Hobbes on natural philosophy as “True Physics” and mixed mathematics

In this paper, 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.     

A study and critique of the teaching of the history of science and technology. Interim report by the committee on undergraduate education of the history of science society (U.S.A.)

A great deal is known about the technical issues surrounding the introduction of Hugo De Vries's mutation theory and the subsequent development of the modern genetical theory of natural selection. But so far little has been done to relate these events to the wider issues of the time. This article suggests that extra-scientific factors played a significant role, and substantiates this by comparing De Vries's respect for the original Darwinian spirit with Thomas Hunt Morgan's use of the mutation theory as part of an attack on the whole philosophy of Darwinism. In particular, it is argued that Morgan's attitude was dictated by his moral objections to the picture of a world dominated by struggle.     

Kant and Newton on the a priori necessity of geometry

In the Transcendental Aesthetic, Kant explicitly rejects Newton’s absolutist position that space is an actually existing thing; however, Kant also concedes that the absolutist successfully preserves the a priori necessity that characterizes our geometrical knowledge of space. My goal in this paper is to explore why the absolutist can explain the a priori necessity of geometry by turning to Newton’s De Gravitatione, an unpublished text in which Newton addresses the essential features associated with our representation of space and the relationship between our geometrical investigation of space and our knowledge of the form of space that is a part of the natural order. Attention to Newton’s account of space in De Gravitatione offers insight into the sense in which absolutist space is a priori and reveals why, in the Aesthetic, Kant could concede a priori geometrical knowledge to his absolutist opponent. What I highlight in particular is that, by Kant’s standards, Newton employs the very constructive method of mathematics that secures the a priori necessity of geometry, even though, as an absolutist, and as emphasized in the arguments of the Aesthetic, Newton fails to provide a metaphysics of space that explains the success of his mathematical method.     

Ernst Cassirer's transcendental account of mathematical reasoning

Cassirer's philosophical agenda revolved around what appears to be a paradoxical goal, that is, to reconcile the Kantian explanation of the possibility of knowledge with the conceptual changes of nineteenth and early twentieth-century science. This paper offers a new discussion of one way in which this paradox manifests itself in Cassirer's philosophy of mathematics. Cassirer articulated a unitary perspective on mathematics as an investigation of structures independently of the nature of individual objects making up those structures. However, this posed the problem of how to account for the applicability of abstract mathematical concepts to empirical reality. My suggestion is that Cassirer was able to address this problem by giving a transcendental account of mathematical reasoning, according to which the very formation of mathematical concepts provides an explanation of the extensibility of mathematical knowledge. In order to spell out what this argument entails, the first part of the paper considers how Cassirer positioned himself within the Marburg neo-Kantian debate over intellectual and sensible conditions of knowledge in 1902–1910. The second part compares what Cassirer says about mathematics in 1910 with some relevant examples of how structural procedures developed in nineteenth-century mathematics.     

Arte rates reguntur: Nautical handbooks in antiquity?

Given the huge number of technical handbooks on multifarious subjects, ranging from astronomy and music to rhetoric, horticulture, and cooking, the absence of ancient nautical handbooks comes as a surprise. Such handbooks did exist in antiquity in some form, likely having been written in the period of the Hellenistic boom of technical texts, but disappearing at some later point, perhaps around the third or fourth century AD. This disappearance could be due to a number of reasons, suggesting that the tastes and needs of the audience(s) for nautical technai were changing. These nautical handbooks may have been superseded by more specialized works, such as treatises on astronomy and mathematics, geography and periploi, and naval tactics, which may have been regarded as being of greater use than an interdisciplinary book on sailing. From a purely aesthetic perspective, many readers will probably not feel inclined to bemoan the loss of all ancient handbooks on navigation, as they will have looked similar to the periploi, containing many imperatives, short main clauses in hypotaxis, and many numerals.     

Maimon's criticism of Kant's doctrine of mathematical cognition and the possibility of metaphysics as a science

The aim of this paper is to discuss Maimon's criticism of Kant's doctrine of mathematical cognition. In particular, we will focus on the consequences of this criticism for the problem of the possibility of metaphysics as a science. Maimon criticizes Kant's explanation of the synthetic a priori character of mathematics and develops a philosophical interpretation of differential calculus according to which mathematics and metaphysics become deeply interwoven. Maimon establishes a parallelism between two relationships: on the one hand, the mathematical relationship between the integral and the differential and on the other, the metaphysical relationship between the sensible and the supersensible. Such a parallelism will be the clue to the Maimonian solution to the Kantian problem of the possibility of metaphysics as a science.     

The Mathematics of the Area Law: Kepler's Successful Proof in Epitome Astronomiae Copernicanae (1621)

Epitome V (1621), and consisted of matching an element of area to an element of time, where each was mathematically determined. His treatment of the area depended solely on the geometry of Euclid's Elements, involving only straight-line and circle propositions – so we have to account for his deliberate avoidance of the sophisticated conic-geometry associated with Apollonius. We show also how his proof could have been made watertight according to modern standards, using methods that lay entirely within his power. The greatest innovation, however, occurred in Kepler's fresh formulation of the measure of time. We trace this concept in relation to early astronomy and conclude that Kepler's treatment unexpectedly entailed the assumption that time varied nonuniformly; meanwhile, a geometrical measure provided the independent variable. Even more surprisingly, this approach turns out to be entirely sound when assessed in present-day terms. Kepler himself attributed the cause of the motion of a single planet around the Sun to a set of `physical' suppositions which represented his religious as well as his Copernican convictions; and we have pared to a minimum – down to four – the number he actually required to achieve this. In the Appendix we use modern mathematics to emphasize the simplicity, both geometrical and kinematical, that objectively characterizes the Sun-focused ellipse as an orbit. Meanwhile we highlight the subjective simplicity of Kepler's own techniques (most of them extremely traditional, some newly created). These two approaches complement each other to account for his success. (Received April 19, 2002) Published online April 2, 2003 Communicated by N. M. Swerdlow     

Jean-Jacques Rousseau: Botanist

It is argued that Hugo de Vries's conversion to Mendelism did not agree with his previous theoretical framework. De Vries regarded the number of offspring expressing a certain character as a hereditary quality, intrinsic to the state of the pangene involved. His was a shortlived conversion since after the ‘rediscovery’ he failed to unify his older views with Mendelism. De Vries was never very much of a Mendelian. The usual stories of the Dutch ‘rediscovery’ need, therefore, a considerable reshaping.     

“The soul of the fact”—Poincaré and proof

Henri Poincaré acquired a reputation in his lifetime for being difficult to read. It was said that he missed out important steps in his arguments, assumed the truth of claims that would be difficult if not impossible to prove, and in short that he lacked rigour. In the years after his death this view coalesced into an exaggerated claim that his work was simply too vague, and has become a cliché. This paper argues that Poincaré was far from indifferent to rigour, and that what characterises his work is an attempt to convey a particular sense of what it is to understand a topic. Throughout his working life Poincaré was concerned to promote the understanding of many domains of mathematics and physics. This is as apparent in his views about geometry, his conventionalism, and his theory of knowledge, as it is in his work on electricity and optics, on number theory, and function theory. It is one of the ways Poincaré discharged his responsibilities as a scientist, and that it accounts not only for a surprising degree of unity in his work but also gives it its distinctive character—at once profound and elusive.     

Galileo and the problem of concentric circles

In 1892, Eliakim Hastings Moore accepted the task of building a mathematics department at the University of Chicago. Working in close conjuction with the other original department members, Oskar Bolza and Heinrich Maschke, Moore established a stimulating mathematical environment not only at the University of Chicago, but also in the Midwest region and in the United States in general. In 1893, he helped organize an international congress of mathematicians. He followed this in 1896 with the organization of the Midwest Section of the New York City-based American Mathematical Society. He became the first editor-in-chief of the Society's Transactions in 1899, and rose to the presidency of the Society in 1901.     

The rhetorical strategy of William Paley’s Natural theology (1802): Part 1, William Paley’s Natural theology in context

This article reconstructs the historical and philosophical contexts of William Paley’s Natural theology (1802). In the wake of the French Revolution, widely believed to be the embodiment of an atheistic political credo, the refutation of the transmutational biological theories of Buffon and Erasmus Darwin was naturally high on Paley’s agenda. But he was also responding to challenges arising from his own moral philosophy, principally the psychological quandary of how men were to be kept in mind of the Creator. It is argued here that Natural theology was the culmination of a complex rhetorical scheme for instilling religious impressions that would increase both the virtue and happiness of mankind. Philosophy formed an integral part of this strategy, but it did not comprise the whole of it. Equally vital were those purely rhetorical aspects of the discourse which, according to Paley, were more concerned with creating ‘impression’. This facet of his writing is explored in part one of this two-part article. Turning to the argumentative side of the scheme, part two examines Paley’s responses to David Hume and Erasmus Darwin in the light of the wider strategy of inculcation at work throughout all his writings.     

The power of images: mathematics and metaphysics in Hobbes's optics

This paper deals with Hobbes's theory of optical images, developed in his optical magnum opus, ‘A Minute or First Draught of the Optiques’ (1646), and published in abridged version in De homine (1658). 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.     

The rhetorical strategy of William Paley’s Natural theology (1802): Part 2, William Paley’s Natural theology and the challenge of atheism

The first part of this two-part article suggested that William Paley’s Natural theology (1802) should be viewed as the culmination of a complex psychological strategy for inculcating religious and moral sentiments. Having focused in Part 1 on Paley’s rhetoric, we now turn our attention to the philosophical part of the programme. This article attempts to settle the vexed question of how far Paley responded to the devastating critique of the teleological argument contained in Hume’s posthumously published Dialogues concerning natural religion (1779). It also identifies tensions that arose in Natural theology between the rhetorical and intellectual sides of the stratagem. In response to Erasmus Darwin’s evolutionary theories, Paley asserted that the divinely designed architecture of nature had remained unchanged since the creation. But the more he emphasized the preordained nature of providence, its effectuation through mechanical dispositions, the less room there appeared to be for particular interventions. Section 2 concentrates on Paley’s efforts to reconcile this model of a law-governed, mechanical universe, with the belief in a personal God who was active in worldly affairs. It therefore challenges the view, long unquestioned in the historical literature, that Paley’s Deity was merely a watchmaker, who had remained idle since the Creation.     

Socially conditioned mathematical change: the case of the French Revolution

This paper examines a historical case of conceptual change in mathematics that was fundamental to its progress. I argue that in this particular case, the change was conditioned primarily by social processes, and these are reflected in the intellectual development of the discipline. Reorganization of mathematicians and the formation of a new mathematical community were the causes of changes in intellectual content, rather than being mere effects. The paper focuses on the French Revolution, which gave rise to revolutionary developments in mathematics. I examine how changes in the political constellation affected mathematicians both individually and collectively, and how a new professional community—with different views on the objects, problems, aims, and values of the discipline—arose. On the basis of this account, I will discuss such Kuhnian themes as the role of the professional community and normal versus revolutionary development.     

Superposition: on Cavalieri’s practice of mathematics

Bonaventura Cavalieri has been the subject of numerous scholarly publications. Recent students of Cavalieri have placed his geometry of indivisibles in the context of early modern mathematics, emphasizing the role of new geometrical objects, such as, for example, linear and plane indivisibles. In this paper, I will complement this recent trend by focusing on how Cavalieri manipulates geometrical objects. In particular, I will investigate one fundamental activity, namely, superposition of geometrical objects. In Cavalieri’s practice, superposition is a means of both manipulating geometrical objects and drawing inferences. Finally, I will suggest that an integrated approach, namely, one which strives to understand both objects and activities, can illuminate the history of mathematics.