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 Fleurys Commentary on the Calculus.
Mathematics and Philosophy at the Turn of the First Millennium sheds light on Abbos 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. Abbos 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.
Scholars and students of the history of philosophy and mathematics will be introduced to a pivotal figure from an often overlooked era. They will be provided with fresh insights into the spread of Neopythagorean doctrines in the early Middle Ages, as they learn how these ideas were transmitted through arithmetic texts and harmonised with theology and natural philosophy. They will also get to know the medieval fraction system and calculus practices.
Mathematics and Philosophy at the Turn of the First Millennium sheds light on Abbos original philosophical system anchored in two central doctrines, which serve as a compass to navigate it: the theory of oneness (henology) and the theory of composition.
Introduction
Chapter 1
Abbo and His Time: European History and Mathematics
1. Politics and (Religious) Culture
2. Liberal Arts
3. Mathematics
3.1 Speculative Arithmetic
3.2 Practical Arithmetic
Chapter 2
Victorious Calculus and Abbos Explanatio
1. Contents of the Calculus
1.1 Praefatio de ratione calculi
1.2 Multiplication Tables
1.3 Supplementary Tables and Texts
2. Victorious, scripulorum calculator: the Calculus and the Reckoning of
Time
3. Two Hypothesis on the Arrival of the Calculus at Fleury
3.1 Hypothesis A: Lupus of Ferričre
3.2 Hypothesis B: Columbanus and the Abbey of Bobbio
4. Abbos Explanatio: Manuscripts and Critical Editions
5. Contents of the Explanatio
5.1 The Prologue (Section I)
5.2 The Theological Premise (Section II)
5.3 The Commentary on on Victoriuss Praefatio (Section III)
5.4 The Commentary on Victoriuouss Multiplication Tables (Section IV)
5.5 On Qualitative Physics (Section V)
6. The Pedagogical Aim and the Role of the Commentator
Chapter 3
A Theological Premise: Number, Measure, and Weight
1. Augustine and Wisd. 11:21
2. Claudianus Mamertus De statu animae
3. Arithmetical Readings of Wisd. 11: 21: Hrabanus Maurus and John Scotus
Eriugena
4. Abbos Tractatus de numero, mensura et pondere
Chapter 4
Henology
1. Neopythagorean Elements: Unity and the Flow of Numbers
2. The Highest Good, individuum, and Unity
3. The Lambda Diagram: a Quadrivial Henology
4. Unity as Ontological Principle
5. Unity as the Object of Arithmetic and Calculus
Chapter 5
The Composition of Reality
1. Natural Compounds
1.1 The Arithmology of Natural Compounds
1.2 Natural Compounds and Change: the Phases of the Moon
2. Artificial Compounds
2.1 The Five Kinds of Inequality Ratios
2.2 Two trinae regulae for Arithmetical Ratios
3. Ontological Composition: the Case of the Earth in Gen. 1:2
4. Division of Non-Material Entities: the Case of the Units of Time
5. Divisibility and Corporality: the Case of the vox
Chapter 6
Arithmetic and Calculus
1. Fractions: Ounces and minutiae
2. Representing Integer Numbers
3. Calculus by Fingers or the loquela digitorum
4. Calculus by Tables
5. Multiplication Rules
Chapter 7
Physics Before the Physics
1. The Early Medieval Concern for Natural Phaenomena
2. Scientific Literature in the Early Medieval Fleury Area
3. Qualitative Physics: from Astronomy to Physiology
4. The Natural Power of Things
Conclusions
Clelia V. Crialesi is a Marie Skodowska-Curie Fellow at SPHERE-CNRS (France). Formerly, she was an FWO Research Fellow at KU Leuven (Belgium) and a Mellon Fellow at PIMS (Canada). Her research focuses on premodern mathematical thought, with publications ranging from Boethian number theory to Euclidean geometry in the late medieval continuum debate.
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