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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).
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