Math 583B Research in Geometric Analysis SOME NOTES ON WRITING MATHEMATICS Spring 2005 Here are some observations about the writing of mathematics that I hope will be useful as you work on the writing assignment for this course Goals and audience As with any written piece mathematical exposition must be written with a particular audience and speci c goals in mind Be sure you have a clear sense of what these are before you start writing The process It is important to bear in mind that writing is a process just like proving a theorem No one pours forth a well organized clear and error free exposition the rst time they sit down to write just as no one produces a complete well structured proof the rst time they think about a problem Most good expository prose has been thoroughly rewritten at least once or twice before it reaches the reader with key sections undergoing perhaps three to ve major revisions To some people this thought makes the prospect of writing seem daunting or even overwhelming but it needn t The idea that much of what you write will eventually be replaced or discarded can be liberating Just sit down and write knowing that anything that doesn t measure up can later be xed You might well nd yourself throwing away the rst several pages you write this is not wasted time since the 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 since the structure of the paper may not become completely clear until later in the process Try starting 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 Put yourself 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 read and comment on Then rewrite again After you think the paper is nished go through it with a ne toothed comb and a sharp razor Sharpen your de nitions statements of theorems and proofs Clarify your logic and your intuitive descriptions Make sure your spelling punctuation and grammar are absolutely correct Omit needless words terminology and symbols Note that rewriting usually means much more than simply correcting errors It means looking critically at what you ve written both locally and globally guring out what works well and what doesn t and doing whatever is necessary to make the whole thing work perfectly Conventions Although you might not believe it after reading some of the mathematical writing 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 in particular that you must write complete sentences organized into paragraphs While many mathematical terms have technical meanings that are di erent from their usage in ordinary English you should still be careful to observe the usual rules regarding parts of speech and subject verb agreement Although you will run across all too many mathematicians who write 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 of asking a native speaker to look over your writing before you submit it 1 2 Precision In mathematical writing more than any other kind precision is of paramount importance Every mathematical statement you make must have a precise mathematical meaning Speci cally every term you use must be well de ned and used properly according to its de nition every mathematical conclusion you reach must be justi ed and every symbol you mention must be either previously de ned or quanti ed in some appropriate way If you write f a 0 do you mean that this is true for every a X or that there exists some a X for which it s true or that it s true for a particular a that you introduced earlier 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 easily comprehensible to your intended audience Don t be stingy with motivation intuitive explanations 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 is done this way and not some other way In addition to providing motivation you should think carefully about how to make your proofs themselves easily understandable The point of writing a mathematical proof is to convince your reader that something is true and why This is not the same as writing a formal logic proof that could be checked by a computer It s all too easy to write a sequence of mathematical statements that are entirely precise and mathematically correct and yet that are incomprehensible to a human being For example if a substantial part of your argument consists of a series of equations be sure to introduce them with a clear explanation of what is to follow and intersperse them at carefully chosen places with some words about what you re doing and why or reasons why one step follows from another The rst person Most authors avoid using the word I in mathematical writing It is standard 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 i if and only if w r t with respect to WLOG without loss of generality therefore These are indispensable for writing on the blackboard and taking notes but should never be used in written mathe matical exposition The only exceptions are abbreviations that would be acceptable in any formal writing such as i e that is or e g for example but if you use these be sure you know the di erence between them Mathematical symbols The feature that most clearly distinguishes