Berkeley's Case Against Realism About Dynamics
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THE THEORY of MATHEMATICAL SUBTRACTION in ARISTOTLE By
THE THEORY OF MATHEMATICAL SUBTRACTION IN ARISTOTLE by Ksenia Romashova Submitted in partial fulfilment of the requirements for the degree of Master of Arts at Dalhousie University Halifax, Nova Scotia August 2019 © Copyright by Ksenia Romashova, 2019 DEDICATION PAGE To my Mother, Vera Romashova Мама, Спасибо тебе за твою бесконечную поддержку, дорогая! ii TABLE OF CONTENTS LIST OF FIGURES .................................................................................................................. iv ABSTRACT ............................................................................................................................... v LIST OF ABBREVIATIONS USED ....................................................................................... vi ACKNOWLEDGEMENTS ..................................................................................................... vii CHAPTER 1 INTRODUCTION ............................................................................................... 1 CHAPTER 2 THE MEANING OF ABSTRACTION .............................................................. 6 2.1 Etymology and Evolution of the Term ................................................................... 7 2.2 The Standard Phrase τὰ ἐξ ἀφαιρέσεως or ‘Abstract Objects’ ............................ 19 CHAPTER 3 GENERAL APPLICATION OF ABSTRACTION .......................................... 24 3.1 The Instances of Aphairein in Plato’s Dialogues ................................................. 24 3.2 The Use of Aphairein in Aristotle’s Topics -
Continuity and Mathematical Ontology in Aristotle
Journal of Ancient Philosophy, vol. 14 issue 1, 2020. DOI: http://dx.doi.org/10.11606/issn.1981-9471.v14i1p30-61 Continuity and Mathematical Ontology in Aristotle Keren Wilson Shatalov In this paper I argue that Aristotle’s understanding of mathematical continuity constrains the mathematical ontology he can consistently hold. On my reading, Aristotle can only be a mathematical abstractionist of a certain sort. To show this, I first present an analysis of Aristotle’s notion of continuity by bringing together texts from his Metaphysica and Physica, to show that continuity is, for Aristotle, a certain kind of per se unity, and that upon this rests his distinction between continuity and contiguity. Next I argue briefly that Aristotle intends for his discussion of continuity to apply to pure mathematical objects such as lines and figures, as well as to extended bodies. I show that this leads him to a difficulty, for it does not at first appear that the distinction between continuity and contiguity can be preserved for abstract mathematicals. Finally, I present a solution according to which Aristotle’s understanding of continuity can only be saved if he holds a certain kind of mathematical ontology. My topic in this paper is Aristotle’s understanding of mathematical continuity. While the idea that continua are composed of infinitely many points is the present day orthodoxy, the Aristotelian understanding of continua as non-punctiform and infinitely divisible was the reigning theory for much of the history of western mathematics, and there is renewed interest in it from current mathematicians and philosophers of mathematics. -
Spinoza and the Sciences Boston Studies in the Philosophy of Science
SPINOZA AND THE SCIENCES BOSTON STUDIES IN THE PHILOSOPHY OF SCIENCE EDITED BY ROBERT S. COHEN AND MARX W. WARTOFSKY VOLUME 91 SPINOZA AND THE SCIENCES Edited by MARJORIE GRENE University of California at Davis and DEBRA NAILS University of the Witwatersrand D. REIDEL PUBLISHING COMPANY A MEMBER OF THE KLUWER ~~~.'~*"~ ACADEMIC PUBLISHERS GROUP i\"lI'4 DORDRECHT/BOSTON/LANCASTER/TOKYO Library of Congress Cataloging-in-Publication Data Main entry under title: Spinoza and the sciences. (Boston studies in the philosophy of science; v. 91) Bibliography: p. Includes index. 1. Spinoza, Benedictus de, 1632-1677. 2. Science- Philosophy-History. 3. Scientists-Netherlands- Biography. I. Grene, Marjorie Glicksman, 1910- II. Nails, Debra, 1950- Ill. Series. Q174.B67 vol. 91 OOI'.Ols 85-28183 101 43.S725J 100 I J ISBN-13: 978-94-010-8511-3 e-ISBN-13: 978-94-009-4514-2 DOl: 10.1007/978-94-009-4514-2 Published by D. Reidel Publishing Company, P.O. Box 17, 3300 AA Dordrecht, Holland. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, Holland. 2-0490-150 ts All Rights Reserved © 1986 by D. Reidel Publishing Company Softcover reprint of the hardcover 1st edition 1986 and copyright holders as specified on appropriate pages within No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner FROM SPINOZA'S LETTER TO OLDENBURG, RIJNSBURG, APRIL, 1662 (Photo by permission of Berend Kolk) TABLE OF CONTENTS ACKNOWLEDGEMENTS ix MARJORIE GRENE I Introduction xi 1. -
Newton.Indd | Sander Pinkse Boekproductie | 16-11-12 / 14:45 | Pag
omslag Newton.indd | Sander Pinkse Boekproductie | 16-11-12 / 14:45 | Pag. 1 e Dutch Republic proved ‘A new light on several to be extremely receptive to major gures involved in the groundbreaking ideas of Newton Isaac Newton (–). the reception of Newton’s Dutch scholars such as Willem work.’ and the Netherlands Jacob ’s Gravesande and Petrus Prof. Bert Theunissen, Newton the Netherlands and van Musschenbroek played a Utrecht University crucial role in the adaption and How Isaac Newton was Fashioned dissemination of Newton’s work, ‘is book provides an in the Dutch Republic not only in the Netherlands important contribution to but also in the rest of Europe. EDITED BY ERIC JORINK In the course of the eighteenth the study of the European AND AD MAAS century, Newton’s ideas (in Enlightenment with new dierent guises and interpre- insights in the circulation tations) became a veritable hype in Dutch society. In Newton of knowledge.’ and the Netherlands Newton’s Prof. Frans van Lunteren, sudden success is analyzed in Leiden University great depth and put into a new perspective. Ad Maas is curator at the Museum Boerhaave, Leiden, the Netherlands. Eric Jorink is researcher at the Huygens Institute for Netherlands History (Royal Dutch Academy of Arts and Sciences). / www.lup.nl LUP Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. 1 Newton and the Netherlands Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. 2 Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. -
The Newton-Leibniz Controversy Over the Invention of the Calculus
The Newton-Leibniz controversy over the invention of the calculus S.Subramanya Sastry 1 Introduction Perhaps one the most infamous controversies in the history of science is the one between Newton and Leibniz over the invention of the infinitesimal calculus. During the 17th century, debates between philosophers over priority issues were dime-a-dozen. Inspite of the fact that priority disputes between scientists were ¡ common, many contemporaries of Newton and Leibniz found the quarrel between these two shocking. Probably, what set this particular case apart from the rest was the stature of the men involved, the significance of the work that was in contention, the length of time through which the controversy extended, and the sheer intensity of the dispute. Newton and Leibniz were at war in the later parts of their lives over a number of issues. Though the dispute was sparked off by the issue of priority over the invention of the calculus, the matter was made worse by the fact that they did not see eye to eye on the matter of the natural philosophy of the world. Newton’s action-at-a-distance theory of gravitation was viewed as a reversion to the times of occultism by Leibniz and many other mechanical philosophers of this era. This intermingling of philosophical issues with the priority issues over the invention of the calculus worsened the nature of the dispute. One of the reasons why the dispute assumed such alarming proportions and why both Newton and Leibniz were anxious to be considered the inventors of the calculus was because of the prevailing 17th century conventions about priority and attitude towards plagiarism. -
Newton and Kant on Absolute Space: from Theology to Transcendental Philosophy
Newton and Kant on Absolute Space: From Theology to Transcendental Philosophy Michael Friedman Abstract I argue that Einstein’s creation of both special and general relativity instantiates Reichenbach’s conception of the relativized a priori. I do this by show- ing how the original Kantian conception actually contributes to the development of Einstein’s theories through the intervening philosophical and scientific work of Helmholtz, Mach, and Poincaré. In my previous work on Newton and Kant I have primarily emphasized methodo- logical issues: why Kant takes the Newtonian Laws of Motion (as well as certain related propositions of what he calls “pure natural science”) as synthetic a priori constitutive principles rather than mere empirical laws, and how this point is inti- mately connected, in turn, with Kant’s conception of absolute space as a regulative idea of reason – as the limit point of an empirical constructive procedure rather than a self-subsistent “container” existing prior to and independently of all perceptible matter. I have also argued that these methodological differences explain the circum- stance that Kant, unlike Newton, asserts that gravitational attraction must be con- ceived as an “action at a distance through empty space,” and even formulates a (rare) criticism of Newton for attempting to leave the question of the “true cause” of gravitational attraction entirely open. In this paper I emphasize the importance of metaphysical and theological issues – about God, his creation of the material world in space, and the consequences different views of such creation have for the metaphysical foundations of physics. I argue, in particular, that Kant’s differences with Newton over these issues constitute an essential part of his radical transforma- tion of the very meaning of metaphysics as practiced by his predecessors. -
Determinism Is False
%&5&3.*/*4. Barry Loewer %FUFSNJOJTNJTBDPOUJOHFOUNFUBQIZTJDBMDMBJNBCPVUUIFGVOEBNFOUBMOBUVSBMMBXT UIBUIPMEJOUIFVOJWFSTF*UTBZT The natural laws and the way things are at time t determine the way things will be at later times. 5IF NBUIFNBUJDJBO 1JFSSF4JNPO -BQMBDF FYQSFTTFE IJT CFMJFG UIBU EFUFS- minism is true this way: 8F PVHIU UP SFHBSE UIF QSFTFOU TUBUF PG UIF VOJWFSTF BT UIF FGGFDU PG JUT antecedent state and as the cause of the state that is to follow. An intel- MJHFODF LOPXJOH BMM UIF GPSDFT BDUJOH JO OBUVSF BU B HJWFO JOTUBOU BT XFMM as the momentary positions of all things in the universe, would be able to comprehend in one single formula the motions of the largest bodies as well as the lightest atoms in the world, provided that its intellect were suf!ciently QPXFSGVMUPTVCKFDUBMMEBUBUPBOBMZTJTUPJUOPUIJOHXPVMECFVODFSUBJO UIF future as well as the past would be present to its eyes. The perfection that the human mind has been able to give to astronomy affords but a feeble outline of such intelligence. 5IF QIZTJDT PG -BQMBDFT EBZ UIF àSTU EFDBEFT PG UIF OJOFUFFOUI DFOUVSZ XBT /FXUPOJBO DMBTTJDBM NFDIBOJDT*TBBD/FXUPOGPSNVMBUFEQSJODJQMFTUIBUIFUIPVHIU FYQSFTTUIFMBXTEFTDSJCJOHIPXGPSDFTEFUFSNJOFUIFNPUJPOTPGCPEJFT F ma) and IPXUIFQPTJUJPOTPGCPEJFTBOEPUIFSGBDUPSTEFUFSNJOFHSBWJUBUJPOBMBOEPUIFSLJOET PGGPSDFT6TJOHUIFTFQSJODJQMFT /FXUPOBOEQIZTJDJTUTGPMMPXJOHIJNXFSFBCMFUP QSFEJDUBOEFYQMBJOUIFNPUJPOTPGDFMFTUJBMBOEUFSSFTUSJBMCPEJFT'PSFYBNQMF UIFTF laws account for the orbits of the planets, the trajectories of cannon balls, and the QFSJPET PG QFOEVMVNT-JLF/FXUPO -BQMBDFEJE OPULOPX BMM UIFGPSDFTUIFSF BSF but he envisioned that, once those forces (and the corresponding force laws) were LOPXO /FXUPOJBO QIZTJDT XPVME CF B complete physical theory. That is, its laws would account for the motions of all material particles. And since he thought that FWFSZUIJOH UIBU FYJTUT JO TQBDF JT DPNQPTFE PG WBSJPVT LJOET PG WFSZ TNBMM NBUFSJBM #"33:-0&8&3 QBSUJDMFT PSBUPNT IFUIPVHIUUIBU/FXUPOJBONFDIBOJDT PODFBMMUIFGPSDFTXFSF LOPXO XPVMECFXIBUUPEBZXFXPVMEDBMMthe theory of everything. -
Newton's Notebook
Newton’s Notebook The Haverford School’s Math & Applied Math Journal Issue I Spring 2017 The Haverford School Newton’s Notebook Spring 2017 “To explain all nature is too difficult a task for any one man or even for any one age. ‘Tis much better to do a little with certainty & leave the rest for others that come after you.” ~Isaac Newton Table of Contents Pure Mathematics: 7 The Golden Ratio.........................................................................................Robert Chen 8 Fermat’s Last Theorem.........................................................................Michael Fairorth 9 Math in Coding............................................................................................Bram Schork 10 The Pythagoreans.........................................................................................Eusha Hasan 12 Transfinite Numbers.................................................................................Caleb Clothier 15 Sphere Equality................................................................................Matthew Baumholtz 16 Interesting Series.......................................................................................Aditya Sardesi 19 Indirect Proofs..............................................................................................Mr. Patrylak Applied Mathematics: 23 Physics in Finance....................................................................................Caleb Clothier 26 The von Bertalanffy Equation..................................................................Will -
Newton As Philosopher
This page intentionally left blank NEWTON AS PHILOSOPHER Newton’s philosophical views are unique and uniquely difficult to categorize. 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 sophisti- cated contributor to natural philosophy, one who engaged with the principal ideas of his most important predecessor, René 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. ANDREW JANIAK is Assistant Professor in the Department of Philosophy, Duke University. He is editor of Newton: Philosophical Writings (2004). NEWTON AS PHILOSOPHER ANDREW JANIAK Duke University CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521862868 © Andrew Janiak 2008 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. -
The Relationship Between the Aristotelian, Newtonian and Holistic Scientific Paradigms and Selected British Detective Fiction 1980 - 2010
THE RELATIONSHIP BETWEEN THE ARISTOTELIAN, NEWTONIAN AND HOLISTIC SCIENTIFIC PARADIGMS AND SELECTED BRITISH DETECTIVE FICTION 1980 - 2010 HILARY ANNE GOLDSMITH A thesis submitted in partial fulfilment of the requirements of the University of Greenwich for the Degree of Doctor of Philosophy July 2010 i ACKNOWLEDGEMENTS I would like to acknowledge the help and support I have received throughout my studies from the academic staff at the University of Greenwich, especially that of my supervisors. I would especially like to acknowledge the unerring support and encouragement I have received from Professor Susan Rowland, my first supervisor. iii ABSTRACT This thesis examines the changing relationship between key elements of the Aristotelian, Newtonian and holistic scientific paradigms and contemporary detective fiction. The work of scholars including N. Katherine Hayles, Martha A. Turner has applied Thomas S. Kuhn’s notion of scientific paradigms to literary works, especially those of the Victorian period. There seemed to be an absence, however, of research of a similar academic standard exploring the relationship between scientific worldviews and detective fiction. Extending their scholarship, this thesis seeks to open up debate in what was perceived to be an under-represented area of literary study. The thesis begins by identifying the main precepts of the three paradigms. It then offers a chronological overview of the developing relationship between these precepts and detective fiction from Sir Arthur Conan Doyle’s The Sign of Four (1890) to P.D.James’s The Black Tower (1975). The present state of this interaction is assessed through a detailed analysis of representative examples of the detective fiction of Reginald Hill, Barbara Nadel, and Quintin Jardine written between 1980 and 2010. -
The Synthesis of Empiricism and Innatism in Berkeley's Doctrine Of
Berkeley Studies 21 (2010) 3 The Synthesis of Empiricism and Innatism in Berkeley’s Doctrine of Notions James Hill Abstract: This essay argues that Berkeley’s doctrine of notions is an account of concept-formation that offers a middle-way between empiricism and innatism, something which Berkeley himself asserts at Siris 308. First, the widespread assumption that Berkeley accepts Locke’s conceptual empiricism is questioned, with particular attention given to Berkeley’s views on innatism and ideas of reflection. Then, it is shown that Berkeley’s doctrine of notions comes very close to the refined form of innatism to be found in Descartes’ later writings and in Leibniz. Finally, it is argued that Berkeley denies a principle common to both empiricism and innatism, namely, that all conceptual knowledge amounts to the perception of ideas. By denying this―at least in the case of the concepts of self, causation, substance, and virtue―Berkeley is able to provide a synthesis of conceptual empiricism and innatism. In Siris, Berkeley offers us a characteristically succinct reflection on his doctrine of notions: [Aristotle] held that the mind of man was a tabula rasa, and that there were no innate ideas. Plato, on the contrary, held original ideas in the mind; that is, notions which never were or can be in the sense, such as being, beauty, goodness, likeness, parity. Some, perhaps, may think the truth to be this: that there are properly no ideas, or passive objects, in the mind but what were derived from sense: but that there are also besides these her own acts or operations; such are notions. -
Abstraction’ in Aristotle [13]
ON THE TERMINOLOGY OF ‘ABSTRACTION’ IN ARISTOTLE [13] If we look at the standard commentaries on Aristotle’s philosophy, we nd the term ‘abstraction’ constantly being used to characterise his epistemology in contrast to that of Plato. Historically speaking, it may have been Aquinas1 who was responsible for making ‘abstractionism’ the by-word for a kind of empiricism which holds that all knowledge derives from the senses. Indeed, the unanimity of the commentators appears to be such that it has congealed into the dogma that Aristotle’s theory of knowledge is basically abstractionist in character.2 In spite of the weight of opinion, I must confess myself to be sceptical about this dogma. Since this may appear heretical to some readers, I hasten to clarify my scepticism. I do not doubt that Aristotle opposed Plato’s theory of recollection with his own claim that all of our knowledge ultimately derives from the senses. What I do question is whether Aristotle ever gives the name ‘abstraction’ to the process by which we grasp universals through sense experience. It is a well-known yet puzzling fact that such terminology does not appear in those few passages where Aristotle actually describes this process (An. Post. II, 19 and Met. I, 1). Now what are we to make of this fact? Perhaps one might object that this is purely a philological issue, but I am convinced that it has more general signi cance for the interpretation of Aristotle. In this paper, therefore, I propose to make a fresh analysis of the ter- minology of ‘abstraction’ as it is used in the Aristotelian corpus.