Jean De Groot

Modern commentators have doubts about the authenticity and cogency of the early propositions of Archimedes’ On Equilibrium of Planes Book 1. Ernst Mach famously said that the proof of Prop. 6, the so-called law of the lever, assumes what is to be proven. Comparing the initial text in Heiberg’s modern edition (1881, 1913) to the first propositions in Eutocius’ commentary on EP 1, J. L. Berggren ([1976]. ‘Spurious Theorems in Archimedes’ Equilibrium of Planes: Book I’, Archive for History of Exact Sciences 16.2 (1976), 87–103.) claimed that the propositions up through Proposition 3 of the standard modern edition are schoolbook additions written by an ancient author inferior to Archimedes. The present paper argues for the logical connectedness of Postulates 1–5 to Props. 1–6, by means of a detailed examination of the course of the argument and a re-examination of Eutocius’ remarks. The paper reinterprets the role of the empirical in the early propositions and offers a reading of the contribution of Archimedes’ mechanics to the method of EP 1.

QUINE'S DOCTRINE of the indeterminacy of translation is made possible by the principle of substitution characteristic of extensional logic. The same characteristic makes it impossible, in philosophy of science, to choose among theoretical models no one of which is obviously best suited to explain the facts. Hilary Putnam achieved a sort of closure to the problem of reference in philosophy of science, when he pointed out the implications of the Skolem-Löwenheim theorem. He said that besides the facts a theory is designed to explain, there are any number of unintended interpretations of the same theory that assign the correct truth values to all sentences of the theory. There can be no sound basis for deciding upon the sole interpretation for a theory. Following Putnam, some philosophers have abandoned the problem of reference in the context of axiomatic systems. Ian Hacking, for instance, prefers a notion of reference "that is not tied by any specific, binding theory about what is referred to." He has turned to the study of approximations and practical models, spurred on by the difficulty of reconciling axiomatic systems to scientific practice.

QUINE'S DOCTRINE of the indeterminacy of translation is made possible by the principle of substitution characteristic of extensional logic. The same characteristic makes it impossible, in philosophy of science, to choose among theoretical models no one of which is obviously best suited to explain the facts. Hilary Putnam achieved a sort of closure to the problem of reference in philosophy of science, when he pointed out the implications of the Skolem-Löwenheim theorem. He said that besides the facts a theory is designed to explain, there are any number of unintended interpretations of the same theory that assign the correct truth values to all sentences of the theory. There can be no sound basis for deciding upon the sole interpretation for a theory. Following Putnam, some philosophers have abandoned the problem of reference in the context of axiomatic systems. Ian Hacking, for instance, prefers a notion of reference "that is not tied by any specific, binding theory about what is referred to." He has turned to the study of approximations and practical models, spurred on by the difficulty of reconciling axiomatic systems to scientific practice.

This paper examines geometrical arguments from Galileo's Mechanics and Two New Sciences to discern the influence of the Aristotelian Mechanical Problems on Galileo's dynamics. A common scientific procedure is found in the Aristotelian author's treatment of the balance and lever and in Galileo's rules concerning motion along inclined planes. This scientific procedure is understood as a development of Eudoxan proportional reasoning, as it was used in Eudoxan astronomy rather than simply as it appears in Euclid's Elements. Topics treated include the significance of the circle in Galileo's demonstrations, the substitution of rectilinear elements for heterogeneous factors like weight and curvilinear distance, and the way in which elements of a motion are used to measure other elements of the same motion. The indirectness of Galileo's proofs, his conception of speed as relative and comparative, and the meaning of his concept of moment all come into clearer focus. Conclusions are drawn about Galilean idealization, and also about the contrast of literal versus figural modes of explanation in Galileo's science.

Examples are presented of Aristotle’s use of non-idealized mathematics. Distinctions Husserl makes in Crisis help to delineate the features of this empiricalmathematics, which include the non-persistence of mathematical aspects of things and the selective application of mathematical traits and proper accidents. In antiquity, non-abstracted mathematics was involved with practical sciences that treat motion. The suggestion is made that these sciences were incorporated by Aristotle into natural philosophy without first being abstracted as pure mathematics—a state of affairs not envisioned by Husserl, for whom science recast natural ontology by means of the idealization of pure mathematics. The relation of empirical mathematics to life-world ontology is considered

Examples are presented of Aristotle’s use of non-idealized mathematics. Distinctions Husserl makes in Crisis help to delineate the features of this empiricalmathematics, which include the non-persistence of mathematical aspects of things and the selective application of mathematical traits and proper accidents. In antiquity, non-abstracted mathematics was involved with practical sciences that treat motion. The suggestion is made that these sciences were incorporated by Aristotle into natural philosophy without first being abstracted as pure mathematics—a state of affairs not envisioned by Husserl, for whom science recast natural ontology by means of the idealization of pure mathematics. The relation of empirical mathematics to life-world ontology is considered.

Originally published in 1991. Philoponus’ long commentary on Aristotle’s definition of light sets up the major concerns, both in optics and theory of light, that is discussed here. Light was of special interest in Neoplatonism because of its being something incorporeal in the world of natural bodies and therefore had a special role in the philosophical analysis of the interpenetration of bodies and also as a paradigm for the soul-body problem. The material investigated in this book contains much about the physiology of vision as well as the propagation of light. Several chapters investigate the philosophical theory and its origins in ‘multiplication of species.’ These issues in the history of science philosophy are looked at further and an analysis is offered of the development of the distinction between Aristotle’s kinesis and energeia. The book covers a substantial amount of the philosophy of mathematical science from the point of view of Philoponus, drawing on more of his works relating to three dimensionality

This paper examines geometrical arguments from Galileo's Mechanics and Two New Sciences to discern the influence of the Aristotelian Mechanical Problems on Galileo's dynamics. A common scientific procedure is found in the Aristotelian author's treatment of the balance and lever and in Galileo's rules concerning motion along inclined planes. This scientific procedure is understood as a development of Eudoxan proportional reasoning, as it was used in Eudoxan astronomy rather than simply as it appears in Euclid's Elements. Topics treated include the significance of the circle in Galileo's demonstrations, the substitution of rectilinear elements for heterogeneous factors like weight and curvilinear distance, and the way in which elements of a motion are used to measure other elements of the same motion. The indirectness of Galileo's proofs, his conception of speed as relative and comparative, and the meaning of his concept of moment all come into clearer focus. Conclusions are drawn about Galilean idealization, and also about the contrast of literal versus figural modes of explanation in Galileo's science

This paper traces the significance of first principles in Greek philosophy to cognitive developments in colonial Greek Italy in the late fifth century BC. Conviction concerning principles comes from the power to make something true by action. Pairing and opposition, the forerunners of metonymy, are shown to structure disparate cultural phenomena—the making of figured numbers, the sundial, and the production, with the aid of device, of fear or panic in the spectators of Greek tragedy. From these starting points, the function of the gnômôn in knowledge is explored.

It is shown that Aristotle’s references to automata in his biological treatises are meant to invoke the principle behind the ancient conception of the lever, i.e. that points on the rotating radius of a circle all move at different speeds proportional to their distances from the center. This principle is mathematical and explains a phenomenon taken as whole. Automata do not signify for him primarily a succession of material movers in contact, the modern model for mechanism. For animal locomotion and embryological development, Aristotle models his dunamis concept on the idea of mechanical potential that the lever principle displays.

In _Aristotle’s Empiricism_, Jean De Groot argues that an important part of Aristotle’s natural philosophy has remained largely unexplored and shows that much of Aristotle’s analysis of natural movement is influenced by the logic and concepts of mathematical mechanics that emerged from late Pythagorean thought. De Groot draws upon the pseudo-Aristotelian_ Physical Problems_ XVI to reconstruct the context of mechanics in Aristotle’s time and to trace the development of kinematic thinking from Archytas to the Aristotelian _Mechanics_. She shows the influence of kinematic thinking on Aristotle’s concept of power or potentiality, which she sees as having a physicalistic meaning originating in the problem of movement. De Groot identifies the source of early mechanical knowledge in kinesthetic awareness of mechanical advantage, showing the relation of Aristotle’s empiricism to more ancient experience. The book sheds light on the classical Greek understanding of imitation and device, as it questions both the claim that Aristotle’s natural philosophy codifies opinions held by convention and the view that the cogency of his scientific ideas depends on metaphysics