The Axiom and Laws of Motion
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How Dualists Should (Not) Respond to the Objection from Energy
c 2019 Imprint Academic Mind & Matter Vol. 17(1), pp. 95–121 How Dualists Should (Not) Respond to the Objection from Energy Conservation Alin C. Cucu International Academy of Philosophy Mauren, Liechtenstein and J. Brian Pitts Faculty of Philosophy and Trinity College University of Cambridge, United Kingdom Abstract The principle of energy conservation is widely taken to be a se- rious difficulty for interactionist dualism (whether property or sub- stance). Interactionists often have therefore tried to make it satisfy energy conservation. This paper examines several such attempts, especially including E. J. Lowe’s varying constants proposal, show- ing how they all miss their goal due to lack of engagement with the physico-mathematical roots of energy conservation physics: the first Noether theorem (that symmetries imply conservation laws), its converse (that conservation laws imply symmetries), and the locality of continuum/field physics. Thus the “conditionality re- sponse”, which sees conservation as (bi)conditional upon symme- tries and simply accepts energy non-conservation as an aspect of interactionist dualism, is seen to be, perhaps surprisingly, the one most in accord with contemporary physics (apart from quantum mechanics) by not conflicting with mathematical theorems basic to physics. A decent objection to interactionism should be a posteri- ori, based on empirically studying the brain. 1. The Objection from Energy Conservation Formulated Among philosophers of mind and metaphysicians, it is widely believed that the principle of energy conservation poses a serious problem to inter- actionist dualism. Insofar as this objection afflicts interactionist dualisms, it applies to property dualism of that sort (i.e., not epiphenomenalist) as much as to substance dualism (Crane 2001, pp. -
Realizing Monads
anDrás kornai Realizing Monads ABSTrACT We reconstruct monads as cyclic finite state automata whose states are what Leibniz calls perceptions and whose transitions are automatically triggered by the next time tick he calls entelechy. This goes some way toward explaining key as- pects of the Monadology, in particular, the lack of inputs and outputs (§7) and the need for universal harmony (§59). 1. INTrODUCTION Monads are important both in category theory and in programming language de- sign, clearly demonstrating that Leibniz is greatly appreciated as the conceptual forerunner of much of abstract thought both in mathematics and in computer sci- ence. Yet we suspect that Leibniz, the engineer, designer of the Stepped reck- oner, would use the resources of the modern world to build a very different kind of monad. Throughout this paper, we confront Leibniz with our contemporary scientific understanding of the world. This is done to make sure that what we say is not “concretely empty” in the sense of Unger (2014). We are not interested in what we can say about Leibniz, but rather in what Leibniz can say about our cur- rent world model. readers expecting some glib demonstration of how our current scientific theories make Leibniz look like a fool will be sorely disappointed. Looking around, and finding a well-built Calculus ratiocinator in Turing ma- chines, pointer machines, and similar abstract models of computation, Leibniz would no doubt consider his Characteristica Universalis well realized in modern proof checkers such as Mizar, Coq, or Agda to the extent mathematical deduc- tion is concerned, but would be somewhat disappointed in their applicability to natural philosophy in general. -
Swimming in Spacetime: Motion by Cyclic Changes in Body Shape
RESEARCH ARTICLE tation of the two disks can be accomplished without external torques, for example, by fixing Swimming in Spacetime: Motion the distance between the centers by a rigid circu- lar arc and then contracting a tension wire sym- by Cyclic Changes in Body Shape metrically attached to the outer edge of the two caps. Nevertheless, their contributions to the zˆ Jack Wisdom component of the angular momentum are parallel and add to give a nonzero angular momentum. Cyclic changes in the shape of a quasi-rigid body on a curved manifold can lead Other components of the angular momentum are to net translation and/or rotation of the body. The amount of translation zero. The total angular momentum of the system depends on the intrinsic curvature of the manifold. Presuming spacetime is a is zero, so the angular momentum due to twisting curved manifold as portrayed by general relativity, translation in space can be must be balanced by the angular momentum of accomplished simply by cyclic changes in the shape of a body, without any the motion of the system around the sphere. external forces. A net rotation of the system can be ac- complished by taking the internal configura- The motion of a swimmer at low Reynolds of cyclic changes in their shape. Then, presum- tion of the system through a cycle. A cycle number is determined by the geometry of the ing spacetime is a curved manifold as portrayed may be accomplished by increasing by ⌬ sequence of shapes that the swimmer assumes by general relativity, I show that net translations while holding fixed, then increasing by (1). -
A Philosophical and Historical Analysis of Cosmology from Copernicus to Newton
University of Central Florida STARS Electronic Theses and Dissertations, 2004-2019 2017 Scientific transformations: a philosophical and historical analysis of cosmology from Copernicus to Newton Manuel-Albert Castillo University of Central Florida Part of the History of Science, Technology, and Medicine Commons Find similar works at: https://stars.library.ucf.edu/etd University of Central Florida Libraries http://library.ucf.edu This Masters Thesis (Open Access) is brought to you for free and open access by STARS. It has been accepted for inclusion in Electronic Theses and Dissertations, 2004-2019 by an authorized administrator of STARS. For more information, please contact [email protected]. STARS Citation Castillo, Manuel-Albert, "Scientific transformations: a philosophical and historical analysis of cosmology from Copernicus to Newton" (2017). Electronic Theses and Dissertations, 2004-2019. 5694. https://stars.library.ucf.edu/etd/5694 SCIENTIFIC TRANSFORMATIONS: A PHILOSOPHICAL AND HISTORICAL ANALYSIS OF COSMOLOGY FROM COPERNICUS TO NEWTON by MANUEL-ALBERT F. CASTILLO A.A., Valencia College, 2013 B.A., University of Central Florida, 2015 A thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts in the department of Interdisciplinary Studies in the College of Graduate Studies at the University of Central Florida Orlando, Florida Fall Term 2017 Major Professor: Donald E. Jones ©2017 Manuel-Albert F. Castillo ii ABSTRACT The purpose of this thesis is to show a transformation around the scientific revolution from the sixteenth to seventeenth centuries against a Whig approach in which it still lingers in the history of science. I find the transformations of modern science through the cosmological models of Nicholas Copernicus, Johannes Kepler, Galileo Galilei and Isaac Newton. -
Orbital Mechanics
Orbital Mechanics These notes provide an alternative and elegant derivation of Kepler's three laws for the motion of two bodies resulting from their gravitational force on each other. Orbit Equation and Kepler I Consider the equation of motion of one of the particles (say, the one with mass m) with respect to the other (with mass M), i.e. the relative motion of m with respect to M: r r = −µ ; (1) r3 with µ given by µ = G(M + m): (2) Let h be the specific angular momentum (i.e. the angular momentum per unit mass) of m, h = r × r:_ (3) The × sign indicates the cross product. Taking the derivative of h with respect to time, t, we can write d (r × r_) = r_ × r_ + r × ¨r dt = 0 + 0 = 0 (4) The first term of the right hand side is zero for obvious reasons; the second term is zero because of Eqn. 1: the vectors r and ¨r are antiparallel. We conclude that h is a constant vector, and its magnitude, h, is constant as well. The vector h is perpendicular to both r and the velocity r_, hence to the plane defined by these two vectors. This plane is the orbital plane. Let us now carry out the cross product of ¨r, given by Eqn. 1, and h, and make use of the vector algebra identity A × (B × C) = (A · C)B − (A · B)C (5) to write µ ¨r × h = − (r · r_)r − r2r_ : (6) r3 { 2 { The r · r_ in this equation can be replaced by rr_ since r · r = r2; and after taking the time derivative of both sides, d d (r · r) = (r2); dt dt 2r · r_ = 2rr;_ r · r_ = rr:_ (7) Substituting Eqn. -
Foundations of Newtonian Dynamics: an Axiomatic Approach For
Foundations of Newtonian Dynamics: 1 An Axiomatic Approach for the Thinking Student C. J. Papachristou 2 Department of Physical Sciences, Hellenic Naval Academy, Piraeus 18539, Greece Abstract. Despite its apparent simplicity, Newtonian mechanics contains conceptual subtleties that may cause some confusion to the deep-thinking student. These subtle- ties concern fundamental issues such as, e.g., the number of independent laws needed to formulate the theory, or, the distinction between genuine physical laws and deriva- tive theorems. This article attempts to clarify these issues for the benefit of the stu- dent by revisiting the foundations of Newtonian dynamics and by proposing a rigor- ous axiomatic approach to the subject. This theoretical scheme is built upon two fun- damental postulates, namely, conservation of momentum and superposition property for interactions. Newton’s laws, as well as all familiar theorems of mechanics, are shown to follow from these basic principles. 1. Introduction Teaching introductory mechanics can be a major challenge, especially in a class of students that are not willing to take anything for granted! The problem is that, even some of the most prestigious textbooks on the subject may leave the student with some degree of confusion, which manifests itself in questions like the following: • Is the law of inertia (Newton’s first law) a law of motion (of free bodies) or is it a statement of existence (of inertial reference frames)? • Are the first two of Newton’s laws independent of each other? It appears that -
16.346 Astrodynamics Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 16.346 Astrodynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lecture 2 The Two Body Problem Continued The Eccentricity Vector or The Laplace Vector µ µe = v × h − r r Explicit Form of the Velocity Vector #3.1 Using the expansion of the triple vector product a × (b × c)=(a · c)b − (a · b)c we have µ h × µe = h × (v × h) − h × r = h2v − (h · v)h − µh × i = h2v − µh i × i r r h r since h and v are perpendicular. Therefore: µ h × µe =⇒ v = i × (e + i ) h h r or hv = i × (e i + i )=e i × i + i × i = e i + i µ h e r h e h r p θ Then since ip = sin f ir + cos f iθ we have hv = e sin f i +(1+e cos f) i µ r θ which is the basic relation for representing the velocity vector in the Hodograph Plane. See Page 1 of Lecture 4 Conservation of Energy hv hv p p 2 1 · = v · v = 2(1 + e cos f)+e 2 − 1=2× − (1 − e 2)=p − µ µ µ r r a which can be written in either of two separate forms each having its own name: 1 µ µ 1 Energy Integral v2 − = − = c 2 r 2a 2 3 2 1 Vis-Viva Integral v 2 = µ − r a The constant c3 is used by Forest Ray Moulton, a Professor at the University of Chicago in his 1902 book “An Introduction to Celestial Mechanics” — the first book on the subject written by an American. -
Leibniz' Dynamical Metaphysicsand the Origins
LEIBNIZ’ DYNAMICAL METAPHYSICSAND THE ORIGINS OF THE VIS VIVA CONTROVERSY* by George Gale Jr. * I am grateful to Marjorie Grene, Neal Gilbert, Ronald Arbini and Rom Harr6 for their comments on earlier drafts of this essay. Systematics Vol. 11 No. 3 Although recent work has begun to clarify the later history and development of the vis viva controversy, the origins of the conflict arc still obscure.1 This is understandable, since it was Leibniz who fired the initial barrage of the battle; and, as always, it is extremely difficult to disentangle the various elements of the great physicist-philosopher’s thought. His conception includes facets of physics, mathematics and, of course, metaphysics. However, one feature is essential to any possible understanding of the genesis of the debate over vis viva: we must note, as did Leibniz, that the vis viva notion was to emerge as the first element of a new science, which differed significantly from that of Descartes and Newton, and which Leibniz called the science of dynamics. In what follows, I attempt to clarify the various strands which were woven into the vis viva concept, noting especially the conceptual framework which emerged and later evolved into the science of dynamics as we know it today. I shall argue, in general, that Leibniz’ science of dynamics developed as a consistent physical interpretation of certain of his metaphysical and mathematical beliefs. Thus the following analysis indicates that at least once in the history of science, an important development in physical conceptualization was intimately dependent upon developments in metaphysical conceptualization. 1. -
Leibniz's Optics and Contingency in Nature
Leibniz's Optics and Contingency in Nature The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation McDonough, Jeffrey K. 2010. Leibniz's Optics and Contingency in Nature. Perspectives on Science 18(4): 432-455. Published Version doi:10.1162/POSC_a_00017 Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:5128571 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#OAP Leibniz’s Optics and Contingency in Nature Jeffrey K. McDonough Department of Philosophy Harvard University 1 Abstract In the late 1670’s to early 1680’s, Leibniz came to hold that the laws of nature are paradigmatically contingent, that they provide the basis for a new argument from design, and that they presuppose the existence of active, goal-directed powers reminiscent of Aristotelian entelechies. In this essay, I argue that the standard view according to which Leibniz forges these signature theses in the domain of physics and opportunistically carries them over to the domain of optics gets things essentially the wrong way around. The crucial nexus of views at the heart of Leibniz’s mature philosophical understanding of the laws of nature has its most intelligible roots in his optical derivations, which appear to have paved the way – both historically and conceptually – for the philosophical significance he assigns to his discoveries in the domain of physics. -
Chapter One Concerning Motion in General
EULER'S MECHANICA VOL. 1. Chapter one. Translated and annotated by Ian Bruce. page 1 Chapter One Concerning Motion In General. [p. 1] DEFINITION 1. 1. Motion is the translation of a body from the place it occupies to another place. Truly rest is a body remaining at the same place. Corollary 1. 2. Therefore the ideas of the body remaining at rest and of moving to other places cannot be entertained together, except that they occupy a place. Whereby the place or position the body occupies shall be a property of the body, that can be said to be possible only for that body, and that the body is either moving or at rest. Corollary 2. 3. And this idea of moving or of being at rest is a property of a body that relates to all bodies. For no body is able to exist that is neither moving nor at rest. DEFINITION 2. 4. The place [occupied by a body] is a part of the immense or boundless space which constitutes the whole world. In this sense the accepted place is accustomed to be called the absolute place [p. 2], in order that it may be distinguished from a relative place, of which mention will soon be made. Corollary 1. 5. Therefore, when a body occupies successively one part and then another, of this immense space, then it is said to be moving ; but if it continues to be present at the same place always, then it is said to be at rest. Corollary 2. 6. Moreover, it is customery to consider fixed boundaries of this space, to which bodies can be referred. -
A Rhetorical Analysis of Sir Isaac Newton's Principia A
A RHETORICAL ANALYSIS OF SIR ISAAC NEWTON’S PRINCIPIA A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS IN THE GRADUATE SCHOOL OF THE TEXAS WOMAN’S UNIVERSITY DEPARTMENT OF ENGLISH, SPEECH, AND FOREIGN LANGUAGES COLLEGE OF ARTS AND SCIENCES BY GIRIBALA JOSHI, B.S., M.S. DENTON, TEXAS AUGUST 2018 Copyright © 2018 by Giribala Joshi DEDICATION Nature and Nature’s Laws lay hid in Night: God said, “Let Newton be!” and all was light. ~ Alexander Pope Dedicated to all the wonderful eighteenth-century Enlightenment thinkers and philosophers! ii ACKNOWLEDGMENTS I would like to acknowledge the continuous support and encouragement that I received from the Department of English, Speech and Foreign Languages. I especially want to thank my thesis committee member Dr. Ashley Bender, and my committee chair Dr. Brian Fehler, for their guidance and feedback while writing this thesis. iii ABSTRACT GIRIBALA JOSHI A RHETORICAL ANALYSIS OF SIR ISAAC NEWTON’S PRINCIPIA AUGUST 2018 In this thesis, I analyze Isaac Newton's Philosophiae Naturalis Principia Mathematica in the framework of Aristotle’s theories of rhetoric. Despite the long-held view that science only deals with brute facts and does not require rhetoric, we learn that science has its own special topics. This study highlights the rhetorical situation of the Principia and Newton’s rhetorical strategies, emphasizing the belief that scientific facts and theories are also rhetorical constructions. This analysis shows that the credibility of the author and the text, the emotional debates before and after the publication of the text, the construction of logical arguments, and the presentation style makes the book the epitome of scientific writing. -
Phoronomy: Space, Construction, and Mathematizing Motion Marius Stan
To appear in Bennett McNulty (ed.), Kant’s Metaphysical Foundations of Natural Science: A Critical Guide. Cambridge University Press. Phoronomy: space, construction, and mathematizing motion Marius Stan With his chapter, Phoronomy, Kant defies even the seasoned interpreter of his philosophy of physics.1 Exegetes have given it little attention, and un- derstandably so: his aims are opaque, his turns in argument little motivated, and his context mysterious, which makes his project there look alienating. I seek to illuminate here some of the darker corners in that chapter. Specifi- cally, I aim to clarify three notions in it: his concepts of velocity, of compo- site motion, and of the construction required to compose motions. I defend three theses about Kant. 1) his choice of velocity concept is ul- timately insufficient. 2) he sided with the rationalist faction in the early- modern debate on directed quantities. 3) it remains an open question if his algebra of motion is a priori, though he believed it was. I begin in § 1 by explaining Kant’s notion of phoronomy and its argu- ment structure in his chapter. In § 2, I present four pictures of velocity cur- rent in Kant’s century, and I assess the one he chose. My § 3 is in three parts: a historical account of why algebra of motion became a topic of early modern debate; a synopsis of the two sides that emerged then; and a brief account of his contribution to the debate. Finally, § 4 assesses how general his account of composite motion is, and if it counts as a priori knowledge.