How to write mathematics

Mathematical prose relies heavily on symbols, but it is important not to use symbols to excess. Effective communication requires complete, grammatically correct sentences. Symbolism should never replace sentences. Mathematical symbols are used to represent complex formulae or longwinded phrases which appear repeatedly in the text. By introducing a new symbol, the author avoids constantly rewriting these formulae (and the reader avoids constantly rereading them). When used judiciously in this fashion, symbolism can enhance clarity. Mathematical symbols should only be used in this way. Introducing new symbols can simplify the exposition, but at a cost: it increases the burden on the reader’s memory. Only introduce new symbols if the benefit outweighs the cost. Any sequence of symbols should read like a fragment of an English sentence (eg. ‘a ∈ B ⊂ C’ means ‘a is an element of B, which is a subset of C’). Thus, mathematical symbols should be surrounded by enough English words to form a complete sentence. If a mathematical assertion is nothing but symbols, it is entirely unreadable. Most important are the stock phrases which indicate the logical role which an assertion plays in your argument. For example: