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)

Given the sharp distinction that follows from Hume’s Fork, the proper epistemic status of propositions of mixed mathematics seems to be a mystery. On the one hand, mathematical propositions concern the relation of ideas. They are intuitive and demonstratively certain. On the other hand, propositions of mixed mathematics, such as in Hume’s own example, the law of conservation of momentum, are also matter of fact propositions. They concern causal relations between species of objects, and, in this sense, they are not (...) intuitive or demonstratively certain, but probable or provable. In this article, I argue that the epistemic status of propositions of mixed mathematics is that of matters of fact. I wish to show that their epistemic status is not a mystery. The reason for this is that the propositions of mixed mathematics are dependent on the Uniformity Principle, unlike the propositions of pure mathematics. (shrink)

This paper aims to reassess the philosophical puzzle of the “applicability of mathematics to physical sciences” as a misunderstood disciplinary interplay. If the border isolating mathematics from the empirical world is based on appropriate criteria, how does one explain the fruitfulness of its systematic crossings in recent centuries? An analysis of the evolution of the criteria used to separate mathematics from experimental sciences will shed some light on this question. In this respect, we will highlight the historical influence of three (...) major disciplinary paradigms. According to the Aristotelian classification of the sciences, the separation of mathematics from physics is based on their respective objects of study. The Baconian system distinguishes these sciences by the type of knowledge involved in each field. Finally, the Whewellian disciplinary layout categorises these disciplines by their respective methods. In this paper, we argue that the cascading effect of such disciplinary categorisations—based successively on ontological, epistemological, and heuristic criteria—resulted in a profound redefinition of the border between mathematics and physics, with the consequence of obscuring the foundations of the interplay between these fields. Our approach intends to put forward such a mechanism as a constitutive piece of the puzzle of the applicability of mathematics. (shrink)

Berichte zur Wissenschaftsgeschichte, EarlyView.

When we examine the history of astronomy up to the end of the seventeenth century by considering the relation between mathematical astronomy and natural philosophy, it has been argued that there were two groups of philosophers and astronomers: instrumentalists and realists. However, this classification is deficient when we consider attitudes toward the explanatory power of mathematics in determining astronomical theories. I offer the solution of dividing realists into two subcategories—mathematical realists and physical realists. Mathematical realists include those who thought mathematics (...) could provide the real motions of celestial bodies. The physical realists, who include Ibn Rushd (Averroës), believed mathematics could not provide us with the real structure of the heavens. To avoid contradictions between the physical and mathematical models, the physical realists held that a mathematical model of the heavens should be designed by considering physics. I argue, in this article, that Francis Bacon can be considered a physical realist. (shrink)

Robert Hooke’s theory of gravitation is a promising case study for probing the fruitfulness of Menachem Fisch’s insistence on the centrality of trading zone mediators for rational change in the history of science and mathematics. In 1679, Hooke proposed an innovative explanation of planetary motions to Newton’s attention. Until the correspondence with Hooke, Newton had embraced planetary models, whereby planets move around the Sun because of the action of an ether filling the interplanetary space. Hooke’s model, instead, consisted in the (...) idea that planets move in the void space under the influence of a gravitational attraction directed toward the sun. There is no doubt that the correspondence with Hooke allowed Newton to conceive a new explanation for planetary motions. This explanation was proposed by Hooke as a hypothesis that needed mathematical development and experimental confirmation. Hooke formulated his new model in a mathematical language which overlapped but not coincided with Newton’s who developed Hooke’s hypothetical model into the theory of universal gravitation as published in the Mathematical Principles of Natural Philosophy (1687). The nature of Hooke’s contributions to mathematized natural philosophy, however, was contested during his own lifetime and gave rise to negative evaluations until the last century. Hooke has been often contrasted to Newton as a practitioner rather than as a “scientist” and unfavorably compared to the eminent Lucasian Professor. Hooke’s correspondence with Newton seems to me an example of the phenomenon, discussed by Fisch in his philosophical works, of the invisibility in official historiography of “trading zone mediators,” namely, of those actors that play a role, crucial but not easily recognized, in promoting rational scientific framework change. (shrink)

: Marin Mersenne was central to the new mathematical approach to nature in Paris in the 1630s and 1640s. Intellectually, he was one of the most enthusiastic practitioners of that program, and published a number of influential books in those important decades. But Mersenne started his career in a rather different way. In the early 1620s, Mersenne was known in Paris primarily as a writer on religious topics, and a staunch defender of Aristotle against attacks by those who would replace (...) him by a new philosophy. In this essay, I would like to examine Mersenne's changing attitude toward Galileo. In the early 1620s, Mersenne lists Galileo among the innovators in natural philosophy whose views should be rejected. However, by the early 1630s, less than a decade later, Mersenne has become one of Galileo's most ardent supporters. How, then, did Mersenne learn to love Galileo? (shrink)

Important to Kuhn's account of scientific change is the observation that when paradigms are in competition with one another, there is a curious breakdown of rational argument and communication between adherents of competing programs. He attributed this to the fact that competing paradigms are incommensurable. The incommensurability thesis centrally involves the claim that there is a deep conceptual gap between competing paradigms in science. In this paper I argue that in one important case of competing paradigms, the Aristotelian explanation of (...) the properties of bodies in terms of matter and form as opposed to the Cartesian mechanist paradigm, where the properties of bodies are explained on the model of machines, there was no such conceptual gap: the notion of a machine was as fully intelligible on the Aristotelian paradigm as it was on the Cartesian. But this does not mean that the debate between the two sides was conducted on purely rational terms. Rational argument breaks down not because of Kuhnian incommensurability, I argue, but because of other cultural factors separating the two camps. (shrink)

The subject of this essay-based dissertation is Hume’s natural philosophy. The dissertation consists of four separate essays and an introduction. These essays do not only treat Hume’s views on the topic of natural philosophy, but his views are placed into a broader context of history of philosophy and science, physics in particular. The introductory section outlines the historical context, shows how the individual essays are connected, expounds what kind of research methodology has been used, and encapsulates the research contributions of (...) the essays. The first essay treats Newton’s experimentalist methodology in gravity research and its relation to Hume’s causal philosophy. It is argued that Hume does not see the relation of cause and effect as being founded on a priori reasoning, similar to the way in which Newton criticized non-empirical hypotheses about the causal properties of gravity. Contrary to Hume’s rules of causation, the universal law does not include a reference either to contiguity or succession, but Hume accepts it in interpreting the force and the law of gravity instrumentally. The second article considers Newtonian and non-Newtonian elements in Hume more broadly. He is sympathetic to many prominently Newtonian themes in natural philosophy, such as experimentalism, critique of hypotheses, inductive proof, and the critique of Leibnizian principles of sufficient reason and intelligibility. However, Hume is not a Newtonian philosopher in many respects: his conceptions regarding space and time, the vacuum, the specifics of causation, the status of mechanism, and the reality of forces differ markedly from Newton’s related conceptions. The third article focuses on Hume’s Fork and the proper epistemic status of propositions of mixed mathematics. It is shown that the epistemic status of propositions of mixed mathematics, such as those concerning laws of nature, is that of matters of fact. The reason for this is that the propositions of mixed mathematics are dependent on the Uniformity Principle. The fourth article analyzes Einstein’s acknowledgement of Hume regarding special relativity. The views of the scientist and the philosopher are juxtaposed, and it is argued that there are two common points to be found in their writings, namely an empiricist theory of ideas and concepts and a relationist ontology regarding space and time. (shrink)