A. J. C. Bo

Akribeia: Certainty and Ontology of Mathematics in Alessandro Piccolomini’s De certitudine mathematicarum

Brill 2025.

This book focuses on a specific moment in the history of early modern akribology (i.e., the philosophical discourse that investigates the traits of exactness, epistemic transparency, and certainty) by examining Alessandro Piccolomini’s De certitudine mathematicarum disciplinarum (Rome, 1547). Álvaro José Campillo Bo offers a technical and historically contextualised analysis of Piccolomini’s treatise, situating it within the intellectual traditions of 16th-century Italy. The study reconstructs Piccolomini’s core argument: that mathematics, while supremely certain, fails to qualify as a genuine scientia due to the phantastic or fictional nature of its objects (quantum phantasiatum) and the resulting absence of proper causal demonstration. Special attention is given to Piccolomini’s use and interpretation of key sources, especially Proclus’ Commentary on Euclid’s Elements and Priscian of Lydia’s In De anima.

C. V. Crialesi

Mathematics and Philosophy at the Turn of the First Millennium
Abbo of Fleury on Calculus

Routledge 2025.

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.

J. Høyrup

The World of the Abbaco.
Abbacus Mathematics Analysed and Situated Historically Between Fibonacci and Stifel

Birkhäuser 2024.

This book offers a critical historical perspective on the development and diffusion of abbacus calculus (from 13th to 17th century). Contrary to popular belief, Høyrup argues that Fibonacci’s role in this tradition has been overstated, shedding light on lesser-known contributors like Dardi of Pisa and Antonio de’ Mazzinghi – yet still providing a well-rounded analysis of Fibonacci’s textbook.
Beyond its historical narrative, the book addresses what happened to the abacus tradition when it went into print, and delves into the divergence of abbacus mathematics in German lands, illustrating how regional influences shaped distinct mathematical traditions.

A. Robert

Le monde mathématique.
Marco Trevisano et la philosophie dans la Venise du Trecento

Paris 2023.

The prevailing scholarly consensus has held that the resurgence of Plato and Pythagoras occurred only in the 15th century. However, this belief is challenged by a remarkable masterpiece from the 14th century by a Venetian nobleman named Marco Trevisano: De Macrocosmo. Even before the Renaissance, Marco Trevisano boldly declares himself a disciple of these two ancient philosophers, delving into the origin of the world and its composition through the lens of mathematics. The significance of De Macrocosmo lies in its innovative content, including the integration of algebra into speculative arithmetic, the concept of unity as number, and the Neopythagorean atomism. These ideas are meticulously presented and thoroughly analyzed in the first comprehensive and systematic exploration by Aurélien Robert, finally bringing this 14th-century thinker out of the shadows.

D. A. Di Liscia

Eine Wiener Exposition zum Traktat De latitutinibus formarum

Edition und Kommentar, Wien 2022.

This book delves into the interpretation of the Tractatus de Latitudinibus Formarum, a widely used text written at the University of Vienna during the 15th century. The doctrine of form latitudes presented in the text aims to explain the numerous variations of qualities. The anonymous author of the commentary discussed in this book addresses the epistemological question of whether the new doctrine of form latitudes constitutes a science. Additionally, the author discusses the well-known theorem of mean velocity, various sophisms in natural philosophy, and the systematic classification of geometrical diagrams for practical use. The edition is based two manuscripts, complete with diagrams, made by Michael Lochmair (d. 1499), a former University of Vienna professor.

A. Axworthy

Motion and Genetic Definitions in the Sixteenth-Century Euclidean Tradition

Birkhäuser 2022.

Angela Axworthhy investigates the different treatments of motion and genetic definitions (i.e., definitions that characterises geometrical objects through its mode of generation rather than through its essential attributes) by seven major 16th-century commentators on Euclid’s Elements: Oronce Fine, Jacques Peletier, Fraçois de Foix-Candale, Henry Billigsley, John Dee, Federico Commandino, and Christoph Clavius. This book is particularly relevant also for historians of medieval geometry and philosophy of mathematics, since the author shows what role the concept of motion (i.e., local motion and generation) played in conceiving geometrical objects from Antiquity to Early Modern time.

D.A. Di Liscia, E.D. Sylla (eds., with the collaboration of P.J.J.M. Bakker)

Quantifying Aristotle.
The Impact, Spread and Decline of the Calculatores Tradition

Brill 2022.

The book gathers 14 essays that shed new light on the link between Aristotelian philosophy and mathematical methods and principles that form the basis of modern science. It surveys the tradition of the Oxford Calculators from its beginnings in the 14th century until Leibniz, exploring how the Calculators’ techniques of quantification expanded the conceptual and methodological limits of Aristotelianism. The authors examine a large number of thinkers and investigate the relationship between various late medieval disciplines.

S. Roudaut

La mesure de l’être.
Le problème de la quantification des formes au Moyen Âge (ca. 1250–1370)

Brill 2022.

This book represents a reference point in the study of mathematical thought in the Middle Ages. The author investigates some of the most relevant questions related to the problem of quantifying forms – for instance, how many times does a body under constant acceleration travel in the second half of its motion the distance covered in the first half of it? Crucial aspects of the scientific thought are explored, diving into the meaning of key-notions (like intensification, degree, and latitude) and showing their use within the scientific, philosophical, and theological context.

I. Caiazzo, C. Macris, A. Robert (eds.)

Brill’s Companion to the Reception of Pythagoras and Pythagoreanism in the Middle Ages and the Renaissance

 

Brill 2021.

The volume is not only dedicated to the ‘Medieval Pythagoras’ in the Latin West, but it also explores the survival of Pythagoreanism in the Arabic, Jewish, and Persian cultures. It is divided into four parts, each one addressing a fundamental theme in the Pythagorean legacy: number theory and the sciences of the quadrivium, ethics, theology and metaphysics, psychology.

L. Corry

Distributivity-Like Results in the Medieval Traditions of Euclid’s Elements:
Between Geometry and Arithmetic

Springer 2021.

This book deals with a specific aspect of Euclidean geometry in medieval mathematical texts, that is, the role and presence of distributivity-like properties. Leo Corry investigates the link between geometry and arithmetic, unveiling the rise of algebraic modes of thought. Islamicate, Latin, and Hebrew mathematical traditions are discussed.