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. (shrink)

Greek antiquity saw the development of two distinct systems of logic: Aristotle’s theory of the categorical syllogism and the Stoic theory of the hypothetical syllogism. Some ancient logicians argued that hypothetical syllogistic is more fundamental than categorical syllogistic on the grounds that the latter relies on modes of propositional reasoning such asreductio ad absurdum. Peripatetic logicians, by contrast, sought to establish the priority of categorical over hypothetical syllogistic by reducing various modes of propositional reasoning to categorical form. In the 17th (...) century, this Peripatetic program of reducing hypothetical to categorical logic was championed by Gottfried Wilhelm Leibniz. In an essay titledSpecimina calculi rationalis, Leibniz develops a theory of propositional terms that allows him to derive the rule ofreductio ad absurdumin a purely categorical calculus in which every proposition is of the formA is B. We reconstruct Leibniz’s categorical calculus and show that it is strong enough to establish not only the rule ofreductio ad absurdum, but all the laws of classical propositional logic. Moreover, we show that the propositional logic generated by the nonmonotonic variant of Leibniz’s categorical calculus is a natural system of relevance logic known as RMI$_{{}_ \to ^\neg }$. (shrink)

Thomas Harriot was the finest English mathematician before Isaac Newton, but his work on the coinage of his country is almost unknown, unlike Newton’s. In the early 1600s Harriot studied several aspects of the gold and silver coins of his time. He investigated the ratio between the values of gold and silver, using data derived from the official weights of the coins; he used hydrostatic weighing to determine the composition of the coins; and he studied the methods used to calculate (...) how metals should be combined in order to produce the required standard of purity. This work required not only arithmetical skill, but also great insight into the relationships between the various quantities, and here his ‘greate invention of algebra’ played its part. The gold and silver coins were critically important in the national economy at that time, because they had intrinsic value and represented the country’s store of wealth. In his lifetime Harriot was supported by the patronage of Henry Percy, the Earl of Northumberland, who was a member of the Privy Council in 1604 when great changes to the English coinage were being made. After Harriot’s death his papers remained under the control of one of his executors, Sir Thomas Aylesbury, who became Master of the Tower Mint in 1635. These facts suggest that his work on the coins was considered to be important for practical reasons, not just academic ones. In this article the interplay between the scholarly work and its application to matters of national significance will be analysed in depth. The conclusions also throw some light on the long-standing mystery surrounding the fate of Harriot’s papers. (shrink)

This paper examines an annotation in Newton’s hand found by H. W. Turnbull in David Gregory’s papers in the Library of the Royal Society. It will be shown that Gregory asked Newton to explain to him how the trajectories of a body accelerated by an inverse-cube force are determined in a corollary in the Principia: an important topic for gravitation theory, since tidal forces are inverse cube. This annotation opens a window on the more hidden mathematical methods which Newton deployed (...) in his magnum opus. The received view according to which the Principia are written in a geometric style with no help from calculus techniques must be revised. (shrink)

The seventeenth century can be viewed as an era of (closely related) innovation in the formal and natural sciences and of paradigmatic diversity in philosophy (due to the coexistence of at least the humanist, the late scholastic, and the early modern tradition). Within this environment, the present study focuses on scholastic logic and, in particular, syllogistic. In seventeenth-century scholastic logic two different approaches to logic can be identified, one represented by the Dominicans Báñez, Poinsot, and Comas del Brugar, the other (...) represented by the Jesuits Hurtado, Arriaga, Oviedo, and Compton. These two groups of authors can be contrasted in three prominent features. First, in the role of the theory of validity, which is either a common basis for all particular theories (in this case, sentential logic and syllogistic), or a set of observations regarding a particular theory (in this case, syllogistic). Second, in the view of syllogistic, which is either an implication of a general theory of validity and a semantics of terms, or an algebra of structured objects. Third, in the role of the scholastic analysis of language in terms ofsuppositio, which either is a semantic underpinning of syllogistic, or it is replaced by a semantics of propositions. (shrink)

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. (shrink)

The aim of this paper is to give a full exposition of Leibniz’s mereological system. My starting point will be his papers on Real Addition, and the distinction between the containment and the part-whole relation. In the first part (§2), I expound the Real Addition calculus; in the second part (§3), I introduce the mereological calculus by restricting the containment relation via the notion of homogeneity which results in the parthood relation (this corresponds to an extension of the Real Addition (...) calculus via what I call the Homogeneity axiom). I analyze in detail such a notion, and argue that it implies a gunk conception of (proper) part. Finally, in the third part (§4), I scrutinize some of the applications of the containment-parthood distinction showing that a number of famous Leibnizian doctrines depend on it. (shrink)