Hobbes: Mathematics

This paper examines Hobbes’s criticisms of Robert Boyle’s air-pump experiments in light of Hobbes’s account in _De Corpore_ and _De Homine_ of the relationship of natural philosophy to geometry. I argue that Hobbes’s criticisms rely upon his understanding of what counts as “true physics.” Instead of seeing Hobbes as defending natural philosophy as “a causal enterprise … [that] as such, secured total and irrevocable assent,” 1 I argue that, in his disagreement with Boyle, Hobbes relied upon his understanding of natural (...) philosophy as a mixed mathematical science. In a mixed mathematical science one can mix facts from experience with causal principles borrowed from geometry. Hobbes’s harsh criticisms of Boyle’s philosophy, especially in the _Dialogus Physicus, sive De natura aeris_, should thus be understood as Hobbes advancing his view of the proper relationship of natural philosophy to geometry in terms of mixing principles from geometry with facts from experience. Understood in this light, Hobbes need not be taken to reject or diminish the importance of experiment/experience; nor should Hobbes’s criticisms in _Dialogus Physicus_ be understood as rejecting experimenting as ignoble and not befitting a philosopher. Instead, Hobbes’s viewpoint is that experiment/experience must be understood within its proper place – it establishes the ‘that’ for a mixed mathematical science explanation. (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)

_ Source: _Volume 29, Issue 1, pp 39 - 65 Since Euclid, optics has been considered a geometrical science, which Aristotle defines as a “mixed” mathematical science. Hobbes follows this tradition and clearly places optics among physical sciences. However, modern scholars point to a confusion between geometry and physics and do not seem to agree about the way Hobbes mixes both sciences. In this paper, I return to this alleged confusion and intend to emphasize the peculiarity of Hobbes’s geometrical optics. (...) This paper suggests that Hobbes’s conception of geometrical optics, as a mixed mathematical science, greatly differs from Descartes’s one, mainly because they do not share the same “mechanical conception of nature.” I will argue that Hobbes and Descartes also have in common the quest for a different kind of geometry for their optics, different from that of the Ancients. I will show that this departure is not recent since Hobbes’s approach is already evident in 1636, when he judges the demonstrations of his contemporary friends, Claude Mydorge and Walter Warner. Finally the paper broadly suggests what is noteworthy in Hobbes’s optics, that is, the importance of the idea of force in his mechanics, although he was not able to conceptualize it in other terms than “quickness.”. (shrink)

This paper deals with Hobbes's theory of optical images, developed in his optical magnum opus, ‘A Minute or First Draught of the Optiques’, and published in abridged version in De homine. The paper suggests that Hobbes's theory of vision and images serves him to ground his philosophy of man on his philosophy of body. Furthermore, since this part of Hobbes's work on optics is the most thoroughly geometrical, it reveals a good deal about the role of mathematics in Hobbes's philosophy. (...) The paper points to some difficulties in the thesis of Shapin and Schaffer, who presented geometry as a ‘paradigm’ for Hobbes's natural philosophy. It will be argued here that Hobbes's application of geometry to optics was dictated by his metaphysical and epistemological principles, not by a blind belief in the power of geometry. Geometry supported causal explanation, and assisted reason in making sense of appearances by helping the philosopher understand the relationships between the world outside us and the images it produces in us. Finally the paper broadly suggests how Hobbes's theory of images may have triggered, by negative example, the flourishing of geometrical optics in Restoration England. (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)

Reason and Ethics in Hobbes's Leviathan JOHN DEIGH HOBBES'S ETHICS teaches the ways of self-preservation. Its lessons are arranged in a system of rules that Hobbes understood to be the laws of nature. These two themes, self-preservation and natural law, have inspired opposing inter- pretations of Hobbes's text. The historically dominant and still prevailing interpretation, which develops the former theme, is that Hobbes's ethics is a form of egoism. A later and less popular interpretation, which develops the latter theme, is (...) that his ethics is a system of absolutes, "a strict deontology. ''1 These two interpretations represent a dispute between what I will call ortho- doxy and dissent in Hobbes scholarship. The dispute chiefly turns on two questions: What is the basis of Hobbes's ethics? and How stringent are the rules it comprises?' Orthodoxy takes Hobbes's ethics to be based on his moral psychology, specifically, his theory of motivation. That theory, on the orthodox reading, is thoroughly egocentric. It holds that all voluntary action springs from self- interest.3 Hence, on this reading, one can attribute to Hobbes the view that a I am grateful to Ed Curley, Bernard Gert, Richard Kraut, Connie Rosati, and Ira Singer for their helpful comments on an earlier draft of this essay. ' A. E. Taylor, "The Ethical Doctrine of Hobbes," Philosophy a 3 : 406-24, esp. 408. ' One can find the orthodox interpretation in Henry Sidgwick's Outline of the History... (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)

The quotation I take above as motto is from the Author's Epistle to the Reader of De Corpore. Immediately after it, Hobbes elaborates the conceit likening six sciences with the six days of divine creation. These are supplemented with divine commandment and final contemplation of "subjection to command." Thus, with some poetic license, all compartments of Hobbes's reiterated ordering of several bodies of science and "Elements of Philosophy" are indicated: De Corpore, and then De Homine and De Cive. Following Hobbes, (...) as I profess, I sunder them that I may acknowledge both their distinction and order; yet I dispute their sufficient derivation, one from another. (shrink)

The object of this article is the study of possibilities and tendencies arising from the use of symbolic language including signs, characters, and symbols in mathematics. Five aspects are discussed: compactness and simultaneity, problem‐solving and generalizations, heuristics and progress, mechanisms and calculations, formalism. This is done primarily by looking at three disciplines, which at the same time are of fundamental importance to theoretical physics: classical algebra, calculus, and vector analysis.Mathematical achievements and statements by eminent mathematicians from antiquity to the present (...) (Archimedes, Leibniz, Newton, Euler, Gauß, Hilbert) show the importance of adequate symbolism for the development of mathematics. The corresponding philosophical discussions (Hobbes, Condillac, Degérando, Apelt, Cassirer, among others) are controversial. (shrink)