Abuse of Notation - Bourbaki

Bourbaki

The term "abuse of language" frequently appears in the writings of Nicolas Bourbaki:

We have made a particular effort always to use rigorously correct language, without sacrificing simplicity. As far as possible we have drawn attention in the text to abuses of language, without which any mathematical text runs the risk of pedantry, not to say unreadability. Bourbaki (1988).

For example:

Let E be a set. A mapping f of E × E into E is called a law of composition on E. By an abuse of language, a mapping of a subset of E × E into E is sometimes called a law of composition not everywhere defined on E. Bourbaki (1988).

In other words, it is an abuse of language to refer to partial functions from E × E to E as "functions from E × E to E that are not everywhere defined." To clarify this, it makes sense to compare the following two sentences.

1. A partial function from A to B is a function f: A' → B, where A' is a subset of A.

2. A function not everywhere defined from A to B is a function f: A' → B, where A' is a subset of A.

If one were to be extremely pedantic, one could say that even the term "partial function" could be called an abuse of language, because a partial function is not a function. (Whereas a continuous function is a function that is continuous.) But the use of adjectives (and adverbs) in this way is standard English practice, although it can occasionally be confusing. Some adjectives, such as "generalized", can only be used in this way. (e.g., a magma is a generalized group.)

The words "not everywhere defined", however, form a relative clause. Since in mathematics relative clauses are rarely used to generalize a noun, this might be considered an abuse of language. As mentioned above, this does not imply that such a term should not be used; although in this case perhaps "function not necessarily everywhere defined" would give a better idea of what is meant, and "partial function" is clearly the best option in most contexts.

Using the term "continuous function not everywhere defined" after having defined only "continuous function" and "function not everywhere defined" is not an example of abuse of language. In fact, as there are several reasonable definitions for this term, this would be an example of woolly thinking or a cryptic writing style.