Mathematics and Philosophy at the Turn of the First Millennium

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

Victoriousâ Calculus and Abboâs 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 Ferri¨re

3.2 Hypothesis B: Columbanus and the Abbey of Bobbio

4. Abboâs 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 Victoriusâs Praefatio (Section III)

5.4 The Commentary on Victoriuousâs 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 Mamertusâ De statu animae

3. Arithmetical Readings of Wisd. 11: 21: Hrabanus Maurus and John Scotus Eriugena

4. Abboâs 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