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Avicenna's Philosophy of Mathematics View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Apollo Avicenna’s Philosophy of Mathematics Mohaammad Saleh Zarepour Faculty of Divinity University of Cambridge This dissertation is submitted for the degree of Doctor of Philosophy Clare Hall January 2019 I hereby declare that this dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration, except where specifically indicated in the text. I further state that no substantial part of my dissertation is the same as any that I have submitted, or, is being concurrently submitted for a degree or diploma or other qualification at the University of Cambridge or any other University or similar institution. This dissertation does not exceed the 80,000-word limit. The second chapter, with slight modifications, has been published in Dialogue: Canadian Philosophical Review. The first and thrid chapters, again with some revisions, are accepted for publication, respectively, in Oriens and Archiv für Geschichte der Philosophie. The final chapter is under review for publication in another journal. Mohammad Saleh Zarepour January 2019 i Avicenna’s Philosophy of Mathematics Mohammad Saleh Zarepour ABSTRACT. 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. ii To Samaneh Ehsaninezhad iii Acknowledgements I thank my supervisor, Tony Street, without whose kind support, encouragement, insightful guidance, and patience in helping me develop my ideas, this dissertation would have remained unwritten. The depth of my gratitude to him cannot be put into words. I am also grateful to the examiners of my first year registration examination, Khaled El-Rouayheb and John Marenbon, whose comments helped me better understand the direction I should follow in this project. Parts of earlier versions of this dissertation have been presented on various occasions, and I have greatly benefited from the feedback and comments of my audiences. Special thanks goes to Peter Adamson, Mohammad Ardeshir, Robert Gleave, Jari Kaukua, Andreas Lammer, Hossein Massoumi Hamedani, Toby Mayer, Seyed N. Mousavian, Reza Pourjavady, Sajjad Rizvi and Ayman Shihadeh; and I also had the opportunity to discuss some of my ideas with Asad Ahmed and Ricardo Strobino while they were visiting Cambridge. I express my gratitude to them all, as well as to Jon McGinnis with whom I had many helpful exchanges. These acknowledgements would not be complete without also thanking my friends Feriel Bouhafa, Daniel De Haan, and Tianyi Zhang, with whom I had many long, enjoyable and fruitful conversations on various aspects of my project as well as on many other issues. Beyond the academic realm, there are others who generously helped me reach this stage of my life. I thank my parents, my sisters, my nephew and my in-laws for supporting me throughout my PhD work and in my life in general. Last, but by no means least, I am deeply grateful to my wife, Samaneh Ehsaninezhad, whose presence in my life has always been heartwarming and inspiring. Without her encouragement this dissertation would have never been completed, and she is the one to whom this work must be dedicated. iv Contents INTRODUCTION 1 1. AGAINST MATHEMATICAL PLATONISM 4 1.1. INTRODUCTION 5 1.2. PLATO’S THEORY OF MATHEMATICAL OBJECTS 7 1.3. AVICENNA’S HISTORIOGRAPHY OF PHILOSOPHY (OF MATHEMATICS) 17 1.4. AVICENNA’S CRITICISM OF ARGUMENTS FOR (SM) AND (PM) 31 1.4.1. (SM-1) IS FALLACIOUS 32 1.4.2. (PM-1) IS FALLACIOUS 36 1.5. AVICENNA’S ARGUMENT AGAINST (SM) 37 1.6. AVICENNA’S ARGUMENT AGAINST (PM) 44 1.7. CONCLUSION 48 2. ON THE NATURE OF MATHEMATICAL OBJECTS 50 2.1. INTRODUCTION 51 2.2. MATHEMATICAL OBJECTS: A GENERAL PICTURE 54 2.3. GEOMETRIC OBJECTS 58 2.4. NUMBERS 61 2.5. ACTUAL AND POTENTIAL PERFECT OBJECTS 68 2.6. CONCLUSION 75 3. ON MATHEMATICAL INFINITY 77 3.1. INTRODUCTION 78 3.2. THE NOTION OF MATHEMATICAL INFINITY 79 3.3. TWO ARGUMENTS AGAINST THE ACTUALITY OF INFINITY 87 3.3.1. THE COLLIMATION ARGUMENT 88 3.3.2. THE LADDER ARGUMENT 93 3.4. AVICENNA’S MAIN ARGUMENT: THE MAPPING ARGUMENT 98 3.4.1. THE STRUCTURE OF THE MAPPING ARGUMENT 103 3.4.2. INSIGHTFUL IDEAS BEHIND THE SURFACE STRUCTURE 109 3.4.3. NON-ORDERED CAN BE ACTUALLY INFINITE 118 3.5. CONCLUSION 126 4. EPISTEMOLOGY OF MATHEMATICS 128 v 4.1. INTRODUCTION 129 4.2. PRELIMINARIES 133 4.3. FORMING MATHEMATICAL CONCEPTS 141 4.4. PERCEIVING PERFECT MATHEMATICAL OBJECTS 153 4.5. KNOWLEDGE OF MATHEMATICAL PROPOSITIONS I: HOW IMAGINATION CONTRIBUTES 158 4.6. KNOWLEDGE OF MATHEMATICAL PROPOSITIONS II: HOW WE GRASP THE PROPOSITIONAL PRINCIPLES 172 4.7. CONCLUSION 182 CONCLUSION 185 REFERENCES 189 vi Introduction Abū ʿAlī Ibn Sīnā (ca. 970-1037)—to whom I refer by his Latinized name ‘Avicenna’—was the most significant philosopher in the Islamic world. His immense influence on the later philosophical traditions, in both the Islamic world and the Latin west, can hardly be exaggerated. Moreover, as a polymath, he wrote on a wide range of (what we now categorize as) scientific topics including medicine, pharmacology, mineralogy, astronomy, and mathematics. Avicenna’s familiarity with some of these sciences clearly had a great impact on how his philosophy was formed. In particular, the traces of his knowledge of medicine and mathematics are easily discernible in such central areas of his philosophy as epistemology and logic. Thus a comprehensive understanding of Avicenna’s philosophy is impossible unless we have a reliable picture of his views about these two sciences. Roughly speaking, we can say that his epistemology combines empiricist and rational elements, with medicine and mathematics respectively serving as paradigm sciences: medicine is often invoked when Avicenna illustrates the role of sense experience in science, while mathematics would seem to be a non-empirical science. 1 While Avicenna’s medicine and the role of medical examples in his empiricism have been studied to some extent, Avicenna’s views on philosophy of mathematics have been largely neglected. While there are some impressive studies on Avicenna’s mathematical works, 2 they mainly concern technical aspects of Avicenna’s treatment of mathematics and have nothing to do with his philosophical views about it. To the best of my knowledge there is only one short book and a few (fewer than five) articles about Avicenna’s philosophy of mathematics, which by no 1 It is well known that Avicenna, following the Aristotelian tradition, considers some sciences, like music and astronomy, to be branches of the mathematical sciences. But, having the modern conception of mathematics in mind, I will focus only on geometry and arithmetic. More precisely, I will focus on what Avicenna (2005, chap. I.3, p. 17, l. 10) calls ‘pure mathematics.’ So, when I speak of Avicenna’s philosophy of mathematics, I speak of his philosophical views concerning three dimensional Euclidean geometry and the arithmetic of natural numbers. 2 See, among others, Al-Daffa and Stroyls (1984), Rashed (1984), Djebbar (1999), and Luther (2004). 1 means provide a systematic and comprehensive picture of his philosophical views about mathematics.1 This is the gap that is intended to be filled by this PhD project.2 When we think about philosophy of mathematics, we are faced with two major questions. One pertains to the ontology of mathematical entities, while the other is related to the epistemology of mathematical concepts and propositions. Thus in this dissertation I deal with some ontological and some epistemological issues related to mathematics.
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