MATH 300 - Math 300A&B Introduction to Mathematical...

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Math 300A&B Introduction to Mathematical Reasoning Fall 2009 Conventions for Writing Mathematical Proofs Writing mathematical proofs is, in many ways, unlike any other kind of writing. Over the years, the mathematical community has agreed upon a number of more-or-less standard conventions for proof writing. This document describes my version of these conventions. Although not every mathematician would agree with everything I recommend here, on the whole these recommendations represent a consensus among the best mathematical writers. Write in paragraph form: First and foremost, remember always that a mathematical proof is designed to communicate the truth of a mathematical statement, and the correctness of your argument, to a human reader . There is an overwhelming consensus that an ordinary prose narrative is much better suited to this purpose than formal symbolic statements. Although you might initially construct your proof as a sequence of terse symbolic statements, when you write it up you should use complete sentences organized into paragraphs. As you read more and more complicated proofs, you will find that paragraph-style proofs are much easier to read and comprehend than symbolic ones or the two-column proofs of high school geometry. Use proper English: All mathematical writing should follow the same conventions of grammar, usage, punctuation, and spelling as any other writing. In addition to writing complete sentences organized into paragraphs, you must use correct punctuation (including a period at the end of every sentence), avoid sentence fragments and run-on sentences, pay attention to subject-verb agreement and parallel structure, and use correct spelling and capitalization. Include motivation: If your proof is at all complicated, or follows an unexpected path, it’s a good idea to include some preliminary discussion that explains such things as why one might expect the theorem to be true, why the proof goes the way it does, and how the result might be used subsequently. Mathematicians call this the motivation , and it’s an essential part of good mathematical exposition. Depending on your purpose, the motivation can come before the statement of a theorem, or at the beginning of a proof, or at a transition between parts of a proof. Identify your audience: Before you begin writing any proof, be sure you’re aware who your audience is and what they already know. For example, if you’re writing a proof as a homework assignment for a course, a good rule of thumb is to write as if you were trying to convince a fellow student in the same class of the truth of the theorem and the correctness of your argument. Assume the reader knows the same background material as you do, but doesn’t know the proof of this particular theorem.
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