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Title: Avicenna's philosophy of mathematics
Author: Zarepour, Mohammad Saleh
ISNI:       0000 0004 7968 4651
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2019
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I discuss four different aspects of Avicenna's philosophical views on mathematics, as scattered across his various works. I first explore the negative aspect of his ontology of mathematics, which concerns the question of what mathematical objects (i.e., numbers and geometrical shapes) are not. Avicenna argues that mathematical objects are not independent immaterial substances. They cannot be fully separated from matter. He rejects what is now called mathematical Platonism. However, his understanding of Plato's view about the nature of mathematical objects differs from both Plato's actual view and the view that Aristotle attributes to Plato. Second, I explore the positive aspect of Avicenna's ontology of mathematics, which is developed in response to the question of what mathematical objects are. He considers mathematical objects to be specific properties of material objects actually existing in the extramental world. Mathematical objects can be separated, in mind, from all the specific kinds of matter to which they are actually attached in the extramental word. Nonetheless, inasmuch as they are subject to mathematical study, they cannot be separated from materiality itself. Even in mind they should be considered as properties of material entities. Third, I scrutinize Avicenna's understanding of mathematical infinity. Like Aristotle, he rejects the infinity of numbers and magnitudes. But he does so by providing arguments that are much more sophisticated than their Aristotelian ancestors. By analyzing the structure of his Mapping Argument against the actuality of infinity, I show that his understanding of the notion of infinity is much more modern than we might expect. Finally, I engage with Avicenna's views on the epistemology of mathematics. He endorses concept empiricism and judgment rationalism regarding mathematics. He believes that we cannot grasp any mathematical concepts unless we first have had some specific perceptual experiences. It is only through the ineliminable and irreplaceable operation of the faculties of estimation and imagination upon some sensible data that we can grasp mathematical concepts. By contrast, after grasping the required mathematical concepts, independently from all other faculties, the intellect alone can prove mathematical theorems. Other faculties, and in particular the cogitative faculty, can assist the intellect in this regard; but the participation of such faculties is merely facilitative and by no means necessary.
Supervisor: Street, Tony Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
Keywords: Avicennna ; Philosophy of Mathematics ; Islamic Philosophy ; History of Philosophy ; Mathematical Objects ; Infinity ; Epistemology of Mathematics