# Speculative arithmetic

By Speculative arithmetic I refer to the research I carried out during my stay as a Mellon Fellow at the Pontifical Institute of Mediaeval Studies in Toronto (2019-2020).

Speculative arithmetic is also the expression used by modern scholars to describe a kind of (medieval) arithmetic that does not focus on practical reckoning, but dwells instead on the status of integer natural numbers and their mutual relations, namely, on the ‘number theory’. Based on the Neopythagorean Nicomachus of Gerasa’s Introduction to Arithmetic, Boethius’ De arithmetica became the most authoritative textbook for the teaching of arithmetic throughout the Middle Ages and beyond. The great success of this work and, consequently, its influence on the medieval Latin thought were determined not only by Boethius’s systematic exposition of the number theory, but also by the Platonic and Neopythagorean doctrines that frame and introduce to the technicalities of this mathematical discipline – prominently, the one concerning the metaphysical role played by numbers in shaping reality.

Many efforts have been made to retrace the reception of this Boethian text in the Middle Ages, especially between the 9th and the 12th century. My previous research can be (humbly) counted among them, as I tried to reconstruct its philosophical influence on the Western medieval thought, by focusing on the lemmatic commentaries on Boethius’ arithmetical text – which means investigating how the teaching of speculative arithmetic developed. With this aim, I started a working-catalogue of those manuscripts preserving these lemmatic commentaries and edited two of them: the so-called Excerptiuncula (which proved to be a short paraphrase of the first book of the De arithmetica that testifies to the presence of this work in the 11th-century Montecassino, and more precisely to Lawrence of Amalfi’s teaching) and an anonymous and more extensive 11th-century commentary (which intertwines speculative arithmetic and psychology, also offering a curious definition of the point constituting the line as indivisibile corpus).

### Tell me why

Why focusing on (sometimes pedantic) commentaries? The reason relies in our still vivid interest in the development of those philosophical traditions that shaped our mindset throughout the centuries. In my case, studying the reception Boethius’s De arithmetica means also retracing the development of certain Neopythagorean doctrines. In an article published in 1978, Gillian Evans stated: “Nowhere is the gap of our knowledge greater that in the field of mathematical arts” [G. R. Evans, Introductions to Boethius’ Arithmetica of the Tenth to the Forutheenth Century, in «History of Science», 16 (1978), 22-41]. When Evans claimed this, she referred especially to the different ways medieval masters introduced their students to texts on mathematics. In my humble opinion, Evans’s statement applies also to the gap in our knowledge of medieval Neopythagoreanism. I shall specify that the gap I am referring to concerns those Neopythagorean principles on which medieval mathematical disciplines (arithmetic in particular) are grounded. And this is particularly true with respect to one specific text, namely, Boethius’s De arithmetica, that conveyed certain Neopythagorean stances to the medieval Latin-speaking world and remarkably shaped the Western philosophical thought throughout the Middle ages. The ‘chain of commentaries’ on this text traces the path along which such Neopythagorean doctrines evolved and interacted with the Latin medieval philosophical tradition.