An algebra and a calculus

One generation develops an area of math and a second generation comes along to tidy up the definitions. Sometimes this leads to odd vocabulary because the new generation wants to retain terms even as it gives them different meanings.

In high school you get used to hearing about “algebra” and “calculus” as subjects, then the first time you hear someone say “an algebra” or “a calculus” it sounds bizarre. Similarly, once you’re used to hearing about logic, statistics, or topology, it sounds exotic to hear someone say “a logic,” “a statistic,” or “a topology,” something like hearing “a speed of light.”

When I signed up for a topology course, visions of “rubber sheet geometry” danced in my head. I was taken aback when the course started out by defining a topology to be a collection of sets closed under finite intersections and arbitrary unions. Where did the rubber sheet go?

Sometimes when a subject takes on an indefinite article, the new definition comes early in the development. For example, “a statistic” is any function of a random sample, a disappointing definition that comes early in a statistics course. But sometimes the definition with the indefinite article comes much later.

You can take a couple courses in algebra in high school and even a college course in algebra (i.e. groups, rings, and fields) without ever hearing the definition of “an algebra.” And you can take several courses in calculus without ever hearing of “a calculus.”

The phrase “an algebra” can have several meanings, and the phrase “a calculus” even more. The core definition of an algebra is a vector space with a bilinear product. But there are more general ideas of an algebra, some so general that they mean “things and operations on things.” (That’s basically universal algebra.)

Often you put an adjective in front of algebra to specify what kind of algebra you’re talking about: Banach algebra, σ-algebra, Boolean algebra, etc. Usually in mathematics, an adjective in front of a noun narrows the definition. For example, an Abelian group is a particular kind of group. But often an algebra with a qualifier, such as a σ-algebra, is not an algebra by what I called the core definition above.

The idea of “a calculus” is even more vague. If it has a formal definition, I’ve never seen it. A calculus is just a subject with a set of useful rules for calculating things. Sometimes there’s a strong analogy to calculus in the sense of differential calculus, as in the calculus of finite differences, a.k.a. umbral calculus. Sometimes the analogy is more of a stretch, as in λ calculus or π calculus.

So what is the point of this post? I don’t really have one. Just throwing out some half-baked musings.

7 thoughts on “An algebra and a calculus

  1. Now I don’t feel so bad about not really knowing what ‘an algebra’ or ‘a calculus’ means when I’m reading papers.

  2. The only calculus worthy of that name is “THE calculus” as it was once called.

  3. Calculus is such an old word, at one point it just meant “a method of calculating”. Then the calculus of differences came along and started to transform the meaning — but not completely.

  4. Actually I think a sigma algebra is an algebra in the usual sense. We view it as a vector space over the field of two elements, with symmetric difference being our addition and the empty set being our 0. Then we let intersection be multiplication and the whole set be our 1. (Finite) union and complement can be defined in terms of intersection and symmetric difference, and then being a sigma-algebra just means that it’s closed with respect to certain limits.

  5. I wonder if not language patterns have influenced the mathematical development to some degree. Mathematicians always look for patterns after all, and once you talk about “a group”, “a space” and so on, it’s very natural to idly ponder what putting “a/an” in front of other terms could mean.

  6. I’ve also heard “calculus” used as an explicit grouping term, generally starting with “the calculus of …”. Such as, “the calculus of 3D kinematics.”

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