# The Philosophy of Mathematics and the Independent ‘Other’.

My PhD thesis was accepted on 15^{th} March 2005.

*Abstract:*

This thesis explores some of the fundamental notions motivating mathematical realism, in particular the idea of a separate, independent mathematical realm. This notion, which at its core is the concept of an 'other', forms the work's primary focus. Various versions of mathematical realism are explored, including structuralism, naturalism and a transcendental approach - all of which attempt to incorporate or engage with an 'other' mathematical realm. Arguing that none of these approaches retains the original intuitive notion of a truly separate 'other', an alternative account is offered, drawing inspiration from Husserl's phenomenology and Derrida's analysis of metaphysics as theology. The account proposes that, in order to retain the concept of an 'other', the independent mathematical realm must be uncircumscribable by any account, and literally beyond comprehension. But, in order to preserve mathematical realism as a philosophy, and in accordance with its fundamental optimism (that the mathematics we access somehow is the mathematical realm and other than us), mathematics must also be within our grasp and able to be fully seen as it is in and of itself. So mathematics must be fully within and fully outside of our grasp (or consciousness). It is argued that this paradox is unavoidable and irreducible, but not necessarily a drawback for realism.

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