There are many examples of confusing terminology in math, along with multiple reasons why they are confusing. Some terms are ambiguous, meaning that they are reused for different concepts. Other terms are “almost ambiguous,” meaning that they are not word-for-word reused, but have such similar names that it would be easy to be confused. These scenarios go from worst to best — the latter case is at least more manageable, if still not desirable.
For each scenario in this list, I try to suggest a remedy, typically via a disambiguation term. Such a term should either (1) already be prevalent in the literature as an alternative or (2) have an easily guessable meaning just based on its name. If we can’t meet either of these conditions and want to introduce a completely new term, then that may be impractical as a replacement for terminology that is already highly entrenched.
This list will probably always be incomplete, and I would love feedback on expanding it.
Ambiguous
- “Algebra” is the name of a subject. It is also the name of a specific algebraic structure, a vector space with a bilinear vector multiplication operation.
- Remedy: the term “
-algebra” is also common for this concept, and we can use this term anyway regardless of whether the field
- Remedy: the term “
- “Variety” is a name for a universal algebra concept, while “algebraic variety” is a name for a totally different algebraic geometry concept (the “algebraic” isn’t just an adjective that asserts more structure.)
- Remedy: we should use the term “equational class” for the universal algebra concept.
- The term “Boolean algebra” refers to a specific algebra, but it also refers to an abstract algebraic structure that encapsulates the properties of this system.
- Remedy: we should use a more explicit term like “abstract Boolean algebra” to distinguish these concepts.
- Historically, a “groupoid” referred to what is now called a “magma,” but it also refers to a separate concept in category theory.
- Remedy: we should use the term “magma” for the algebraic structure.
- A “distribution” can mean a probability distribution or a mathematical concept that informally can be described as a “generalized function.”
- Remedy: we should use “probability distribution” in the former case and “Schwartz distribution” for the latter case.
- A “positive homogeneous” function is not a function that is homogeneous and positive. The term “positive homogenous” (all together) has a specific meaning.
- Remedy: an undesirable solution could be to use “homogeneous positive” for a function that is homogeneous and positive (just switch around the terms.) Another still undesirable solution would be to use “positively homogeneous.” I propose the following: call a function
a
-homogeneous function if there exists a constant
such that for all
we have
. (I am not sure whether this notation is already in use.) We can then replace the term “positive homogeneous” by “
-homogeneous.”
- Remedy: an undesirable solution could be to use “homogeneous positive” for a function that is homogeneous and positive (just switch around the terms.) Another still undesirable solution would be to use “positively homogeneous.” I propose the following: call a function
Almost Ambiguous
- The subject of “functional analysis” is not the analysis of functionals. It is the study of topological notions on vector spaces, which strongly generalize functionals. However, a “functional” still refers to the specific concept, and the analysis of functionals is instead called the “calculus of variations.”
- Remedy: instead of the term “functional analysis,” we should use “vector space analysis.” (Another good proposal could be “linear analysis,” but I have seen this term used already to mean other things.)
- A “dynamical system” is a general term for a system that evolves over time, while a “dynamic equation” has a specific technical meaning as an equation involving the Hilger derivative of time-scale calculus. Even more confusingly, we can speak of “systems of dynamic equations,” which mean a different thing than “dynamical systems.”
- Remedy: instead of the term “dynamic equation,” we should have a different name. I propose “time-scale differential equation.”
- “One-to-one correspondence” refers to a type of function, but a different type than a “one-to-one function.”
- Remedy: we should use the terms “injection,” “surjection,” and “bijection,” which are more clearly different.
- A “homomorphism” is a separate concept from a “homeomorphism” (the only difference is the letter “e.”)
- Remedy: instead of “homeomorphism” (with an “e”) we should use “topological isomorphism.”
Last updated March 22, 2022 (published later.)
