Clelia V. Crialesi

This book takes readers through an exploration of fundamental discussions that redefined mathematics and its philosophical significance in the centuries foregoing modernity. From William of Auvergne’s paradoxes of infinity to Christoph Clavius’ interpretation of Euclidean principles, it examines the evolving understanding of central issues among which continuity, the existence of mathematical objects such as numbers, and the way humans can make true statements regarding such things. Each chapter sheds light on how premodern scholars bridged mathematics and philosophy, forging concepts and approaches that continued to influence early modern thought. A compelling read for historians, philosophers, and anyone intrigued by the origins and enduring legacy of mathematical ideas as both tools for inquiry and objects of reflection.

At the turn of the first millennium, scientific and philosophical knowledge was far from dormant. Arithmetic, with its diverse calculation techniques and number theory, served as a bridge to philosophy, theology, and the study of the physical world. Even something as simple as a series of multiplication tables could unlock a profound knowledge of both the divine realm and natural phenomena. Such is the case with Abbo of Fleury’s Commentary on the Calculus. Mathematics and Philosophy at the Turn of the First Millennium sheds light on Abbo’s original philosophical system anchored in two central doctrines, which serve as a compass to navigate it: the theory of unity (henology) and the theory of composition. Yet, the Commentary on the Calculus covers much more. The present study, thus, explores an eclectic range of topics – from water clocks to barleycorns, constellations to human voice, synodic month to the human lifespan, and numbers to God. Abbo’s work is an ambitious attempt to tie together the study of both the visible and invisible realms, what can be measured and what cannot, what can be quantified and what exceeds quantification.

This paper presents a taxonomy of divisibilism, a philosophical perspective advocating for the infinite divisibility of continua. The taxonomy is founded on various conceptualizations of indivisibles, enabling the identification of two types of divisibilism: ‘moderate’ and ‘strong’. The former denies indivisibles as constituent parts of magnitudes, whereas the latter rejects indivisibles as even intrinsic elements (such as limits or junctions) of magnitudes. The paper proceeds to demonstrate how Gregory of Rimini falls into the second category, utilizing geometry and non-entitism as primary tools to dismiss the very possibility of the existence of indivisibles. Within the framework of this ‘strong divisibilism’, the paper also examines Gregory’s interpretation of mathematical items (e.g. points, lines, surfaces) against the backdrop of one of his main sources, that is, Augustine. Lastly, the paper delves into Gregory’s interpretation of the fundamental Aristotelian divisibilist claim, namely, that a continuum is infinitely divisible. It highlights how Gregory’s stance ultimately conflicts with Aristotelian premises, particularly concerning his assertion that the division of all continua is made into infinitely many actual parts.

This article deals with Robert Grosseteste’s account of ‘spatial differences’, such as ‘up’, ‘down’, ‘right’, ‘left’, ‘before’, and ‘behind’. More specifically, attention is focused on Grosseteste’s De differentiis localibus, which is a concise scientific treatise arguing for the objectiveness of the differences of place pertaining to all living bodies, including heavenly ones. The article has a two-fold goal: to present the contents of such an understudied opuscule, and to check if there is some compelling reliance on any of the Latin versions of Aristotle’s On the Heavens. Such an analysis reveals that Grosseteste’s reading of Aristotle’s On the Heavens is angled by Averroes’ Long Commentary on the Physics, on which Grosseteste relies as well to build his conception of mathematical and natural differences.

This article deals with the first Latin reception of Pseudo-Aristotle’s On Indivisible Lines and its impact on the medieval debate about the continuum. Robert Grosseteste’s and Albert the Great’s references to this pseudo-Aristotelian text show that it could be regarded as a source for where to find information about the indivisibilist tenet, as well as an expansion of Aristotle’s anti-atomistic critiques scattered throughout his authentic works. The use of On Indivisible Lines made by Henry of Harclay and Adam of Wodeham confirms this trend: the reading of this text could be twofold according to the tenet defended. While Henry argues against Pseudo-Aristotle to defend indivisibilism, Adam expands on pseudo-Aristotelian arguments to show the incongruities implied by indivisibilism.

This paper focuses on the exegetical proposal of the Tractatus de sex dierum operibus by Thierry of Chartres and it is tasked with analyzing the twofold interpretative framework adopted by the Cancelor: first, the accordance between the narration of Genesis and the heuristic models of physical and cosmological causality; second, the mathematical theology, which revises the work of creation according to an arithmological approach. The study is divided into two parts which follow the structure of the Tractatus. In the first part, I analyse the physical plausibility of Christian cosmogony and subsequently the conception of matter with regard to both William of Conches’ inordinatum and the hermetic locus mundi. In the second part, I examine how the numerical discourse interprets the creatural unfolding and dependence on God, as well as the creation per verbum and the individuality of each created being. My purpose is threefold: first, to test the inner philosophical coherence of the Tractatus; second, to interpret specific theoretical points at the light of Thierry’s commentaries on Boethius’ De Trinitate, and finally, to relate the arithmetical issues of the Tractatus to Thierry's Commentum super Arithmeticam Boethii.