The role of mathematics as a foundation of other
sciences is well acknowledged since the dawn of
modern science. This fundamental and important
role is then not only limited to the natural and
physical sciences but also further extended to
the social domains of humanities. This article
reviews the many definitions of mathematics as
put forth by philosophers and mathematicians,
how these definitions are understood, and how they subsequently influence the development
of mathematics itself as well as other disciplines
influenced by mathematics. These definitions
are: mathematics as the science of teaching
and learning (ʿulūm al-taʿālim); mathematics as
abstraction or the construction of abstracts;
mathematics as logic; mathematics as a formal
system; mathematics as the science of quantity and
structure, mathematics=ontology; and mathematics
as intellectual exercise (riyāḍiyyāt). Based on the
aforementioned definitions, this article attempts
to analyse the ontological status of mathematics
and mathematical objects based on the degrees of
existence (marātib al-wujūd) framework which orders
existence into five levels, namely essential existence
(wujūd dhātī), concrete existence (wujūd ‘aynī), mental
existence (wujūd dhihnī), verbal existence (wujūd lafẓī),
dan formal or written existence (wujūd khaṭṭī).