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)

This chapter provides an overview of Thomas Hobbes's materialistic philosophy of mathematics. Hobbes's mathematical ontology rejects the seventeenth century's received view of the subject and his proposed first principles departed quite significantly from the tradition. Hobbes's understanding of geometry as a generalized science of material bodies puts him at odds with the traditional notion that the objects of geometrical investigation are radically distinct from the realm of material things. Hobbes's methodology holds that demonstrative knowledge must be based on definitions that (...) identify the causes of things. Hobbes was quite hostile to the algebraic methods characteristic of Descartes's analytic geometry, and he found fault with some presentations of the “method of indivisibles” that is an important precursor to the calculus. Hobbes approved of Bonaventura Cavalieri's method while dismissing John Wallis's approach as incoherent. The chapter investigates Hobbes's controversy with Wallis. Any discussion of Hobbes's mathematics takes place against the background of his controversy with Wallis. (shrink)

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.

Hobbes and Wallis's "battle of the books" illuminates the intimate relationship between science and crucial seventeenth-century debates over the limits of sovereign power and the existence of God.

"The development of the calculus during the 17th century was successful in mathematical practice, but raised questions about the nature of infinitesimals: were they real or rather fictitious? This collection of essays, by scholars from Canada, the US, Germany, United Kingdom and Switzerland, gives a comprehensive study of the controversies over the nature and status of the infinitesimal. Aside from Leibniz, the scholars considered are Hobbes, Wallis, Newton, Bernoulli, Hermann, and Nieuwentijt. The collection also contains newly discovered marginalia of Leibniz (...) to the writings of Hobbes."--BOOK JACKET. (shrink)

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. Jesseph challenges the prevailing view that Berkeley's mathematical writings are peripheral to his philosophy and argues that mathematics is in fact central to his thought, developing out of his critique of abstraction. Jesseph's argument situates Berkeley's ideas within the larger historical and intellectual context of (...) the Scientific Revolution. Jesseph begins with Berkeley's radical opposition to the received view of mathematics in the philosophy of the late seventeenth and early eighteenth centuries, when mathematics was considered a "science of abstractions." Since this view seriously conflicted with Berkeley's critique of abstract ideas, Jesseph contends that he was forced to come up with a nonabstract philosophy of mathematics. Jesseph examines Berkeley's unique treatments of geometry and arithmetic and his famous critique of the calculus in The Analyst. By putting Berkeley's mathematical writings in the perspective of his larger philosophical project and examining their impact on eighteenth-century British mathematics, Jesseph makes a major contribution to philosophy and to the history and philosophy of science. (shrink)

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 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)

Berkeley's philosophy has been much studied and discussed over the years, and a growing number of scholars have come to the realization that scientific and mathematical writings are an essential part of his philosophical enterprise. The aim of this volume is to present Berkeley's two most important scientific texts in a form which meets contemporary standards of scholarship while rendering them accessible to the modern reader. Although editions of both are contained in the fourth volume of the Works, these lack (...) adequate introductions and do not provide com plete and corrected texts. The present edition contains a complete and critically established text of both De Motu and The Analyst, in addi tion to a new translation of De Motu. The introductions and notes are designed to provide the background necessary for a full understanding of Berkeley's account of science and mathematics. Although these two texts are very different, they are united by a shared a concern with the work of Newton and Leibniz. Berkeley's De Motu deals extensively with Newton's Principia and Leibniz's Specimen Dynamicum, while The Analyst critiques both Leibnizian and Newto nian mathematics. Berkeley is commonly thought of as a successor to Locke or Malebranche, but as these works show he is also a successor to Newton and Leibniz. (shrink)

PrefaceList of AbbreviationsChapter One: The Mathematical Career of the Monster of MalmesburyChapter Two: The Reform of Mathematics and of the UniversitiesIdeological Origins of the DisputeChapter Three: De Corpore and the Mathematics of MaterialismChapter Four: Disputed FoundationsHobbes vs. Wallis on the Philosophy of MathematicsChapter Five: The "Modern Analytics" and the Nature of DemonstrationChapter Six: The Demise of Hobbesian GeometryChapter Seven: The Religion, Rhetoric, and Politics of Mr. Hobbes and Dr. WallisChapter Eight: Persistence in ErrorWhy Was Hobbes So Resolutely Wrong?Appendix: Selections from (...) Hobbes's Mathematical WritingsReferencesIndex Copyright © Libri GmbH. All rights reserved. (shrink)

: 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)

This article examines Hobbes’s conception of mathematical method, situating his methodological writings in the context of disputed mathematical issues of the seventeenth century. After a brief exposition of the Hobbesian philosophy of mathematics, it investigates Hobbes’s attempts to resolve three important mathematical controversies of the seventeenth century: the debates over the status of analytic geometry, disputes over the nature of ratios, and the problem of the “angle of contact” between a curve and tangent. In the course of these investigations, Hobbes’s (...) account of mathematics and its method is contrasted with the those of Descartes, Isaac Barrow, John Wallis, and Christopher Clavius. (shrink)

The dispute that raged between Thomas Hobbes and John Wallis from 1655 until Hobbes's death in 1679 was one of the most intense of the ‘battles of the books’ in seventeenth-century intellectual life. The dispute was principally centered on geometric questions, but it also involved questions of religion and politics. This paper investigates the origins of the dispute and argues that Wallis’s primary motivation was not so much to refute Hobbes’s geometry as to demolish his reputation as an authority in (...) political, philosophical, and religious matters. It also highlights the very different conceptions of geometrical methodology employed by the two disputants. In the end, I argue that, although Wallis was successful in showing the inadequacies of Hobbes’s geometric endeavours, he failed in his quest to discredit the Hobbesian philosophy in toto. (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.

Early in his mathematical career Leibniz discovered some important methods and results but had to recognize that his findings had been anticipated by other mathematicians such as Pierre de Fermat, James Gregory, Isaac Newton, François Regnauld, John Wallis, etc. This paper investigates the cases of Isaac Barrow and Pietro Mengoli who, earlier than Leibniz, had been familiar with the characteristic triangle, transmutations methods, the inverse connection between determining tangents and areas of curves or the sums of the reciprocal figurate numbers, (...) and the harmonic triangle. To what extent was Leibniz aware of the results and publications of his predecessors? How did he assess their methods and results? Why did Leibniz never acknowledge any influence of these two mathematicians on his own studies? After publication of Leibniz’s manuscripts concerning the prehistory and early history of the calculus in the Academy Edition these questions can be investigated on the solid foundation of original texts. (shrink)

This is a puzzling book. On the one hand, Stoneham insists that “we cannot appreciate the contributions made by philosophers like Berkeley without coming to terms with the full breadth and detail of his thought”. On the other hand, his interpretive efforts are directed almost exclusively at the Three Dialogues between Hylas and Philonous—a work Berkeley intended as a popular recasting of his doctrines and one that scholars generally regard as conspicuously lacking the “full breadth and detail” of his philosophy. (...) There is an evident tension between the goal of thoroughly examining Berkeley’s thought and the chosen means of focusing on the Three Dialogues. This tension arises, in large part, from the fact that Stoneham’s intended audience is not Berkeley scholars, but rather people seriously interested in philosophy and its history, yet lacking significant familiarity with Berkeley’s thought or its background in early modern philosophy. As a way to introduce advanced undergraduates or graduate students to Berkeley, the book is a reasonably successful effort, although there are some significant shortcomings. (shrink)

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)

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)

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.

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.

This is a puzzling book. On the one hand, Stoneham insists that “we cannot appreciate the contributions made by philosophers like Berkeley without coming to terms with the full breadth and detail of his thought”. On the other hand, his interpretive efforts are directed almost exclusively at the Three Dialogues between Hylas and Philonous—a work Berkeley intended as a popular recasting of his doctrines and one that scholars generally regard as conspicuously lacking the “full breadth and detail” of his philosophy. (...) There is an evident tension between the goal of thoroughly examining Berkeley’s thought and the chosen means of focusing on the Three Dialogues. This tension arises, in large part, from the fact that Stoneham’s intended audience is not Berkeley scholars, but rather people seriously interested in philosophy and its history, yet lacking significant familiarity with Berkeley’s thought or its background in early modern philosophy. As a way to introduce advanced undergraduates or graduate students to Berkeley, the book is a reasonably successful effort, although there are some significant shortcomings. (shrink)