If you are interested a proper respect for my

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If you are interested, a proper respect for my predecessors means that I must explain briefly how my work depends on theirs. Now let me sketch the plan of my paper. Section 2 contains definitions and preliminary computations. Section 3 is devoted to the proof of the key lemma in the case of the real line. Section 4 extends the lemma to general locally compact Abelian groups. The details are very technical and not needed if we only wish to prove the main theorem for standard groups like the line and 14
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the circle. Section 5 contains the proof of the main theorem. Section 6 contains examples showing that our main theorem is, in some sense, best possible. Academic convention prevents you from being quite as direct as this but the crustacean style requires you to get as close as possible. As a rule of thumb the introduction should occupy about 1/7 of a mathematical paper but in a short paper it may well occupy rather more and in a long paper rather less. If you only wish to write your introduction once you will have to make it the last thing you write. It is very rare for mathematical paper to turn out as planned and the introduction describes what you have written and not what you wished to write. My personal preference is to write the introduction first and then rewrite it repeatedly as the exposition progresses. This requires more work but makes sure that you keep the general plan constantly in mind. 4.7 The conclusion The introduction had to be an advertisement to tempt the customer to buy and a map to help the explorer navigate. At the end of your essay the customer has already bought the goods and the explorer has completed the journey. There is no logical reason why we should trouble with an ending just as there is no logical reason why we should say thank you at the end of a transaction or goodbye at the end of a meeting but there are human reasons why we should. Your reader has worked hard to understand your work. Can you not reward him or her with some final insight or a tantalising open question so that your essay ends on a high note? Or, if this is not possible, should you not review the path you have taken together? Remember the preacher’s advice ‘First I tell them what I’m going to say. Then I tell them. Then I tell them what I’ve said.’ What applies to the essay as a whole applies to its component sections. I am tempted to reverse my advice on the crustacean style. Last paragraphs of sections summarise the section, last sentences of paragraphs summarise the paragraph. This advice should not be taken literally but I am sure that most mathematics lecture courses would be improved if mathematicians took more trouble with the beginning and ending of each lecture. 15
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4.8 A note on omissions In a talk it may well be desirable to omit long calculations and long lists of conditions which are not required to understand the central idea of the proof.
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