Douglas Jesseph

Galileo, Hobbes, and the Book of Nature

This paper investigates the inuence of Galileos natural philosophy on the philosophical and methodological doctrines of Thomas Hobbes. In particular, I argue that what Hobbes took away from his encounter with Galileo was the fundamental idea that the world is a mechanical system in which everything can be understood in terms of mathematically-specifiable laws of motion. After tracing the history of Hobbess encounters with Galilean science (through the Welbeck group connected with William Cavendish, earl of Newcastle and the Mersenne circle in Paris), I argue that Hobbess 1655 treatise De Corpore is deeply indebted to Galileo. More specifically, I show that Hobbess mechanistic theory of mind owes a significant debt to Galileo while his treatment of the geometry of parabolic figures in chapter 16 of De Corpore was taken almost straight out of the account of accelerated motion Two New Sciences

Leibniz on The Elimination of Infinitesimals

My aim in this paper is to consider Leibniz’s response to concerns raised about the foundations of his differential calculus, and specifically with his doctrine that infinitesimals are “fictions,” albeit fictions so well-founded that their use will never lead to error. I begin with a very brief sketch of the traditional conception of rigorous demonstration and the methodological disputes engendered by the advent of the Leibnizian calculus differentialis. I then examine Leibniz’s claim that infinitesimal magnitudes are fictions, and consider two strategies he employed in the attempt to show that such fictions are acceptable.

Hobbes on the Ratios of Motions and Magnitudes: The Central Task of De Corpore, Part III

Hobbes intended and expected De Corpore to secure his place among the foremost mathematicians of his era. This is evident from the content of Part III of the work, which contains putative solutions to the most eagerly sought mathematical results of the seventeenth century. It is well known that Hobbes failed abysmally in his attempts to solve problems of this sort, but it is not generally understood that the mathematics of De Corpore is closely connected with the work of some of seventeenth-century Europe’s most important mathematicians. This paper investigates the connection between the main mathematical chapters of De Corpore and the work of Galileo Galilei, Bonaventura Cavalieri, and Gilles Personne de Roberval. I show that Hobbes’s approach in Chapter 16 borrows heavily from Galileo’s Two New Sciences, while his treatment of “deficient figures’ in Chapter 17 is nearly identical in method to Cavalieri’s Exercitationes Geometricae Sex. Further, I argue that Hobbes’s attempt to determine the arc length of the parabola in Chapter 18 is intended to use Roberval’s methods to generate a more general result than one that Roberval himself had achieved in the 1640s (when he and Hobbes were both active in the circle of mathematicians around Marin Mersenne). I claim Hobbes was convinced that his first principles had led him to discover a “method of motion” that he mistakenly thought could solve any geometric problem with elementary constructions.

Hobbes on ‘Conatus’: a Study in the Foundations of Hobbesian Philosophy

This paper will deal with the notion of conatus (endeavor) and the role it plays in Hobbes’s program for natural philosophy. As defined by Hobbes, the conatus of a body is essentially its instantaneous motion, and he sees this as the means to account for a variety of phenomena in both natural philosophy and mathematics. Although I foucs principally on Hobbesian physics, I will also consider the extent to which Hobbes’s account of conatus does important explanatory work in his theory of human perception, psychology, and political philosophy. I argue that, in the end, there are important limitations in Hobbes’s account of conatus, but that Leibniz adapted the concept in important ways in developing his science of dynamics.

Hobbes’s Theory of Space

Philosophy in the seventeenth century is often, and with reason, characterized as a collection of grand systems devoted to an all-encompassing account of the world and its workings.

Machines, Mechanism, and the Development of Mechanics: Contemporary Understandings

The early modern period is a source of nearly inexhaustible studies for historians and philosophers of science, and it is reasonably safe to conclude that it will remain so for the foreseeable future. Yet there have been signiacant changes in historiographic tendencies over the last few decades, and it is useful to pause and reoect on where we are in our understanding of the scientiac and mathematical developments from the late sixteenth through the early eighteenth centuries. The period was once commonly known as the “Scientiac Revolution,” (a coinage due to the great Alexandre Koyre), and much scholarship from the twentieth century stressed the abrupt and seemingly irreversible shift in scientiac outlook between the publication of Copernicus’s De Revolutionibus orbium coeliestium in 1543 and Newton’s Philosopiae Naturalis Principia Mathematica in 1687. The fo-

Scientia in Hobbes

Thomas Hobbes is much better remembered for his political philosophy than his epistemology. Nevertheless, his theory of knowledge is an important contribution to the development of epistemology in the early modern period. The centerpiece of Hobbes’s theory of knowledge is his account of scientia, or deductively structured systematic knowledge that is grounded in the consideration of causes.

Descartes, Pascal, and the Epistemology of Mathematics: The Case of the Cycloid

This paper deals with the very different attitudes that Descartes and Pascal had to the cycloidthe curve traced by the motion of a point on the periphery of a circle as the circle rolls across a right line. Descartes insisted that such a curve was merely mechanical and not truly geometric, and so was of no real mathematical interest. He nevertheless responded to enquiries from Mersenne, who posed the problems of determining its area and constructing its tangent. Pascal, in contrast, saw the cycloid as a paradigm of geometric intelligibility, and he made it the focus of a series of challenge problems he posed to the mathematical world in 1658. After dealing with some of the history of the cycloid (including the work of Galileo, Mersenne, and Torricelli), I trace this difference in attitude to an underlying difference in the mathematical epistemologies of Descartes and Pascal. For Descartes, the truly geometric is that which can be expressed in terms of finite ratios between right lines, which in turn are expressible as closed polynomial equations. As Descartes pointed out, this means that ratios between straight and curved lines are not geometrically admissible, and curves (such as the cycloid) that require them must be banished from geometry. Pascal, in contrast, thought that the scope of geometry included curves such as the cycloid, which are to be studied by employing infinitesimal methods and ratios between curved and straight lines.

Rigorous proof and the history of mathematics: Comments on Crowe

Duhems portrayal of the history of mathematics as manifesting calm and regular development is traced to his conception of mathematical rigor as an essentially static concept. This account is undermined by citing controversies over rigorous demonstration from the eighteenth and twentieth centuries.