Monica Ugaglia

ArgumentThe studies of Bonaventura Cavalieri’s indivisibles by Giusti, Andersen, Mancosu and others provide a comprehensive picture of Cavalieri’s mathematics, as well as of the mathematical objections to it as formulated by Paul Guldin and other critics. Issues that have been studied in less detail concern the theological underpinnings of the contemporary debate over indivisibles, its historical roots, the geopolitical situation at the time, and its relation to the ultimate suppression of Cavalieri’s religious order. We analyze sources from the seventeenth through twenty-first centuries to investigate this relationship.

In this chapter, we offer a reconstruction of the evolution of Leibniz’s thought concerning the problem of the infinite divisibility of bodies, the tension between actuality, unassignability, and syncategorematicity, and the closely related question of the possibility of infinitesimal quantities, both in physics and in mathematics.Some scholars have argued that syncategorematicity is a mature acquisition, to which Leibniz resorts to solve the question of his infinitesimals – namely the idea that infinitesimals are just signs for Archimedean exhaustions, and their unassignability is a nominalist maneuver. On the contrary, we show that syncategorematicity, as a traditional idea of classical scholasticism, is a feature of young Leibniz’s thinking, from which he moves away in order to solve the same problem, as he gains mathematical knowledge.We have divided Leibniz’s path toward his mature view of infinitesimals into five phases, which are especially significant for reconstructing the entire evolution. In our reconstruction, an important role is played by Leibniz’s text De Quadratura Arithmetica. Based on this and other texts, we dispute the thesis that fictionality coincides with syncategorematicity, and that unassignability can be bypassed. (In this chapter, we employ “syncategorematic” as a shorthand for “eventually identifiable with a procedure of exhaustion” and, as a consequence, “involving only assignable quantities.” We also identify syncategorematicity with potentiality, as suggested by some part of the scholastics, and in Sect. 2.2, we show that the two characterizations are equivalent, provided that “potentiality” is intended in the correct way: namely as in the unending iterative procedures of Greek mathematics.) On the contrary, we maintain that unassignability, as incompatible with the principle of harmony, is the ultimate reason for the fictionality of infinitesimals.

Aristotle posits that time, as defined by the “number of motion in respect of before and after” (_Physics_ IV 11.219b1-2), is an inherent property of motion itself rather than a prerequisite. This implies the possibility of identifying time-independent properties of natural motions. One such critical feature, crucial to understanding the basic meaning of time, is the presence of an inherent order of before and after within motion, regardless of time. The concept of a non-temporal before and after within motion is now accepted or seriously considered in scholarly discussions. However, it remains one of the most confusing and difficult issues in the interpretation of Aristotle’s conception of time. How can we conceive of motion apart from time? Moreover, how does Aristotle seemingly take this aspect for granted? In order to address these issues, I propose to combine philosophical inquiry with grammatical and mathematical reflections. I will examine the analysis of motion alongside the verbs used to describe it, with a keen focus on aspect. Hence I will draw connections between the relationship of motion and time and that of physical and mathematical objects.

In his 1676 text De Quadratura Arithmetica, Leibniz distinguished infinita terminata from infinita interminata. The text also deals with the notion, originating with Desargues, of the point of intersection at infinite distance for parallel lines. We examine contrasting interpretations of these notions in the context of Leibniz’s analysis of asymptotes for logarithmic curves and hyperbolas. We point out difficulties that arise due to conflating these notions of infinity. As noted by Rodríguez Hurtado et al., a significant difference exists between the Cartesian model of magnitudes and Leibniz’s search for a qualitative model for studying perspective, including ideal points at infinity. We show how respecting the distinction between these notions enables a consistent interpretation thereof.

The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility innature, rather than inmathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part–whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, arementis fictiones, i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything inrerum natura. Unlike infinite wholes, infinitesimals—as well as imaginary roots and other well-founded fictions—may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality.

In this paper the way of doing physics typical of mathematical physics is contrasted with the way of doing physics theorised, and practised, by Aristotle, which is not extraneous to mathematics but deals with it in a completely different manner: not as a demonstrative tool but as a reservoir of analogies. These two different uses are the tangible expression of two different underlying metaphysics of mathematics: two incommensurable metaphysics, which give rise to two incommensurable physics. In the first part of this paper this incommensurability is analysed, then Aristotle’s way of using mathematics is clarified in relation to some controversial mathematical passages, and finally the relation between mathematics and the building of theories is discussed.

Aristotle is committed to finitism in mathematics. But there are certain uses of infinity in mathematics which are indispensable, even in the mathematics of his day. So we had better understand Aristotle’s finitism in a way that is compatible with the mathematics of his time (unless we are willing to ascribe complete naivete to him about these mathematics).In particular, we have to address the issue of infinitely extendible lines, which are used in Greek mathematics – for example, in Euclid’s definition of parallel lines. Aristotle denies that such lines exist: like any other object which exceeds the fixed and finite size of the cosmos, infinitely extendible lines are definitely excluded from Aristotle’s physics. Moreover, due to Aristotle’s immanentism, they are excluded from his mathematics, too.But how can Aristotle do mathematics without infinitely extendible lines? In the following I will suggest a possible solution, based on an analysis of the procedure of converse increasing, which Aristotle introduces and discusses in Ph. III 6. This procedure is both infinite – it is infinitely iterable – and does not require the existence of any infinite (or infinitely extendible) magnitudes.The procedure is interesting, but it must be handled with care. On the one hand, if one fails to acknowledge its subtle mathematical content, one also risks compromising the philosophical interpretation, by ascribing to Aristotle gratuitous naiveties. On the other hand, if one overstates this mathematical content, one risks ascribing to him anachronistic, ‘non-Euclidean’ intents. At the end of the paper I will discuss two cases in point.

Aristotle’s way of conceiving the relationship between mathematics and other branches of scientific knowledge is completely different from the way a contemporary scientist conceives it. This is one of the causes of the fact that we look at the mathematical passage we find in Aristotle’s works with the wrong expectation. We expect to find more or less stringent proofs, while for the most part Aristotle employs mere analogies. Indeed, this is the primary function of mathematics when employed in a philosophical context: not a demonstrative tool, but a purely analogical model. In the case of the geometrical examples discussed in this paper, the diagrams are not conceived as part of a formalized proof, but as a work in progress. Aristotle is not interested in the final diagram but in the construction viewed in its process of development; namely in the figure a geometer draws, and gradually modifies, when he tries to solve a problem. The way in which the geometer makes use of the elements of his diagram, and the relation between these elements and his inner state of knowledge is the real feature which interests Aristotle. His goal is to use analogy in order to give the reader an idea of the states of mind involved in a more general process of knowing.

The paper discusses Aristotle’s conception of motion and proposes a new perspective for a reading. Of course, motion can be located between two states of a movable, between potentiality and actuality, but it can also be located between (the state of) a movable and (that of) its mover. It is also argued that this relational perspective is the correct way to understand Aristotle’s definition of motion. When Aristotle defines motion as “the actuality of what potentially is, qua such” he does not simply mean that motion is “the actuality of the movable”: by being the actuality of the movable “qua movable”, motion is also the actuality of the mover. Indeed, in its proper meaning, ‘movable’ does not refer to a per se entity. Instead it is a relative term, always standing for “movable by the mover”; the qua-clause is intended to stress this relational aspect.

The passage has been an object of scholarly debate: the lack of independent sources on the mathematical construction described by Aristotle, the terseness of the formulation and the resulting syntactical ambiguities make the exact interpretation of the text quite difficult, as already noted by Philoponus. What does it mean that the gnomons are ‘placed round the one and without’ (περὶ τὸ ἓν καὶ χωρίς)? And in what sense is this an indication of the even being ‘cut off, enclosed (ἐναπολαμβανόμενον), and limited (περαινόμενον) by the odd’? The aim of the present paper is to discuss the available interpretations of the passage and to propose a new one.

Summary This paper presents the main features of the treatise on magnetism written by the Jesuit Leonardo Garzoni (1543?92). The treatise was believed to be lost, but a copy of it has been recently recovered. The treatise is briefly described and analysed. The results of a comparison between Garzoni's treatise, Della Porta's Magia Naturalis (1589), and Gilbert's De Magnete (1600) are also summarized. As claimed in the seventeenth century by Niccolò Cabeo and Niccolò Zucchi, the treatise contains quite a lot of the material to be found subsequently in the Magia Naturalis and in the De Magnete. Most importantly, the treatise presents so many interesting features, well before Gilbert's work, which make it the first example of a modern treatment of magnetic phenomena

Mathematics is a subtle and pervasive presence in Aristotle’s physics: not as a demonstrative tool, but as model, in the sense of an analogy. In this paper the particular case of Aristotle’s theory of local motion is considered, which I claim has been constructed by analogy to hydrostatics (understood as a branch of mathematics). Theoretical and empirical evidence is offered in support of the thesis, together with an analysis of the possible sources of the theory and of its distorted reception by Late–Antiquity Scholastic.