34 some papers are just long proofs if a paper is

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3.4 Some papers are just long proofs If a paper is just a long proof then we should treat it as such. First we must identify the central theorem. Having done so we try to prove it as before. Now the point at which we say ‘Ah, I did not think of that’ may be a reference to a previous lemma. We now know (or think we know) the 9
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point of the lemma and we try to prove it. By repeated use of this technique we can identify the structure of lemmas and definitions which support the theorem and, ultimately, obtain the full proof of the main theorem. Much of mathematics is automatic writing, only by trying to do as much of the proof yourself as you can will you identify the key steps which are not automatic. 3.5 But all papers have context Mathematicians ask two questions about theorems — how and why. How do you prove it and why should you prove it. Thus, given a theorem, we may ask: (1) What simpler results does it generalise? (2) How can you use it prove other things? Can you give examples? (3) Does it generalise? If not, what is the obstacle? What are the counter- examples which demonstrate the obstacle? (4) What is the next step in the development of the subject? What are the open questions? Even if the paper does not consider these questions your essay should do so. 3.6 A possible moral If you are a great mathematician like Kolmogorov you may have so many ideas that you have no time to spend in presenting them. In any case, if you are a great mathematician, people will not grudge the work required to read your papers. If you are a mathematician with nothing to say then no matter how clearly you write and how inviting your presentation, people will not read your papers. If you are a middle ranking mathematician (particularly if you are start- ing out in your profession) the number of people who actually read your papers will depend on how clearly written they are as well as on what they say. Most middle ranking mathematicians (and quite a lot of high ranking mathematicians) have papers which they feel have been unfairly neglected. Sometimes their opinion is at fault (we are not the best judges of our own children), often it is a question of fashion or bad luck, but sometimes the paper just fails to communicate its point. Erd˝os says ‘Everyone writes. Nobody reads.’ Creative mathematicians are more interested in their own ideas than in other people’s. Reading math- ematics is hard. You have to write so as to catch and hold the attention of an unwilling audience. You have read other people’s papers. Can you do better? 10
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4 How to write your essay Much of what follows is a more or less lightly modified version of Marj Batch- elor’s advice. 4.1 Your purpose Your only purpose should be to teach your readers a little bit about your subject. You may think that your sole purpose is to obtain an alpha for your Part III exam and you may argue that the only person who is likely to read this essay will be your assessor. However, this one of the only exercises
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