  The definition of who is or what makes a ``mathematician" is an important and
urgent issue to be addressed in the mathematics community. Too often, a
narrower definition of who is considered a mathematician (and what is
considered mathematics) is used to exclude people from the discipline -- both
explicitly and implicitly. However, using a narrow definition of a
mathematician allows us to examine and challenge systemic barriers that exist
in certain spaces of the community. This paper explores and illuminates
tensions between narrow and broad definitions and how they can be used to
promote both inclusion and exclusion simultaneously. In this article, we
present a framework of definitions based on identity, function, and
qualification and exploring several different meanings of ``mathematician". By
interrogating various definitions, we highlight their risks and opportunities,
with an emphasis on implications for broadening and/or narrowing participation
of underrepresented groups.
