In this first modern, critical assessment of the place of mathematics in Berkeley's philosophy and Berkeley's place in the history of mathematics, Douglas M. Jesseph provides a bold reinterpretation of Berkeley's work.
The dissertation is a detailed analysis of Berkeley's writings on mathematics, concentrating on the link between his attack on the theory of abstract ideas and his philosophy of mathematics. Although the focus is on Berkeley's works, I also trace the important connections between Berkeley's views and those of Isaac Barrow, John Wallis, John Keill, and Isaac Newton . The basic thesis I defend is that Berkeley's philosophy of mathematics is a natural extension of his views on abstraction. The first chapter (...) is devoted to a consideration of Berkeley's treatment of abstraction, including his arguments against the doctrine of abstract ideas and his own account of how the explanatory ideas traditionally assigned to abstract ideas can be filled by a non-abstractionist account of human knowledge. In chapter two I investigate the details of Berkeley's proposed new foundations for geometry, showing how his rejection of abstract ideas led him to a critique of the traditional conception of geometry . Of particular importance in this context is Berkeley's denial of infinite divisibility and his attempts to show that a satisfactory account of geometry does not require that geometric magnitudes be infinitely divisible. Chapter three is concerned with Berkeley's treatment of arithmetic and algebra. Here I argue that Berkeley's denial of the claim that arithmetic is the science of abstract ideas of number ultimately results in his advocacy of a strongly nominalistic conception of arithmetic which has strong similarities to modern fomalism. In chapter four I discuss Berkeley's famous critique of the calculus in The Analyst and other works, concluding that his criticism of the calculus is essentially correct, although his attempted explanation of the success of infinitesimal methods is unconvincing. (shrink)
_ Source: _Volume 30, Issue 1, pp 58 - 82 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. 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. (shrink)
This paper analyzes Berkeley's arguments for the existence of God in the Principles of Human Knowledge, Three Dialogues, and Alciphron. Where most scholarship has interpreted Berkeley as offering three quite distinct attempted proofs of God's existence, I argue that these are all variations on the strategy of inference to the best explanation. I also consider how this reading of Berkeley connects his conception of God to his views about causation and explanation.
: This paper investigates the influence of Galileo's 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 Hobbes's encounters with Galilean science (through the "Welbeck group" connected with William Cavendish, earl of Newcastle and the "Mersenne (...) circle" in Paris), I argue that Hobbes's 1655 treatise De Corpore is deeply indebted to Galileo. More specifically, I show that Hobbes's 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. (shrink)
_ Source: _Volume 29, Issue 1, pp 66 - 85 This paper will deal with the notion of _conatus_ 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. (shrink)
Duhem's 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.
It is argued that, contrary to the standard accounts of the development of infinitesimal mathematics, the leading mathematicians of the seventeenth century were deeply concerned with the rigor of their methods. examples are taken from the work of cavalieri and leibniz, with further material drawn from guldin, barrow, and wallis.
This paper deals with the very different attitudes that Descartes and Pascal had to the cycloid—the 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 , 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 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. (shrink)
This chapter examines the views of seventeenth-century British philosophers on the notion of logic and demonstrative knowledge, particularly Francis Bacon, Thomas Hobbes, and John Locke, offering an overview of traditional Aristotelianism in relation to logic and describing Bacon's approach to demonstration and logic. It also analyzes the contribution of the Cambridge Platonists and evaluates the influence of Cartesianism. The chapter concludes that theorizing about logic and demonstrative knowledge followed an arc familiar from other branches of philosophy such as metaphysics or (...) the philosophy of science. (shrink)
The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmetic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with (...) the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics. (shrink)
This is a dictionary of Descartes and Cartesian philosophy, primarily covering philosophy in the 17th century, with a chronology and biography of Descartes's life and times and a bibliography of primary and secondary works related to Descartes and to Cartesians.
The A to Z of Descartes and Cartesian Philosophy includes a chronology, an introduction, a bibliography, and cross-reference dictionary entries Descartes's writings, concepts, and findings, as well as entries on those who supported him, those who criticized him, those who corrected him, and those who together formed one of the major movements in philosophy, Cartesianism.