Mathematics is a subject in which the jargon is required to be especially rigorous and unified. Like other bloggers this month, I will discuss how jargon, terminology, and other wordly matters play out in mathematics.
One of the first important distinctions in mathematics is that certain words are used as in normal English, while others are used with specific technical meaning. For example, the term group is used in common parlance to denote some specific set of objects, people, etc. that have some common characteristic or purpose. In mathematics, a group is a particular type of mathematical structure. Mathematicians use different words (like set, family, or collection) to refer to groups (in the common English sense) of things that are not endowed with this structure. It is even important to split hairs when using such words. The term set is defined in a specific way in mathematics, but it means precisely what it might mean in English. However, there are restrictions on what a set could be. Russell’s Paradox is one example of how something that could be called a set in English is not a set in mathematics. And so a new word is created to describe this kind of thing – it is not a set, it is a class. Fellow GSNB blogger Michael notes here the confusion that may arise if common English terms collide with technical terms.
Mathematics often borrows or redefines terms from common parlance, from other fields of study, or even from different areas of mathematics. The important consideration is whether a reader would understand – generally erring on the side of caution, defining words and phrases as needed. Mathematical exposition is peppered with definitions in a way others may find only in dictionaries. But if an astronomer were to read about a syzygy in a physics paper that requires some mathematics of that sort, he or she would be confused without having a mathematical definition handy to differentiate this from the astronomical definition of syzygy. It’s hard to fight the urge not to define terms throughout this blog post!
An important quality of mathematical exposition is not just mathematical fluency, but clarity. Terminology should be used judiciously. This is important because the terminology does not just describe or expound the content, it is itself content. Carolyn, another GSNB blogger, discusses here the construction of meaningless strings that may sound very impressive if not read very carefully. This is true in mathematics too, where a randomly-generated paper was recently accepted for publication, albeit to a for-profit journal that seems to lack appropriate peer-review. And, importantly, anyone with mathematical training could have spotted this paper as fraudulent without any particular specialty or knowledge. It’s virtually impossible to “bluff” in this way in mathematics.
Abbreviations, wordplay, and figurative terminology are all useful in mathematics, but provide further barriers to understanding. The term “subadditive” describes a function where f(x+y)≤f(x)+f(y). The “sub” gives you ≤ and the “additive” gives you the + sign. It’s a definition one can parse from the word itself, given experience with mathematical terminology. But if this were abbreviated, it would lose that meaning. For example, a directed acyclic graph becomes a DAG; a partially ordered set, a poset; a universal Turing machine, a UTM. A separable completely metrizable topological space is called a Polish space (because they were first studied by Poles). Similarly, a ring without an identity element is called a rng (the i is removed, since there is no identity). And if acronyms and wordplay were not enough, terms like “potatoes” and “squiggles” can be used to describe something with a more precise meaning in an informal context. Mathematicians describe structures as “well behaved” or “badly behaved,” and even the term “almost always” has a precise, technical meaning. There is no snake involved in the snake lemma.
In the end, mathematical terminology is important because these terms, ideas, structures, etc. make up the building blocks of the theory. But they should be explained clearly, concisely, and precisely, and so with the audience in mind. Technical language should be an accessory to normal language, allowing us to make more refined and meaningful statements. Mathematical jargon allows us to use technical (and even non-technical) language to make progress, not to obfuscate it.
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