathematical Method
出版年: 20040405
页数: 288
定价: USD 18.95
装帧: Paperback
丛书: Princeton Science Library
ISBN: 9780691119663
内容简介 · · · · · ·
A perennial bestseller by eminent mathematician G. Polya, "How to Solve It" will show anyone in any field how to think straight. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out  from building a bridge to winning a game of anagrams. Generatio...
A perennial bestseller by eminent mathematician G. Polya, "How to Solve It" will show anyone in any field how to think straight. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out  from building a bridge to winning a game of anagrams. Generations of readers have relished Polya's deft  indeed, brilliant  instructions on stripping away irrelevancies and going straight to the heart of the problem. In this bestselling classic, George Polya revealed how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out  from building a bridge to winning a game of anagrams.Generations of readers have relished Polya's deft instructions on stripping away irrelevancies and going straight to the heart of a problem. "How to Solve It" popularized heuristics, the art and science of discovery and invention. It has been in print continuously since 1945 and has been translated into twentythree different languages. Polya was one of the most influential mathematicians of the twentieth century. He made important contributions to a great variety of mathematical research: from complex analysis to mathematical physics, number theory, probability, geometry, astronomy, and combinatorics. He was also an extraordinary teacher  he taught until he was ninety  and maintained a strong interest in pedagogical matters throughout his long career.In addition to "How to Solve It", he published a twovolume work on the topic of problem solving, "Mathematics of Plausible Reasoning", also with Princeton. Polya is one of the most frequently quoted mathematicians, and the following statements from "How to Solve It" make clear why: "My method to overcome a difficulty is to go around it." "Geometry is the science of correct reasoning on incorrect figures." "In order to solve this differential equation you look at it till a solution occurs to you."
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上高中时在甘肃省图书馆里偶然看到了这本书，当时也只是想通过这本书能提高自己的考试的分数，读的时候也是囫囵吞枣的看了一遍，发现也没有提高我的考试成绩……只是觉得这本书很有意思！ 近两年开始关注思想问题，发现数学是很奇妙的，人类的很多活动实际是基于数学的，数学实... (展开)天才教师 Polya
A Modern Discussion about Heuristic
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simoncos (给岁月以生命。)
First, we have to understand the problem; we have to see clearly what is required. Second, we have to see how the various items are connected, how the unknown is linked to the data, in order to obtain the idea of the solution, to make a plan. Third, we carry out our plan. Fourth, we look back at the completed solution, we review and discuss it. loc. 712720 The worst may happen if the student e...20190527 19:45
First, we have to understand the problem; we have to see clearly what is required. Second, we have to see how the various items are connected, how the unknown is linked to the data, in order to obtain the idea of the solution, to make a plan. Third, we carry out our plan. Fourth, we look back at the completed solution, we review and discuss it.
loc. 712720
The worst may happen if the student embarks upon computations or constructions without having understood the problem.
loc. 725726
The teacher can make the problem interesting by making it concrete.
loc. 771772
Good ideas are based on past experience and formerly acquired knowledge.
loc. 813814
The materials necessary for solving a mathematical problem are certain relevant items of our formerly acquired mathematical knowledge, as formerly solved problems, or formerly proved theorems. Thus, it is often appropriate to start the work with the question: Do you know a related problem?
loc. 816819
The difficulty is that there are usually too many problems which are somewhat related to our present problem, that is, have some point in common with it. How can we choose the one, or the few, which are really useful? There is a suggestion that puts our finger on an essential common point: Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.
loc. 820825
The foregoing questions, well understood and seriously considered, very often help to start the right train of ideas; but they cannot help always, they cannot work magic. If they do not work, we must look around for some other appropriate point of contact, and explore the various aspects of our problem; we have to vary, to transform, to modify the problem. Could you restate the problem? Some of the questions of our list hint specific means to vary the problem, as generalization, specialization, use of analogy, dropping a part of the condition, and so on; the details are important but we cannot go into them now. Variation of the problem may lead to some appropriate auxiliary problem: If you cannot solve the proposed problem try to solve first some related problem.
loc. 829838
11. Carrying out the plan. To devise a plan, to conceive the idea of the solution is not easy. It takes so much to succeed; formerly acquired knowledge, good mental habits, concentration upon the purpose, and one more thing: good luck. To carry out the plan is much easier; what we need is mainly patience.
loc. 904908
We may convince ourselves of the correctness of a step in our reasoning either “intuitively” or “formally.”
loc. 918919
In certain cases, the teacher may emphasize the difference between “seeing” and “proving”: Can you see clearly that the step is correct? But can you also prove that the step is correct?
loc. 924926
The students will find looking back at the solution really interesting if they have made an honest effort, and have the consciousness of having done well. Then they are eager to see what else they could accomplish with that effort, and how they could do equally well another time. The teacher should encourage the students to imagine cases in which they could utilize again the procedure used, or apply the result obtained. Can you use the result, or the method, for some other problem?
loc. 9961002
The teacher can ask several questions about the result which the students may readily answer with “Yes”; but an answer “No” would show a serious flaw in the result.
loc. 10131014
“Did you use all the data? Do all the data a, b, c appear in your formula for the diagonal?”
“Length, width, and height play the same role in our question; our problem is symmetric with respect to a, b, c. Is the expression you obtained for the diagonal symmetric in a, b, c? Does it remain unchanged when a, b, c are interchanged?”
“Our problem is a problem of solid geometry: to find the diagonal of a parallelepiped with given dimensions a, b, c. Our problem is analogous to a problem of plane geometry: to find the diagonal of a rectangle with given dimensions a, b. Is the result of our ‘solid’ problem analogous to the result of the ‘plane’ problem?”
“If the height c decreases, and finally vanishes, the parallelepiped becomes a parallelogram. If you put c = 0 in your formula, do you obtain the correct formula for the diagonal of the rectangular parallelogram?” “If the height c increases, the diagonal increases. Does your formula show this?”
“If all three measures a, b, c of the parallelepiped increase in the same proportion, the diagonal also increases in the same proportion. If, in your formula, you substitute 12a, 12b, 12c for a, b, c respectively, the expression of the diagonal, owing to this substitution, should also be multiplied by 12. Is that so?”
“If a, b, c are measured in feet, your formula gives the diagonal measured in feet too; but if you change all measures into inches, the formula should remain correct. Is that so?” (The two last questions are essentially equivalent; see TEST BY DIMENSION.)
loc. 10151051
Let us go back to the situation as it presented itself at the beginning of section 10 when the question was asked: Do you know a related problem? Instead of this, with the best intention to help the students, the question may be offered: Could you apply the theorem of Pythagoras? The intention may be the best, but the question is about the worst. We must realize in what situation it was offered; then we shall see that there is a long sequence of objections against that sort of “help.”
(1) If the student is near to the solution, he may understand the suggestion implied by the question; but if he is not, he quite possibly will not see at all the point at which the question is driving. Thus the question fails to help where help is most needed.
(2) If the suggestion is understood, it gives the whole secret away, very little remains for the student to do.
(3) The suggestion is of too special a nature. Even if the student can make use of it in solving the present problem, nothing is learned for future problems. The question is not instructive.
(4) Even if he understands the suggestion, the student can scarcely understand how the teacher came to the idea of putting such a question. And how could he, the student, find such a question by himself? It appears as an unnatural surprise, as a rabbit pulled out of a hat; it is really not instructive.
loc. 11741190
What can I do? Make your grasp quite secure. Carry through in detail all the algebraic or geometric operations which you have recognized previously as feasible. Convince yourself of the correctness of each step by formal reasoning, or by intuitive insight, or both ways if you can. If your problem is very complex you may distinguish “great” steps and “small” steps, each great step being composed of several small ones. Check first the great steps, and get down to the smaller ones afterwards.
loc. 16161619
Analogy is used on very different levels. People often use vague, ambiguous, incomplete, or incompletely clarified analogies, but analogy may reach the level of mathematical precision.
loc. 16631665
But two questions may be easier to answer than just one question—provided that the two questions are intelligently connected.
loc. 16811682
The feeling that harmonious simple order cannot be deceitful guides the discoverer both in the mathematical and in the other sciences, and is expressed by the Latin saying: simplex sigillum veri (simplicity is the seal of truth).
loc. 18551857
There is a oneone correspondence between the objects of the two systems S and S′, preserving certain relations. That is, if such a relation holds between the objects of one system, the same relation holds between the corresponding objects of the other system. Such a connection between two systems is a very precise sort of analogy; it is called isomorphism (or holohedral isomorphism).
loc. 18801886
There is a onemany correspondence between the objects of the two systems S and S′ preserving certain relations. Such a connection (which is important in various branches of advanced mathematical study, especially in the Theory of Groups, and need not be discussed here in detail) is called merohedral isomorphism (or homomorphism; homoiomorphism would be, perhaps, a better term). Merohedral isomorphism may be considered as another very precise sort of analogy.]
loc. 18861892
If a tricky auxiliary line appears abruptly in the figure, without any motivation, and solves the problem surprisingly, intelligent students and readers are disappointed; they feel that they are cheated.
loc. 20492050
3. Profit. The profit that we derive from the consideration of an auxiliary problem may be of various kinds. We may use the result of the auxiliary problem. Thus, in example 1, having found by solving the quadratic equation for y that y is equal to 4 or to 9, we infer that x2 = 4 or x2 = 9 and derive hence all possible values of x. In other cases, we may use the method of the auxiliary problem. Thus, in example 2, the auxiliary problem is a problem of plane geometry; it is analogous to, but simpler than, the original problem which is a problem of solid geometry. It is reasonable to introduce an auxiliary problem of this kind in the hope that it will be instructive, that it will give us opportunity to familiarize ourselves with certain methods, operations, or tools, which we may use afterwards for our original problem.
loc. 21002117
Consider the following theorems: A. In any equilateral triangle, each angle is equal to 60°. B. In any equiangular triangle, each angle is equal to 60°. These two theorems are not identical. They contain different notions; one is concerned with equality of the sides, the other with equality of the angles of a triangle. But each theorem follows from the other. Therefore, the problem to prove A is equivalent to the problem to prove B.
loc. 21442150
advantage in introducing, as an auxiliary problem, the problem to prove B. The theorem B is a little easier to prove than A and, what is more important, we may foresee that B is easier than A, we may judge so, we may find plausible from the outset that B is easier than A. In fact, the theorem B, concerned only with angles, is more “homogeneous” than the theorem A which is concerned with both angles and sides. The passage from the original problem to the auxiliary problem is called convertible reduction, or bilateral reduction, or equivalent reduction if these two problems, the original and the auxiliary, are equivalent. Thus, the reduction of A to B (see above) is convertible and so is the reduction in example 1. Convertible reductions are, in a certain respect, more important and more desirable than other ways to introduce auxiliary problems, but auxiliary problems which are not equivalent to the original problem may also be very useful; take example 2.
loc. 21512165
Chains of equivalent auxiliary problems are frequent in mathematical reasoning. We are required to solve a problem A; we cannot see the solution, but we may find that A is equivalent to another problem B. Considering B we may run into a third problem C equivalent to B. Proceeding in the same way, we reduce C to D, and so on, until we come upon a last problem L whose solution is known or immediate. Each problem being equivalent to the preceding, the last problem L must be equivalent to our original problem A. Thus we are able to infer the solution of the original problem A from the problem L which we attained as the last link in a chain of auxiliary problems.
loc. 21662173
If, from a proposed problem, we pass either to a more ambitious or to a less ambitious auxiliary problem we call the step a unilateral reduction. There are two kinds of unilateral reduction, and both are, in some way or other, more risky than a bilateral or convertible reduction.
loc. 22252229
We may test this result by SPECIALIZATION. In fact, if b = a the frustum becomes a prism and the formula yields a2h; and if b = 0 the frustum becomes a pyramid and the formula yields . We may apply the TEST BY DIMENSION. In fact, the expression has as dimension the cube of a length. Again, we may test the formula by variation of the data. In fact, if any one of the positive quantities a, b or h increases the value of the expression increases.
loc. 23092326
Tests of this sort can be applied not only to the final result but also to intermediate results. They are so useful that it is worth while preparing for them; see VARIATION OF THE PROBLEM, 4. In order to be able to use such tests, we may find advantage in generalizing a “problem in numbers” and changing it into a “problem in letters”; see GENERALIZATION
loc. 23262332
When the solution that we have finally obtained is long and involved, we naturally suspect that there is some clearer and less roundabout solution: Can you derive the result differently?
loc. 23542356
To find a new problem which is both interesting and accessible, is not so easy;
loc. 24442444
Good problems and mushrooms of certain kinds have something in common; they grow in clusters. Having found one, you should look around; there is a good chance that there are some more quite near.
loc. 24462448
Given the three dimensions (length, breadth, and height) of a rectangular parallelepiped, find the diagonal. If we know the solution of this problem, we can easily solve any of the following problems (of which the first two were almost stated in section 14). Given the three dimensions of a rectangular parallelepiped, find the radius of the circumscribed sphere. The base of a pyramid is a rectangle of which the center is the foot of the altitude of the pyramid. Given the altitude of the pyramid and the sides of its base, find the lateral edges. Given the rectangular coordinates (x1, y1, z1), (x2, y2, z2) of two points in space, find the distance of these points.
loc. 24512469
In fact, the solution of our original problem consists essentially in establishing a relation among four quantities, the three dimensions of the parallelepiped and its diagonal. If any three of these four quantities are given, we can calculate the fourth from the relation. Thus, we can solve the new problem.
loc. 24792482
We have here a pattern to derive easily solvable new problems from a problem we have solved: we regard the original unknown as given and one of the original data as unknown. The relation connecting the unknown and the data is the same in both problems, the old and the new. Having found this relation in one, we can use it also in the other.
loc. 24832487
The mathematical experience of the student is incomplete if he never had an opportunity to solve a problem invented by himself.
loc. 25122514
1. We may use provisional and merely plausible arguments when devising the final and rigorous argument as we use scaffolding to support a bridge during construction. When, however, the work is sufficiently advanced we take off the scaffolding, and the bridge should be able to stand by itself. In the same way, when the solution is sufficiently advanced, we brush aside all kinds of provisional and merely plausible arguments, and the result should be supported by rigorous argument alone.
loc. 25232527
The order in which we work out the details of the argument may be very different from the order in which we invented them; and the order in which we write down the details in a definitive exposition may be still different. Euclid’s Elements present the details of the argument in a rigid systematic order which was often imitated and often criticized.
loc. 25422545
3. In Euclid’s exposition all arguments proceed in the same direction: from the data toward the unknown in “problems to find,” and from the hypothesis toward the conclusion in “problems to prove.” Any new element, point, line, etc., has to be correctly derived from the data or from elements correctly derived in foregoing steps. Any new assertion has to be correctly proved from the hypothesis or from assertions correctly proved in foregoing steps. Each new element, each new assertion is examined when it is encountered first, and so it has to be examined just once; we may concentrate all our attention upon the present step, we need not look behind us, or look ahead. The very last new element whose derivation we have to check, is the unknown. The very last assertion whose proof we have to examine, is the conclusion. If each step is correct, also the last one, the whole argument is correct.
loc. 25462556
The Euclidean way of exposition can be highly recommended, without reservation, if the purpose is to examine the argument in detail. Especially, if it is our own argument, and it is long and complicated, and we have not only found it but have also surveyed it on large lines so that nothing is left but to examine each particular point in itself, then nothing is better than to write out the whole argument in the Euclidean way. The Euclidean way of exposition, however, cannot be recommended without reservation if the purpose is to convey an argument to a reader or to a listener who never heard of it before. The Euclidean exposition is excellent to show each particular point but not so good to show the main line of the argument. THE INTELLIGENT READER can easily see that each step is correct but has great difficulty in perceiving the source, the purpose, the connection of the whole argument. The reason for this difficulty is that the Euclidean exposition fairly often proceeds in an order exactly opposite to the natural order of invention. (Euclid’s exposition follows rigidly the order of “synthesis”; see PAPPUS, especially comments 3, 4, 5.)
loc. 25562568
Carrying out our plan, we check each step. Checking our step, we may rely on intuitive insight or on formal rules. Sometimes the intuition is ahead, sometimes the formal reasoning. It is an interesting and useful exercise to do it both ways. Can you see clearly that the step is correct? Yes, I can see it clearly and distinctly. Intuition is ahead; but could formal reasoning overtake it? Can you also PROVE that it is correct? Trying to prove formally what is seen intuitively and to see intuitively what is proved formally is an invigorating mental exercise. Unfortunately, in the classroom there is not always enough time for it. The example, discussed in sections 12 and 14, is typical in this respect.
loc. 25972608
A condition is called redundant if it contains superfluous parts. It is called contradictory if its parts are mutually opposed and inconsistent so that there is no object satisfying the condition. Thus, if a condition is expressed by more linear equations than there are unknowns, it is either redundant or contradictory; if the condition is expressed by fewer equations than there are unknowns, it is insufficient to determine the unknowns; if the condition is expressed by just as many equations as there are unknowns it is usually just sufficient to determine the unknowns but may be, in exceptional cases, contradictory or insufficient.
loc. 26142622
Look at the unknown! And try to think of a familiar problem having the same or a similar unknown. This suggests starting the work from the unknown. Look at the data! Could you derive something useful from the data? This suggests starting the work from the data. It appears that starting the reasoning from the unknown is usually preferable (see PAPPUS and WORKING BACKWARDS). Yet the alternative start, from the data, also has chances of success, must often be tried, and deserves illustration.
loc. 26362646
Decomposing and recombining are important operations of the mind. You examine an object that touches your interest or challenges your curiosity: a house you intend to rent, an important but cryptic telegram, any object whose purpose and origin puzzle you, or any problem you intend to solve. You have an impression of the object as a whole but this impression, possibly, is not definite enough. A detail strikes you, and you focus your attention upon it. Then, you concentrate upon another detail; then, again, upon another. Various combinations of details may present themselves and after a while you again consider the object as a whole but you see it now differently. You decompose the whole into its parts, and you recombine the parts into a more or less different whole.
loc. 27182727
Therefore, let us, first of all, understand the problem as a whole. Having understood the problem, we shall be in a better position to judge which particular points may be the most essential. Having examined one or two essential points we shall be in a better position to judge which further details might deserve closer examination. Let us go into detail and decompose the problem gradually, but not further than we need to.
loc. 27342738
Could you change the unknown, or the data, or both if necessary, so that the new unknown and the new data are nearer to each other?
loc. 28352836
An interesting way of changing both the unknown and the data is interchanging the unknown with one of the data.
loc. 28362837
The procedure that we have just applied has a certain interest; solving problems of geometric construction, we can often follow successfully its pattern: Reduce the problem to the construction of a point, and construct the point as an intersection of two loci.
loc. 29202922
Keep only a part of the condition, drop the other part. Doing so, we weaken the condition of the proposed problem, we restrict less the unknown.
loc. 29242926
Doing so, we weaken the condition of the proposed problem, we restrict less the unknown. How far
loc. 29252926
Definition of a term is a statement of its meaning in other terms which are supposed to be well known.
loc. 30063007
1. Technical terms in mathematics are of two kinds. Some are accepted as primitive terms and are not defined. Others are considered as derived terms and are defined in due form; that is, their meaning is stated in primitive terms and in formerly defined derived terms. Thus, we do not give a formal definition of such primitive notions as point, straight line, and plane.3 Yet we give formal definitions of such notions as “bisector of an angle” or “circle” or “parabola.”
loc. 30083017
“Construct a point P on the given straight line c at equal distances from the given point F and the given straight line d.” “Observe the progress from the original statement to your restatement. The original statement of the problem was full of unfamiliar technical terms, parabola, focus, directrix; it sounded just a little pompous and inflated. And now, nothing remains of those unfamiliar technical terms; you have deflated the problem. Well done!”
loc. 31143124
4. Elimination of technical terms is the result of the work in the foregoing example. We started from a statement of the problem containing certain technical terms (parabola, focus, directrix) and we arrived finally at a restatement free of those terms. In order to eliminate a technical term we
loc. 31263130
In order to eliminate a technical term we must know its definition; but it is not enough to know the definition, we must use it.
loc. 31293130
5. Definitions and known theorems. If we know the name “parabola” and have some vague idea of the shape of the curve but do not know anything else about it, our knowledge is obviously insufficient to solve the problem proposed as example, or any other serious geometric problem about the parabola. What kind of knowledge is needed for such a purpose? The science of geometry may be considered as consisting of axioms, definitions, and theorems. The parabola is not mentioned in the axioms which deal only with such primitive terms as point, straight line, and so on. Any geometric argumentation concerned with the parabola, the solution of any problem involving it, must use either its definition or theorems about it. To solve such a problem, we must know, at least, the definition but it is better to know some theorems too.
loc. 31463156
“You can undertake without hope and persevere without success.” Thus may speak an inflexible will, or honor and duty, or a nobleman with a noble cause. This sort of determination, however, would not do for the scientist, who should have some hope to start with, and some success to go on. In scientific work, it is necessary to apportion wisely determination to outlook. You do not take up a problem, unless it has some interest; you settle down to work seriously if the problem seems instructive; you throw in your whole personality if there is a great promise. If your purpose is set, you stick to it, but you do not make it unnecessarily difficult for yourself. You do not despise little successes, on the contrary, you seek them: If you cannot solve the proposed problem try to solve first some related problem.
loc. 32343243
With respect to devising a plan and obtaining a general idea of the solution two opposite faults are frequent. Some students rush into calculations and constructions without any plan or general idea; others wait clumsily for some idea to come and cannot do anything that would accelerate its coming.
loc. 32663269
That is, if we discard any part of the hypothesis, the theorem ceases to be true. Therefore, if the proof neglects to use any part of the hypothesis, the proof must be wrong. Does the proof use the whole hypothesis?
loc. 33113314
Many a guess has turned out to be wrong but nevertheless useful in leading to a better one. No idea is really bad, unless we are uncritical. What is really bad is to have no idea at all.
loc. 33883390
If we decide ourselves to examine this statement, the situation changes. Originally, we had a “problem to find.” After having formulated our guess, we have a “problem to prove”; we have to prove or disprove the theorem formulated.
loc. 34353437
Notes: 1) WEAKER PROBLEM
This theorem appears more accessible than the former; it is, of course, weaker.
loc. 34443445
even if your problem is not a problem of geometry, you may try to draw a figure. To find a lucid geometric representation for your nongeometrical problem could be an important step toward the solution.
loc. 36263629
1. If, by some chance, we come across the sum 1 + 8 + 27 + 64 = 100 we may observe that it can be expressed in the curious form 13 + 23 + 33 + 43 = 102. Now, it is natural to ask ourselves: Does it often happen that a sum of successive cubes as 13 + 23 + 33 + · · · + n3 is a square? In asking this, we generalize. This generalization is a lucky one; it leads from one observation to a remarkable general law. Many results were found by lucky generalizations in mathematics, physics, and the natural sciences. See INDUCTION AND MATHEMATICAL INDUCTION.
loc. 36333653
The more general problem may be easier to solve.
loc. 36713671
In mathematics as in the physical sciences we may use observation and induction to discover general laws. But there is a difference. In the physical sciences, there is no higher authority than observation and induction but in mathematics there is such an authority: rigorous proof.
loc. 39143917
The foregoing proof may serve as a pattern in many similar cases. What are the essential lines of this pattern? The assertion we have to prove must be given in advance, in precise form. The assertion must depend on an integer n. The assertion must be sufficiently “explicit” so that we have some possibility of testing whether it remains true in the passage from n to the next integer n + 1.
loc. 39954003
If we succeed in testing this effectively, we may be able to use our experience, gained in the process of testing, to conclude that the assertion must be true for n + 1 provided it is true for n. When we are so far it is sufficient to know that the assertion is true for n = 1; hence it follows for n = 2; hence it follows for n = 3. and so on; passing from any integer to the next, we prove the assertion generally.
loc. 40034012
Unfortunately, the accepted technical term is “mathematical induction.” This name results from a random circumstance. The precise assertion that we have to prove may come from any source, and it is immaterial from the logical viewpoint what the source is. Now, in many cases, as in the case we discussed here in detail, the source is induction, the assertion is found experimentally, and so the proof appears as a mathematical complement to induction; this explains the name.
loc. 40174021
The precise assertion that we have to prove may come from any source, and it is immaterial from the logical viewpoint what the source is. Now, in many cases, as in the case we discussed here in detail, the source is induction, the assertion is found experimentally, and so the proof appears as a mathematical complement to induction; this explains the name.
loc. 40184021
Inventor’s paradox. The more ambitious plan may have more chances of success. This sounds paradoxical. Yet, when passing from one problem to another, we may often observe that the new, more ambitious problem is easier to handle than the original problem. More questions may be easier to answer than just one question. The more comprehensive theorem may be easier to prove, the more general problem may be easier to solve.
loc. 40394044
Lemma means “auxiliary theorem.” The word is of Greek origin; a more literal translation would be “what is assumed.” We are trying to prove a theorem, say, A. We are led to suspect another theorem, say, B; if B were true we could perhaps, using it, prove A. We assume B provisionally, postponing its proof, and go ahead with the proof of A. Such a theorem B is assumed, and is an auxiliary theorem to the originally proposed theorem A. Our little story is fairly typical and explains the present meaning of the word “lemma.”
loc. 40824097
To know or not to know a formerly solved problem with the same unknown may make all the difference between an easy and a difficult problem.
loc. 41964197
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 on teaching, especially on the teaching of mathematics.
loc. 42524256
2. Some mathematical symbols, as +, −, =, and several others, have a fixed traditional meaning, but other symbols, as the small and capital letters of the Roman and Greek alphabets, are used in different meanings in different problems. When we face a new problem, we must choose certain symbols, we have to introduce suitable notation. There is something analogous in the use of ordinary language. Many words are used in different meanings in different contexts; when precision is important, we have to choose our words carefully. An important step in solving a problem is to choose the notation. It should be done carefully. The time we spend now on choosing the notation may be well repaid by the time we save later by avoiding hesitation and confusion. Moreover, choosing the notation carefully, we have to think sharply of the elements of the problem which must be denoted. Thus, choosing a suitable notation may contribute essentially to understanding the problem.
loc. 44074419
An important step in solving a problem is to choose the notation. It should be done carefully. The time we spend now on choosing the notation may be well repaid by the time we save later by avoiding hesitation and confusion. Moreover, choosing the notation carefully, we have to
loc. 44154418
“Now analysis is of two kinds; the one is the analysis of the ‘problems to prove’ and aims at establishing true theorems; the other is the analysis of the ‘problems to find’ and aims at finding the unknown.
loc. 46384639
which transforms the condition into 8z2 − 54z + 85 = 0. Here the analysis ends, provided that the problemsolver is acquainted with the solution of quadratic equations. What is the synthesis? Carrying through, step by step, the calculations whose possibility was foreseen by the analysis. The problemsolver needs no new idea to finish his problem, only some patience and attention in calculating the various unknowns. The order of calculation is opposite to the order of invention; first z is found (z = 5/2, 17/4), then y (y = 2, 1/2, 4, 1/4), and finally the originally required x (x = 1, −1, 2, −2). The synthesis retraces the steps of the analysis, and it is easy to see in the present case why it does so.
loc. 47324750
Here the analysis ends, provided that the problemsolver is acquainted with the solution of quadratic equations. What is the synthesis? Carrying through, step by step, the calculations whose possibility was foreseen by the analysis. The problemsolver needs no new idea to finish his problem, only some patience and attention in calculating the various unknowns.
loc. 47374740
4. Nonmathematical illustration. A primitive man wishes to cross a creek; but he cannot do so in the usual way because the water has risen overnight. Thus, the crossing becomes the object of a problem; “crossing the creek” is the x of this primitive problem. The man may recall that he has crossed some other creek by walking along a fallen tree. He looks around for a suitable fallen tree which becomes his new unknown, his y. He cannot find any suitable tree but there are plenty of trees standing along the creek; he wishes that one of them would fall. Could he make a tree fall across the creek? There is a great idea and there is a new unknown; by what means could he tilt the tree over the creek? This train of ideas ought to be called analysis if we accept the terminology of Pappus. If the primitive man succeeds in finishing his analysis he may become the inventor of the bridge and of the axe. What will be the synthesis? Translation of ideas into actions. The finishing act of the synthesis is walking along a tree across the creek.
loc. 47514766
the analysis consists in thoughts, the synthesis in acts. There is another difference; the order is reversed. Walking across the creek is the first desire from which the analysis starts and it is the last act with which the synthesis ends.
loc. 47684770
Analysis comes naturally first, synthesis afterwards; analysis is invention, synthesis, execution; analysis is devising a plan, synthesis carrying through the plan.
loc. 47744776
The paraphrase preserves and even emphasizes certain curious phrases of the original: “assume what is required to be done as already done, what is sought as found, what you have to prove as true.”
loc. 47774779
The paraphrase uses twice the important phrase “provided that all our derivations are convertible”;
loc. 48134813
You should ask no question, make no suggestion, indiscriminately, following some rigid habit.
loc. 48514851
1. An impressive practical problem is the construction of a dam across a river. We need no special knowledge to understand this problem. In almost prehistoric times, long before our modern age of scientific theories, men built dams of some sort in the valley of the Nile, and in other parts of the world, where the crops depended on irrigation. Let us visualize the problem of constructing an important modern dam. What is the unknown? Many unknowns are involved in a problem of this kind: the exact location of the dam, its geometric shape and dimensions, the materials used in its construction, and so on. What is the condition? We cannot answer this question in one short sentence because there are many conditions. In so large a project it is necessary to satisfy many important economic needs and to hurt other needs as little as possible. The dam should provide electric power, supply water for irrigation or the use of certain communities, and also help to control floods. On the other hand, it should disturb as little as possible navigation, or economically important fishlife, or beautiful scenery; and so forth. And, of course, it should cost as little as possible and be constructed as quickly as possible. What are the data? The multitude of desirable data is tremendous. We need topographical data concerning the vicinity of the river and its tributaries; geological data important for the solidity of foundations, possible leakage, and available materials of construction; meteorological data about annual precipitation and the height of floods; economic data concerning the value of ground which will be flooded, cost of materials and labor; and so on. Our example shows that unknowns, data, and conditions are more complex and less sharply defined in a practical problem than in a mathematical problem.
loc. 48634884
2. In order to solve a problem, we need a certain amount of previously acquired knowledge. The modern engineer has a highly specialized body of knowledge at his disposal, a scientific theory of the strength of materials, his own experience, and the mass of engineering experience stored in special technical literature. We cannot avail ourselves of such special knowledge here but we may try to imagine what was in the mind of an ancient Egyptian dambuilder.
loc. 48844889
He has seen, of course, various other, perhaps smaller, dams: banks of earth or masonry holding back the water. He has seen the flood, laden with all sorts of debris, pressing against the bank. He might have helped to repair the cracks and the erosion left by the flood. He might have seen a dam break, giving way under the impact of the flood. He has certainly heard stories about dams withstanding the test of centuries or causing catastrophe by an unexpected break. His mind may have pictured the pressure of the river against the surface of the dam and the strain and stress in its interior. Yet the Egyptian dambuilder had no precise, quantitative, scientific concepts of fluid pressure or of strain and stress in a solid body. Such concepts form an essential part of the intellectual equipment of a modern engineer. Yet the latter also uses much knowledge which has not yet quite reached a precise, scientific level; what he knows about erosion by flowing water, the transportation of silt, the plasticity and other not quite clearly circumscribed properties of certain materials, is knowledge of a rather empirical character. Our example shows that the knowledge needed and the concepts used are more complex and less sharply defined in practical problems than in mathematical problems.
loc. 48894903
The data of his problem are, strictly speaking, inexhaustible. For instance, he would like to know a little more about the geologic nature of the ground on which the foundations must be laid, but eventually he must stop collecting geologic data although a certain margin of uncertainty unavoidably remains.
loc. 49264928
In setting up and in solving mathematical problems derived from practical problems, we usually content ourselves with an approximation.
loc. 49484950
We gain much in simplicity and do not lose a great deal in accuracy.
loc. 49614961
What is a bright idea? An abrupt and momentous change of our outlook, a sudden reorganization of our mode of conceiving the problem, a just emerging confident prevision of the steps we have to take in order to attain the solution.
loc. 51085109
Reductio ad absurdum shows the falsity of an assumption by deriving from it a manifest absurdity. “Reduction to an absurdity” is a mathematical procedure but it has some resemblance to irony which is the favorite procedure of the satirist. Irony adopts, to all appearance, a certain opinion and stresses it and overstresses it till it leads to a manifest absurdity.
loc. 51985201
Indirect proof establishes the truth of an assertion by showing the falsity of the opposite assumption. Thus, indirect proof has some resemblance to a politician’s trick of establishing a candidate by demolishing the reputation of his opponent.
loc. 52025205
If we wish to set up an equation, we have to express in mathematical language that all parts of the condition are satisfied, although we do not know yet whether it is actually possible to satisfy all these parts simultaneously.
loc. 52905292
In general, a problem is a “routine problem” if it can be solved either by substituting special data into a formerly solved general problem, or by following step by step, without any trace of originality, some wellworn conspicuous example.
loc. 54605461
Teaching the mechanical performance of routine mathematical operations and nothing else is well under the level of the cookbook because kitchen recipes do leave something to the imagination and judgment of the cook but mathematical recipes do not.
loc. 54685469
Rules of style. The first rule of style is to have something to say. The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time.
loc. 54865489
Rules of teaching. The first rule of teaching is to know what you are supposed to teach. The second rule of teaching is to know a little more than what you are supposed to teach.
loc. 54905492
Separate the various parts of the condition. Can you write them down? We often have opportunity to ask this question when we are SETTING UP EQUATIONS.
loc. 55105513
Setting up equations is like translation from one language into another (NOTATION, 1). This comparison, used by Newton in his Arithmetica Universalis, may help to clarify the nature of certain difficulties often felt both by students and by teachers.
loc. 55155520
In order to translate a sentence from English into French two things are necessary. First, we must understand thoroughly the English sentence. Second, we must be familiar with the forms of expression peculiar to the French language. The situation is very similar when we attempt to express in mathematical symbols a condition proposed in words. First, we must understand thoroughly the condition. Second, we must be familiar with the forms of mathematical expression.
loc. 55235528
An English sentence is relatively easy to translate into French if it can be translated word for word. But there are English idioms which cannot be translated into French word for word. If our sentence contains such idioms, the translation becomes difficult; we have to pay less attention to the separate words, and more attention to the whole meaning; before translating the sentence, we may have to rearrange it.
loc. 55295533
In all cases, easy or difficult, we have to understand the condition, to separate the various parts of the condition, and to ask: Can you write them down? In easy cases, we succeed without hesitation in dividing the condition into parts that can be written down in mathematical symbols; in difficult cases, the appropriate division of the condition is less obvious.
loc. 55405545
If you take a heuristic conclusion as certain, you may be fooled and disappointed; but if you neglect heuristic conclusions altogether you will make no progress at all.
loc. 57215722
In fact, to solve a problem is, essentially, to find the connection between the data and the unknown. Moreover
loc. 57315734
Indeed, analogy is one of the main sources of invention. If other means fail, we should try to imagine an analogous problem. Therefore, if such a problem emerges spontaneously, by its own accord, we naturally feel elated; we feel that we are approaching the solution.
loc. 57445747
Yes, signs may misguide us in any single case, but they guide us right in the majority of them. A hunter may misinterpret now and then the traces of his game but he must be right on the average, otherwise he could not make a living by hunting.
loc. 58085810
The signs that convince the inventor that his idea is good, the indications that guide us in our everyday affairs, the circumstantial evidence of the lawyer, the inductive evidence of the scientist, statistical evidence invoked in many and diverse subjects—all these kinds of evidence agree in two essential points. First, they do not have the certainty of a strict demonstration. Second, they are useful in acquiring essentially new knowledge, and even indispensable to any not purely mathematical or logical knowledge, to any knowledge concerned with the physical world. We could call the reasoning that underlies this kind of evidence “heuristic reasoning” or “inductive reasoning” or (if we wish to avoid stretching the meaning of existing terms) “plausible reasoning.” We accept here the last term.
loc. 58675875
The conclusion is not fully expressed and is not fully supported by the premises. The direction is expressed and is implied by the premises, the magnitude is not.
loc. 59145916
The direction is expressed and is implied by the premises, the magnitude is not. For any reasonable person, the premises involve that
loc. 59155917
If, however, we find that the general statement is verified even in the extreme case, the inductive evidence derived from this verification will be strong, just because the prospect of refutation was strong. Thus, we are tempted to reshape the saying from which we started: “Prospective exceptions test the rule.”
loc. 60136016
Now, whatever the data may be, the required solution must apply and we do not see yet how to fit the same solution to all these possibilities. Out of such feeling of “too much variety” this question and answer may eventually emerge: Could you imagine a more accessible related problem? A more special problem?
loc. 60616064
Mathematics being a very abstract science should be presented very concretely.
loc. 61356135
In a more general acceptance of the word, a whole is termed symmetric if it has interchangeable parts. There are many kinds of symmetry; they differ in the number of interchangeable parts, and in the operations which exchange the parts.
loc. 61816182
Symmetry, in a general sense, is important for our subject. If a problem is symmetric in some ways we may derive some profit from noticing its interchangeable parts and it often pays to treat those parts which play the same role in the same fashion
loc. 61906192
The dimension of a product is the product of the dimensions of its factors, and there is a similar rule about powers. Replacing the quantities by their dimensions on both sides of the formula that we are testing, we obtain
loc. 63126315
The future mathematician learns, as does everybody else, by imitation and practice. He should look out for the right model to imitate. He should observe a stimulating teacher. He should compete with a capable friend. Then, what may be the most important, he should read not only current textbooks but good authors till he finds one whose ways he is naturally inclined to imitate.
loc. 64146419
He should enjoy and seek what seems to him simple or instructive or beautiful. He should solve problems, choose the problems which are in his line, meditate upon their solution, and invent new problems. By these means, and by all other means, he should endeavor to make his first important discovery: he should discover his likes and his dislikes, his taste, his own line.
loc. 64206423
If he cannot summon up real desire for solving the problem he would do better to leave it alone. The open secret of real success is to throw your whole personality into your problem.
loc. 64416443
The open secret of real success is to throw your whole personality into your problem.
loc. 64426443
The intelligent listener to a mathematical lecture has the same wishes. If he cannot see that the present step of the argument is correct and even suspects that it is, possibly, incorrect, he may protest and ask a question. If he cannot see any purpose in the present step, nor suspect any reason for it, he usually cannot even formulate a clear objection, he does not protest, he is just dismayed and bored, and loses the thread of the argument.
loc. 64486452
The insect, the mouse, and the man follow it; but if one follows it with more success than the others it is because he varies his problem more intelligently.
loc. 64926494
If our work progresses, there is something to do, there are new points to examine, our attention is occupied, our interest is alive. But if we fail to make progress, our attention falters, our interest fades, we get tired of the problem, our thoughts begin to wander, and there is danger of losing the problem altogether. To escape from this danger we have to set ourselves a new question about the problem.
loc. 65176522
The new question unfolds untried possibilities of contact with our previous knowledge, it revives our hope of making useful contacts. The new question reconquers our interest by varying the problem, by showing some new aspect of it.
loc. 65226525
2. Logical system. Geometry, as presented in Euclid’s Elements, is not a mere collection of facts but a logical system. The axioms, definitions, and propositions are not listed in a random sequence but disposed in accomplished order. Each proposition is so placed that it can be based on the foregoing axioms, definitions, and propositions. We may regard the disposition of the propositions as Euclid’s main achievement and their logical system as the main merit of the Elements. Euclid’s geometry is not only a logical system but it is the first and greatest example of such a system, which other sciences have tried, and are still trying, to imitate. Should other sciences—especially those very far from geometry, as psychology, or jurisprudence—imitate Euclid’s rigid logic? This is a debatable question; but nobody can take part in the debate with competence who is not acquainted with the Euclidean system.
loc. 67276738
the facts must be presented in some connection and in some sort of system, since isolated items are laboriously acquired and easily forgotten.
loc. 67596760
Any sort of connection that unites the facts simply, naturally, and durably, is welcome here. The system need not be founded on logic, it must only be designed to aid the memory effectively; it must be what is called a mnemotechnic system.
loc. 67606763
If the calculus is presented according to modern standards of rigor, it demands proofs of a certain degree of difficulty and subtlety (“epsilonproofs”). But engineers study the calculus in view of its application and have neither enough time nor enough training or interest to struggle through long proofs or to appreciate subtleties. Thus, there is a strong temptation to cut out all the proofs. Doing so, however, we reduce the calculus to the level of the cookbook. The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. The cookbook may serve its purpose perfectly. In fact, it need not have any sort of logical or mnemotechnic system since recipes are written or printed and not retained in memory.
loc. 67776785
5. Incomplete proofs. The best way of handling the dilemma between too heavy proofs and the level of the cookbook may be to make reasonable use of incomplete proofs.
loc. 67916793
In short, incomplete proofs may be used as a sort of mnemotechnic device (but, of course, not as substitutes for complete proofs) when the aim is tolerable coherence of presentation and not strictly logical consistency. It is very dangerous to advocate incomplete proofs. Possible abuse, however, may be kept within bounds by a few rules. First, if a proof is incomplete, it must be indicated as such, somewhere and somehow. Second, an author or a teacher is not entitled to present an incomplete proof for a theorem unless he knows very well a complete proof for it himself. And it may be confessed that to present an incomplete proof in good taste is not easy at all.
loc. 68416848
There are many shrewd and some subtle remarks in proverbs but, obviously, there is no scientific system free of inconsistencies and obscurities in them. On the contrary, many a proverb can be matched with another proverb giving exactly opposite advice, and there is a great latitude of interpretation. It would be foolish to regard proverbs as an authoritative source of universally applicable wisdom but it would be a pity to disregard the graphic description of heuristic procedures provided by proverbs.
loc. 68586862
It is true, we have discovered the appropriate sequence in retrogressive order but all that is left to do is to reverse the process and start from the point which we reached last of all in the analysis (as Pappus says).
loc. 70677070
Notes:
1) 反方向,类似走迷宫
Going around an obstacle is what we do in solving any kind of problem; the experiment has a sort of symbolic value. The hen acted like people who solve their problem muddling through, trying again and again, and succeeding eventually by some lucky accident without much insight into the reasons for their success. The dog who scratched and jumped and barked before turning around solved his problem about as well as we did ours about the two containers.
loc. 71307133
回应 20190527 19:45 
旺三 (一蓑烟雨任平生。)
一、理解问题 第一步：你必须理解问题。 什么是未知的？数据有哪些？条件是什么？ 是否可能满足条件？条件是否足够决定未知？或是不充分？或是冗余？或是矛盾？ 画图，介绍适当的符号。 分离条件的不同部分，能否写下来？ 二、设计计划 第二步：找到数据和未知之间的联系，如果找不到某个中间的联，你可能被要求考虑副问题。你最终需要一个解决方案的计划。 你是否遇到过？是否看到过形式稍有不同的同样的问题？ 是否知道一个相...20190526 19:36
一、理解问题
第一步：你必须理解问题。
什么是未知的？数据有哪些？条件是什么？
是否可能满足条件？条件是否足够决定未知？或是不充分？或是冗余？或是矛盾？
画图，介绍适当的符号。
分离条件的不同部分，能否写下来？
二、设计计划
第二步：找到数据和未知之间的联系，如果找不到某个中间的联，你可能被要求考虑副问题。你最终需要一个解决方案的计划。
你是否遇到过？是否看到过形式稍有不同的同样的问题？
是否知道一个相关的问题？是否知道一个用得上的定理？
看未知量，试着想一个有相同或相似未知量的熟悉的问题。
如果有一个和你的问题相关且解决过的问题，你是否能用上？是否能用它的结果？是否能用它的方法？为了使它用得上，是否要引入一些辅助元素？
是否能重新描述问题？能否再差异化地描述它？回到定义上。
如果你不能解决提出的问题，试着先解决一些相关的问题。能否想象一个更好理解的相关问题？一个更通用的问题？一个更特别的问题？一个类似的问题？你能否解决部分问题？只保留部分条件，放下其它部分；未知量还要多远才能决定，能有多少不同？能否从数据得出一些有用的东西？能否想到别的适合决定未知量的数据？能否更改未知量或数据，或必要时两者，使新未知量和新数据离得更近？是否使用了所有数据？是否使用了所有条件？是否考虑了问题涉及的所有必要的概念？
三、执行计划
步骤三：执行计划
执行解决方案的计划，检查每一步。你能否清楚看出该步骤是证券的？能否证明它是正确的？
四、回看
步骤四：检验获得的解决方案。
能否检查你的结果？能否检查参数？
能否用不同地方法得出结果？能否一眼看出来？
能否将结果或方法用于其它问题？
回应 20190526 19:36 
枯草[已注销] (Subtillion)
页码其实是xvi。傻缺豆瓣。。只好写成016了。 second. have you seen it before? or have you seen the same problem in a slightly different form?” 哎呀怎么办啊。看到这个话就想笑啊。。。 http://www.sdxxb.cn/shuxue/ShowArticle.asp?ArticleID=463120120815 04:30
页码其实是xvi。傻缺豆瓣。。只好写成016了。
”哎呀怎么办啊。看到这个话就想笑啊。。。 http://www.sdxxb.cn/shuxue/ShowArticle.asp?ArticleID=4631second. have you seen it before? or have you seen the same problem in a slightly different form?
回应 20120815 04:30

枯草[已注销] (Subtillion)
页码其实是xvi。傻缺豆瓣。。只好写成016了。 second. have you seen it before? or have you seen the same problem in a slightly different form?” 哎呀怎么办啊。看到这个话就想笑啊。。。 http://www.sdxxb.cn/shuxue/ShowArticle.asp?ArticleID=463120120815 04:30
页码其实是xvi。傻缺豆瓣。。只好写成016了。
”哎呀怎么办啊。看到这个话就想笑啊。。。 http://www.sdxxb.cn/shuxue/ShowArticle.asp?ArticleID=4631second. have you seen it before? or have you seen the same problem in a slightly different form?
回应 20120815 04:30 
simoncos (给岁月以生命。)
First, we have to understand the problem; we have to see clearly what is required. Second, we have to see how the various items are connected, how the unknown is linked to the data, in order to obtain the idea of the solution, to make a plan. Third, we carry out our plan. Fourth, we look back at the completed solution, we review and discuss it. loc. 712720 The worst may happen if the student e...20190527 19:45
First, we have to understand the problem; we have to see clearly what is required. Second, we have to see how the various items are connected, how the unknown is linked to the data, in order to obtain the idea of the solution, to make a plan. Third, we carry out our plan. Fourth, we look back at the completed solution, we review and discuss it.
loc. 712720
The worst may happen if the student embarks upon computations or constructions without having understood the problem.
loc. 725726
The teacher can make the problem interesting by making it concrete.
loc. 771772
Good ideas are based on past experience and formerly acquired knowledge.
loc. 813814
The materials necessary for solving a mathematical problem are certain relevant items of our formerly acquired mathematical knowledge, as formerly solved problems, or formerly proved theorems. Thus, it is often appropriate to start the work with the question: Do you know a related problem?
loc. 816819
The difficulty is that there are usually too many problems which are somewhat related to our present problem, that is, have some point in common with it. How can we choose the one, or the few, which are really useful? There is a suggestion that puts our finger on an essential common point: Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.
loc. 820825
The foregoing questions, well understood and seriously considered, very often help to start the right train of ideas; but they cannot help always, they cannot work magic. If they do not work, we must look around for some other appropriate point of contact, and explore the various aspects of our problem; we have to vary, to transform, to modify the problem. Could you restate the problem? Some of the questions of our list hint specific means to vary the problem, as generalization, specialization, use of analogy, dropping a part of the condition, and so on; the details are important but we cannot go into them now. Variation of the problem may lead to some appropriate auxiliary problem: If you cannot solve the proposed problem try to solve first some related problem.
loc. 829838
11. Carrying out the plan. To devise a plan, to conceive the idea of the solution is not easy. It takes so much to succeed; formerly acquired knowledge, good mental habits, concentration upon the purpose, and one more thing: good luck. To carry out the plan is much easier; what we need is mainly patience.
loc. 904908
We may convince ourselves of the correctness of a step in our reasoning either “intuitively” or “formally.”
loc. 918919
In certain cases, the teacher may emphasize the difference between “seeing” and “proving”: Can you see clearly that the step is correct? But can you also prove that the step is correct?
loc. 924926
The students will find looking back at the solution really interesting if they have made an honest effort, and have the consciousness of having done well. Then they are eager to see what else they could accomplish with that effort, and how they could do equally well another time. The teacher should encourage the students to imagine cases in which they could utilize again the procedure used, or apply the result obtained. Can you use the result, or the method, for some other problem?
loc. 9961002
The teacher can ask several questions about the result which the students may readily answer with “Yes”; but an answer “No” would show a serious flaw in the result.
loc. 10131014
“Did you use all the data? Do all the data a, b, c appear in your formula for the diagonal?”
“Length, width, and height play the same role in our question; our problem is symmetric with respect to a, b, c. Is the expression you obtained for the diagonal symmetric in a, b, c? Does it remain unchanged when a, b, c are interchanged?”
“Our problem is a problem of solid geometry: to find the diagonal of a parallelepiped with given dimensions a, b, c. Our problem is analogous to a problem of plane geometry: to find the diagonal of a rectangle with given dimensions a, b. Is the result of our ‘solid’ problem analogous to the result of the ‘plane’ problem?”
“If the height c decreases, and finally vanishes, the parallelepiped becomes a parallelogram. If you put c = 0 in your formula, do you obtain the correct formula for the diagonal of the rectangular parallelogram?” “If the height c increases, the diagonal increases. Does your formula show this?”
“If all three measures a, b, c of the parallelepiped increase in the same proportion, the diagonal also increases in the same proportion. If, in your formula, you substitute 12a, 12b, 12c for a, b, c respectively, the expression of the diagonal, owing to this substitution, should also be multiplied by 12. Is that so?”
“If a, b, c are measured in feet, your formula gives the diagonal measured in feet too; but if you change all measures into inches, the formula should remain correct. Is that so?” (The two last questions are essentially equivalent; see TEST BY DIMENSION.)
loc. 10151051
Let us go back to the situation as it presented itself at the beginning of section 10 when the question was asked: Do you know a related problem? Instead of this, with the best intention to help the students, the question may be offered: Could you apply the theorem of Pythagoras? The intention may be the best, but the question is about the worst. We must realize in what situation it was offered; then we shall see that there is a long sequence of objections against that sort of “help.”
(1) If the student is near to the solution, he may understand the suggestion implied by the question; but if he is not, he quite possibly will not see at all the point at which the question is driving. Thus the question fails to help where help is most needed.
(2) If the suggestion is understood, it gives the whole secret away, very little remains for the student to do.
(3) The suggestion is of too special a nature. Even if the student can make use of it in solving the present problem, nothing is learned for future problems. The question is not instructive.
(4) Even if he understands the suggestion, the student can scarcely understand how the teacher came to the idea of putting such a question. And how could he, the student, find such a question by himself? It appears as an unnatural surprise, as a rabbit pulled out of a hat; it is really not instructive.
loc. 11741190
What can I do? Make your grasp quite secure. Carry through in detail all the algebraic or geometric operations which you have recognized previously as feasible. Convince yourself of the correctness of each step by formal reasoning, or by intuitive insight, or both ways if you can. If your problem is very complex you may distinguish “great” steps and “small” steps, each great step being composed of several small ones. Check first the great steps, and get down to the smaller ones afterwards.
loc. 16161619
Analogy is used on very different levels. People often use vague, ambiguous, incomplete, or incompletely clarified analogies, but analogy may reach the level of mathematical precision.
loc. 16631665
But two questions may be easier to answer than just one question—provided that the two questions are intelligently connected.
loc. 16811682
The feeling that harmonious simple order cannot be deceitful guides the discoverer both in the mathematical and in the other sciences, and is expressed by the Latin saying: simplex sigillum veri (simplicity is the seal of truth).
loc. 18551857
There is a oneone correspondence between the objects of the two systems S and S′, preserving certain relations. That is, if such a relation holds between the objects of one system, the same relation holds between the corresponding objects of the other system. Such a connection between two systems is a very precise sort of analogy; it is called isomorphism (or holohedral isomorphism).
loc. 18801886
There is a onemany correspondence between the objects of the two systems S and S′ preserving certain relations. Such a connection (which is important in various branches of advanced mathematical study, especially in the Theory of Groups, and need not be discussed here in detail) is called merohedral isomorphism (or homomorphism; homoiomorphism would be, perhaps, a better term). Merohedral isomorphism may be considered as another very precise sort of analogy.]
loc. 18861892
If a tricky auxiliary line appears abruptly in the figure, without any motivation, and solves the problem surprisingly, intelligent students and readers are disappointed; they feel that they are cheated.
loc. 20492050
3. Profit. The profit that we derive from the consideration of an auxiliary problem may be of various kinds. We may use the result of the auxiliary problem. Thus, in example 1, having found by solving the quadratic equation for y that y is equal to 4 or to 9, we infer that x2 = 4 or x2 = 9 and derive hence all possible values of x. In other cases, we may use the method of the auxiliary problem. Thus, in example 2, the auxiliary problem is a problem of plane geometry; it is analogous to, but simpler than, the original problem which is a problem of solid geometry. It is reasonable to introduce an auxiliary problem of this kind in the hope that it will be instructive, that it will give us opportunity to familiarize ourselves with certain methods, operations, or tools, which we may use afterwards for our original problem.
loc. 21002117
Consider the following theorems: A. In any equilateral triangle, each angle is equal to 60°. B. In any equiangular triangle, each angle is equal to 60°. These two theorems are not identical. They contain different notions; one is concerned with equality of the sides, the other with equality of the angles of a triangle. But each theorem follows from the other. Therefore, the problem to prove A is equivalent to the problem to prove B.
loc. 21442150
advantage in introducing, as an auxiliary problem, the problem to prove B. The theorem B is a little easier to prove than A and, what is more important, we may foresee that B is easier than A, we may judge so, we may find plausible from the outset that B is easier than A. In fact, the theorem B, concerned only with angles, is more “homogeneous” than the theorem A which is concerned with both angles and sides. The passage from the original problem to the auxiliary problem is called convertible reduction, or bilateral reduction, or equivalent reduction if these two problems, the original and the auxiliary, are equivalent. Thus, the reduction of A to B (see above) is convertible and so is the reduction in example 1. Convertible reductions are, in a certain respect, more important and more desirable than other ways to introduce auxiliary problems, but auxiliary problems which are not equivalent to the original problem may also be very useful; take example 2.
loc. 21512165
Chains of equivalent auxiliary problems are frequent in mathematical reasoning. We are required to solve a problem A; we cannot see the solution, but we may find that A is equivalent to another problem B. Considering B we may run into a third problem C equivalent to B. Proceeding in the same way, we reduce C to D, and so on, until we come upon a last problem L whose solution is known or immediate. Each problem being equivalent to the preceding, the last problem L must be equivalent to our original problem A. Thus we are able to infer the solution of the original problem A from the problem L which we attained as the last link in a chain of auxiliary problems.
loc. 21662173
If, from a proposed problem, we pass either to a more ambitious or to a less ambitious auxiliary problem we call the step a unilateral reduction. There are two kinds of unilateral reduction, and both are, in some way or other, more risky than a bilateral or convertible reduction.
loc. 22252229
We may test this result by SPECIALIZATION. In fact, if b = a the frustum becomes a prism and the formula yields a2h; and if b = 0 the frustum becomes a pyramid and the formula yields . We may apply the TEST BY DIMENSION. In fact, the expression has as dimension the cube of a length. Again, we may test the formula by variation of the data. In fact, if any one of the positive quantities a, b or h increases the value of the expression increases.
loc. 23092326
Tests of this sort can be applied not only to the final result but also to intermediate results. They are so useful that it is worth while preparing for them; see VARIATION OF THE PROBLEM, 4. In order to be able to use such tests, we may find advantage in generalizing a “problem in numbers” and changing it into a “problem in letters”; see GENERALIZATION
loc. 23262332
When the solution that we have finally obtained is long and involved, we naturally suspect that there is some clearer and less roundabout solution: Can you derive the result differently?
loc. 23542356
To find a new problem which is both interesting and accessible, is not so easy;
loc. 24442444
Good problems and mushrooms of certain kinds have something in common; they grow in clusters. Having found one, you should look around; there is a good chance that there are some more quite near.
loc. 24462448
Given the three dimensions (length, breadth, and height) of a rectangular parallelepiped, find the diagonal. If we know the solution of this problem, we can easily solve any of the following problems (of which the first two were almost stated in section 14). Given the three dimensions of a rectangular parallelepiped, find the radius of the circumscribed sphere. The base of a pyramid is a rectangle of which the center is the foot of the altitude of the pyramid. Given the altitude of the pyramid and the sides of its base, find the lateral edges. Given the rectangular coordinates (x1, y1, z1), (x2, y2, z2) of two points in space, find the distance of these points.
loc. 24512469
In fact, the solution of our original problem consists essentially in establishing a relation among four quantities, the three dimensions of the parallelepiped and its diagonal. If any three of these four quantities are given, we can calculate the fourth from the relation. Thus, we can solve the new problem.
loc. 24792482
We have here a pattern to derive easily solvable new problems from a problem we have solved: we regard the original unknown as given and one of the original data as unknown. The relation connecting the unknown and the data is the same in both problems, the old and the new. Having found this relation in one, we can use it also in the other.
loc. 24832487
The mathematical experience of the student is incomplete if he never had an opportunity to solve a problem invented by himself.
loc. 25122514
1. We may use provisional and merely plausible arguments when devising the final and rigorous argument as we use scaffolding to support a bridge during construction. When, however, the work is sufficiently advanced we take off the scaffolding, and the bridge should be able to stand by itself. In the same way, when the solution is sufficiently advanced, we brush aside all kinds of provisional and merely plausible arguments, and the result should be supported by rigorous argument alone.
loc. 25232527
The order in which we work out the details of the argument may be very different from the order in which we invented them; and the order in which we write down the details in a definitive exposition may be still different. Euclid’s Elements present the details of the argument in a rigid systematic order which was often imitated and often criticized.
loc. 25422545
3. In Euclid’s exposition all arguments proceed in the same direction: from the data toward the unknown in “problems to find,” and from the hypothesis toward the conclusion in “problems to prove.” Any new element, point, line, etc., has to be correctly derived from the data or from elements correctly derived in foregoing steps. Any new assertion has to be correctly proved from the hypothesis or from assertions correctly proved in foregoing steps. Each new element, each new assertion is examined when it is encountered first, and so it has to be examined just once; we may concentrate all our attention upon the present step, we need not look behind us, or look ahead. The very last new element whose derivation we have to check, is the unknown. The very last assertion whose proof we have to examine, is the conclusion. If each step is correct, also the last one, the whole argument is correct.
loc. 25462556
The Euclidean way of exposition can be highly recommended, without reservation, if the purpose is to examine the argument in detail. Especially, if it is our own argument, and it is long and complicated, and we have not only found it but have also surveyed it on large lines so that nothing is left but to examine each particular point in itself, then nothing is better than to write out the whole argument in the Euclidean way. The Euclidean way of exposition, however, cannot be recommended without reservation if the purpose is to convey an argument to a reader or to a listener who never heard of it before. The Euclidean exposition is excellent to show each particular point but not so good to show the main line of the argument. THE INTELLIGENT READER can easily see that each step is correct but has great difficulty in perceiving the source, the purpose, the connection of the whole argument. The reason for this difficulty is that the Euclidean exposition fairly often proceeds in an order exactly opposite to the natural order of invention. (Euclid’s exposition follows rigidly the order of “synthesis”; see PAPPUS, especially comments 3, 4, 5.)
loc. 25562568
Carrying out our plan, we check each step. Checking our step, we may rely on intuitive insight or on formal rules. Sometimes the intuition is ahead, sometimes the formal reasoning. It is an interesting and useful exercise to do it both ways. Can you see clearly that the step is correct? Yes, I can see it clearly and distinctly. Intuition is ahead; but could formal reasoning overtake it? Can you also PROVE that it is correct? Trying to prove formally what is seen intuitively and to see intuitively what is proved formally is an invigorating mental exercise. Unfortunately, in the classroom there is not always enough time for it. The example, discussed in sections 12 and 14, is typical in this respect.
loc. 25972608
A condition is called redundant if it contains superfluous parts. It is called contradictory if its parts are mutually opposed and inconsistent so that there is no object satisfying the condition. Thus, if a condition is expressed by more linear equations than there are unknowns, it is either redundant or contradictory; if the condition is expressed by fewer equations than there are unknowns, it is insufficient to determine the unknowns; if the condition is expressed by just as many equations as there are unknowns it is usually just sufficient to determine the unknowns but may be, in exceptional cases, contradictory or insufficient.
loc. 26142622
Look at the unknown! And try to think of a familiar problem having the same or a similar unknown. This suggests starting the work from the unknown. Look at the data! Could you derive something useful from the data? This suggests starting the work from the data. It appears that starting the reasoning from the unknown is usually preferable (see PAPPUS and WORKING BACKWARDS). Yet the alternative start, from the data, also has chances of success, must often be tried, and deserves illustration.
loc. 26362646
Decomposing and recombining are important operations of the mind. You examine an object that touches your interest or challenges your curiosity: a house you intend to rent, an important but cryptic telegram, any object whose purpose and origin puzzle you, or any problem you intend to solve. You have an impression of the object as a whole but this impression, possibly, is not definite enough. A detail strikes you, and you focus your attention upon it. Then, you concentrate upon another detail; then, again, upon another. Various combinations of details may present themselves and after a while you again consider the object as a whole but you see it now differently. You decompose the whole into its parts, and you recombine the parts into a more or less different whole.
loc. 27182727
Therefore, let us, first of all, understand the problem as a whole. Having understood the problem, we shall be in a better position to judge which particular points may be the most essential. Having examined one or two essential points we shall be in a better position to judge which further details might deserve closer examination. Let us go into detail and decompose the problem gradually, but not further than we need to.
loc. 27342738
Could you change the unknown, or the data, or both if necessary, so that the new unknown and the new data are nearer to each other?
loc. 28352836
An interesting way of changing both the unknown and the data is interchanging the unknown with one of the data.
loc. 28362837
The procedure that we have just applied has a certain interest; solving problems of geometric construction, we can often follow successfully its pattern: Reduce the problem to the construction of a point, and construct the point as an intersection of two loci.
loc. 29202922
Keep only a part of the condition, drop the other part. Doing so, we weaken the condition of the proposed problem, we restrict less the unknown.
loc. 29242926
Doing so, we weaken the condition of the proposed problem, we restrict less the unknown. How far
loc. 29252926
Definition of a term is a statement of its meaning in other terms which are supposed to be well known.
loc. 30063007
1. Technical terms in mathematics are of two kinds. Some are accepted as primitive terms and are not defined. Others are considered as derived terms and are defined in due form; that is, their meaning is stated in primitive terms and in formerly defined derived terms. Thus, we do not give a formal definition of such primitive notions as point, straight line, and plane.3 Yet we give formal definitions of such notions as “bisector of an angle” or “circle” or “parabola.”
loc. 30083017
“Construct a point P on the given straight line c at equal distances from the given point F and the given straight line d.” “Observe the progress from the original statement to your restatement. The original statement of the problem was full of unfamiliar technical terms, parabola, focus, directrix; it sounded just a little pompous and inflated. And now, nothing remains of those unfamiliar technical terms; you have deflated the problem. Well done!”
loc. 31143124
4. Elimination of technical terms is the result of the work in the foregoing example. We started from a statement of the problem containing certain technical terms (parabola, focus, directrix) and we arrived finally at a restatement free of those terms. In order to eliminate a technical term we
loc. 31263130
In order to eliminate a technical term we must know its definition; but it is not enough to know the definition, we must use it.
loc. 31293130
5. Definitions and known theorems. If we know the name “parabola” and have some vague idea of the shape of the curve but do not know anything else about it, our knowledge is obviously insufficient to solve the problem proposed as example, or any other serious geometric problem about the parabola. What kind of knowledge is needed for such a purpose? The science of geometry may be considered as consisting of axioms, definitions, and theorems. The parabola is not mentioned in the axioms which deal only with such primitive terms as point, straight line, and so on. Any geometric argumentation concerned with the parabola, the solution of any problem involving it, must use either its definition or theorems about it. To solve such a problem, we must know, at least, the definition but it is better to know some theorems too.
loc. 31463156
“You can undertake without hope and persevere without success.” Thus may speak an inflexible will, or honor and duty, or a nobleman with a noble cause. This sort of determination, however, would not do for the scientist, who should have some hope to start with, and some success to go on. In scientific work, it is necessary to apportion wisely determination to outlook. You do not take up a problem, unless it has some interest; you settle down to work seriously if the problem seems instructive; you throw in your whole personality if there is a great promise. If your purpose is set, you stick to it, but you do not make it unnecessarily difficult for yourself. You do not despise little successes, on the contrary, you seek them: If you cannot solve the proposed problem try to solve first some related problem.
loc. 32343243
With respect to devising a plan and obtaining a general idea of the solution two opposite faults are frequent. Some students rush into calculations and constructions without any plan or general idea; others wait clumsily for some idea to come and cannot do anything that would accelerate its coming.
loc. 32663269
That is, if we discard any part of the hypothesis, the theorem ceases to be true. Therefore, if the proof neglects to use any part of the hypothesis, the proof must be wrong. Does the proof use the whole hypothesis?
loc. 33113314
Many a guess has turned out to be wrong but nevertheless useful in leading to a better one. No idea is really bad, unless we are uncritical. What is really bad is to have no idea at all.
loc. 33883390
If we decide ourselves to examine this statement, the situation changes. Originally, we had a “problem to find.” After having formulated our guess, we have a “problem to prove”; we have to prove or disprove the theorem formulated.
loc. 34353437
Notes: 1) WEAKER PROBLEM
This theorem appears more accessible than the former; it is, of course, weaker.
loc. 34443445
even if your problem is not a problem of geometry, you may try to draw a figure. To find a lucid geometric representation for your nongeometrical problem could be an important step toward the solution.
loc. 36263629
1. If, by some chance, we come across the sum 1 + 8 + 27 + 64 = 100 we may observe that it can be expressed in the curious form 13 + 23 + 33 + 43 = 102. Now, it is natural to ask ourselves: Does it often happen that a sum of successive cubes as 13 + 23 + 33 + · · · + n3 is a square? In asking this, we generalize. This generalization is a lucky one; it leads from one observation to a remarkable general law. Many results were found by lucky generalizations in mathematics, physics, and the natural sciences. See INDUCTION AND MATHEMATICAL INDUCTION.
loc. 36333653
The more general problem may be easier to solve.
loc. 36713671
In mathematics as in the physical sciences we may use observation and induction to discover general laws. But there is a difference. In the physical sciences, there is no higher authority than observation and induction but in mathematics there is such an authority: rigorous proof.
loc. 39143917
The foregoing proof may serve as a pattern in many similar cases. What are the essential lines of this pattern? The assertion we have to prove must be given in advance, in precise form. The assertion must depend on an integer n. The assertion must be sufficiently “explicit” so that we have some possibility of testing whether it remains true in the passage from n to the next integer n + 1.
loc. 39954003
If we succeed in testing this effectively, we may be able to use our experience, gained in the process of testing, to conclude that the assertion must be true for n + 1 provided it is true for n. When we are so far it is sufficient to know that the assertion is true for n = 1; hence it follows for n = 2; hence it follows for n = 3. and so on; passing from any integer to the next, we prove the assertion generally.
loc. 40034012
Unfortunately, the accepted technical term is “mathematical induction.” This name results from a random circumstance. The precise assertion that we have to prove may come from any source, and it is immaterial from the logical viewpoint what the source is. Now, in many cases, as in the case we discussed here in detail, the source is induction, the assertion is found experimentally, and so the proof appears as a mathematical complement to induction; this explains the name.
loc. 40174021
The precise assertion that we have to prove may come from any source, and it is immaterial from the logical viewpoint what the source is. Now, in many cases, as in the case we discussed here in detail, the source is induction, the assertion is found experimentally, and so the proof appears as a mathematical complement to induction; this explains the name.
loc. 40184021
Inventor’s paradox. The more ambitious plan may have more chances of success. This sounds paradoxical. Yet, when passing from one problem to another, we may often observe that the new, more ambitious problem is easier to handle than the original problem. More questions may be easier to answer than just one question. The more comprehensive theorem may be easier to prove, the more general problem may be easier to solve.
loc. 40394044
Lemma means “auxiliary theorem.” The word is of Greek origin; a more literal translation would be “what is assumed.” We are trying to prove a theorem, say, A. We are led to suspect another theorem, say, B; if B were true we could perhaps, using it, prove A. We assume B provisionally, postponing its proof, and go ahead with the proof of A. Such a theorem B is assumed, and is an auxiliary theorem to the originally proposed theorem A. Our little story is fairly typical and explains the present meaning of the word “lemma.”
loc. 40824097
To know or not to know a formerly solved problem with the same unknown may make all the difference between an easy and a difficult problem.
loc. 41964197
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 on teaching, especially on the teaching of mathematics.
loc. 42524256
2. Some mathematical symbols, as +, −, =, and several others, have a fixed traditional meaning, but other symbols, as the small and capital letters of the Roman and Greek alphabets, are used in different meanings in different problems. When we face a new problem, we must choose certain symbols, we have to introduce suitable notation. There is something analogous in the use of ordinary language. Many words are used in different meanings in different contexts; when precision is important, we have to choose our words carefully. An important step in solving a problem is to choose the notation. It should be done carefully. The time we spend now on choosing the notation may be well repaid by the time we save later by avoiding hesitation and confusion. Moreover, choosing the notation carefully, we have to think sharply of the elements of the problem which must be denoted. Thus, choosing a suitable notation may contribute essentially to understanding the problem.
loc. 44074419
An important step in solving a problem is to choose the notation. It should be done carefully. The time we spend now on choosing the notation may be well repaid by the time we save later by avoiding hesitation and confusion. Moreover, choosing the notation carefully, we have to
loc. 44154418
“Now analysis is of two kinds; the one is the analysis of the ‘problems to prove’ and aims at establishing true theorems; the other is the analysis of the ‘problems to find’ and aims at finding the unknown.
loc. 46384639
which transforms the condition into 8z2 − 54z + 85 = 0. Here the analysis ends, provided that the problemsolver is acquainted with the solution of quadratic equations. What is the synthesis? Carrying through, step by step, the calculations whose possibility was foreseen by the analysis. The problemsolver needs no new idea to finish his problem, only some patience and attention in calculating the various unknowns. The order of calculation is opposite to the order of invention; first z is found (z = 5/2, 17/4), then y (y = 2, 1/2, 4, 1/4), and finally the originally required x (x = 1, −1, 2, −2). The synthesis retraces the steps of the analysis, and it is easy to see in the present case why it does so.
loc. 47324750
Here the analysis ends, provided that the problemsolver is acquainted with the solution of quadratic equations. What is the synthesis? Carrying through, step by step, the calculations whose possibility was foreseen by the analysis. The problemsolver needs no new idea to finish his problem, only some patience and attention in calculating the various unknowns.
loc. 47374740
4. Nonmathematical illustration. A primitive man wishes to cross a creek; but he cannot do so in the usual way because the water has risen overnight. Thus, the crossing becomes the object of a problem; “crossing the creek” is the x of this primitive problem. The man may recall that he has crossed some other creek by walking along a fallen tree. He looks around for a suitable fallen tree which becomes his new unknown, his y. He cannot find any suitable tree but there are plenty of trees standing along the creek; he wishes that one of them would fall. Could he make a tree fall across the creek? There is a great idea and there is a new unknown; by what means could he tilt the tree over the creek? This train of ideas ought to be called analysis if we accept the terminology of Pappus. If the primitive man succeeds in finishing his analysis he may become the inventor of the bridge and of the axe. What will be the synthesis? Translation of ideas into actions. The finishing act of the synthesis is walking along a tree across the creek.
loc. 47514766
the analysis consists in thoughts, the synthesis in acts. There is another difference; the order is reversed. Walking across the creek is the first desire from which the analysis starts and it is the last act with which the synthesis ends.
loc. 47684770
Analysis comes naturally first, synthesis afterwards; analysis is invention, synthesis, execution; analysis is devising a plan, synthesis carrying through the plan.
loc. 47744776
The paraphrase preserves and even emphasizes certain curious phrases of the original: “assume what is required to be done as already done, what is sought as found, what you have to prove as true.”
loc. 47774779
The paraphrase uses twice the important phrase “provided that all our derivations are convertible”;
loc. 48134813
You should ask no question, make no suggestion, indiscriminately, following some rigid habit.
loc. 48514851
1. An impressive practical problem is the construction of a dam across a river. We need no special knowledge to understand this problem. In almost prehistoric times, long before our modern age of scientific theories, men built dams of some sort in the valley of the Nile, and in other parts of the world, where the crops depended on irrigation. Let us visualize the problem of constructing an important modern dam. What is the unknown? Many unknowns are involved in a problem of this kind: the exact location of the dam, its geometric shape and dimensions, the materials used in its construction, and so on. What is the condition? We cannot answer this question in one short sentence because there are many conditions. In so large a project it is necessary to satisfy many important economic needs and to hurt other needs as little as possible. The dam should provide electric power, supply water for irrigation or the use of certain communities, and also help to control floods. On the other hand, it should disturb as little as possible navigation, or economically important fishlife, or beautiful scenery; and so forth. And, of course, it should cost as little as possible and be constructed as quickly as possible. What are the data? The multitude of desirable data is tremendous. We need topographical data concerning the vicinity of the river and its tributaries; geological data important for the solidity of foundations, possible leakage, and available materials of construction; meteorological data about annual precipitation and the height of floods; economic data concerning the value of ground which will be flooded, cost of materials and labor; and so on. Our example shows that unknowns, data, and conditions are more complex and less sharply defined in a practical problem than in a mathematical problem.
loc. 48634884
2. In order to solve a problem, we need a certain amount of previously acquired knowledge. The modern engineer has a highly specialized body of knowledge at his disposal, a scientific theory of the strength of materials, his own experience, and the mass of engineering experience stored in special technical literature. We cannot avail ourselves of such special knowledge here but we may try to imagine what was in the mind of an ancient Egyptian dambuilder.
loc. 48844889
He has seen, of course, various other, perhaps smaller, dams: banks of earth or masonry holding back the water. He has seen the flood, laden with all sorts of debris, pressing against the bank. He might have helped to repair the cracks and the erosion left by the flood. He might have seen a dam break, giving way under the impact of the flood. He has certainly heard stories about dams withstanding the test of centuries or causing catastrophe by an unexpected break. His mind may have pictured the pressure of the river against the surface of the dam and the strain and stress in its interior. Yet the Egyptian dambuilder had no precise, quantitative, scientific concepts of fluid pressure or of strain and stress in a solid body. Such concepts form an essential part of the intellectual equipment of a modern engineer. Yet the latter also uses much knowledge which has not yet quite reached a precise, scientific level; what he knows about erosion by flowing water, the transportation of silt, the plasticity and other not quite clearly circumscribed properties of certain materials, is knowledge of a rather empirical character. Our example shows that the knowledge needed and the concepts used are more complex and less sharply defined in practical problems than in mathematical problems.
loc. 48894903
The data of his problem are, strictly speaking, inexhaustible. For instance, he would like to know a little more about the geologic nature of the ground on which the foundations must be laid, but eventually he must stop collecting geologic data although a certain margin of uncertainty unavoidably remains.
loc. 49264928
In setting up and in solving mathematical problems derived from practical problems, we usually content ourselves with an approximation.
loc. 49484950
We gain much in simplicity and do not lose a great deal in accuracy.
loc. 49614961
What is a bright idea? An abrupt and momentous change of our outlook, a sudden reorganization of our mode of conceiving the problem, a just emerging confident prevision of the steps we have to take in order to attain the solution.
loc. 51085109
Reductio ad absurdum shows the falsity of an assumption by deriving from it a manifest absurdity. “Reduction to an absurdity” is a mathematical procedure but it has some resemblance to irony which is the favorite procedure of the satirist. Irony adopts, to all appearance, a certain opinion and stresses it and overstresses it till it leads to a manifest absurdity.
loc. 51985201
Indirect proof establishes the truth of an assertion by showing the falsity of the opposite assumption. Thus, indirect proof has some resemblance to a politician’s trick of establishing a candidate by demolishing the reputation of his opponent.
loc. 52025205
If we wish to set up an equation, we have to express in mathematical language that all parts of the condition are satisfied, although we do not know yet whether it is actually possible to satisfy all these parts simultaneously.
loc. 52905292
In general, a problem is a “routine problem” if it can be solved either by substituting special data into a formerly solved general problem, or by following step by step, without any trace of originality, some wellworn conspicuous example.
loc. 54605461
Teaching the mechanical performance of routine mathematical operations and nothing else is well under the level of the cookbook because kitchen recipes do leave something to the imagination and judgment of the cook but mathematical recipes do not.
loc. 54685469
Rules of style. The first rule of style is to have something to say. The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time.
loc. 54865489
Rules of teaching. The first rule of teaching is to know what you are supposed to teach. The second rule of teaching is to know a little more than what you are supposed to teach.
loc. 54905492
Separate the various parts of the condition. Can you write them down? We often have opportunity to ask this question when we are SETTING UP EQUATIONS.
loc. 55105513
Setting up equations is like translation from one language into another (NOTATION, 1). This comparison, used by Newton in his Arithmetica Universalis, may help to clarify the nature of certain difficulties often felt both by students and by teachers.
loc. 55155520
In order to translate a sentence from English into French two things are necessary. First, we must understand thoroughly the English sentence. Second, we must be familiar with the forms of expression peculiar to the French language. The situation is very similar when we attempt to express in mathematical symbols a condition proposed in words. First, we must understand thoroughly the condition. Second, we must be familiar with the forms of mathematical expression.
loc. 55235528
An English sentence is relatively easy to translate into French if it can be translated word for word. But there are English idioms which cannot be translated into French word for word. If our sentence contains such idioms, the translation becomes difficult; we have to pay less attention to the separate words, and more attention to the whole meaning; before translating the sentence, we may have to rearrange it.
loc. 55295533
In all cases, easy or difficult, we have to understand the condition, to separate the various parts of the condition, and to ask: Can you write them down? In easy cases, we succeed without hesitation in dividing the condition into parts that can be written down in mathematical symbols; in difficult cases, the appropriate division of the condition is less obvious.
loc. 55405545
If you take a heuristic conclusion as certain, you may be fooled and disappointed; but if you neglect heuristic conclusions altogether you will make no progress at all.
loc. 57215722
In fact, to solve a problem is, essentially, to find the connection between the data and the unknown. Moreover
loc. 57315734
Indeed, analogy is one of the main sources of invention. If other means fail, we should try to imagine an analogous problem. Therefore, if such a problem emerges spontaneously, by its own accord, we naturally feel elated; we feel that we are approaching the solution.
loc. 57445747
Yes, signs may misguide us in any single case, but they guide us right in the majority of them. A hunter may misinterpret now and then the traces of his game but he must be right on the average, otherwise he could not make a living by hunting.
loc. 58085810
The signs that convince the inventor that his idea is good, the indications that guide us in our everyday affairs, the circumstantial evidence of the lawyer, the inductive evidence of the scientist, statistical evidence invoked in many and diverse subjects—all these kinds of evidence agree in two essential points. First, they do not have the certainty of a strict demonstration. Second, they are useful in acquiring essentially new knowledge, and even indispensable to any not purely mathematical or logical knowledge, to any knowledge concerned with the physical world. We could call the reasoning that underlies this kind of evidence “heuristic reasoning” or “inductive reasoning” or (if we wish to avoid stretching the meaning of existing terms) “plausible reasoning.” We accept here the last term.
loc. 58675875
The conclusion is not fully expressed and is not fully supported by the premises. The direction is expressed and is implied by the premises, the magnitude is not.
loc. 59145916
The direction is expressed and is implied by the premises, the magnitude is not. For any reasonable person, the premises involve that
loc. 59155917
If, however, we find that the general statement is verified even in the extreme case, the inductive evidence derived from this verification will be strong, just because the prospect of refutation was strong. Thus, we are tempted to reshape the saying from which we started: “Prospective exceptions test the rule.”
loc. 60136016
Now, whatever the data may be, the required solution must apply and we do not see yet how to fit the same solution to all these possibilities. Out of such feeling of “too much variety” this question and answer may eventually emerge: Could you imagine a more accessible related problem? A more special problem?
loc. 60616064
Mathematics being a very abstract science should be presented very concretely.
loc. 61356135
In a more general acceptance of the word, a whole is termed symmetric if it has interchangeable parts. There are many kinds of symmetry; they differ in the number of interchangeable parts, and in the operations which exchange the parts.
loc. 61816182
Symmetry, in a general sense, is important for our subject. If a problem is symmetric in some ways we may derive some profit from noticing its interchangeable parts and it often pays to treat those parts which play the same role in the same fashion
loc. 61906192
The dimension of a product is the product of the dimensions of its factors, and there is a similar rule about powers. Replacing the quantities by their dimensions on both sides of the formula that we are testing, we obtain
loc. 63126315
The future mathematician learns, as does everybody else, by imitation and practice. He should look out for the right model to imitate. He should observe a stimulating teacher. He should compete with a capable friend. Then, what may be the most important, he should read not only current textbooks but good authors till he finds one whose ways he is naturally inclined to imitate.
loc. 64146419
He should enjoy and seek what seems to him simple or instructive or beautiful. He should solve problems, choose the problems which are in his line, meditate upon their solution, and invent new problems. By these means, and by all other means, he should endeavor to make his first important discovery: he should discover his likes and his dislikes, his taste, his own line.
loc. 64206423
If he cannot summon up real desire for solving the problem he would do better to leave it alone. The open secret of real success is to throw your whole personality into your problem.
loc. 64416443
The open secret of real success is to throw your whole personality into your problem.
loc. 64426443
The intelligent listener to a mathematical lecture has the same wishes. If he cannot see that the present step of the argument is correct and even suspects that it is, possibly, incorrect, he may protest and ask a question. If he cannot see any purpose in the present step, nor suspect any reason for it, he usually cannot even formulate a clear objection, he does not protest, he is just dismayed and bored, and loses the thread of the argument.
loc. 64486452
The insect, the mouse, and the man follow it; but if one follows it with more success than the others it is because he varies his problem more intelligently.
loc. 64926494
If our work progresses, there is something to do, there are new points to examine, our attention is occupied, our interest is alive. But if we fail to make progress, our attention falters, our interest fades, we get tired of the problem, our thoughts begin to wander, and there is danger of losing the problem altogether. To escape from this danger we have to set ourselves a new question about the problem.
loc. 65176522
The new question unfolds untried possibilities of contact with our previous knowledge, it revives our hope of making useful contacts. The new question reconquers our interest by varying the problem, by showing some new aspect of it.
loc. 65226525
2. Logical system. Geometry, as presented in Euclid’s Elements, is not a mere collection of facts but a logical system. The axioms, definitions, and propositions are not listed in a random sequence but disposed in accomplished order. Each proposition is so placed that it can be based on the foregoing axioms, definitions, and propositions. We may regard the disposition of the propositions as Euclid’s main achievement and their logical system as the main merit of the Elements. Euclid’s geometry is not only a logical system but it is the first and greatest example of such a system, which other sciences have tried, and are still trying, to imitate. Should other sciences—especially those very far from geometry, as psychology, or jurisprudence—imitate Euclid’s rigid logic? This is a debatable question; but nobody can take part in the debate with competence who is not acquainted with the Euclidean system.
loc. 67276738
the facts must be presented in some connection and in some sort of system, since isolated items are laboriously acquired and easily forgotten.
loc. 67596760
Any sort of connection that unites the facts simply, naturally, and durably, is welcome here. The system need not be founded on logic, it must only be designed to aid the memory effectively; it must be what is called a mnemotechnic system.
loc. 67606763
If the calculus is presented according to modern standards of rigor, it demands proofs of a certain degree of difficulty and subtlety (“epsilonproofs”). But engineers study the calculus in view of its application and have neither enough time nor enough training or interest to struggle through long proofs or to appreciate subtleties. Thus, there is a strong temptation to cut out all the proofs. Doing so, however, we reduce the calculus to the level of the cookbook. The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. The cookbook may serve its purpose perfectly. In fact, it need not have any sort of logical or mnemotechnic system since recipes are written or printed and not retained in memory.
loc. 67776785
5. Incomplete proofs. The best way of handling the dilemma between too heavy proofs and the level of the cookbook may be to make reasonable use of incomplete proofs.
loc. 67916793
In short, incomplete proofs may be used as a sort of mnemotechnic device (but, of course, not as substitutes for complete proofs) when the aim is tolerable coherence of presentation and not strictly logical consistency. It is very dangerous to advocate incomplete proofs. Possible abuse, however, may be kept within bounds by a few rules. First, if a proof is incomplete, it must be indicated as such, somewhere and somehow. Second, an author or a teacher is not entitled to present an incomplete proof for a theorem unless he knows very well a complete proof for it himself. And it may be confessed that to present an incomplete proof in good taste is not easy at all.
loc. 68416848
There are many shrewd and some subtle remarks in proverbs but, obviously, there is no scientific system free of inconsistencies and obscurities in them. On the contrary, many a proverb can be matched with another proverb giving exactly opposite advice, and there is a great latitude of interpretation. It would be foolish to regard proverbs as an authoritative source of universally applicable wisdom but it would be a pity to disregard the graphic description of heuristic procedures provided by proverbs.
loc. 68586862
It is true, we have discovered the appropriate sequence in retrogressive order but all that is left to do is to reverse the process and start from the point which we reached last of all in the analysis (as Pappus says).
loc. 70677070
Notes:
1) 反方向,类似走迷宫
Going around an obstacle is what we do in solving any kind of problem; the experiment has a sort of symbolic value. The hen acted like people who solve their problem muddling through, trying again and again, and succeeding eventually by some lucky accident without much insight into the reasons for their success. The dog who scratched and jumped and barked before turning around solved his problem about as well as we did ours about the two containers.
loc. 71307133
回应 20190527 19:45 
旺三 (一蓑烟雨任平生。)
一、理解问题 第一步：你必须理解问题。 什么是未知的？数据有哪些？条件是什么？ 是否可能满足条件？条件是否足够决定未知？或是不充分？或是冗余？或是矛盾？ 画图，介绍适当的符号。 分离条件的不同部分，能否写下来？ 二、设计计划 第二步：找到数据和未知之间的联系，如果找不到某个中间的联，你可能被要求考虑副问题。你最终需要一个解决方案的计划。 你是否遇到过？是否看到过形式稍有不同的同样的问题？ 是否知道一个相...20190526 19:36
一、理解问题
第一步：你必须理解问题。
什么是未知的？数据有哪些？条件是什么？
是否可能满足条件？条件是否足够决定未知？或是不充分？或是冗余？或是矛盾？
画图，介绍适当的符号。
分离条件的不同部分，能否写下来？
二、设计计划
第二步：找到数据和未知之间的联系，如果找不到某个中间的联，你可能被要求考虑副问题。你最终需要一个解决方案的计划。
你是否遇到过？是否看到过形式稍有不同的同样的问题？
是否知道一个相关的问题？是否知道一个用得上的定理？
看未知量，试着想一个有相同或相似未知量的熟悉的问题。
如果有一个和你的问题相关且解决过的问题，你是否能用上？是否能用它的结果？是否能用它的方法？为了使它用得上，是否要引入一些辅助元素？
是否能重新描述问题？能否再差异化地描述它？回到定义上。
如果你不能解决提出的问题，试着先解决一些相关的问题。能否想象一个更好理解的相关问题？一个更通用的问题？一个更特别的问题？一个类似的问题？你能否解决部分问题？只保留部分条件，放下其它部分；未知量还要多远才能决定，能有多少不同？能否从数据得出一些有用的东西？能否想到别的适合决定未知量的数据？能否更改未知量或数据，或必要时两者，使新未知量和新数据离得更近？是否使用了所有数据？是否使用了所有条件？是否考虑了问题涉及的所有必要的概念？
三、执行计划
步骤三：执行计划
执行解决方案的计划，检查每一步。你能否清楚看出该步骤是证券的？能否证明它是正确的？
四、回看
步骤四：检验获得的解决方案。
能否检查你的结果？能否检查参数？
能否用不同地方法得出结果？能否一眼看出来？
能否将结果或方法用于其它问题？
回应 20190526 19:36

simoncos (给岁月以生命。)
First, we have to understand the problem; we have to see clearly what is required. Second, we have to see how the various items are connected, how the unknown is linked to the data, in order to obtain the idea of the solution, to make a plan. Third, we carry out our plan. Fourth, we look back at the completed solution, we review and discuss it. loc. 712720 The worst may happen if the student e...20190527 19:45
First, we have to understand the problem; we have to see clearly what is required. Second, we have to see how the various items are connected, how the unknown is linked to the data, in order to obtain the idea of the solution, to make a plan. Third, we carry out our plan. Fourth, we look back at the completed solution, we review and discuss it.
loc. 712720
The worst may happen if the student embarks upon computations or constructions without having understood the problem.
loc. 725726
The teacher can make the problem interesting by making it concrete.
loc. 771772
Good ideas are based on past experience and formerly acquired knowledge.
loc. 813814
The materials necessary for solving a mathematical problem are certain relevant items of our formerly acquired mathematical knowledge, as formerly solved problems, or formerly proved theorems. Thus, it is often appropriate to start the work with the question: Do you know a related problem?
loc. 816819
The difficulty is that there are usually too many problems which are somewhat related to our present problem, that is, have some point in common with it. How can we choose the one, or the few, which are really useful? There is a suggestion that puts our finger on an essential common point: Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.
loc. 820825
The foregoing questions, well understood and seriously considered, very often help to start the right train of ideas; but they cannot help always, they cannot work magic. If they do not work, we must look around for some other appropriate point of contact, and explore the various aspects of our problem; we have to vary, to transform, to modify the problem. Could you restate the problem? Some of the questions of our list hint specific means to vary the problem, as generalization, specialization, use of analogy, dropping a part of the condition, and so on; the details are important but we cannot go into them now. Variation of the problem may lead to some appropriate auxiliary problem: If you cannot solve the proposed problem try to solve first some related problem.
loc. 829838
11. Carrying out the plan. To devise a plan, to conceive the idea of the solution is not easy. It takes so much to succeed; formerly acquired knowledge, good mental habits, concentration upon the purpose, and one more thing: good luck. To carry out the plan is much easier; what we need is mainly patience.
loc. 904908
We may convince ourselves of the correctness of a step in our reasoning either “intuitively” or “formally.”
loc. 918919
In certain cases, the teacher may emphasize the difference between “seeing” and “proving”: Can you see clearly that the step is correct? But can you also prove that the step is correct?
loc. 924926
The students will find looking back at the solution really interesting if they have made an honest effort, and have the consciousness of having done well. Then they are eager to see what else they could accomplish with that effort, and how they could do equally well another time. The teacher should encourage the students to imagine cases in which they could utilize again the procedure used, or apply the result obtained. Can you use the result, or the method, for some other problem?
loc. 9961002
The teacher can ask several questions about the result which the students may readily answer with “Yes”; but an answer “No” would show a serious flaw in the result.
loc. 10131014
“Did you use all the data? Do all the data a, b, c appear in your formula for the diagonal?”
“Length, width, and height play the same role in our question; our problem is symmetric with respect to a, b, c. Is the expression you obtained for the diagonal symmetric in a, b, c? Does it remain unchanged when a, b, c are interchanged?”
“Our problem is a problem of solid geometry: to find the diagonal of a parallelepiped with given dimensions a, b, c. Our problem is analogous to a problem of plane geometry: to find the diagonal of a rectangle with given dimensions a, b. Is the result of our ‘solid’ problem analogous to the result of the ‘plane’ problem?”
“If the height c decreases, and finally vanishes, the parallelepiped becomes a parallelogram. If you put c = 0 in your formula, do you obtain the correct formula for the diagonal of the rectangular parallelogram?” “If the height c increases, the diagonal increases. Does your formula show this?”
“If all three measures a, b, c of the parallelepiped increase in the same proportion, the diagonal also increases in the same proportion. If, in your formula, you substitute 12a, 12b, 12c for a, b, c respectively, the expression of the diagonal, owing to this substitution, should also be multiplied by 12. Is that so?”
“If a, b, c are measured in feet, your formula gives the diagonal measured in feet too; but if you change all measures into inches, the formula should remain correct. Is that so?” (The two last questions are essentially equivalent; see TEST BY DIMENSION.)
loc. 10151051
Let us go back to the situation as it presented itself at the beginning of section 10 when the question was asked: Do you know a related problem? Instead of this, with the best intention to help the students, the question may be offered: Could you apply the theorem of Pythagoras? The intention may be the best, but the question is about the worst. We must realize in what situation it was offered; then we shall see that there is a long sequence of objections against that sort of “help.”
(1) If the student is near to the solution, he may understand the suggestion implied by the question; but if he is not, he quite possibly will not see at all the point at which the question is driving. Thus the question fails to help where help is most needed.
(2) If the suggestion is understood, it gives the whole secret away, very little remains for the student to do.
(3) The suggestion is of too special a nature. Even if the student can make use of it in solving the present problem, nothing is learned for future problems. The question is not instructive.
(4) Even if he understands the suggestion, the student can scarcely understand how the teacher came to the idea of putting such a question. And how could he, the student, find such a question by himself? It appears as an unnatural surprise, as a rabbit pulled out of a hat; it is really not instructive.
loc. 11741190
What can I do? Make your grasp quite secure. Carry through in detail all the algebraic or geometric operations which you have recognized previously as feasible. Convince yourself of the correctness of each step by formal reasoning, or by intuitive insight, or both ways if you can. If your problem is very complex you may distinguish “great” steps and “small” steps, each great step being composed of several small ones. Check first the great steps, and get down to the smaller ones afterwards.
loc. 16161619
Analogy is used on very different levels. People often use vague, ambiguous, incomplete, or incompletely clarified analogies, but analogy may reach the level of mathematical precision.
loc. 16631665
But two questions may be easier to answer than just one question—provided that the two questions are intelligently connected.
loc. 16811682
The feeling that harmonious simple order cannot be deceitful guides the discoverer both in the mathematical and in the other sciences, and is expressed by the Latin saying: simplex sigillum veri (simplicity is the seal of truth).
loc. 18551857
There is a oneone correspondence between the objects of the two systems S and S′, preserving certain relations. That is, if such a relation holds between the objects of one system, the same relation holds between the corresponding objects of the other system. Such a connection between two systems is a very precise sort of analogy; it is called isomorphism (or holohedral isomorphism).
loc. 18801886
There is a onemany correspondence between the objects of the two systems S and S′ preserving certain relations. Such a connection (which is important in various branches of advanced mathematical study, especially in the Theory of Groups, and need not be discussed here in detail) is called merohedral isomorphism (or homomorphism; homoiomorphism would be, perhaps, a better term). Merohedral isomorphism may be considered as another very precise sort of analogy.]
loc. 18861892
If a tricky auxiliary line appears abruptly in the figure, without any motivation, and solves the problem surprisingly, intelligent students and readers are disappointed; they feel that they are cheated.
loc. 20492050
3. Profit. The profit that we derive from the consideration of an auxiliary problem may be of various kinds. We may use the result of the auxiliary problem. Thus, in example 1, having found by solving the quadratic equation for y that y is equal to 4 or to 9, we infer that x2 = 4 or x2 = 9 and derive hence all possible values of x. In other cases, we may use the method of the auxiliary problem. Thus, in example 2, the auxiliary problem is a problem of plane geometry; it is analogous to, but simpler than, the original problem which is a problem of solid geometry. It is reasonable to introduce an auxiliary problem of this kind in the hope that it will be instructive, that it will give us opportunity to familiarize ourselves with certain methods, operations, or tools, which we may use afterwards for our original problem.
loc. 21002117
Consider the following theorems: A. In any equilateral triangle, each angle is equal to 60°. B. In any equiangular triangle, each angle is equal to 60°. These two theorems are not identical. They contain different notions; one is concerned with equality of the sides, the other with equality of the angles of a triangle. But each theorem follows from the other. Therefore, the problem to prove A is equivalent to the problem to prove B.
loc. 21442150
advantage in introducing, as an auxiliary problem, the problem to prove B. The theorem B is a little easier to prove than A and, what is more important, we may foresee that B is easier than A, we may judge so, we may find plausible from the outset that B is easier than A. In fact, the theorem B, concerned only with angles, is more “homogeneous” than the theorem A which is concerned with both angles and sides. The passage from the original problem to the auxiliary problem is called convertible reduction, or bilateral reduction, or equivalent reduction if these two problems, the original and the auxiliary, are equivalent. Thus, the reduction of A to B (see above) is convertible and so is the reduction in example 1. Convertible reductions are, in a certain respect, more important and more desirable than other ways to introduce auxiliary problems, but auxiliary problems which are not equivalent to the original problem may also be very useful; take example 2.
loc. 21512165
Chains of equivalent auxiliary problems are frequent in mathematical reasoning. We are required to solve a problem A; we cannot see the solution, but we may find that A is equivalent to another problem B. Considering B we may run into a third problem C equivalent to B. Proceeding in the same way, we reduce C to D, and so on, until we come upon a last problem L whose solution is known or immediate. Each problem being equivalent to the preceding, the last problem L must be equivalent to our original problem A. Thus we are able to infer the solution of the original problem A from the problem L which we attained as the last link in a chain of auxiliary problems.
loc. 21662173
If, from a proposed problem, we pass either to a more ambitious or to a less ambitious auxiliary problem we call the step a unilateral reduction. There are two kinds of unilateral reduction, and both are, in some way or other, more risky than a bilateral or convertible reduction.
loc. 22252229
We may test this result by SPECIALIZATION. In fact, if b = a the frustum becomes a prism and the formula yields a2h; and if b = 0 the frustum becomes a pyramid and the formula yields . We may apply the TEST BY DIMENSION. In fact, the expression has as dimension the cube of a length. Again, we may test the formula by variation of the data. In fact, if any one of the positive quantities a, b or h increases the value of the expression increases.
loc. 23092326
Tests of this sort can be applied not only to the final result but also to intermediate results. They are so useful that it is worth while preparing for them; see VARIATION OF THE PROBLEM, 4. In order to be able to use such tests, we may find advantage in generalizing a “problem in numbers” and changing it into a “problem in letters”; see GENERALIZATION
loc. 23262332
When the solution that we have finally obtained is long and involved, we naturally suspect that there is some clearer and less roundabout solution: Can you derive the result differently?
loc. 23542356
To find a new problem which is both interesting and accessible, is not so easy;
loc. 24442444
Good problems and mushrooms of certain kinds have something in common; they grow in clusters. Having found one, you should look around; there is a good chance that there are some more quite near.
loc. 24462448
Given the three dimensions (length, breadth, and height) of a rectangular parallelepiped, find the diagonal. If we know the solution of this problem, we can easily solve any of the following problems (of which the first two were almost stated in section 14). Given the three dimensions of a rectangular parallelepiped, find the radius of the circumscribed sphere. The base of a pyramid is a rectangle of which the center is the foot of the altitude of the pyramid. Given the altitude of the pyramid and the sides of its base, find the lateral edges. Given the rectangular coordinates (x1, y1, z1), (x2, y2, z2) of two points in space, find the distance of these points.
loc. 24512469
In fact, the solution of our original problem consists essentially in establishing a relation among four quantities, the three dimensions of the parallelepiped and its diagonal. If any three of these four quantities are given, we can calculate the fourth from the relation. Thus, we can solve the new problem.
loc. 24792482
We have here a pattern to derive easily solvable new problems from a problem we have solved: we regard the original unknown as given and one of the original data as unknown. The relation connecting the unknown and the data is the same in both problems, the old and the new. Having found this relation in one, we can use it also in the other.
loc. 24832487
The mathematical experience of the student is incomplete if he never had an opportunity to solve a problem invented by himself.
loc. 25122514
1. We may use provisional and merely plausible arguments when devising the final and rigorous argument as we use scaffolding to support a bridge during construction. When, however, the work is sufficiently advanced we take off the scaffolding, and the bridge should be able to stand by itself. In the same way, when the solution is sufficiently advanced, we brush aside all kinds of provisional and merely plausible arguments, and the result should be supported by rigorous argument alone.
loc. 25232527
The order in which we work out the details of the argument may be very different from the order in which we invented them; and the order in which we write down the details in a definitive exposition may be still different. Euclid’s Elements present the details of the argument in a rigid systematic order which was often imitated and often criticized.
loc. 25422545
3. In Euclid’s exposition all arguments proceed in the same direction: from the data toward the unknown in “problems to find,” and from the hypothesis toward the conclusion in “problems to prove.” Any new element, point, line, etc., has to be correctly derived from the data or from elements correctly derived in foregoing steps. Any new assertion has to be correctly proved from the hypothesis or from assertions correctly proved in foregoing steps. Each new element, each new assertion is examined when it is encountered first, and so it has to be examined just once; we may concentrate all our attention upon the present step, we need not look behind us, or look ahead. The very last new element whose derivation we have to check, is the unknown. The very last assertion whose proof we have to examine, is the conclusion. If each step is correct, also the last one, the whole argument is correct.
loc. 25462556
The Euclidean way of exposition can be highly recommended, without reservation, if the purpose is to examine the argument in detail. Especially, if it is our own argument, and it is long and complicated, and we have not only found it but have also surveyed it on large lines so that nothing is left but to examine each particular point in itself, then nothing is better than to write out the whole argument in the Euclidean way. The Euclidean way of exposition, however, cannot be recommended without reservation if the purpose is to convey an argument to a reader or to a listener who never heard of it before. The Euclidean exposition is excellent to show each particular point but not so good to show the main line of the argument. THE INTELLIGENT READER can easily see that each step is correct but has great difficulty in perceiving the source, the purpose, the connection of the whole argument. The reason for this difficulty is that the Euclidean exposition fairly often proceeds in an order exactly opposite to the natural order of invention. (Euclid’s exposition follows rigidly the order of “synthesis”; see PAPPUS, especially comments 3, 4, 5.)
loc. 25562568
Carrying out our plan, we check each step. Checking our step, we may rely on intuitive insight or on formal rules. Sometimes the intuition is ahead, sometimes the formal reasoning. It is an interesting and useful exercise to do it both ways. Can you see clearly that the step is correct? Yes, I can see it clearly and distinctly. Intuition is ahead; but could formal reasoning overtake it? Can you also PROVE that it is correct? Trying to prove formally what is seen intuitively and to see intuitively what is proved formally is an invigorating mental exercise. Unfortunately, in the classroom there is not always enough time for it. The example, discussed in sections 12 and 14, is typical in this respect.
loc. 25972608
A condition is called redundant if it contains superfluous parts. It is called contradictory if its parts are mutually opposed and inconsistent so that there is no object satisfying the condition. Thus, if a condition is expressed by more linear equations than there are unknowns, it is either redundant or contradictory; if the condition is expressed by fewer equations than there are unknowns, it is insufficient to determine the unknowns; if the condition is expressed by just as many equations as there are unknowns it is usually just sufficient to determine the unknowns but may be, in exceptional cases, contradictory or insufficient.
loc. 26142622
Look at the unknown! And try to think of a familiar problem having the same or a similar unknown. This suggests starting the work from the unknown. Look at the data! Could you derive something useful from the data? This suggests starting the work from the data. It appears that starting the reasoning from the unknown is usually preferable (see PAPPUS and WORKING BACKWARDS). Yet the alternative start, from the data, also has chances of success, must often be tried, and deserves illustration.
loc. 26362646
Decomposing and recombining are important operations of the mind. You examine an object that touches your interest or challenges your curiosity: a house you intend to rent, an important but cryptic telegram, any object whose purpose and origin puzzle you, or any problem you intend to solve. You have an impression of the object as a whole but this impression, possibly, is not definite enough. A detail strikes you, and you focus your attention upon it. Then, you concentrate upon another detail; then, again, upon another. Various combinations of details may present themselves and after a while you again consider the object as a whole but you see it now differently. You decompose the whole into its parts, and you recombine the parts into a more or less different whole.
loc. 27182727
Therefore, let us, first of all, understand the problem as a whole. Having understood the problem, we shall be in a better position to judge which particular points may be the most essential. Having examined one or two essential points we shall be in a better position to judge which further details might deserve closer examination. Let us go into detail and decompose the problem gradually, but not further than we need to.
loc. 27342738
Could you change the unknown, or the data, or both if necessary, so that the new unknown and the new data are nearer to each other?
loc. 28352836
An interesting way of changing both the unknown and the data is interchanging the unknown with one of the data.
loc. 28362837
The procedure that we have just applied has a certain interest; solving problems of geometric construction, we can often follow successfully its pattern: Reduce the problem to the construction of a point, and construct the point as an intersection of two loci.
loc. 29202922
Keep only a part of the condition, drop the other part. Doing so, we weaken the condition of the proposed problem, we restrict less the unknown.
loc. 29242926
Doing so, we weaken the condition of the proposed problem, we restrict less the unknown. How far
loc. 29252926
Definition of a term is a statement of its meaning in other terms which are supposed to be well known.
loc. 30063007
1. Technical terms in mathematics are of two kinds. Some are accepted as primitive terms and are not defined. Others are considered as derived terms and are defined in due form; that is, their meaning is stated in primitive terms and in formerly defined derived terms. Thus, we do not give a formal definition of such primitive notions as point, straight line, and plane.3 Yet we give formal definitions of such notions as “bisector of an angle” or “circle” or “parabola.”
loc. 30083017
“Construct a point P on the given straight line c at equal distances from the given point F and the given straight line d.” “Observe the progress from the original statement to your restatement. The original statement of the problem was full of unfamiliar technical terms, parabola, focus, directrix; it sounded just a little pompous and inflated. And now, nothing remains of those unfamiliar technical terms; you have deflated the problem. Well done!”
loc. 31143124
4. Elimination of technical terms is the result of the work in the foregoing example. We started from a statement of the problem containing certain technical terms (parabola, focus, directrix) and we arrived finally at a restatement free of those terms. In order to eliminate a technical term we
loc. 31263130
In order to eliminate a technical term we must know its definition; but it is not enough to know the definition, we must use it.
loc. 31293130
5. Definitions and known theorems. If we know the name “parabola” and have some vague idea of the shape of the curve but do not know anything else about it, our knowledge is obviously insufficient to solve the problem proposed as example, or any other serious geometric problem about the parabola. What kind of knowledge is needed for such a purpose? The science of geometry may be considered as consisting of axioms, definitions, and theorems. The parabola is not mentioned in the axioms which deal only with such primitive terms as point, straight line, and so on. Any geometric argumentation concerned with the parabola, the solution of any problem involving it, must use either its definition or theorems about it. To solve such a problem, we must know, at least, the definition but it is better to know some theorems too.
loc. 31463156
“You can undertake without hope and persevere without success.” Thus may speak an inflexible will, or honor and duty, or a nobleman with a noble cause. This sort of determination, however, would not do for the scientist, who should have some hope to start with, and some success to go on. In scientific work, it is necessary to apportion wisely determination to outlook. You do not take up a problem, unless it has some interest; you settle down to work seriously if the problem seems instructive; you throw in your whole personality if there is a great promise. If your purpose is set, you stick to it, but you do not make it unnecessarily difficult for yourself. You do not despise little successes, on the contrary, you seek them: If you cannot solve the proposed problem try to solve first some related problem.
loc. 32343243
With respect to devising a plan and obtaining a general idea of the solution two opposite faults are frequent. Some students rush into calculations and constructions without any plan or general idea; others wait clumsily for some idea to come and cannot do anything that would accelerate its coming.
loc. 32663269
That is, if we discard any part of the hypothesis, the theorem ceases to be true. Therefore, if the proof neglects to use any part of the hypothesis, the proof must be wrong. Does the proof use the whole hypothesis?
loc. 33113314
Many a guess has turned out to be wrong but nevertheless useful in leading to a better one. No idea is really bad, unless we are uncritical. What is really bad is to have no idea at all.
loc. 33883390
If we decide ourselves to examine this statement, the situation changes. Originally, we had a “problem to find.” After having formulated our guess, we have a “problem to prove”; we have to prove or disprove the theorem formulated.
loc. 34353437
Notes: 1) WEAKER PROBLEM
This theorem appears more accessible than the former; it is, of course, weaker.
loc. 34443445
even if your problem is not a problem of geometry, you may try to draw a figure. To find a lucid geometric representation for your nongeometrical problem could be an important step toward the solution.
loc. 36263629
1. If, by some chance, we come across the sum 1 + 8 + 27 + 64 = 100 we may observe that it can be expressed in the curious form 13 + 23 + 33 + 43 = 102. Now, it is natural to ask ourselves: Does it often happen that a sum of successive cubes as 13 + 23 + 33 + · · · + n3 is a square? In asking this, we generalize. This generalization is a lucky one; it leads from one observation to a remarkable general law. Many results were found by lucky generalizations in mathematics, physics, and the natural sciences. See INDUCTION AND MATHEMATICAL INDUCTION.
loc. 36333653
The more general problem may be easier to solve.
loc. 36713671
In mathematics as in the physical sciences we may use observation and induction to discover general laws. But there is a difference. In the physical sciences, there is no higher authority than observation and induction but in mathematics there is such an authority: rigorous proof.
loc. 39143917
The foregoing proof may serve as a pattern in many similar cases. What are the essential lines of this pattern? The assertion we have to prove must be given in advance, in precise form. The assertion must depend on an integer n. The assertion must be sufficiently “explicit” so that we have some possibility of testing whether it remains true in the passage from n to the next integer n + 1.
loc. 39954003
If we succeed in testing this effectively, we may be able to use our experience, gained in the process of testing, to conclude that the assertion must be true for n + 1 provided it is true for n. When we are so far it is sufficient to know that the assertion is true for n = 1; hence it follows for n = 2; hence it follows for n = 3. and so on; passing from any integer to the next, we prove the assertion generally.
loc. 40034012
Unfortunately, the accepted technical term is “mathematical induction.” This name results from a random circumstance. The precise assertion that we have to prove may come from any source, and it is immaterial from the logical viewpoint what the source is. Now, in many cases, as in the case we discussed here in detail, the source is induction, the assertion is found experimentally, and so the proof appears as a mathematical complement to induction; this explains the name.
loc. 40174021
The precise assertion that we have to prove may come from any source, and it is immaterial from the logical viewpoint what the source is. Now, in many cases, as in the case we discussed here in detail, the source is induction, the assertion is found experimentally, and so the proof appears as a mathematical complement to induction; this explains the name.
loc. 40184021
Inventor’s paradox. The more ambitious plan may have more chances of success. This sounds paradoxical. Yet, when passing from one problem to another, we may often observe that the new, more ambitious problem is easier to handle than the original problem. More questions may be easier to answer than just one question. The more comprehensive theorem may be easier to prove, the more general problem may be easier to solve.
loc. 40394044
Lemma means “auxiliary theorem.” The word is of Greek origin; a more literal translation would be “what is assumed.” We are trying to prove a theorem, say, A. We are led to suspect another theorem, say, B; if B were true we could perhaps, using it, prove A. We assume B provisionally, postponing its proof, and go ahead with the proof of A. Such a theorem B is assumed, and is an auxiliary theorem to the originally proposed theorem A. Our little story is fairly typical and explains the present meaning of the word “lemma.”
loc. 40824097
To know or not to know a formerly solved problem with the same unknown may make all the difference between an easy and a difficult problem.
loc. 41964197
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 on teaching, especially on the teaching of mathematics.
loc. 42524256
2. Some mathematical symbols, as +, −, =, and several others, have a fixed traditional meaning, but other symbols, as the small and capital letters of the Roman and Greek alphabets, are used in different meanings in different problems. When we face a new problem, we must choose certain symbols, we have to introduce suitable notation. There is something analogous in the use of ordinary language. Many words are used in different meanings in different contexts; when precision is important, we have to choose our words carefully. An important step in solving a problem is to choose the notation. It should be done carefully. The time we spend now on choosing the notation may be well repaid by the time we save later by avoiding hesitation and confusion. Moreover, choosing the notation carefully, we have to think sharply of the elements of the problem which must be denoted. Thus, choosing a suitable notation may contribute essentially to understanding the problem.
loc. 44074419
An important step in solving a problem is to choose the notation. It should be done carefully. The time we spend now on choosing the notation may be well repaid by the time we save later by avoiding hesitation and confusion. Moreover, choosing the notation carefully, we have to
loc. 44154418
“Now analysis is of two kinds; the one is the analysis of the ‘problems to prove’ and aims at establishing true theorems; the other is the analysis of the ‘problems to find’ and aims at finding the unknown.
loc. 46384639
which transforms the condition into 8z2 − 54z + 85 = 0. Here the analysis ends, provided that the problemsolver is acquainted with the solution of quadratic equations. What is the synthesis? Carrying through, step by step, the calculations whose possibility was foreseen by the analysis. The problemsolver needs no new idea to finish his problem, only some patience and attention in calculating the various unknowns. The order of calculation is opposite to the order of invention; first z is found (z = 5/2, 17/4), then y (y = 2, 1/2, 4, 1/4), and finally the originally required x (x = 1, −1, 2, −2). The synthesis retraces the steps of the analysis, and it is easy to see in the present case why it does so.
loc. 47324750
Here the analysis ends, provided that the problemsolver is acquainted with the solution of quadratic equations. What is the synthesis? Carrying through, step by step, the calculations whose possibility was foreseen by the analysis. The problemsolver needs no new idea to finish his problem, only some patience and attention in calculating the various unknowns.
loc. 47374740
4. Nonmathematical illustration. A primitive man wishes to cross a creek; but he cannot do so in the usual way because the water has risen overnight. Thus, the crossing becomes the object of a problem; “crossing the creek” is the x of this primitive problem. The man may recall that he has crossed some other creek by walking along a fallen tree. He looks around for a suitable fallen tree which becomes his new unknown, his y. He cannot find any suitable tree but there are plenty of trees standing along the creek; he wishes that one of them would fall. Could he make a tree fall across the creek? There is a great idea and there is a new unknown; by what means could he tilt the tree over the creek? This train of ideas ought to be called analysis if we accept the terminology of Pappus. If the primitive man succeeds in finishing his analysis he may become the inventor of the bridge and of the axe. What will be the synthesis? Translation of ideas into actions. The finishing act of the synthesis is walking along a tree across the creek.
loc. 47514766
the analysis consists in thoughts, the synthesis in acts. There is another difference; the order is reversed. Walking across the creek is the first desire from which the analysis starts and it is the last act with which the synthesis ends.
loc. 47684770
Analysis comes naturally first, synthesis afterwards; analysis is invention, synthesis, execution; analysis is devising a plan, synthesis carrying through the plan.
loc. 47744776
The paraphrase preserves and even emphasizes certain curious phrases of the original: “assume what is required to be done as already done, what is sought as found, what you have to prove as true.”
loc. 47774779
The paraphrase uses twice the important phrase “provided that all our derivations are convertible”;
loc. 48134813
You should ask no question, make no suggestion, indiscriminately, following some rigid habit.
loc. 48514851
1. An impressive practical problem is the construction of a dam across a river. We need no special knowledge to understand this problem. In almost prehistoric times, long before our modern age of scientific theories, men built dams of some sort in the valley of the Nile, and in other parts of the world, where the crops depended on irrigation. Let us visualize the problem of constructing an important modern dam. What is the unknown? Many unknowns are involved in a problem of this kind: the exact location of the dam, its geometric shape and dimensions, the materials used in its construction, and so on. What is the condition? We cannot answer this question in one short sentence because there are many conditions. In so large a project it is necessary to satisfy many important economic needs and to hurt other needs as little as possible. The dam should provide electric power, supply water for irrigation or the use of certain communities, and also help to control floods. On the other hand, it should disturb as little as possible navigation, or economically important fishlife, or beautiful scenery; and so forth. And, of course, it should cost as little as possible and be constructed as quickly as possible. What are the data? The multitude of desirable data is tremendous. We need topographical data concerning the vicinity of the river and its tributaries; geological data important for the solidity of foundations, possible leakage, and available materials of construction; meteorological data about annual precipitation and the height of floods; economic data concerning the value of ground which will be flooded, cost of materials and labor; and so on. Our example shows that unknowns, data, and conditions are more complex and less sharply defined in a practical problem than in a mathematical problem.
loc. 48634884
2. In order to solve a problem, we need a certain amount of previously acquired knowledge. The modern engineer has a highly specialized body of knowledge at his disposal, a scientific theory of the strength of materials, his own experience, and the mass of engineering experience stored in special technical literature. We cannot avail ourselves of such special knowledge here but we may try to imagine what was in the mind of an ancient Egyptian dambuilder.
loc. 48844889
He has seen, of course, various other, perhaps smaller, dams: banks of earth or masonry holding back the water. He has seen the flood, laden with all sorts of debris, pressing against the bank. He might have helped to repair the cracks and the erosion left by the flood. He might have seen a dam break, giving way under the impact of the flood. He has certainly heard stories about dams withstanding the test of centuries or causing catastrophe by an unexpected break. His mind may have pictured the pressure of the river against the surface of the dam and the strain and stress in its interior. Yet the Egyptian dambuilder had no precise, quantitative, scientific concepts of fluid pressure or of strain and stress in a solid body. Such concepts form an essential part of the intellectual equipment of a modern engineer. Yet the latter also uses much knowledge which has not yet quite reached a precise, scientific level; what he knows about erosion by flowing water, the transportation of silt, the plasticity and other not quite clearly circumscribed properties of certain materials, is knowledge of a rather empirical character. Our example shows that the knowledge needed and the concepts used are more complex and less sharply defined in practical problems than in mathematical problems.
loc. 48894903
The data of his problem are, strictly speaking, inexhaustible. For instance, he would like to know a little more about the geologic nature of the ground on which the foundations must be laid, but eventually he must stop collecting geologic data although a certain margin of uncertainty unavoidably remains.
loc. 49264928
In setting up and in solving mathematical problems derived from practical problems, we usually content ourselves with an approximation.
loc. 49484950
We gain much in simplicity and do not lose a great deal in accuracy.
loc. 49614961
What is a bright idea? An abrupt and momentous change of our outlook, a sudden reorganization of our mode of conceiving the problem, a just emerging confident prevision of the steps we have to take in order to attain the solution.
loc. 51085109
Reductio ad absurdum shows the falsity of an assumption by deriving from it a manifest absurdity. “Reduction to an absurdity” is a mathematical procedure but it has some resemblance to irony which is the favorite procedure of the satirist. Irony adopts, to all appearance, a certain opinion and stresses it and overstresses it till it leads to a manifest absurdity.
loc. 51985201
Indirect proof establishes the truth of an assertion by showing the falsity of the opposite assumption. Thus, indirect proof has some resemblance to a politician’s trick of establishing a candidate by demolishing the reputation of his opponent.
loc. 52025205
If we wish to set up an equation, we have to express in mathematical language that all parts of the condition are satisfied, although we do not know yet whether it is actually possible to satisfy all these parts simultaneously.
loc. 52905292
In general, a problem is a “routine problem” if it can be solved either by substituting special data into a formerly solved general problem, or by following step by step, without any trace of originality, some wellworn conspicuous example.
loc. 54605461
Teaching the mechanical performance of routine mathematical operations and nothing else is well under the level of the cookbook because kitchen recipes do leave something to the imagination and judgment of the cook but mathematical recipes do not.
loc. 54685469
Rules of style. The first rule of style is to have something to say. The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time.
loc. 54865489
Rules of teaching. The first rule of teaching is to know what you are supposed to teach. The second rule of teaching is to know a little more than what you are supposed to teach.
loc. 54905492
Separate the various parts of the condition. Can you write them down? We often have opportunity to ask this question when we are SETTING UP EQUATIONS.
loc. 55105513
Setting up equations is like translation from one language into another (NOTATION, 1). This comparison, used by Newton in his Arithmetica Universalis, may help to clarify the nature of certain difficulties often felt both by students and by teachers.
loc. 55155520
In order to translate a sentence from English into French two things are necessary. First, we must understand thoroughly the English sentence. Second, we must be familiar with the forms of expression peculiar to the French language. The situation is very similar when we attempt to express in mathematical symbols a condition proposed in words. First, we must understand thoroughly the condition. Second, we must be familiar with the forms of mathematical expression.
loc. 55235528
An English sentence is relatively easy to translate into French if it can be translated word for word. But there are English idioms which cannot be translated into French word for word. If our sentence contains such idioms, the translation becomes difficult; we have to pay less attention to the separate words, and more attention to the whole meaning; before translating the sentence, we may have to rearrange it.
loc. 55295533
In all cases, easy or difficult, we have to understand the condition, to separate the various parts of the condition, and to ask: Can you write them down? In easy cases, we succeed without hesitation in dividing the condition into parts that can be written down in mathematical symbols; in difficult cases, the appropriate division of the condition is less obvious.
loc. 55405545
If you take a heuristic conclusion as certain, you may be fooled and disappointed; but if you neglect heuristic conclusions altogether you will make no progress at all.
loc. 57215722
In fact, to solve a problem is, essentially, to find the connection between the data and the unknown. Moreover
loc. 57315734
Indeed, analogy is one of the main sources of invention. If other means fail, we should try to imagine an analogous problem. Therefore, if such a problem emerges spontaneously, by its own accord, we naturally feel elated; we feel that we are approaching the solution.
loc. 57445747
Yes, signs may misguide us in any single case, but they guide us right in the majority of them. A hunter may misinterpret now and then the traces of his game but he must be right on the average, otherwise he could not make a living by hunting.
loc. 58085810
The signs that convince the inventor that his idea is good, the indications that guide us in our everyday affairs, the circumstantial evidence of the lawyer, the inductive evidence of the scientist, statistical evidence invoked in many and diverse subjects—all these kinds of evidence agree in two essential points. First, they do not have the certainty of a strict demonstration. Second, they are useful in acquiring essentially new knowledge, and even indispensable to any not purely mathematical or logical knowledge, to any knowledge concerned with the physical world. We could call the reasoning that underlies this kind of evidence “heuristic reasoning” or “inductive reasoning” or (if we wish to avoid stretching the meaning of existing terms) “plausible reasoning.” We accept here the last term.
loc. 58675875
The conclusion is not fully expressed and is not fully supported by the premises. The direction is expressed and is implied by the premises, the magnitude is not.
loc. 59145916
The direction is expressed and is implied by the premises, the magnitude is not. For any reasonable person, the premises involve that
loc. 59155917
If, however, we find that the general statement is verified even in the extreme case, the inductive evidence derived from this verification will be strong, just because the prospect of refutation was strong. Thus, we are tempted to reshape the saying from which we started: “Prospective exceptions test the rule.”
loc. 60136016
Now, whatever the data may be, the required solution must apply and we do not see yet how to fit the same solution to all these possibilities. Out of such feeling of “too much variety” this question and answer may eventually emerge: Could you imagine a more accessible related problem? A more special problem?
loc. 60616064
Mathematics being a very abstract science should be presented very concretely.
loc. 61356135
In a more general acceptance of the word, a whole is termed symmetric if it has interchangeable parts. There are many kinds of symmetry; they differ in the number of interchangeable parts, and in the operations which exchange the parts.
loc. 61816182
Symmetry, in a general sense, is important for our subject. If a problem is symmetric in some ways we may derive some profit from noticing its interchangeable parts and it often pays to treat those parts which play the same role in the same fashion
loc. 61906192
The dimension of a product is the product of the dimensions of its factors, and there is a similar rule about powers. Replacing the quantities by their dimensions on both sides of the formula that we are testing, we obtain
loc. 63126315
The future mathematician learns, as does everybody else, by imitation and practice. He should look out for the right model to imitate. He should observe a stimulating teacher. He should compete with a capable friend. Then, what may be the most important, he should read not only current textbooks but good authors till he finds one whose ways he is naturally inclined to imitate.
loc. 64146419
He should enjoy and seek what seems to him simple or instructive or beautiful. He should solve problems, choose the problems which are in his line, meditate upon their solution, and invent new problems. By these means, and by all other means, he should endeavor to make his first important discovery: he should discover his likes and his dislikes, his taste, his own line.
loc. 64206423
If he cannot summon up real desire for solving the problem he would do better to leave it alone. The open secret of real success is to throw your whole personality into your problem.
loc. 64416443
The open secret of real success is to throw your whole personality into your problem.
loc. 64426443
The intelligent listener to a mathematical lecture has the same wishes. If he cannot see that the present step of the argument is correct and even suspects that it is, possibly, incorrect, he may protest and ask a question. If he cannot see any purpose in the present step, nor suspect any reason for it, he usually cannot even formulate a clear objection, he does not protest, he is just dismayed and bored, and loses the thread of the argument.
loc. 64486452
The insect, the mouse, and the man follow it; but if one follows it with more success than the others it is because he varies his problem more intelligently.
loc. 64926494
If our work progresses, there is something to do, there are new points to examine, our attention is occupied, our interest is alive. But if we fail to make progress, our attention falters, our interest fades, we get tired of the problem, our thoughts begin to wander, and there is danger of losing the problem altogether. To escape from this danger we have to set ourselves a new question about the problem.
loc. 65176522
The new question unfolds untried possibilities of contact with our previous knowledge, it revives our hope of making useful contacts. The new question reconquers our interest by varying the problem, by showing some new aspect of it.
loc. 65226525
2. Logical system. Geometry, as presented in Euclid’s Elements, is not a mere collection of facts but a logical system. The axioms, definitions, and propositions are not listed in a random sequence but disposed in accomplished order. Each proposition is so placed that it can be based on the foregoing axioms, definitions, and propositions. We may regard the disposition of the propositions as Euclid’s main achievement and their logical system as the main merit of the Elements. Euclid’s geometry is not only a logical system but it is the first and greatest example of such a system, which other sciences have tried, and are still trying, to imitate. Should other sciences—especially those very far from geometry, as psychology, or jurisprudence—imitate Euclid’s rigid logic? This is a debatable question; but nobody can take part in the debate with competence who is not acquainted with the Euclidean system.
loc. 67276738
the facts must be presented in some connection and in some sort of system, since isolated items are laboriously acquired and easily forgotten.
loc. 67596760
Any sort of connection that unites the facts simply, naturally, and durably, is welcome here. The system need not be founded on logic, it must only be designed to aid the memory effectively; it must be what is called a mnemotechnic system.
loc. 67606763
If the calculus is presented according to modern standards of rigor, it demands proofs of a certain degree of difficulty and subtlety (“epsilonproofs”). But engineers study the calculus in view of its application and have neither enough time nor enough training or interest to struggle through long proofs or to appreciate subtleties. Thus, there is a strong temptation to cut out all the proofs. Doing so, however, we reduce the calculus to the level of the cookbook. The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. The cookbook may serve its purpose perfectly. In fact, it need not have any sort of logical or mnemotechnic system since recipes are written or printed and not retained in memory.
loc. 67776785
5. Incomplete proofs. The best way of handling the dilemma between too heavy proofs and the level of the cookbook may be to make reasonable use of incomplete proofs.
loc. 67916793
In short, incomplete proofs may be used as a sort of mnemotechnic device (but, of course, not as substitutes for complete proofs) when the aim is tolerable coherence of presentation and not strictly logical consistency. It is very dangerous to advocate incomplete proofs. Possible abuse, however, may be kept within bounds by a few rules. First, if a proof is incomplete, it must be indicated as such, somewhere and somehow. Second, an author or a teacher is not entitled to present an incomplete proof for a theorem unless he knows very well a complete proof for it himself. And it may be confessed that to present an incomplete proof in good taste is not easy at all.
loc. 68416848
There are many shrewd and some subtle remarks in proverbs but, obviously, there is no scientific system free of inconsistencies and obscurities in them. On the contrary, many a proverb can be matched with another proverb giving exactly opposite advice, and there is a great latitude of interpretation. It would be foolish to regard proverbs as an authoritative source of universally applicable wisdom but it would be a pity to disregard the graphic description of heuristic procedures provided by proverbs.
loc. 68586862
It is true, we have discovered the appropriate sequence in retrogressive order but all that is left to do is to reverse the process and start from the point which we reached last of all in the analysis (as Pappus says).
loc. 70677070
Notes:
1) 反方向,类似走迷宫
Going around an obstacle is what we do in solving any kind of problem; the experiment has a sort of symbolic value. The hen acted like people who solve their problem muddling through, trying again and again, and succeeding eventually by some lucky accident without much insight into the reasons for their success. The dog who scratched and jumped and barked before turning around solved his problem about as well as we did ours about the two containers.
loc. 71307133
回应 20190527 19:45 
旺三 (一蓑烟雨任平生。)
一、理解问题 第一步：你必须理解问题。 什么是未知的？数据有哪些？条件是什么？ 是否可能满足条件？条件是否足够决定未知？或是不充分？或是冗余？或是矛盾？ 画图，介绍适当的符号。 分离条件的不同部分，能否写下来？ 二、设计计划 第二步：找到数据和未知之间的联系，如果找不到某个中间的联，你可能被要求考虑副问题。你最终需要一个解决方案的计划。 你是否遇到过？是否看到过形式稍有不同的同样的问题？ 是否知道一个相...20190526 19:36
一、理解问题
第一步：你必须理解问题。
什么是未知的？数据有哪些？条件是什么？
是否可能满足条件？条件是否足够决定未知？或是不充分？或是冗余？或是矛盾？
画图，介绍适当的符号。
分离条件的不同部分，能否写下来？
二、设计计划
第二步：找到数据和未知之间的联系，如果找不到某个中间的联，你可能被要求考虑副问题。你最终需要一个解决方案的计划。
你是否遇到过？是否看到过形式稍有不同的同样的问题？
是否知道一个相关的问题？是否知道一个用得上的定理？
看未知量，试着想一个有相同或相似未知量的熟悉的问题。
如果有一个和你的问题相关且解决过的问题，你是否能用上？是否能用它的结果？是否能用它的方法？为了使它用得上，是否要引入一些辅助元素？
是否能重新描述问题？能否再差异化地描述它？回到定义上。
如果你不能解决提出的问题，试着先解决一些相关的问题。能否想象一个更好理解的相关问题？一个更通用的问题？一个更特别的问题？一个类似的问题？你能否解决部分问题？只保留部分条件，放下其它部分；未知量还要多远才能决定，能有多少不同？能否从数据得出一些有用的东西？能否想到别的适合决定未知量的数据？能否更改未知量或数据，或必要时两者，使新未知量和新数据离得更近？是否使用了所有数据？是否使用了所有条件？是否考虑了问题涉及的所有必要的概念？
三、执行计划
步骤三：执行计划
执行解决方案的计划，检查每一步。你能否清楚看出该步骤是证券的？能否证明它是正确的？
四、回看
步骤四：检验获得的解决方案。
能否检查你的结果？能否检查参数？
能否用不同地方法得出结果？能否一眼看出来？
能否将结果或方法用于其它问题？
回应 20190526 19:36 
枯草[已注销] (Subtillion)
页码其实是xvi。傻缺豆瓣。。只好写成016了。 second. have you seen it before? or have you seen the same problem in a slightly different form?” 哎呀怎么办啊。看到这个话就想笑啊。。。 http://www.sdxxb.cn/shuxue/ShowArticle.asp?ArticleID=463120120815 04:30
页码其实是xvi。傻缺豆瓣。。只好写成016了。
”哎呀怎么办啊。看到这个话就想笑啊。。。 http://www.sdxxb.cn/shuxue/ShowArticle.asp?ArticleID=4631second. have you seen it before? or have you seen the same problem in a slightly different form?
回应 20120815 04:30
论坛 · · · · · ·
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电子版有售 · · · · · ·
其他版本有售 · · · · · ·
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5 有用 c h l 20120815
这书看完part 1就给跪了。。。要是我能在中学时代养成这么一套思路，得少做多少无用功啊。。。然，我本人资质十分有限所以即使少做很多无用功余下的努力也基本上都是无用功这件事儿就略过不表了
1 有用 yuzusodadada 20180910
写的真好。觉得自己小学搞奥数的时候真是智力黄金顶峰时代
1 有用 TairAlex 20130806
在图书馆翻了一遍（今天实在时昏昏欲睡），要是高中时看就好了，可惜一切都晚了。
1 有用 可爱的肖恩 20090214
好书
1 有用 又萌又逗又傻 20161221
大部分的问题，都可以按照数学题来对待，寻找相似的解决方法和答案，可能是最有效的办法
0 有用 xing 20190131
The essence of learning solving problem is the logic flow that he delivered rather than concrete examples. Learning from a master and get a few insights is much better than absorbing the whole picture... The essence of learning solving problem is the logic flow that he delivered rather than concrete examples. Learning from a master and get a few insights is much better than absorbing the whole picture of mediocre men (展开)
1 有用 yuzusodadada 20180910
写的真好。觉得自己小学搞奥数的时候真是智力黄金顶峰时代
0 有用 御手洗洁 20190213
和我想的不太一样......
0 有用 HungYueh_ 20170305
老感觉读起来没啥收获，读中文版再看看。
0 有用 O_o 20170210
一套深入剖析解决问题的方法论指导..启发式引导,激发探索兴趣