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APU MATH 583B - Notes on Writing Mathematics

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Math 583B Research in Geometric Analysis Spring 2005SOME NOTES ON WRITING MATHEMATICSHere are some observations about the writing of mathematics that I hope will be useful asyou work on the writing assignment for this course.Goals and audience: As with any written piece, mathematical exposition must be writtenwith a particular audience and specific goals in mind. Be sure you have a clear sense ofwhat these are before you start writing.The process: It is important to bear in mind that writing is a process, just like provinga theorem. No one pours forth a well-organized, clear, and error-free exposition the firsttime they sit down to write, just as no one produces a complete, well-structured proof thefirst time they think about a problem. Most good expository prose has been thoroughlyrewritten at least once or twice before it reaches the reader, with key sections undergoingperhaps three to five major revisions. To some people, this thought makes the prospect ofwriting seem daunting or even overwhelming, but it needn’t: The idea that much of whatyou write will eventually be replaced or discarded can be liberating. Just sit down andwrite, knowing that anything that doesn’t measure up can later be fixed. You might wellfind yourself throwing away the first several pages you write—this is not wasted time, sincethe trial and error process helps you immensely in clarifying what you really want to say.When you begin writing a draft, the introduction may not be the best place to start, sincethe structure of the paper may not become completely clear until later in the process. Trystarting somewhere in the middle, with whichever part of the paper is clearest in your mind.As soon as you have a section or more in relatively coherent form, sit back and read it. Putyourself in the mind of your audience, and see if it makes complete sense. Then rewrite.When you have something you think is close to acceptable, give it to someone else to readand comment on. Then rewrite again.After you think the paper is finished, go through it with a fine-toothed comb and a sharprazor. Sharpen your definitions, statements of theorems, and proofs. Clarify your logicand your intuitive descriptions. Make sure your spelling, punctuation, and grammar areabsolutely correct. Omit needless words, terminology, and symbols.Note that “rewriting” usually means much more than simply correcting errors. It meanslooking critically at what you’ve written both locally and globally, figuring out what workswell and what doesn’t, and doing whatever is necessary to make the whole thing workperfectly.Conventions: Although you might not believe it after reading some of the mathematicalwriting that has made it into print, mathematical writing should follow the same conven-tions of grammar, usage, punctuation, and spelling as any other writing. This means, inparticular, that you must write complete sentences organized into paragraphs. While manymathematical terms have technical meanings that are different from their usage in ordinaryEnglish, you should still be careful to observe the usual rules regarding parts of speech andsubject-verb agreement. Although you will run across (all too many) mathematicians whowrite ungrammatical sentences like “Suppose f is an onto map,” don’t you do it!If you are not a native English speaker, it would be a good idea to cultivate the habit ofasking a native speaker to look over your writing before you submit it.12Precision: In mathematical writing more than any other kind, precision is of paramountimportance. Every mathematical statement you make must have a precise mathematicalmeaning. Specifically, every term you use must be well defined, and used properly accordingto its definition; every mathematical conclusion you reach must be justified; and everysymbol you mention must be either previously defined or quantified in some appropriateway. If you write f(a) > 0, do you mean that this is true for every a ∈ X, or that thereexists some a ∈ X for which it’s true, or that it’s true for a particular a that you introducedearlier in the argument?Ask yourself these two key questions about each mathematical sentence you write:• What exactly does this mean?• Why exactly is this true?You don’t necessarily have to include the entire answer to the second question in your paper,but make sure you let the reader know where the answer can be found.Clarity: Just as important as mathematical precision is making sure your writing is easilycomprehensible to your intended audience. Don’t be stingy with motivation—intuitiveexplanations of what the result means, what led up to it historically, why it’s interesting,why we should expect it to be true, what are the main ideas in the proof, why the proof isdone this way and not some other way.In addition to providing motivation, you should think carefully about how to make yourproofs themselves easily understandable. The point of writing a mathematical proof is toconvince your reader that something is true and why. This is not the same as writinga formal-logic proof that could be checked by a computer. It’s all too easy to write asequence of mathematical statements that are entirely precise and mathematically correct,and yet that are incomprehensible to a human being. For example, if a substantial partof your argument consists of a series of equations, be sure to introduce them with a clearexplanation of what is to follow, and intersperse them at carefully chosen places with somewords about what you’re doing and why, or reasons why one step follows from another.The first person: Most authors avoid using the word “I” in mathematical writing. It isstandard practice to use “we” whenever it can reasonably be interpreted as referring to“the writer and the reader.” Thus: “We will prove the theorem by induction on n,” and“Because f is injective, we see that x1= x2.” But if you’re really referring only to yourself,it’s better to go ahead and use “I”: “I learned this technique from Richard Melrose.”Abbreviations: There are a host of abbreviations that we use frequently in informal math-ematical communication: “s.t.” or “3” (such that), “iff” (if and only if), “w.r.t.” (withrespect to), “WLOG” (without loss of generality), “∴” (therefore). These are indispensablefor writing on the blackboard and taking notes, but should never be used in written mathe-matical exposition. The only exceptions


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