Common words that have a technical meaning in math

Mathematical writing is the opposite of business writing in at least one respect. Math uses common words as technical terms, whereas business coins technical terms to refer to common ideas.

There are a few math terms I use fairly often and implicitly assume readers understand. Perhaps the most surprising is almost as in “almost everywhere.” My previous post, for example, talks about something being true for “almost all x.”

The term “almost” sounds vague but it actually has a precise technical meaning. A statement is true almost everywhere, or holds for almost all x, if the set of points where it doesn’t hold has measure zero.

For example, almost all real numbers are irrational. There are infinitely many rational numbers, and so there are a lot of exceptions to the statement “all real numbers are irrational,” but the set of exceptions has measure zero [1].

An event is almost certain if it occurs with probability 1; the exceptions (if there are any) have measure zero with respect to the probability measure.

In common parlance, you might use ball and sphere interchangeably, but in math they’re different. In a normed vector space, the set of all points of norm no more than r is the ball of radius r. The set of all points with norm exactly r is the sphere of radius r. A sphere is the surface of a ball.

The word smooth typically means “infinitely differentiable,” or depending on context, differentiable as many times as you need. Often there’s no practical loss of generality in assuming something is infinitely differentiable when you only need to know, for example, that it only needs three derivatives [2]. For example, a manifold whose charts are once differentiable can always be altered slightly to be infinitely differentiable.

The words regular and normal are used throughout mathematics as technical terms, and their meaning changes completely depending on context. For example, in topology regular and normal are two kinds of separation axioms. They tell you whether a topology has enough open sets to separate a point from a closed set or separate two closed sets from each other.

When I use normal I’m most often talking about a normal (i.e. Gaussian) probability distribution. I don’t think I use regular as a technical term that often, but when I do it probably means something like smooth, but more precise. A regularity result in differential equations, for example, tells you what sort of desirable properties a solution has: whether it’s a classical solution or only a weak solution, whether it’s continuous or differentiable, etc.

While I’m giving a sort of reader’s guide to my terminology, log always refers to natural log and trig functions are always in radians unless noted otherwise. More on that here.

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The footnotes below are much more technical than the text above.

[1] Here’s a proof that any countable set of points has measure zero. Pick any ε > 0. Put an open interval of width ε/2 around the first point, an interval of width ε/4 around the second point, an interval of width ε/8 around the third point etc. This covers the countable set of points with a cover of measure ε, and since ε as arbitrary, the set of points must have measure 0.

The irrational numbers are uncountable, but that’s not why they have positive measure. A countable set has measure zero, but a set of measure zero may be uncountable. For example, the Cantor set is uncountable but has measure zero. Or to be more precise, I should say the standard Cantor set has measure zero. There are other Cantor sets, i.e. sets homoemorphic to the standard Cantor set, that have positive measure. This shows that “measure zero” is not a topological property.

[2] I said above that often it doesn’t matter how many times you can differentiate a function, but partial differential equations are an exception to that rule. There you’ll often you’ll care exactly how many (generalized) derivatives a solution has. And you’ll obsess over exactly which powers of the function or its derivatives are integrable. The reason is that a large part of the theory revolves around embedding theorems, whether this function space embeds in that function space. The number of derivatives a function has and the precise exponents p for the Lebesgue spaces they live in matters a great deal. Existence and uniqueness of solutions can hang on such fine details.