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Mathematics demands precise expression. If you get the quantifiers in the wrong order, then you totally change the meaning.
In mathematics, you are used to dealing with arbitrary rules. Therefore, you should be able to write grammatical prose, for grammar is just a set of (somewhat) arbitrary rules to follow. Here are some examples of rules to keep in mind.
Make sure that the subject of a sentence agrees with the verb, and that the subject is the one you intend. For example, it is wrong to write, "Integrating by parts, the expression becomes ...", for it is not the expression that is doing the integrating. This is an example of a dangling participle. It would be correct to write, "Integrating by parts, we find that the expression becomes ...".
Make sure that the antecedents of pronouns are clear. "Let H be a subgroup of a group G which is solvable." Does "which" refer to G or to H?
In writing mathematics, it is a good idea to minimize the use of pronouns. Consider a sentence beginning, "If f is a differentiable function and L is a linear operator, then it ...." The reader hiccups at the word "it": is "it" the function f, the operator L, or the beginning of a phrase like "it is easy to see"? Your writing is easier for the reader to follow if you repeat the noun or the symbol instead of using a pronoun.
Rules do change over time.
A long-dead grammarian declared that one should not split infinitives. In other words, the mission of the Starship Enterprise should not be "to boldly go where no one has gone before", but rather "boldly to go ...". Many modern grammarians feel that splitting infinitives is acceptable. If you do split infinitives, do so deliberately and with the understanding that some of your readers may wince.
Similarly, many high school teachers hold fast to the arbitrary rule of not ending a sentence with a preposition. What are they thinking of?
Modern authorities feel that terminal prepositions are allowable in moderation. The third edition of Fowler's Modern English Usage declares that "in formal writing, it is desirable to avoid placing a preposition at the end of a clause or sentence, where it has the appearance of being stranded. But there are many circumstances in which a preposition may or even must be placed late".
The Math 696 course
pages were last modified April 5, 2005. | | | Grammar |