# Proofs are written in English

Do not worry overly much about the names of these rules of inference. For this class, if you use the inferences correctly, that is good enough. You don't have to remember which one is modus ponens and which one is modus tollens. In fact, if you flip through almost any math book, you will not find any mention of "modus ponens" or "disjunctive syllogism" in the proofs (with the possible exception of books on logic). This is because applying a rule of inference is such a small step in the reasoning process that it would not be helpful to the reader to point out that you have just used a particular rule.

Mathematical proofs are usually written in paragraph form and complete sentences. Even a sentence that uses mathematical notation should be a complete sentence when read aloud. For example:

If the vertex xA, then we color x red.
This sentence should be read aloud as "If the vertex x is an element of A, then we color x red." The verb in the "if" clause is the ∈ symbol.

Most of the rules for writing you learned in English classes are applicable in mathematical writing. For example, when you start a new topic, you should start a new paragraph. You should also avoid run-on sentences, they are hard to decipher. About the only exception is the advice that you vary the vocabulary. You might have been told in a writing class to use different words to describe an object so you do not repeatedly use the same word. In mathematical writing, you should actually stick to the same terminology. For example, although the terms "vertex" and "node" are used interchangeably, you should stick to one or the other and not use both words in the same writing. Thus, in mathematical writing, you do not need to wrack your brains to think up "exciting" adjectives and adverbs. For example, a vertex without any edges is called an "isolated" vertex. You will always call this an isolated vertex. You won't ever have to describe the vertex as "lonely", "alienated", "dejected" or "outcast".

Instead, concentrate on the 4 C's of mathematical writing: try to be clear, concise, convincing and correct.

• Being clear means that the reader understands your intended meaning. It is helpful to give names to the objects you are talking about and to use the names consistently. It is also important to use mathematical notation correctly.

• Being concise means getting to the point. Don't introduce things that are not relevant to the proof. If the proof is complicated, break it down into smaller steps. Try to consolidate cases so there are not too many of them.

• Being convincing to a semi-skeptical reader is the main point of writing a proof down. Proofs often have a key idea and the rest is just "window dressing" to set up the proof. Make sure that this key idea is not lost in the window dressing. If there are subtle points in the proof, warn the reader. Try to anticipate any questions or objections that the reader may have and answer them.

• Being correct is of course the final and most important point. A well-written proof that is clear, concise and convincing is of no use if the proof is actually wrong! Read and re-read your proof to look for bugs. Be paranoid!