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[ Polya,G ] Carrying out To conceive and plan and to carry it though are two different things. This is true also of mathematical problems in a certain sense; between carrying out the plan of the solution, and conceiving it, there are certain differences in the character of the work. 1. We may use provisional and merely plausible arguments when nerving the final and rigorous argument as we use scaffolding to support a bridge during construction. When however, the work is sufficiently advanced we take of the scaffolding, and the bridge should be able to stand by itself. In the same way, when the solution is sufficiently advance, we brush aside all kinds of provisional and merely plausible arguments, and the result should be supported by rigorous argument alone. Devising the plan of the solution, we should not be too afraid of merely plausible, heuristic reasoning. Anything is right that leads us to there right idea. But we have no change this standpoint when we start carrying out the plan and then we should accept only conclusive, strict arguments. Carrying out your plan to the solution, check each step. Can you see clearly that the step is correct? The more painstakingly we check our steps when carrying out the plan, the more freely we may use heuristic reasoning when devising it. 2. We should give some consideration to the order in which we work out the details of our plan, especially if our problem is complex, we should not omit and detail, we should understand the relation of the detail before us to the whole problem, we should not lose sight of the connection of the major steps. Therefor, we should proceed in proper order. In particular, it is not reasonable, to check minor details before we have good reasons to believe that the major steps of the argument are sound. If there is a break in the main line of the argument, checking this or that secondary detail would be useless anyhow. The order in which we work out the details of the argument may be very different form the order in which we invented them,; and the order in which we write down the details in a definitive exposition may be sill different. Euclid's ' elements present the details of the argument in a rigid systematic order which was often imitated and often criticised. 3. In Euclid's exposition all argument proceed in the same direction; from the data toward th 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 form 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. 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 recommenced, without reservation if the propose 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 no 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). 4. Let us sum up. Euclid's manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion is perfect for checking the argument in detail but far from being perfect for making understandable the main line of the argument. It is highly desirable that the students should examine their own arguments in the Euclidean manner, proceeding from the data to the unknown, and checking each step although nothing of this kind should be too rigidly enforced. It is not so desirable that the teacher should present many proofs in the pure Euclidean manner, although though the Euclidean presentation may be very useful after a discussion in which, as is recommended by the present book, the students guided by the teacher discover the main idea of the solution as independently as possibly. Also durable seems to be the manner adopted by some textbooks in which an intuitive sketch of the main idea is presented first and the details in the Euclidean order of exposition afterwards. 5. Wishing to satisfy himself that his proposition is true, th conscientious mathematician tries to see it intuitively and to give a formal proof. Can you see clearly that it is correct? Can you prove that it is correct? The conscientious mathematician acts in this respect like the lady who is a conscientious shopper. Wishing to satisfy herself of the quality of a fabric, she wants to see it and to touch it. Intuitive insight and formal proof are two different ways of perceiving the truth, comparable to the perception of a material object through two different senses, slight and touch. Intuitive insight may rush far ahead of formal proof. Any intelligent student, without any systematic knowledge of solid geometry, can see as soon as he has clearly understood the terms that two starlight lines parallel to the same straight line are parallel to each other (the three lines may or may not be in the he same plane). Yet the proof of this statement, as given in proposition 9 of the 11th book of Euclid's Elements, needs a long, careful and ingenious preparation. Formal manipulation of logical rules and algebraic formula may get far ahead of intuition. Almost everybody can see at once that 3 straight lines, taken at random divide the plane into 7 parts(look at the only finite part, the triangle inclined by the 3 line). Scarcely anybody is able to see, even straining his attention to the utmost, that 5 planes, taken at random, divide space into 26 parts. Yet it can be rigidly proved that the right number is actually 26, and the proof is not even long or difficult. 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 right? 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.