Hobbes: Philosophy of Mathematics

Offers comprehensive treatment of Thomas Hobbes’s thought, providing readers with different ways of understanding Hobbes as a systematic philosopher As one of the founders of modern political philosophy, Thomas Hobbes is best known for his ideas regarding the nature of legitimate government and the necessity of society submitting to the absolute authority of sovereign power. Yet Hobbes produced a wide range of writings, from translations of texts by Homer and Thucydides, to interpretations of Biblical books, to works devoted to geometry, (...) optics, morality, and religion. Hobbes viewed himself as presenting a unified method for theoretical and practical science—an interconnected system of philosophy that provides many entry points into his thought. A Companion to Hobbes is an expertly curated collection of essays offering close textual engagement with the thought of Thomas Hobbes in his major works while probing his ideas regarding natural philosophy, mathematics, human nature, civil philosophy, religion, and more. The Companion discusses the ways in which scholars have tried to understand the unity and diversity of Hobbes’s philosophical system and examines the reception of the different parts of Hobbes’s philosophy by thinkers such as René Descartes, Margaret Cavendish, David Hume, and Immanuel Kant. Presenting a diversity of fresh perspectives by both emerging and established scholars, this volume: Provides a comprehensive treatment of Hobbes’s thought in his works, including Elements of Law, Elements of Philosophy, and Leviathan Explores the connecting points between Hobbes’ metaphysics, epistemology, mathematics, natural philosophy, morality, and civil philosophy Offers readers strategies for understanding how the parts of Hobbes’s philosophical system fit together Examines Hobbes’s philosophy of mathematics and his attempts to understand geometrical objects and definitions Considers Hobbes’s philosophy in contexts such as the natural state of humans, gender relations, and materialist worldviews Challenges conceptions of Hobbes’s moral theory and his views about the rights of sovereigns Part of the acclaimed Blackwell Companions to Philosophy series, A Companion to Hobbes is an invaluable resource for scholars and advanced students of Early modern thought, particularly those from disciplines such as History of Philosophy, Political Philosophy, Intellectual History, History of Politics, Political Theory, and English. (shrink)

This chapter examines the role of experience in Thomas Hobbes’ science of politics. Although Hobbes claims for his own formulation of civil philosophy a kind of definitiveness and certainty that only geometry has among the sciences, and although both geometry and civil philosophy are supposed to be the products of reason, where reason excludes experience (sense and memory), the necessity of establishing and submitting to the commonwealth is open to a certain sort of confirmation from experience. This is not because (...) Hobbes concedes cognitive authority to sense and memory after all, but because civil philosophy has a rhetorical purpose that a certain kind of appeal to experience helps to achieve. (shrink)

I offer an alternative account of the relationship of Hobbesian geometry to natural philosophy by arguing that mixed mathematics provided Hobbes with a model for thinking about it. In mixed mathematics, one may borrow causal principles from one science and use them in another science without there being a deductive relationship between those two sciences. Natural philosophy for Hobbes is mixed because an explanation may combine observations from experience (the ‘that’) with causal principles from geometry (the ‘why’). My argument shows (...) that Hobbesian natural philosophy relies upon suppositions that bodies plausibly behave according to these borrowed causal principles from geometry, acknowledging that bodies in the world may not actually behave this way. First, I consider Hobbes's relation to Aristotelian mixed mathematics and to Isaac Barrow's broadening of mixed mathematics in Mathematical Lectures (1683). I show that for Hobbes maker's knowledge from geometry provides the ‘why’ in mixed-mathematical explanations. Next, I examine two explanations from De corpore Part IV: (1) the explanation of sense in De corpore 25.1-2; and (2) the explanation of the swelling of parts of the body when they become warm in De corpore 27.3. In both explanations, I show Hobbes borrowing and citing geometrical principles and mixing these principles with appeals to experience. (shrink)

Accounts of Hobbes’s ‘system’ of sciences oscillate between two extremes. On one extreme, the system is portrayed as wholly axiomtic-deductive, with statecraft being deduced in an unbroken chain from the principles of logic and first philosophy. On the other, it is portrayed as rife with conceptual cracks and fissures, with Hobbes’s statements about its deductive structure amounting to mere window-dressing. This paper argues that a middle way is found by conceiving of Hobbes’s _Elements of Philosophy_ on the model of a (...) mixed-mathematical science, not the model provided by Euclid’s _Elements of Geometry_. I suggest that Hobbes is a test case for understanding early-modern system-construction more generally, as inspired by the structure of the applied mathematical sciences. This approach has the additional virtue of bolstering, in a novel way, the thesis that the transformation of philosophy in the long seventeenth century was heavily indebted to mathematics, a thesis that has increasingly come under attack in recent years. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Infinitesimal Differences: Controversies between Leibniz and his ContemporariesFrançoise Monnoyeur-BroitmanUrsula Goldenbaum and Douglas Jesseph, editors. Infinitesimal Differences: Controversies between Leibniz and his Contemporaries. Berlin-New York: Walter de Gruyter, 2008. Pp. vi + 327. Cloth, $109.00.Leibniz is well known for his formulation of the infinitesimal calculus. Nevertheless, the nature and logic of his discovery are seldom questioned: does it belong more to mathematics or metaphysics, and how is it connected (...) to his physics? This book, composed of fourteen essays, investigates the nature and foundation of the calculus, its relationship to the physics of force and principle of continuity, and its overall method and metaphysics. The Leibnizian calculus is presented in its origin and context together with its main contributors: Archimedes, Cavalieri, Wallis, Hobbes, Pascal, Huygens, Bernoulli, and Nieuwentijt.Many of us know and probably have used the Leibnizian formula: to calculate the area under a curve; this book considers the origin, nature, method, and metaphysics of this formula. The most fascinating question discussed in it is the fiction introduced to define the infinitesimal calculation: in his mature period, Leibniz himself referred to his calculus as “a well-founded fiction.” We learn that he gradually gave up the idea of an actual infinite in favor of describing infinitely small quantities. As a result, the quantities became fictive and finite or indefinite instead of real and infinite, and the process of calculation shifted towards mathematics rather than metaphysics.In the first essay, Richard Arthur compares the calculus of Leibniz and Newton and demonstrates how it was based for both on the development of continuously varying quantity. He also shows how the understanding of this quantity derived from the Archimedean axiom that excludes the existence of actual infinite quantity. Ursula Goldenbaum next demonstrates how Hobbes, with his conatus considered as point and instant, had a direct impact on the Leibnizian conception of continuity; to confirm her interpretation, she includes at the end of her article the “Marginalia or Leibniz comments” on Hobbes. Samuel Levey establishes a connection between the notion of the infinitesimal as a “well-founded fiction” and the law of continuity, arguing that the law of continuity, understood as an extension of Archimedes’ principle, gives a finite foundation to the infinitesimal calculation. O. Bradley Bassler discusses the idea of the differential in terms of infinitely small quantities and the use of proportions between infinitely small and finite quantities in order to measure [End Page 527] the infinitely small; here, differentials are also presented as fictions. Fritz Nagel discusses the methodological sophistication of the calculus and its opponents. His main questions concern the different grades of infinity, quantities regarded as zero, and the infinitesimal as a fiction that can be calculated. Douglas Jesseph considers the reality of the infinitesimal and the “well-founded fiction,” showing how Leibniz methodically constructs infinitesimal quantities without reality by returning to Hobbes’ notion of conatus. Hobbes is, for Jesseph, the initiator of these infinitely small quantities without existence.A second group of essays examines the connection between algebra and geometry in the calculus. Philip Beeley demonstrates how John Wallis, on the path of Cavalieri’s method of the indivisibles, found a way to transform geometric problems into sums of arithmetic sequences. Siegmund Probst expresses the different phases in the discovery of the Leibnizian method. He mentions that Leibniz coined the term ‘infinitesimal’ in 1673, and points out that he did not need the method of indivisibles to find a curve from its given arc, but only a series expansion using differences of higher order. Emily Grosholz looks at how, thanks to his algebraic and geometrical notations of curves, Leibniz could formulate the equation of his infinitesimal calculus. According to Grosholz, Leibniz exploits the ambiguity of notations in order to express the geometrical and dynamical aspect of the calculus. Herbert Breger underlines the novelty and abstract nature of Leibniz’s new method of calculation, showing how the theory of infinitesimals became an algebraic calculus independent of geometry.A final group of articles studies the relationship between the calculus and Leibniz’s physics. François Duchesneau shows how the differential and integral calculus provided a model for the integrative laws of Leibniz’s... (shrink)

In the central chapters of Leviathan, Hobbes offers a demonstration of the "true doctrine of the laws of nature," which is identified with the "science of virtue andvice" and the "true moral philosophy." In his deduction of the laws of nature, Hobbes attempts to mimic the science of geometry, which he says is the "only science God had hitherto bestowed on mankind. "In this paper, I discuss some of the problems associated with Hobbes's application of the method of geometry to (...) civil philosophy. After locating the root of these problems in Hobbes's in ability to recognize the distinction between formal and applied sciences, I discuss a possible solution. According to this solution, Hobbes's "science of morality" is considered to be a formal science that is applied to the world by an act of human creation. (shrink)

The canonical view of Hobbes the mathematician is of a doddering old man tilting at quadrature. That picture arises from his twenty-five year debate with John Wallis, an English algebraist, Savile Professor of Geometry at Oxford, and a founding fellow of the Royal Society of London. In 1655, Hobbes attempted to square the circle, Wallis promptly proved him wrong, and their vitriolic exchange was born ;Hobbes is always portrayed as the clear loser. Recently, however, scholars have noticed both flashes of (...) insight in Hobbes's mathematical writings and his excellent reputation among mathematicians on the Continent during the 1640s. Yet the overall picture remains confusing. Hobbes's apparent errors ranged from very subtle to downright bizarre, even when he discussed works by his former colleagues. ;Drawing on the history of mathematics, education, and popular culture, I argue that Hobbes's reputation rests on two systematic errors: misrepresentation of the events of his life, based in his reputation as a purely political philosopher, and a misunderstanding of the nature of mathematics that conceals the deep diversity among early-modern mathematicians. Once those errors are addressed, Hobbes looks considerably more knowledgeable---although still incorrect. ;I begin by reviewing the lives of both Hobbes and Wallis, contending that when the debate started 1655, Hobbes had extensive training in formal mathematics while Wallis had virtually none. Then, by examining four mathematical traditions that flourished in the mid-1600s, I show that terms such as of "mathematical," and "demonstration" have settled meanings only in reference to a single practice. Finally, I explore issues from the Hobbes-Wallis debate, arguing that they center on terms whose meaning had not been generally established, and that Wallis touted an algebraic form of applied mathematics while Hobbes defended the standards of classical geometry. ;Neither Hobbes nor Wallis can be accurately described as right. Hobbes had genuine technical limitations but a firm grasp of methodological issues, while Wallis's technical brilliance was offset by his frequent inability to appreciate the requirements of rigor and universality. Reexamining the debate in light of those differences reveals a wealth of issues in early-modern mathematics, Hobbes studies, and historiography. (shrink)

My thesis examines Thomas Hobbes's attempt to develop a mathematical account of nature. I argue that Hobbes's conception of how we should think quantitatively about the world was deeply indebted to the ideas of his ancient and medieval predecessors. These ideas were often amenable to Hobbes's vision of a demonstrative, geometrically-based science. However, he was forced to adapt the ancient and medieval models to the demands of his own thoroughgoing materialism. This hybrid resulted in a distinctive, if only partially successful, (...) approach to the problems of the new mechanical philosophy. (shrink)

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.

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)

Our understanding of Hobbes is pivotal to our grasp of modernity, and yet we misunderstand him. Specifically, we have misconstrued the way his philosophy exemplifies modern political reason. Contrary to the conventional view, Hobbes should not be thought of as the first modern political scientist. His works are not abstract schema devoted to explaining the logic of commonwealths or the actions of rationally self-interested actors. Rather than explanatory, Hobbes's science was didactic: it was written to craft and fashion obedient selves. (...) His works teach political duties and/or strive to persuade the sovereign that his doctrine ought to be taught in the schools---to the exclusion of all others. Hobbes's promise of a peaceable commonwealth required that his doctrine become the political education for England's elite. I illustrate parallels between Hobbes and contemporaries also interested in reeducating England . To reinforce and complement this reading, I defend a new interpretation of his science. Hobbes's approach to geometry, I argue, attaches certainty to what we can make. He uses mathematics to pursue a constructed, rather than a natural, order. He was, therefore, more concerned with imposing an order he designed himself than with predicting nature's. With this new reading I hope to change our perceptions of the origins of modern science and the relationship between science and politics in modernity. My approach is both philosophical and historical, and I hope it will interest persons concerned with modernity, the construction of the modern subject, political education, politics and literature, the history of science, and critiques of modern political rationalism. (shrink)

Hobbes ' geometrical disputes are significant since they highlight several important strands in his thought - issues concerning the right to make definitions, his anti-clericalism, the maker's knowledge argument and his objections to algebra. These are examined, and the foundational position, according to Hobbes, of geomentry in relation to philosophy, science and technology, explained and discussed.

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)

Until recently, historians of mathematics usually agreed in refusing to consider the numerous geometrical publications of Thomas Hobbes as a contribution to the development of mathematics in the seventeenth century. From time to time, one could find statements that although Hobbes did not find new theorems he undoubtedly had profound insights into the logical foundations of mathematics, but these occasional remarks did not encourage historians to go deeper into Hobbes's mathematical thought. In the end, the general conclusion was that Hobbes's (...) preoccupation with squaring the circle, doubling the cube (starting when the philosopher was more than forty years of age), and challenging Euclid's definitions were better ignored, at least in the history of science. In particular, his controversy with the Savilian professors Seth Ward and John Wallis was seen as a ‘deplorable affair’, liable only to damage the reputation of the protagonists. (shrink)

Hobbes' philosophy of geometry was eccentric to contemporary movements and worsted in specific controversy. But he laid down stipulations defining geometry and its method which might provide a significant and workable alternative "meta-geometry". Some of these are isolated and reinterpreted here, especially those concerned with describing magnitudes, motions and quantities, and with his use of proportions. Rather than refutation of commentaries and historical rehash, the effort here is to isolate definitive texts and to offer a reinterpretation of their arguments in (...) succinct form showing their coherence and their interdependence with his other philosophic principles. (shrink)