Clavius, Proclus, and the Limits of Interpretation: Snapshot-Idealization Versus Projectionism

Abstract

It is hard to overestimate the role played by Christopher Clavius (1538–1612), professor of mathematics at the Collegio Romano from 1565 until his death, in the establishment of mathematics as an incontestable part of the Jesuit curriculum and body of thought. Besides his direct influence on the constitution of the mathematical part(s) of the Ratio studiorum, Clavius’s mathematical writings, ranging from introductory textbooks to commentaries to polemical pamphlets, almost immediately began to serve as standard references, despite their — occasionally — controversial character. Today, Clavius’s contentious stand towards the older school at the Collegio embodied in the person of Benedictus Pereira, who held an extremely negative view on mathematics, is regarded as a decisive factor in the sixteenth-century renaissance of mathematics, both within and outside the Society. Nowadays, most scholars seem to agree that the philosophical framework of Clavius’s institutional struggle can be found in the Prolegomena to his edition of Euclid’s Elements, first published in Rome in 1574. According to some, the general outlines of this preface reflect a philosophy of mathematics that is largely indebted to the one presented by the Neoplatonist Proclus (410/412 – 485) in the double prologue of his commentary on the first book of Euclid’s Elements. Without wanting to obscure the importance of Clavius’s revolutionary project, I will try to sketch a different picture of Clavius’s philosophy of mathematics, both on the ontological and on the epistemological level. In this re-evaluation, the limits of both Proclus’s influence and the interpretation of his commentary on Euclid will be my primary focus. Clavius’s views on the nature of mathematical objects will be shown to be incompatible with the two conventional modes of interpretation found in the ancient commentators, namely the projectionism maintained by Proclus on the one hand, and the constructive abstractionism of writers such as Ammonius and Philoponus on the other. For Proclus, mathematical objects are extended and divisible projections of the unextended and indivisible ideas with which our understanding is equipped. The imagination acts both as the faculty that projects these ideas and as the imaginary screen on which they are projected. The constructive abstractionism read in Ammonius, for example, sees mathematical entities as existing only in the mind of the mathematician who in thought has separated, i.e. abstracted, them from matter. Clavius’s position will lay bare the need to invoke a third category, namely that of a ‘snapshot-idealization’: in mathematics, according to Clavius, sensible objects are indeed considered free from matter, but they are not attributed any properties that they do not perfectly instantiate as sensible objects. As in a photograph, certain