Some Remarks on Writing Mathematical Proofs

1 some Remarks on Writing Mathematical ProofsJohn M. LeeUniversity of Washington Mathematics DepartmentWriting Mathematical Proofs is, in many ways, unlike any other kind of Writing . Over the years, themathematical community has agreed upon a number of more-or-less standard conventions for proofwriting. This note describes my version of these conventions. Although not every mathematicianwould agree with everything I recommend here, on the whole these recommendations represent aconsensus among the best Mathematical writers. And though most of these guidelines are statedas hard and fast rules, every rule admits exceptions, and experienced Mathematical writers mightencounter situations that call for different choices. But novice proof writers will generally benefitfrom following these guidelines Considerations About Mathematical Writing Identify your you begin Writing about mathematics (or about anything,for that matter), know who your audience is and what they already know.

2 For example,if you re Writing a proof as a homework assignment for a course, a good rule of thumb isto write as if you were trying to convince a fellow student in the same class of the truthof the theorem and the correctness of your argument assume the reader knows the samebackground material as you do, but doesn t know the proof of this particular theorem. If youare Writing for publication, who is the intended audience? Write in paragraph always that a Mathematical proof is designed tocommunicateto a human reader. You are communicating the truth of a Mathematical state-ment, the correctness of your argument, and hopefully some insight about what the resultmeans and why it s true. There is an overwhelming consensus that an ordinary prose narra-tive is much better suited to this purpose than formal symbolic statements.

3 Although youmight initially construct your proof as a sequence of terse symbolic statements, when youwrite it up you should use complete sentences organized intoparagraphs. As you read in-creasingly complicated Proofs , you ll find that paragraph-style Proofs are much easier to readand comprehend than symbolic ones or the two-column Proofs of high school geometry. Use proper Writing should follow the same conventions ofgram-mar, usage, punctuation, and spelling as any other addition to Writing completesentences organized into paragraphs, you must use correct punctuation (including a period atthe end of every sentence); avoid sentence fragments, run-on sentences, and dangling modi-fiers; pay attention to subject-verb agreement and parallelstructure; and use correct spellingand capitalization.

4 Violating the conventions of standardEnglish will, at the very least, tripup careful readers; and at worst it can make your meaning impossible to decipher. It s anexcellent idea to find a good book on grammar and usage and makefriends with it. Write you may feel that some of the Mathematical Writing you ve read isdeliberately opaque, the goal of good Mathematical writingshould be to produce prose thatis clear enough to be easily comprehensible to the intended audience. Don t be stingy withintuitive explanations of what s going on and why. If the structure of your proof is anythingRevised September 12, 2019. Copies available at 2010, 2012, 2019 John M. Lee. This note is distributed under the Creative Commons Attribution-ShareAlike License ( ).

5 1other than a simple direct proof , state at the beginning whattype of proof you re using ( wewill prove the contrapositive or we will prove this by induction, for example). Include the Mathematical ideas you are trying to convey are at all com-plicated, or follow an unexpected path, it s wise to includesome preliminary discussion thatexplains such things as why things are defined as they are, whyone might expect the theoremto be true, how one might have been led to the proof , why the proof is structured the way itis, and how the result might be used subsequently. Mathematicians call this themotivation,and it s an essential part of good Mathematical on your purpose, mo-tivation might be inserted before the statement of a theorem, or at the beginning of a proof ,or at transition points between parts of Proofs , or all of theabove.

6 Use the first person singular authors avoid using the word I inmathematical Writing . It is standard practice to use we whenever it can reasonably beinterpreted as referring to the writer and the reader. Thus: We will prove the theorem byinduction onn, and because ABCis equilateral, we see thatAB=BC=CA. But ifyou re really referring only to yourself, it s better to go ahead and use I so you don t soundlike the Queen of England: I learned this technique from Richard Melrose. Avoid most are many abbreviations that we use frequently in infor-mal Mathematical communication: (such that), (with respect to), and (without loss of generality) are some of the most common. These are indispensable for writingon the blackboard and taking notes, but should usually be avoided in written mathematicalexposition, especially in formal contexts: they might saveyou a few seconds of Writing time,but they make your text ugly and cryptic and are likely to cause readers to waste consider-ably more time deciphering what you wrote than the time you saved.

7 The only exceptionsare abbreviations that would be acceptable in any formal Writing , such as (id est, whichmeans that is ) or (exempli gratia, which means for example ); but if you use these,be sure you know the difference between them!One abbreviation that deserves special mention is iff (if and only if). some mathematicalwriters use this routinely, even in quite formal Writing . But my opinion is that, like theother abbreviations mentioned above, it actually acts as a hindrance to understanding inmathematical prose, because it s likely to briefly trip up your readers as they formulate yoursentences in their minds. Thus it should be reserved for the blackboard and your notes. sure to read what you ve written from beginning to end after you thinkyou re all finished.

8 You ll be amazed how many silly mistakesyou can catch that Considerations for proof Writing State what you re you re Writing a proof , you should always precede it witha precise statment, in one or more English sentences, of the theorem you are proving. Thisapplies even if you re Writing a proof as a homework assignment for a course. Depending onyour instructor s preference, you might do this by copying the problem statement verbatim,by summarizing the problem statement, or by paraphrasing the problem in the form of atheorem statement. My preference is the latter. For example, suppose you re assigned thefollowing homework problem:Prove that ifxis a real number, thenx2 you copy this verbatim into your homework paper, it s likely to look as if you re commanding2the reader to do something.

9 Instead, your solution might be clearest if you start with astatement like this:Theorem:Ifxis a real number, thenx2 0. Label your theorem you state should be clearly labeled with an identifyingtag such asTheorem. With computer typesetting programs like TEX or Microsoft Word,the usual convention is to set the word Theorem in boldface, with the statement of thetheorem itself italicized. In handwritten Proofs , just underline the word Theorem. In some contexts, the word Theorem might be replaced by Proposition, Corollary, or , these all mean the same thing (a Mathematical statement that can be proved fromassumptions and previously proved results), but your choice of label can alert the readerabout the role that the result plays in the current context.

10 In modern usage, atheoremisan important result; apropositionis a result that is interesting in its own right, but not asimportant as a theorem; alemmais a result that might not be interesting in itself, but isuseful for proving another theorem; and acorollaryis a result that follows easily from sometheorem or proposition, usually the immediately precedingone. Show where your Proofs begin and proof should begin with the wordProof,and end with a distinctive symbol such as the square at the endof this paragraph. In olderbooks, ends of Proofs are frequently marked with the Latin abbreviation QED (quod eratdemonstrandum, that which was to be proved ), but this is rapidly going outof style. Write with Mathematical Writing more than any other kind, precisionisparamount.