Newton and Leibniz had profound disagreements concerning metaphysics and the relationship of mathematics to natural philosophy, as well as deeply opposed attitudes towards analysis. Nevertheless, or so I shall argue, despite these deeply held and distracting differences in their background assumptions and metaphysical views, there was a considerable consilience in their positions on the status of infinitesimals. In this paper I compare the foundation Newton provides in his Method Of First and Ultimate Ratios (sketched at some time between 1671 and (...) 1684, and published in the Principia of 1687) with that provided independently by Leibniz in his unpublished manuscript De quadratura arithmetica (1675-6) as well as in later writings. I argue that both appeal to a version of the Archimedean Axiom to underwrite their use of infinitesimal techniques, which must be interpreted as a shorthand for rigorously finitist methods. (shrink)

In this paper I challenge the usual interpretations of Newton's and Leibniz's views on the nature of space and the relativity of motion. Newton's ‘relative space’ is not a reference frame; and Leibniz did not regard space as defined with respect to actual enduring bodies. Newton did not subscribe to the relativity of intertial motions; whereas Leibniz believed no body to be at rest, and Newton's absolute motion to be a useful fiction. A more accurate rendering of the opposition between (...) them, I argue, leads to a wholly different understanding of Leibniz's theory of space, one which is not susceptible to the objections Newton had raised against Descartes regarding the representation of motion. This in turn suggests a new approach for contemporary theory of space, one which neither hypostatizes space nor tries to reduce it to relations among actual things. * This work was generously supported by the National Endowment for the Humanities, with a Fellowship for College Teachers and Independent Scholars (FB-26897-89), and also by a sabbatical leave from my institution, Middlebury College. Iam very grateful to various members of faculty of York University for their appreciative reception of an earlier one-week-old version of this paper. ‘Relative Space in Newton and Leibniz’, read to the Department of Philosophy there in January 1990, and to Robert Rynasiewicz for criticisms of an extract read at the 1991 History of Science meeting. (shrink)

The insights described by Nevin & Grace may be summarized without reference to the Newtonian concepts they suggest. The metaphor to Newtonian mechanics seems dubious in three ways: (1) extensions seem to lead to paradoxes; (2) many well-known phenomena are ignored; (3) the Newtonian concepts seem difficult to reconcile with the larger framework of evolutionary theory.

In this paper we argue that the primary issue in Descartes’ Principles of Philosophy, Part II, articles 1-40, is the problem of individuating bodies. We demonstrate that Descartes departs from the traditional quest for a principle of individuation, moving to a different strategy with the more modest aim of constructing bodies adequate to the needs of his cosmology. In doing this he meets with a series of difficulties, and this is precisely the challenge that Newton took up. We show that (...) Descartes’ questions and his strategy influenced not only Newton’s account of physical bodies, but also the structure of his mechanics. (shrink)

Presenting the history of space-time physics, from Newton to Einstein, as a philosophical development DiSalle reflects our increasing understanding of the connections between ideas of space and time and our physical knowledge. He suggests that philosophy's greatest impact on physics has come about, less by the influence of philosophical hypotheses, than by the philosophical analysis of concepts of space, time, and motion and the roles they play in our assumptions about physical objects and physical measurements. This way of thinking leads (...) to new interpretations of the work of Newton and Einstein and the connections between them. It also offers new ways of looking at old questions about a priori knowledge, the physical interpretation of mathematics, and the nature of conceptual change. Understanding Space-Time will interest readers in philosophy, history and philosophy of science, and physics, as well as readers interested in the relations between physics and philosophy. (shrink)

Here, I shall argue that Van Helmont needs to be added to the list of sources on which Newton drew when formulating his doctrine of absolute time. This by no means implies that Van Helmont is the factual source of Newton's views on absolute time (I have found no clear-cut evidence in support of this claim). It is by no means my aim to debunk the importance of the other sources, but rather to broaden them. Different authors help to explain (...) different aspects of Newton's conception of absolute time. (shrink)

The Reeh-Schlieder theorem asserts the vacuum and certain other states to be spacelike superentangled relative to local fields. This motivates an inquiry into the physical status of various concepts of localization. It is argued that a covariant generalization of Newton-Wigner localization is a physically illuminating concept. When analyzed in terms of nonlocally covariant quantum fields, creating and annihilating quanta in Newton-Wigner localized states, the vacuum is seen to not possess the spacelike superentanglement that the Reeh-Schlieder theorem displays relative to local (...) fields, and to be locally empty as well as globally empty. Newton-Wigner localization is then shown to be physically interpretable in terms of a covariant generalization of the center of energy, the two localizations being identical if the system has no internal angular momentum. Finally, some of the counterintuitive features of Newton-Wigner localization are shown to have close analogues in classical special relativity. (shrink)

The traditional absolutist-relationist debate is still clearly formulable in the context of General Relativity Theory (GTR), despite the important differences between Einstein's theory and the earlier context of Newtonian physics. This paper answers recent arguments by Robert Rynasiewicz against the significance of the debate in the GTR context. In his (1996) (‘Absolute vs. Relational Spacetime: An Outmoded Debate?’), Rynasiewicz argues that already in the late nineteenth century, and even more so in the context of General Relativity theory, the terms of (...) the original Descartes–Newton–Leibniz dispute about space are not to be found. Nineteenth-century ether theories of electromagnetism, and the metric field of GTR, he claims, do not lend themselves to being interpreted clearly as either absolute space à la Newton, or relational structures à la either Descartes or Leibniz. I argue that, while in some imaginable theories Rynasiewicz's claim that the classical debate dissolves would be correct, in fact in the most important historical theories he discusses, this is not the case. In particular, I argue that in both Lorentz's ether theory and General relativity theory, there is a clear and compelling way to establish connections to the classical absolutist-relationist disputes, and that in both these theories it is the absolutist position that is prima facie victorious. To support my arguments and give a clear overview of the whole debate, I end by offering definitional sketches of relationism and absolutism (substantivalism) about spacetime in the context of contemporary physics. The sketches show the clear connections between these views today and their ancestors in Newton and Leibniz. But at the same time, they indicate how both views are not just claims about existing physical theories, but rather also bets about how future physics will clarify the ontological picture. (shrink)

Where should we begin our story? Many books start with Newton, but Newton was responding to both Galileo1 and especially (for our purposes) Descartes. But Galileo and Descartes themselves were writing in the context of late Aristotelianism, and so were trained in and critical of that rich school of thought, so if we want to fully understand their work we would need to understand scholastic views on space and motion (see Grant, 1974, Murdoch and Sylla, 1978 and Ariew and Gabbey, (...) 1998). But late scholasticism itself is the result of a long history tracing from Plato and Aristotle through Jewish, Arabic, Islamic and European thought. And of course Plato and Aristotle are explicitly reacting to their predecessors and contemporaries. In other words, we could start the story as early in recorded thought as we like. (shrink)

Since antiquity, natural philosophers have struggled to comprehend the nature of three tightly interconnected concepts: space, time, and motion. A proper understanding of motion, in particular, has been seen to be crucial for deciding questions about the natures of space and time, and their interconnections. Since the time of Newton and Leibniz, philosophers’ struggles to comprehend these concepts have often appeared to take the form of a dispute between absolute conceptions of space, time and motion, and relational conceptions. This article (...) guides the reader through some of the history of these philosophical struggles. Rather than taking sides in the (alleged) ongoing debates, or reproducing the standard dialectic recounted in most introductory texts, we have chosen to scrutinize carefully the history of the thinking of the canonical participants in these debates — principally Descartes, Newton, Leibniz, Mach and Einstein. Readers interested in following up either the historical questions or current debates about the natures of space, time and motion will find ample links and references scattered through the discussion and in the Other Internet Resources section below. (shrink)

Newton's arguments for the immobility of the parts of absolute space have been claimed to licence several proposals concerning his metaphysics. This paper clarifies Newton, first distinguishing two distinct arguments. Then, it demonstrates, contrary to Nerlich ([2005]), that Newton does not appeal to the identity of indiscernibles, but rather to a view about de re representation. Additionally, DiSalle ([1994]) claims that one argument shows Newton to be an anti-substantivalist. I agree that its premises imply a denial of a kind of (...) substantivalism, but I show that they are inconsistent with Newton's core doctrine that not all motion is the relative motions of bodies, and so conclude that they are not part of his considered views on space. The Arguments The Identity Argument 2.1 Identity of indiscernibles for individuals 2.2 Identity of indiscernibles for worlds and states 2.3 Representation de re Kinematic Relationism Conclusion CiteULike    Connotea    Del.icio.us    What's this? (shrink)

The two books discussed here make important contributions to our understanding of the role of spacetime concepts in physical theories and how that understanding has changed during the evolution of physics. Both emphasize what can be called a ‘dynamical’ account, according to which geometric structures should be understood in terms of their roles in the laws governing matter and force. I explore how the books contribute to such a project; while generally sympathetic, I offer criticisms of some historical claims concerning (...) Newton, and argue that the dynamical account does not undercut ontological issues as the books claim. *Received January 2009; revised March 2009. †To contact the author, please write to: Department of Philosophy, 1423 University Hall MC 267, University of Illinois at Chicago, 601 S. Morgan Street, Chicago, IL 60607; e‐mail: huggett@uic.edu. (shrink)

Newton's philosophical views are unique and uniquely difficult to categorise. In the course of a long career from the early 1670s until his death in 1727, he articulated profound responses to Cartesian natural philosophy and to the prevailing mechanical philosophy of his day. Newton as Philosopher presents Newton as an original and sophisticated contributor to natural philosophy, one who engaged with the principal ideas of his most important predecessor, Rene; Descartes, and of his most influential critic, G. W. Leibniz. Unlike (...) Descartes and Leibniz, Newton was systematic and philosophical without presenting a philosophical system, but over the course of his life, he developed a novel picture of nature, our place within it, and its relation to the creator. This rich treatment of his philosophical ideas, the first in English for thirty years, will be of wide interest to historians of philosophy, science, and ideas. (shrink)

: Newton's critics argued that his treatment of gravity in the Principia saddles him with a substantial dilemma. If he insists that gravity is a real force, he must invoke action at a distance because of his explicit failure to characterize the mechanism underlying gravity. To avoid distant action, however, he must admit that gravity is not a real force, and that he has therefore failed to discover the actual cause of the phenomena associated with it. A reinterpretation of Newton's (...) distinction between the "mathematical" and the "physical" treatment of force indicates how he can reject each horn of this dilemma. (shrink)

A striking feature of Newton’s thought is the very broad reach of his empiricism, potentially extending even to immaterial substances, including God, minds, and should one exist, a non-perceiving immaterial medium. Yet Newton is also drawn to certain metaphysical principles—most notably the principle that matter cannot act where it is not—and this second, rationalist feature of his thought is most pronounced in his struggle to discover ‘gravity’s cause’. The causal problem remains vexing, for he neither invokes primary causation, nor accepts (...) action at a distance by locating active powers in matter. To the extent that he is drawn to metaphysical principles, then, the causal problem is that of discovering some non-perceiving immaterial medium. Yet Newton’s thought has a third striking feature, one with roots in the other two: he allows that substances of different kinds might simultaneously occupy the very same region of space. I elicit the implications of these three features. For Newton to insist upon all three would transform the causal question about gravity into an insoluble problem about apportioning active powers. More seriously, it would undermine his means of individuating substances, provoking what I call ‘Newton’s Substance Counting Problem’. (shrink)

The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmatic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with (...) the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics. (shrink)

: This article is concerned with Newton's appropriation of Descartes' ontology of true and immutable natures in developing his theory of infinitely extended space. It contends that unless the part played by the Platonic distinction between "being a nature" and "having a nature" in Newton's thinking is properly appreciated the foundation of his doctrine of space in relation to God will not be fully understood. It also contends that Newton's Platonism is consistent with his empiricism once the mediating role is (...) made clear that the geometry of moving loci play in grounding his intuitions concerning infinite natures. (shrink)

Paragraph 6 of Newtons Scholium argues that the parts of space cannot move. A premise of the argument – that parts have individuality only through an order of position – has drawn distinguished modern support yet little agreement among interpretations of the paragraph. I argue that the paragraph offers an a priori, metaphysical argument for absolute motion, an argument which is invalid. That order of position is powerless to distinguish one part of Euclidean space from any other has gone virtually (...) unremarked. It remains uncertain what the import of the paragraph is but it is not close to apparently similar arguments of Leibniz. (shrink)

The implications for the substantivalist–relationalist controversy of Barbour and Bertotti's successful implementation of a Machian approach to dynamics are investigated. It is argued that in the context of Newtonian mechanics, the Machian framework provides a genuinely relational interpretation of dynamics and that it is more explanatory than the conventional, substantival interpretation. In a companion paper (Pooley [2002a]), the viability of the Machian framework as an interpretation of relativistic physics is explored. 1 Introduction 2 Newton versus Leibniz 3 Absolute space versus (...) an affine connection 4 Anti-relationalist arguments 5 Rehabilitating relationalism 6 Dynamics on the relative configuration space 7 Intrinsic particle dynamics 8 Conclusion. (shrink)

In this paper I interpret Newton’s speculative treatment of gravity as a relational, accidental property of matter that arises through what Newton calls “the shared action” of two bodies of matter. In doing so, I expand and extend on a hint by Howard Stein. However, in developing the details of my interpretation I end up disagreeing with Stein’s claim that for Newton a single body can generate a gravity/force field. I argue that when Newton drafted the first edition of the (...) Principia in the mid 1680s, he thought that (at least a part of) the cause of gravity is the disposition inherent in any individual body, but that the force of gravity is the actualization of that disposition; a necessary condition for the actualization of the disposition is the actual obtaining of a relation between two bodies having the disposition. The cause of gravity is not essential to matter because God could have created matter without that disposition. Nevertheless, at least a part of the cause of gravity inheres in individual bodies and were there one body in the universe it would inhere in that body. On the other hand, the force of gravity is neither essential to matter nor inherent in matter, because (to repeat) it is the actualization of a shared disposition. A lone part-less particle would, thus, not generate a gravity field. Seeing this allows us to helpfully distinguish among a) accepting gravity as causally real; b) the cause(s) (e.g. the qualities of matter) of the properties of gravity; c) making claims about the mechanism or medium by which gravity is transmitted. This will help clarify what Newton could have meant when he insisted that gravity is a real force. I present my argument in opposition to Andrew Janiak’s influential and fine 2007 paper. Along the way, I call attention to my disagreement with Janiak on a number of secondary issues (e.g. Janiak’s attribution to Newton of a distinction between ‘local’ and ‘distant’ action; Janiak’s reading of the “Letter to Bentley,” etc). (shrink)

This presentation will investigate the parts of space, and its relationship with metrical structure, in Newton’s natural philosophy. The historical background to Newton’s claims will form an important part of the investigation, in addition to an assessment of the recent articles by Nerlich, Huggett, Maudlin, DiSalle, Torretti, McGuire, and several others, on this subject. While various aspects of these previous contributions will prove informative, it will be argued that the underlying goals of Newton’s pronouncements on the parts of space, including (...) their ontological implications for absolute and/or substantival space, have largely eluded prior analysis. (shrink)

This essay will investigate the interrelationship between the parts of space and topological and metrical structure in Newton’s natural philosophy, as well as its influence in contemporary spacetime debates. The historical background to Newton’s claims will form an important part of the investigation, in addition to an assessment of the recent articles by Nerlich, Huggett, Torretti, DiSalle, and several others, on this subject. While various aspects of these previous contributions will prove informative, it will be argued that the underlying goals (...) of Newton’s pronouncements on the relationship among the parts of space, including their methodological and ontological implications for absolute space and contemporary spacetime theories, have largely eluded prior analysis. (shrink)

This paper explores the possibility of constructing a Cartesian space-time that can resolve the dilemma posed by a famous argument from Newton's early essay, De gravitatione. In particular, Huygens' concept of a center-of-mass reference frame is utilized in an attempt to reconcile Descartes' relationalist theory of space and motion with both the Cartesian analysis of bodily impact and conservation law for quantity of motion. After presenting a modern formulation of a Cartesian space-time employing Huygens' frames, a series of Newtonian counter-replies (...) are developed in order to estimate the viability of this relationalist project. (shrink)