The word "abelian" is generally not capitalized in mathematical contexts because it has become so deeply ingrained and institutionalized within the field that it functions more as a descriptive adjective for a mathematical property than a direct proper noun derived from the mathematician Niels Henrik Abel.
A Rare Exception Among Mathematical Adjectives
While many mathematical terms derived from the names of mathematicians typically retain capitalization (e.g., Riemannian geometry, Euclidean space, Gaussian distribution), "abelian" stands out as a rare exception. Unlike terms that explicitly honor an individual by consistently capitalizing their name, "abelian" is often spelled with a lowercase 'a'.
The Institutionalization of Abel's Name
The lack of capitalization for "abelian" serves as a tacit acknowledgment of the profound degree to which Norwegian mathematician Niels Henrik Abel's name and his contributions have been institutionalized into fundamental mathematical concepts. This means:
- Deep Integration: Concepts associated with "abelian," such as an abelian group or an abelian category, are considered foundational and ubiquitous across various branches of mathematics.
- Beyond Honorific: The term has transcended being merely an honorific. It functions as a standard descriptor for a specific type of mathematical structure or property, much like a common adjective.
- Ubiquitous Usage: Its widespread and routine use in advanced mathematics has led to its linguistic evolution, where its origin as a proper name becomes secondary to its functional meaning.
This convention reflects how certain fundamental ideas, even those originating from specific individuals, eventually become common vocabulary within a discipline.
