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)

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)

: Although Galileo's struggle to mathematize the study of nature is well known and oft discussed, less discussed is the form this struggle takes in relation to Galileo's first new science, the science of the second day of the Discorsi. This essay argues that Galileo's first science ought to be understood as the science of matter—not, as it is usually understood, the science of the strength of materials. This understanding sheds light on the convoluted structure of the Discorsi's first day. (...) It suggests that the day's meandering discussions of the continuum, infinity, the vacuum, and condensation and rarefaction establish that a formal treatment of the "eternal and necessary" properties of matter is possible; i.e., that matter as such can be considered mathematically. This would have been a necessary, and indeed revolutionary, preliminary to the mathematical science of the second day because matter itself was thought in the Aristotelian tradition to be responsible for the departure of natural bodies from the unchanging and thus mathematizable character of abstract objects. In addition, the first day establishes that when considered physically, these properties account for matter's force of cohesion and resistance to fracture. This essay closes by showing that this dual style of reasoning accords with the conceptual structure of mixed 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)

Historians of fourteenth-century science have long recognized the extraordinary work at both Oxford and Paris in which natural philosophy was becoming highly mathematical. The movement to subject natural philosophy to a mathematical analysis and to quantify such qualities as heat, color, and of course speed surely stands as one of the most significant aspects of late medieval science. Yet as Edith Sylla has observed, because qualities and quantities pertain to different categories in Aristotelian theory, one might expect Aristotelian theorists to (...) avoid quantifying qualities. Even more serious still, the very task of quantifying physical qualities exposes a tension in the nature of science that was discussed first by Aristotle in his Posterior Analytics. (shrink)

The ways in which the Aristotelian sciences are related to each other has been discussed in the literature, with some focus on the subalternate sciences. While it is acknowledged that Aristotle, and Plato as well, was concerned as well with how the arts were related to one another, less attention has been paid to Aristotle's views on relationships among the arts. In this paper, I argue that Aristotle's account of the subalternate sciences helps shed light on how Aristotle saw the (...) art of rhetoric relating to dialectic and politics. Initial motivation for comparing rhetoric with the subalternate sciences is Aristotle's use of the language of boundary transgression, germane to the Posterior Analytics, when discussing rhetoric’s boundaries, as well as the language of "over" and "under" found in APo. First, I discuss three passages in Rhetoric Book I and argue that Garver's (1988) account cannot be correct. Second, I look to the subalternate sciences, especially focusing on optics and the distinction between "unqualified" optics and mathematical optics. Third, I discuss rhetoric's dependence on both dialectic and politics. (shrink)

Historians have long sought putative connections between different areas of Newton’s scientific work, while recently scholars have argued that there were causal links between even more disparate fields of his intellectual activity. In this paper I take an opposite approach, and attempt to account for certain tensions in Newton’s ‘scientific’ work by examining his great sensitivity to the disciplinary divisions that both conditioned and facilitated his early investigations in science and mathematics. These momentous undertakings, exemplified by research that he wrote (...) up in two separate notebooks, obey strict distinctions between approaches appropriate to both new and old ‘natural philosophy’ and those appropriate to the mixed mathematical sciences. He retained a fairly rigid demarcation between them until the early eighteenth century. At the same time as Newton presented the ‘mathematical principles’ of natural philosophy in his magnum opus of 1687, he remained equally committed to a separate and more private world or ontology that he publicly denigrated as hypothetical or conjectural. This is to say nothing of the worlds implicit in his work on mathematics and alchemy. He did not lurch from one overarching ontological commitment to the next but instead simultaneously—and often radically—developed generically distinct concepts and ontologies that were appropriate to specific settings and locations as well as to relevant styles of argument. Accordingly I argue that the concepts used by Newton throughout his career were intimately bound up with these appropriate generic or quasi-disciplinary ‘structures’. His later efforts to bring together active principles, aethers and voids in various works were not failures that resulted from his ‘confusion’ but were bold attempts to meld together concepts or ontologies that belonged to distinct enquiries. His analysis could not be ‘coherent’ because the structures in which they appeared were fundamentally incompatible.Author Keywords: Connectionism; Incoherence; Appropriateness; Setting; Discipline; Genre; Structure. (shrink)

The research that I have carried out on the sense of hearing in the Aristotelian ambit is based on a personal interest in the medical aspects that can be found in the treaties of the Stagirite. If, on the one hand, there has always been very deep attention by the scholars to the phenomenon of perception, and still there is, on the other hand, although not ignored, hearing remains perhaps somewhat neglected or, however, not sufficiently investigated so far, despite its (...) importance; but, above all, the effort to consider all its various implications together and the whole Corpus Aristotelicum has not been accomplished up to today. This is the intent that I have purposed and which I hope to have at least partially achieved. Of course, it was a constant need to refer also to the other senses and, in general, to Aristotle's theory of perception; I have, however, tried to point out every possible aspects, which could explain the peculiarities of hearing. Without claiming in any way to draw up a rank list of the senses, since each of them, according to the different points of view, seems more or less to aspire to a kind of primacy, I was able to demonstrate the extraordinary importance of hearing and, within certain limits, his preferability. After an introductory chapter on the position of hearing among the other senses, I have proposed a rather long one, in which I more specifically dealt with the anatomical, physiological and pathological aspects (and, finally, even some physical-mathematical considerations, that I think of great philosophical and scientific relevance); then, it has been given ample space (and could not be otherwise) to the complex issue of the relationship between hearing and language, which made it possible to highlight, thanks to the comparative approach of the Aristotelian investigations in the biological ambit, the differences between sound, voice and speech, and to show the key role of hearing in order to learning (also animal) and intellectual development (especially human). In a final articulated chapter I approached what I consider the further possible implications of hearing, always clinging to the Aristotelian treatment: in addition to the physical phenomena of echo, roar, resonance and the psychophysicist ones of acoustic illusions, I could not exempt myself from talking about music, also in "technical" terms (as my limited skills in the field have allowed me to), but especially because of its educational features examined in the paragraph Emotions, morality and education, where hearing is reported as undisputed protagonist in the ethical, aesthetic, rhetorical, political ambits; the chapter, and the dissertation itself, are closed by some remarks about the relations of hearing with memory and recollection and with dream: apart from the latter, where sight seems to dominate, with the first two the auditory role is on the contrary of primary importance, despite still today people are often inclined to mention visual memory more frequently. Even from these few notes I think I can state that it was really worthwhile to dedicate a research to the sense of hearing, especially in regard to a Corpus so broad, rich, varied and complex, such as the Aristotelian one. (shrink)

RésuméLe présent article propose la liste des œuvres médicales composées par Ibn Bājja, donne une présentation synthétique de celles qui nous ont été transmises et étudie le métacommentaire au commentaire de Galien sur lesAphorismesd'Hippocrate (Sharḥ fī al-Fuṣūl). Ce texte montre une influence profonde d'al- Fārābī, en particulier dans sa conception de l'expérience médicale, qui remonte à la façon dont ce dernier construit l'expérience (tajriba) comme le procédé inductif, décrit par Aristote dans lesSeconds Analytiques, produisant les prémisses de la démonstration. Sur (...) cette base, Ibn Bājja défend la pertinence d'une médecine moins scolastique et plus empirique, une position dont Ibn Rushd se fera l'écho. Des similitudes entre le texte d'Ibn Bājja et leK. al-Kulliyyāt fī al-ṭibbd'Ibn Rushd suggèrent que celui-ci avait lu leSharḥ fī al-Fuṣūl. Cette œuvre, en outre, pourrait bien attester que la dissection humaine était pratiquée à l'époque d'Ibn Bājja. (shrink)

Both Pierre Duhem and his successors emphasized that medieval scholastics created a science of mechanics by bringing both observation and mathematical techniques to bear on natural effects. Recent research into medieval and early modern science has suggested that Aristotle's subalternate sciences also were used in this program, although the degree to which the theory of subalternation had been modified is still not entirely clear. This paper focuses on the English tradition of subalternation between 1310 and 1350, and concludes with a (...) discussion of the theory advanced by Thomas Claxton early in the fifteenth century. (shrink)