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We investigate the form of mathematical structuralism that acknowledges the existence of structures and their distinctive structural elements. This form of structuralism has been subject to criticisms recently, and our view is that the problems raised are resolved by proper, mathematics-free theoretical foundations. Starting with an axiomatic theory of abstract objects, we identify a mathematical structure as an abstract object encoding the truths of a mathematical theory. From such foundations, we derive consequences that address the main questions and issues that have arisen. Namely, elements of different structures are different. A structure and its elements ontologically depend on each other. There are no haecceities and each element of a structure must be discernible within the theory. These consequences are not developed piecemeal but rather follow from our definitions of basic structuralist concepts. * This is a preprint of a paper published in Mind, 123/489 (2014): 39-78. We use red font to make a small correction to footnote 14 and to revise the infix notation in two formulas on page 36 (to make the formatting more consistent).

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Ibn 'Arabî (1165-1240) can be considered the greatest of all Muslim philosophers, provided we understand philosophy in the broad, modern sense and not simply as the discipline of falsafa, whose outstanding representatives are Avicenna and, many would say, Mullâ Sadrâ. Western scholarship and much of the later Islamic tradition have classified Ibn 'Arabî as a "Sufi", though he himself did not; his works cover the whole gamut of Islamic sciences, not least Koran commentary, Hadith (sayings of Muhammad), jurisprudence, principles of jurisprudence, theology, philosophy, and mysticism. Unlike al-Ghazâlî, whose range of work is similar to Ibn 'Arabî, he did not usually write in specific genres, but tended rather to integrate and synthesize the sciences in the context of thematic works, ranging in length from one or two folios to several thousand pages. Nor did he depart from the highest level of discourse, or repeat himself in different works. The later Sufi tradition called him al-Shaykh al-Akbar, the Greatest Master, a title that was understood to mean that no one else has been or will be able to unpack the multi-layered significance of the sources of the Islamic tradition with such detail and profundity.