In theory see littlewoods essay mathematics with a

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In theory (see Littlewood’s essay Mathematics with a minimum of raw material in [9]) two sentences could deserve a College Research Fellowship (and so, certainly a PhD). In the same way 2000 words, if they were the right words, could well constitute an α + essay but, in practice, much less than 5000 words would probably be considered a bit light-weight. When I explain this point students invariably ask ‘Do you count formu- lae?’ This shows that they do not understand what I have just said. A figure like 6000 words is only a guideline. Nobody is going to actually count words so nobody cares what you do about formulae and diagrams. (Most students use too many formulae and not enough diagrams, but that is another mat- ter.) I give a method for estimating the number of words in your essay in essay on other, educational grounds and, in fact, the top candidates almost invariably do an essay. 3 Unless your assessor suggests otherwise. This is one of the things you should discuss at your first meeting. It is unlikely that your assessor agrees with everything in these notes and since your task is to please him or her, your assessor’s views are much more important than mine. 4
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Section 2.1. 1.3 Shakespeare it is not The essay is not an exercise in fine English but a preparation for writing mathematics papers. (The only way to learn to ride a bicycle is to ride a bicycle.) If English is not your first language you should rest assured that this will not handicap you 4 . The assessors are interested in clear thought and clear mathematics however expressed. For good or ill, English is likely to remain the lingua franca of mathematics for your lifetime. You may have to write your mathematics in English in the future. You should seize the opportunity to practice now. More generally the less you want to do the essay, the more important it is that you do it. The next time you write a substantial piece of mathematics it will probably be a paper for publication or a PhD thesis for examination. Take this one available opportunity for a dress rehearsal. 1.4 Nor is it research Sometimes students ask why the essay could not be replaced by a research project. This question makes three basic assumptions: (1) Reading mathematics is easy. (2) Writing mathematics is easy. (3) Mathematical research is easy. Assumption (3) is clearly false. A PhD is a three year apprenticeship and the test of a good PhD is that it should contain one good and original idea. In my opinion (and in the opinion of most mathematicians I know) assumptions (1) and (2) are also false. The essay gives you the opportunity to practice reading and writing mathematics. Of course, the assessor will be delighted if you find a new proof of some result or make a clever application of the method you describe but such things are not required for a good essay 5 .
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