Implicit notational conventions

[Reprinted from a Google buzz article from Mar 20, 2010.]

Like any other human language, mathematical notation has a number of implicit conventions which are usually not made explicit in the formal description of the langauge. These conventions serve a useful purpose by conveying additional contextual data beyond the formal logical content of the mathematical sentences.

A good example of this is the naming conventions for variables. While in principle any symbol can be used for one type of variable, in practice individual symbols have pre-existing connotations that make it more natural to assign them to specific variable types. For instance, one usually denotes:

• x to denote a real number
• z to denote a complex number
• n to denote a natural number

A mathematical argument involving a complex number x, a natural number z, and a real number n would read very strangely. The most famous example of this is perhaps the use of the symbol ε in analysis; an analysis argument involving a quantity ε which was very large or negative would cause a lot of unnecessary cognitive dissonance. In contrast, by sticking to the conventional roles that each symbol plays, the notational structure of an argument is reinforced and made easier to remember; a reader who has temporarily forgotten the definition of, say, z, in an argument can at least guess that it should be a complex number, which can assist in recalling what the actual definition is.

As another example from analysis, when stating an inequality such as f(x) ≤ C or |S| ≤ N, it is customary that the left-hand side represents an “unknown” that one wishes to control, and the right-hand side represents a more “known” quantity that one is better able to control; thus for instance f(x) ≤ C is preferable to C ≥ f(x), despite the logical equivalence of the two statements. This is why analysts make a significant distinction between “upper bounds” and “lower bounds”; the two are not symmetric, because in both cases one is bounding an unknown quantity by a known quantity. (Another relevant convention in analysis here is that it is preferable to bound non-negative quantities rather than non-positive ones, thus for instance f(x) ≥ 0 is preferable to -f(x) ≤ 0.)

Continuing the above example, if the known bound C is itself the sum of several terms, e.g. C = A + B, then it is customary to put the “main” term first and the “error” terms later; thus for instance f(x) ≤ A + B is preferable to f(x) ≤ B + A. By adhering to this standard convention, one conveys useful information as to which terms are considered main terms and which ones considered error terms.

There are of course many other implied conventions, both in analysis or in other fields of mathematics; unfortunately, there is no exhaustive listing of such conventions, and one generally has to pick them up through extensive reading of a subject.

Some followup discussion, on the specific connotations assigned to round numbers such as 100, can be found here.