Mathematics and Jargon

by Willie Wong

Each profession has its own set of special language: some, like the medical profession, rely on long and precisely defined words, often with Latin, Greek, or German origins, to describe objects and events that we do not encounter on the everyday (I doubt anyone actually found the need to use pneumonoultramicroscopicsilicovolcanoconiosis in everyday conversation [it is a lung disease caused by inhalation of metallic dust]); some, like the legal profession, attach a preferred meaning to common everyday words (in addition to special terminology), and put certain heft on syntactical and grammatical specification to say exactly what they mean.

In the natural sciences, the tendency has been toward the coinage of words to refer to new objects or ideas. My guess is that this has to do with the earliest natural philosophers taking a cue from Adam and giving a name to everything they have not seen before. In mathematics (which, by the way, while may be arguably natural by certain definitions, is not a science), on the other hand, the tendency has been toward usurping everyday words for specialist purposes. Here I’ll list a few:

Algebra seems the worst offender (probably because some early algebraists have some sort of taxonomist mindset in classifying things): set, category, group, ring, field, associative, commute, variety, simple, ideal … (and even the name algebra itself has been usurped for a kind of algebraic structure)

Prime in number theory, manifold in geometry, inject in elementary analysis. Sometimes the same word carries vastly different meanings even within different, but related, branches of mathematics. Take for example the word “normal”. In studying vector spaces, sometimes the word is used in place of “normed” as in having a way to measure length. In linear algebra, normal can be taken to mean perpendicular, but the normal form of a linear operator is its presentation in a special basis. In many fields normal can mean “normalized”, in the sense that an object is rescaled to a standard size. In probability theory, a normal distribution is a Gaussian one. In functional analysis, a normal operator is one that commutes with its adjoint. In algebra it denotes a specific kind of field extensions (yes my Galois theory is a bit spotty now), as well as a special kind of subgroup. And I am sure there are more definitions! The word is certainly overloaded (a computer programming parlance).

So why did this topic come to mind? Last Friday I went to a somewhat interesting talk, from which I certainly picked up plenty of “culture”. But I was definitely “duped” into going to the talk. The title was “Global stability of compressible flow around swept parabolic bodies.” Now, in my field of specialty (partial differential equations), “stability” has a very precise meaning (rather, there are three different, but all closely related, and all very precise, meanings). It refers to the study of perturbation of solutions. A solution is stable, roughly speaking, if a very small perturbation of the current conditions will lead to a not-too-different future. It is, in some sense, the opposite of mathematical chaos. In some cases, the stability may be even strong enough that perturbations will disperse or dissipate, somehow reconciling the two futures.

Now, this talk was part of the Fluids Seminar. My personal, albeit limited, exposure to fluids suggests that those people should “like” partial differential equations. After all, a lot of modern theory of evolutionary differential equations found its roots in the study of water waves by 19th century mathematicians. Furthermore, one of the Clay Millennium Prize problems is about the Navier-Stokes equation, a fundamental descriptor of fluid flow. So I somewhat expected the talk to involve stability as I understand it. Maybe not completely mathematically rigorously: perhaps in a relaxed, physics-type of stability, or perhaps in a indicative, numerical-simulation-way of stability.

Let’s just say I was wrong.