Your object is to keep the reader interested and to

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Your object is to keep the reader interested and to ensure that he or she can understand the general sweep of the argument without bothering with all the details. For example if your topic is Theorem A and Corollary B but the proof of Theorem A depends on Lemmas 1, 2 and 3 which have long and complicated proofs, it may be best to order your outline in the following way. Section 1. Introduction Section 2. Statements of Lemmas 1, 2 and 3. Section 3. Statement and proof of Theorem A. Section 4. Statement and proof of Corollary B. Section 5. Discussion of Corollary B. Sections 6, 7, 8, . . . Proofs of Lemmas 1, 2 and 3. In this way the reader learns as much as is needed to get to the point in as brief a way as possible. Details of proofs are sentenced to the end of the paper to be consulted if needed. This technique is particularly important when giving talks. Always put your important points at the beginning of your talk. Observation shows that halfway through the average seminar most of the audience are asleep, catching up on correspondence, thinking about their own mathematics or trying to prove your result by a slicker method. In 13
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addition all talks take 50% longer to give than expected (even after allowing for this rule) so if you leave important results until last you will have to engage in an undignified scramble to reach them. 4.5 Navigation Remember that few people read mathematics papers straight through. You must make it easy for your reader to skip bits (with the intention, of course, of coming back to them later) or to refer back to some previous point without rereading the whole essay. Good layout will help but the necessary signposts should be incorporated into your prose. If you make sure that new informa- tion is never buried in the middle of a paragraph your readers can hop about your essay with the confidence that all they need will be evident. 4.6 The introduction For many mathematicians the introduction is the place where they dump a survey of the literature and all the definitions and trivial remarks they can find. If you are sure that everybody will have to read your paper or you expect that nobody will, this is the easiest way to construct a paper. If you think that you have something worthwhile to say but are modest enough to doubt whether your potential readers know this, you will use your introduction as an advertisement and a map. You would like to address your reader as follows:- Since you have read my title and my abstract I can assume that you are interested to hear what I have to offer. My main theorem is the following. You may need the following two defini- tions to understand it. It is important because it does so and so. You may also be interested in the following corollary and in the lemma I use to establish the theorem. If you are not interested in these then I am afraid you will not be interested in the rest and we part company with no hard feelings.
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