# Paolo Rossini

## Giordano Bruno’s Theory of Minima and the Concept of Measurable Magnitude

Paper Presented at the International Conference “Humours, Mixtures and Corpuscles”

#### Background

In Bruno’s thought, mathematics and philosophy cannot be separated from each other. For an interpreter, this entails the impossibility to comprehend Bruno’s mathematical reasoning without a prior understanding of his metaphysics, as well as of his overall philosophical project. is has also determined how Bruno’s mathematics has been approached in recent years. e earlier interpreters (Olschki, Koyrè) have opted for a strictly mathematical viewpoint, which has mainly high- lighted the inconsistencies of Bruno’s mathematics. In the last decades, instead, a more philosophical approach has prevailed, thanks to the studies conducted mainly by Aquilecchia, Boenker-Vallon, Muslow, De Bernart, Luethy, Seidengart and Ma eoli. Generally speaking, these studies have assessed Bruno’s mathemat- ics by placing it against its historical and philosophical background. is research project is designed to reach the same objective. It is structured in three parts.

e rst part deals with the way in which Bruno’s mathematics is related to that of his contemporaries. In particular, I consider two di erent conceptions of mathematics which emerge in the sixteenth century. e rst conception is that resulting from the aestio de certitudine mathematicarum. e second con- ception is that developed within the Ramist circle. Both these conceptions are informed by a vivid debate on the role and the epistemological status of mathe- matics, as well as on the ontological nature of mathematical beings. Philosophy of mathematics, so to say, is a ma er of dispute for Bruno as well. Hence, I argue that this debate might be also extended to Bruno’s mathematics, allowing thereby for a comparison among three, di erent, contemporary positions: Bruno’s, Ramist, and that related to the aestio. To this extent, this comparison would permit to place Bruno into the historical context of sixteenth century mathematics.

e second part of my research is devoted to the sources of Bruno’s math- ematics. As has already been proved, Bruno mainly relies on Euclid and Pro- clus. Also Plotinus’s On Numbers can be regarded as a primary source of Bruno’s mathematics. In addition, Bruno reframes insights taken from the works of Cu- sanus, Bovelles and Boethius. Insofar as the mathematics of Cusanus, Bovelles and Boethius is modelled a er the Pythagorean teachings, the same also applies to Bruno?s mathematics. I would like to deepen this issue of Bruno’s Pythagore- anism by taking into account another philosophical tradition, namely, Lullism. It is well known that Bruno was an expert of Llull’s doctrine. He wrote a number of Lullian works, and was deeply interested in Llull’s Ars. Accordingly, Bruno schol- ars have made an e ort to determine the impact of Lullism on Bruno’s thought. However, in these investigations, Bruno’s mathematics has been neglected. I in- tend to ll this gap by comparing Bruno’s and Llull’s mathematics. is compari- son con rms the above perspective on Bruno’s Pythagoreanism, for Llull’s math- ematics presents a number of Pythagorean elements, too. e goal of this rst part is, then, to propose Llull as another channel though which Pythagoreanism might have come to in uence Bruno’s mathematics.

e third part addresses the mathematical procedures outlined in Bruno’s works. If one reads Bruno’s De triplici minimo—in which his atomistic theory is presented from a geometrical point of view—one might have the impression that Bruno is more interested in physics, rather than in mathematics. For that reason, Bruno’s naturalism has been o en taken as the reason why his mathematics is a ected by a number of contradictions. For example, it remains undetermined whether for Bruno a point is extended or not; or whether the points in a line are in nite or nite. Nevertheless, in some other writings—namely, the four dialogues on the proportional compass invented by Fabrizio Mordente—Bruno seems truly interested in solving mathematical problems, such as the quadrature of the circle. To this end, he makes use of mathematical procedures which I aim to analyze, in order to assess their applicability.

#### Bruno’s Concept of Measurable Magnitude: Abstract

As is known, early modern science was characterized by an increasing tendency toward quanti cation. Gradually, quantitative measurements entered all scien- ti c disciplines, ranging from physics to medicine to biology. What is less known, however, is that the concept of measurable magnitude was introduced into geom- etry only in the sixteenth century. Indeed, in classical Greek mathematics mag- nitudes were not strictly measurable nor operable. at said, much remains to be learned about the concept of measurable magnitude and how it is related to the quanti cation of science. Such an analysis would shed light on the complex interplay between advancements in early modern mathematics and the scienti c revolution.

A remarkable a empt to introduce measurability into geometry was made by Giordano Bruno with his theory of minima, whose mathematical and philosoph- ical underpinnings I explore in this paper. With his theory of minima, Bruno aimed to provide a universal theory of measurement in accordance with his view that “nature is a numerable number and a measurable magnitude.” To this end, he designed the concept of indivisible minimum to serve as unit of measurement for all magnitudes. More precisely, Bruno conceived magnitudes as made up of minima, composing thereby the geometrical continuum out of indivisible parts. Hence, theory of minima (and the related concept of measurable magnitude) pre- supposed an atomistic view of geometrical objects. In the paper, I outline this atomistic background, and show some examples of how Bruno employs the no- tion of minimum to cope with problems of measurement. Finally, in the light of this analysis, I try to frame Bruno’s theory of minima within the broader context of the scienti c revolution.