Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than the (...) ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

Whether aristotle wrote a work on mathematics as he did on physics is not known, and sources differ. this book attempts to present the main features of aristotle's philosophy of mathematics. methodologically, the presentation is based on aristotle's "posterior analytics", which discusses the nature of scientific knowledge and procedure. concerning aristotle's views on mathematics in particular, they are presented with the support of numerous references to his extant works. his criticism of his predecessors is added at the end.

It is widely known that Aristotle rules out the existence of actual infinities but allows for potential infinities. However, precisely why Aristotle should deny the existence of actual infinities remains somewhat obscure and has received relatively little attention in the secondary literature. In this paper I investigate the motivations of Aristotle’s finitism and offer a careful examination of some of the arguments considered by Aristotle both in favour of and against the existence of actual infinities. I argue that Aristotle has (...) good reason to resist the traditional arguments offered in favour of the existence of the infinite and that, while there is a lacuna in his own ‘logical’ arguments against actual infinities, his arguments against the existence of infinite magnitude and number are valid and more well grounded than commonly supposed. (shrink)

One of the chief innovations in medieval adaptations of Aristotelian psychology was the expansion of Aristotle's notion of imagination orphantasiato include a variety of distinct perceptual powers known collectively as the internal senses. Amongst medieval philosophers in the Arabic world, Avicenna offers one of the most complex and sophisticated accounts of the internal senses. Within his list of internal senses, Avicenna includes a faculty known as “estimation”, to which various functions are assigned in a wide variety of contexts. Although many (...) philosophers in the Arabic world as well as in the Latin West accepted Avicenna's positing of an estimative faculty, Avicenna's best-known critics, al-Ghazâlî and Averroes, found Avicenna's arguments in support of a distinct estimative faculty problematic. For different reasons, both Averroes and Ghazâlî raised the basic question of whether one needed to posit a distinct faculty of estimation to supplement the perceptual abilities of the other internal senses, and whether the notion of an estimative power as defined by Avicenna was internally coherent. Such criticisms suggest that the Avicennian conception of estimation is not entirely unambiguous, and that a correct understanding of Avicenna's motivations for delineating an estimative power requires a careful study of the diverse activities assigned to it throughout Avicenna's philosophical writings. (shrink)

Jonathan Lear; XII*—Aristotelian Infinity, Proceedings of the Aristotelian Society, Volume 80, Issue 1, 1 June 1980, Pages 187–210, https://doi.org/10.1093/aris.

Al-Kindi was the first philosopher of the Islamic world. He lived in Iraq and studied in Baghdad, where he became attached to the caliphal court. In due course he would become an important figure at court: a tutor to the caliph's son, and a central figure in the translation movement of the ninth century, which rendered much of Greek philosophy, science, and medicine into Arabic. Al-Kindi's wide-ranging intellectual interests included not only philosophy but also music, astronomy, mathematics, and medicine. Through (...) deep engagement with Greek tradition al-Kindi developed original theories on key issues in the philosophy of religion, metaphysics, physical science, and ethics. He is especially known for his arguments against the world's eternity, and his innovative use of Greek ideas to explore the idea of God's unity and transcendence. Despite al-Kindi's historical and philosophical importance no book has presented a complete, in-depth look at his thought until now. In this accessible introduction to al-Kindi's works, Peter Adamson surveys what is known of his life and examines his method and his attitude towards the Greek tradition, as well as his subtle relationship with the Muslim intellectual culture of his day. Above all the book focuses on explaining and evaluating the ideas found in al-Kindi's wide-ranging philosophical corpus, including works devoted to science and mathematics. Throughout, Adamson writes in language that is both serious and engaging, academic and approachable. This book will be of interest to experts in the field, but it requires no knowledge of Greek or Arabic, and is also aimed at non-experts who are simply interested in one of the greatest of Islamic philosophers. (shrink)

In Physics, Aristotle starts his positive account of the infinite by raising a problem: “[I]f one supposes it not to exist, many impossible things result, and equally if one supposes it to exist.” His views on time, extended magnitudes, and number imply that there must be some sense in which the infinite exists, for he holds that time has no beginning or end, magnitudes are infinitely divisible, and there is no highest number. In Aristotle's view, a plurality cannot escape having (...) bounds if all of its members exist at once. Two interesting, and contrasting, interpretations of Aristotle's account can be found in the work of Jaako Hintikka and of Jonathan Lear. Hintikka tries to explain the sense in which the infinite is actually, and the sense in which its being is like the being of a day or a contest. Lear focuses on the sense in which the infinite is only potential, and emphasizes that an infinite, unlike a day or a contest, is always incomplete. (shrink)

This volume introduces the major classical Arabic philosophers through substantial selections from the key works (many of which appear in translation for the first time here) in each of the fields—including logic, philosophy of science, natural philosophy, metaphysics, ethics, and politics—to which they made significant contributions. -/- An extensive Introduction situating the works within their historical, cultural, and philosophical contexts offers support to students approaching the subject for the first time, as well as to instructors with little or no formal (...) training in Arabic thought. A glossary, select bibliography, and index are also included. (shrink)

Few philosophers that have been studied as much as Ibn Sīnā have been as much misunderstood. His extraordinary ability to reflect upon and write in a variety of styles about seemingly every topic in every domain has steered his thought from philosophy and theology to mysticism and esoterism. Instead of helping us to learn and understand better Ibn Sīnā than he has previously been understood, the recent surge of Avicennan studies only adds more confusion to the already complex social context (...) which he was living in. (shrink)

The systematic comparison of Avicenna’s Ilāhiyyāt of the Šifā' with Aristotle’s Metaphysics , accomplished for the first time in the present volume, provides a detailed account of Avicenna’s reworking of the epistemological profile and contents of the Metaphysics and a comprehensive investigation of this latter’s transmission in pre-Avicennian Greek and Arabic philosophy.

Bowin begins with an apparent paradox about Aristotelian infinity: Aristotle clearly says that infinity exists only potentially and not actually. However, Aristotle appears to say two different things about the nature of that potential existence. On the one hand, he seems to say that the potentiality is like that of a process that might occur but isn't right now. Aristotle uses the Olympics as an example: they might be occurring, but they aren't just now. On the other hand, Aristotle says (...) that infinity "exists in actuality as a process that is now occurring" (234). Bowin makes clear that Aristotle doesn't explicitly solve this problem, so we are left to work out the best reading we can. His proposed solution is that "infinity must be...a per se accident...of number and magnitude" (250). (Bryn Mawr Classical Review 2008.07.47). (shrink)

This chapter shows how al-Kindī interweaves ideas from Greek cosmology to give a theory that can explain the efficacy of astrology and how God’s providence is dispersed by means of heavenly influence. A concrete example is found in al-Kindī’s works on meteorology, since he thinks that weather is produced by celestial causation. The mechanics of this causation are explained differently in different works, which leads to a consideration of the authenticity of On Rays, which is ascribed to al-Kindī, and its (...) place in his corpus. Finally, the chapter considers whether al-Kindī’s account commits him to determinism, and whether he thinks universal causal determinism is compatible with human freedom. (shrink)

This paper is on Aristotle's conception of the continuum. It is argued that although Aristotle did not have the modern conception of real numbers, his account of the continuum does mirror the topology of the real number continuum in modern mathematics especially as seen in the work of Georg Cantor. Some differences are noted, particularly as regards Aristotle's conception of number and the modern conception of real numbers. The issue of whether Aristotle had the notion of open versus closed intervals (...) is discussed. Finally, it is suggested that one reason there is a common structure between Aristotle's account of the continuum and that found in Cantor's definition of the real number continuum is that our intuitions about the continuum have their source in the experience of the real spatiotemporal world. A plea is made to consider Aristotle's abstractionist philosophy of mathematics anew. (shrink)

Avicenna's discussion of space is found in his comments on Aristotle's account of place. Aristotle identified four candidates for place: a body's matter, form, the occupied space, or the limits of the containing body, and opted for the last. Neoplatonic commentators argued contra Aristotle that a thing's place is the space it occupied. Space for these Neoplatonists is something possessing dimensions and distinct from any body that occupies it, even if never devoid of body. Avicenna argues that this Neoplatonic notion (...) of space is untenable on the basis of three arguments. In general he maintains that bodies' impenetrability is explained by reference to dimensionality. Consequently, if it is dimensionality that explains impenetrability, and yet as the Neoplatonists hold space inherently possesses dimensions, material bodies could never interpenetrate space and so occupy it and thus bodies could never have a place. The conclusion is patently false. In additions Avicenna argues that the method used to arrive at the possibility of space is illicit, and so Neoplatonist cannot show that space is even possible. Thus, concludes Avicenna, Aristotle's initial account must be correct. The paper outlines the historical context of this debate and then treats Avicenna's arguments against space in detail. (shrink)

Some authors have proposed that Avicenna considers mathematical objects, i.e., geometric shapes and numbers, to be mental existents completely separated from matter. In this paper, I will show that this description, though not completely wrong, is misleading. Avicenna endorses, I will argue, some sort of literalism, potentialism, and finitism.

Al-Q, mathematician of the 10th century, examines critically two arguments in the 6th book of the Aristotelian Physics. This critic does not follow the method of the philosophers, with doctrinal amendments, but with a mathematical and experimental style. For understanding of this critical examination and its influence, it is necessary to situate it in the mathesis of al-Q and to produce its mechanical presuppositions. This is the purpose of the author of this paper.

RésuméCet article se penche sur la réception des théories avicenniennes du mouvement au VIe/XIIe siècle. Avicenne a conçu des façons innovantes de comprendre le mouvement, répondant à la fois aux défis et conditions établis par la tradition philosophique antérieure et à ceux qui naissent de sa critique interne. Le mouvement est pour lui soit le mode d’être entre deux termes, soit le passage ou l'intervalle, le premier étant le type de mouvement extra-mentalement réel, tandis que le second est un produit (...) de l'imagination. Au VIe/XIIe siècle, la critique implicite conduite par certains savants prééminents aboutit, dans certains cercles, à l'adoption de la thèse selon laquelle le mouvement-passage est extra-mentalement réel. Cette position était jugée viable aussi bien par les tenants de l'atomisme que par ceux de la théorie de la continuité des corps. (shrink)

In his day, al-Kindi was the only philosopher of pure Arab descent, and became known as “the philosopher of the Arabs.” He was one of the first Arab scholars interested in a scientific rather than theological viewpoint, and played a key role in bringing Greek learning into the orbit of Islam. al-Kindi wrote over three hundred fifty treatises, for the most part short studies on special topics in science and philosophy. Nicholas Rescher assembles this annotated bibliography, listing of over three (...) hundred items, to assist students and scholars through the maze of publications related to al-Kindi. (shrink)