# Math Terminology and Naming of Things

By Xah Lee. Date:

I love math, i love math symbols, and i love math terms. Here are some of the terms i love:

• group, ring, field.
• algebra ( algebraic geometry, algebraic topology, algebraic number …)
• Lattice
• isomorphism
• homomorphism
• isometry (rigid transformation)
• homeomorphism (continuous transformation)
• affine: (affine geometry, Affine transformation)
• homotopy
• topology
• homogeneous (space)
• manifold (n-dimensional shaped space)
• polytope (n-dimensional “polygons”)
• polynomial
• projective (geometry, plane)
• linear: (linear algebra, linear transformation)
• function, functional, functor, map, bijection
• iff (if and only if)
• complex: (Complex number)
• degree (angle, polynomial)
• valence (network nodes)
• genus (of graphs)
• matrix, vector, vector space

I thought that i could create a list of a few terms i dislike as well.

• calculus and analysis (better: limit theory, continuity theory, infinitesimal theory)
• lambda calculus (better: formal logic theory, lambda logic theory)
• complex numbers. (Better: couple numbers, plane numbers)
• complex analysis, theory of functions. (continuity theory with plane numbers)
• graph theory (better: network theory)
• abscissa, ordinate (effete. Better: x-axis, y-axis)
• abelian (better: commutative)
• elliptic functions. (planar periodic functions)
• differentials (spetre of departed souls)
• determinants (much ado and determining not much)

Most of these i dislike because of the misleading nature of the terms. (such is sometimes termed “misnomer”). But some i dislike because i dislike the concept behind it, such as differentials or determinants. In particular, these necessarily came from a particular philosophy of mathematics i hate. Specifically, “differentials” is from physics that does not have a sound connection as a pure math atomic concept, and “determinants” i dislike because it is ad hoc process which doesn't have a solid connection to math.

The naming of things has significant effect on how things are understood. Not directly as in teaching a specific field, but with massive indirect consequences in communication that implicitly shape people's thoughts at large. More generally, this is a problem that stems from natural languages. Languages itself, has massive indirect effect on how people think. The other generic reason that jargons are unnecessarily used is because the human nature of class differentiation.

Imagine if the entire corpus of mathematics employ proper, clear, logical, names, where upon seeing a term one can often infer its context, ontology (category and relations), and meaning. (A significant portion of math terms, if not majority, are not named in this way.) To any college math student and most critically the general public, it would be like as if a fog around math has suddenly been lifted, and the relations and their hierarchies and categories and the contexts they are used would suddenly all become clear. With confusion and unnecessary mysteries gone, efforts and directions in math would change massively in a implicit way. This is the type of improvement hard to realize in advance but couldn't live without afterwards.

Mathematicians, like other professionals, tend to use jargon unnecessarily. (one good thing is that mathematicians tend to create less unnecessary jargons, and their jargons have exacting definitions comparatively speaking (in contrast to the “computer science” and Info Tech industry.).) The tendency to employ or overuse jargons may be inevitable, since tech jargons not only serve communication needs for professionals, but its use is inescapably also a social function, giving airs of reconditeness and keeping others out. The massive communication revolution in technology (internet, cellphone) is making a lot changes to jargons in society.

## The Terminology “Implicit Surface”

I always disliked the terminology “implicit surface” (or “implicit curve”). A implicit surface is the surface of a equation of 3 variables, for example, “x^2+y^2+z^2-1==0”. They are so-called implicit because the word suggests that the equation implicitly defines a surface, namely, the set of its solutions are points in 3D space, typically smoothly connected, thus a surface. To call this “implicit” is rather silly, because math surfaces and manifolds are in a sense all implicit; they are mostly a plotting scheme for visualization, or as rigorous definition to geometric entities (For example, algebraic variety, manifold). The word “implicit” in “implicit surface” is mostly used to contrast it with “parametric surface”. In parametric surface, a formula explicitly gives points on a surface. The force of the term “implicit surface” is in its contrast to the so-called “parametric surface”.

Curves and surfaces, in general, are predominantly visual elements. In the context of math, they are objects studied in geometry. The essential quality of curves and surfaces, and in general geometry, is the visual aspect. Characterizing a surface by implicit or parametric, is a characterization of the formula, of which a visual object is constructed. In essence, it is a characterization of a visualization scheme. The characterization has little to do with the character of the surface itself. Surfaces defined as roots of polynomial or as point-generating formula are just particular methods of plotting for visualization.

Today i found a much better term for the so-called implicit surface: Isosurface. This term is much better because the term gives the essential info about the surface, namely, it is a surface of Level set. Compare the term isosurface and implicit surface. The former gives clear info about a nature of the surface, while the “implicit” doesn't do much.

Of course, the reason that “implicit surface” or “implicit curves” is popular due to historical development. Namely, it is not a parametric surface, at a time of 1800s or 1900s a surface is pretty much just parametric or implicit. But i think as today, math knowledge and communication efficiency and the information exchanged have advanced some few hundred times, isosurface will become much more popular, as it should.

When i was studying calculus in about 1992, the textbook has a section named implicit curves or implicitly defined curves. I wonder today's calculus books still use that term.

### The Terminology “Parametric Equation”

Another terminology i find to be not good quality is “Parametric Equation”. A parametric equation, means a set of functions as a scheme of plotting a surface, or as a way to define a math entity of manifold.

I find “Parametric Equation” not being a quality terminology, is because the use of the term Equation in “Parametric Equation”. Parametric Equation, are really not equations. They are, in its context, more like functions, or a set of definition of functions. Further, the phrase “Parametric Equation” does not make much sense logically, since equations usually don't have parameters, and if they do, it's often called a family of equations, where the parameters are the constants.

I haven't thought deep about a better term, but so far on my website's math expositions, i use the term “Parametric formula” instead.

Of course, the reason “Parametric Equation” became popular, is due to historical developments. Namely, the phrase immediate connote the concept of parameters, which is the intrinsic nature of parametric curves and surfaces. Namely, the surface is generated by varying parameters. The reason that “equation” is part of the phrase is because, during 1800s or 1900s, the distinction of equation and definition is not clearly made, because the study of logic or formal systems have barely started. In particular, there is no computer languages, computer algebra systems, or theorem verification or proving systems. Even today, the notation for equality and definition are often still the same: “=”. (but in computing languages, notation for equation is necessarily made distinct from definitions. Equation is often denoted by “==” while definition is “=”. Since 1990s or so, some mathematicians started to use “:=” for definition to distinguish it from equality, and “===” is sometimes used for identity.)

Also note, from logic point of view, the concept of “equation”, “identity”, “definition” are all distinct. In traditional math notation, they all still use the same symbol “=”.

“group theory” should be called “symmetry theory”