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# Math Terminology and Naming of Things

Xah Lee,

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 (⁖ 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 text book 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's 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 “=”.

## Chinese Math Terms

Here are some math terms in Chinese. Terminology in different languages give us a fresh view of their meaning.

There are two sets of math terminology in Chinese. One is used in China, one in Taiwan. Most terms are same for both, but not always. China's science terminology tends to be more logical and makes sense in Chinese itself. Many of math and computer terms used in Taiwan tend to be transliterations of English terms. (that is, by sound) So in general, science terminologies of People's Republic of China are better, but not always.

(the Taiwan government is a mess. They purportedly pride themselves in “guarding” traditional Chinese values (refused to adopt the more scientific simplified characters reformed by People's Republic of China (PRC)), yet they promiscuously borrow English terms (and Western fashion and everything else Western), and around 2000 invented their own Romanization system from China's PinYin system so that they can “just be different”. (PinYin system was already a widely used international standard.) The situation of Romanization in Taiwan now is that sometimes you'll see PinYin system, while other times Taiwan's own invention, very confusing to foreigners.)

Some of the Chinese math terms are direct translations of English ones. For example, graph theory in Chinese is 圖論, which literally means “picture theory”. This stupidity is in part due to the stupidity in English term itself. The proper term should be Network Theory (網論). The Chinese term for Topology (拓樸) is very bad. It is a translation of the English term by sound.

In Taiwan, university math students often use English text books. I think math communities in Taiwan are more familiar with English terminologies than in PRC. This is not necessarily a bad thing. From a efficiency point of view, it'd be better if the whole world uses just one language. Translation can be considered as a extraneous activity.

Multiple languages is also problematic in that it creates huge political enmity. On the other hand, since the world as is not a single language world, and language is not purely a tool of communication because it is a cultural artifact, therefore steps towards one single language in the already multi-lingual world may not necessarily be a good thing as it reduces diversity in a complex context.

• discrete math 離散數學
• finite math 有限數學
• Category Theory 範論
• graph theory 圖論 (network theory 網論)
• number theory 數論
• algebra 代數, group 群, field 體, ring theory 環論
• combinatorics 組合學
• game theory 對局論
• logic 邏輯
• cryptography 密碼學
• knot theory 結論
• tiling 鑲嵌
• symmetry
• Computational Geometry 計算幾何
• Catastrophe theory 劇變論
• linear algebra 線性代數 • linear transformation
• chaos 混沌
• Dynamical Systems 動力系統
• complex analysis 復變分析
• set theory 集論
• manifold 流形
• topology 拓樸
• Fractals geometry 碎形幾何
• derivative 微分 • integral 積分
• differential equations 微分方程, partial differential equations 偏微分方程
• analysis 分析, calculus 微積分 • multi-variable analysis 多變量分析
• probability 機率論
• statistics  統計
• Non-Euclidean geometry 非歐幾何, elliptical, hyperbolic
• Plane Curves 平面曲線
• Expression Transformation (形式轉化), Pattern Matching (形式配對)
• Matrix 矩陣, vector 向量, tensor 张量
• Elliptic curves 橢圓曲線
• prime number 質數
• Fermat's Last Theorem 費馬最後定理
• functional programing, 函数式程序设计, function 函數
• variable 變數?自變數?應變數?
• parameter
• argument
• polyhedrons 多面體, Uniform Polyhedrons 均勻多面體,polytope
• dimensions 次元,次方