DOCSLIB.ORG

Newton and Leibniz Isaac Newton (1642/43 – 1726/27)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Newton and Leibniz Isaac Newton (1642/43 – 1726/27) Newton and Leibniz Isaac Newton (1642/43 – 1726/27) • born in rural Lincolnshire, son of a rich farmer • went to a good school • student at Cambridge 1661–65, 67–68 • Isaac Barrow resigned his chair 1669 in favor of N, N dodged the requirement to become a priest • wrote religious tracts in the 1690s • warden of the Royal Mint 1696, cracked down on counterfeiters • knighted 1705 by Queen Anne Newton’s key works • (Philosophiae Naturalis) Principia Mathematica (1687) foundation of Newtonian physics (mechanics, laws of motion, gravitation, planetary motions) • Opticks (1704) theory of light (as particles) and color • De analysi (1711), de methodis (1736) Newton’s foundations of calculus, which he already used in Principia • plus works on theology, occultism, alchemy Characteristics of Newton’s approach to calculus • Keep in mind that the concept of function did not exist! • Newton came to consider power series by looking at the binomial theorem and generalizing it for arbitrary rational exponents • Newton thought of variables x as motions (“fluents”) with respect to time, and would call its derivative ẋ “fluxion”. • He used infinitesimals “x+ẋo” for infinitesimal movements to derive rules of differentiation and integration, reverted to power series when explicit integration was hard • Solved extremal problems, computed curvatures and areas Why did Newton not publish his results? • He did write about his methods in the beginning of Principia. • He was probably aware that the “infinitesimals” weren’t rigidly defined and was afraid of criticism. • He generally wasn’t fast to publish. • He was wary that people might steal his results. Gottfried Wilhelm Leibniz (1646–1716) • born in Leipzig, son of a philosophy professor • studied philosophy in Leipzig 1661– 64 (PhD), law 1665–66 (promotion) • landed some administrative/political jobs due to his connections • met Huygens in Paris 1672, who embarrassed L by exposing his patchy knowledge in math, commenced self-study • courtier at duke’s court in Hanover 1676 with vast freedom • died fallen out of favor, unmarked grave for 50 years. Leibniz’s key works • Dissertatio de arte combinatorica (1666), a philosophical work seeking to encode all thought symbolically • Théodicée (1712), central philosophical work • Several shorter works on philosophy, most important his theory of monads (“Monadologie”, 1714) • Several papers on calculus in Acta eruditorum, first one “a new method for maxima and minima, and the same for tangents, unhindered by either fractional or irrational powers, and a unique kind of calculus for that, by G. G. L.” 1684 Leibniz’s approach to calculus • a discrete, arithmetic approach: an-a1 = (an-an-1)+(an-1-an-2)+…+(a2-a1) • wrote dx for a small difference, ∫ y for a sum of a sequence, so that ∫ dx = x • found differentiation rules such as d(xy)=xdy+ydx • fundamental theorem was obvious • “arithmetic quadrature of the circle” = series expression for π • like Newton, was aware of the shortcomings of his infinitesimals The controversy • Newton found the results first, but Leibniz published first • Completely different approaches • Newton accused Leibniz of plagiarism, especially after L’s visit to London • Newton made the Royal Society condemn Leibniz, halting all communication between British and Continental mathematicians • Leibniz’s notation proved more flexible, resulting in English mathematics lagging behind for a whole century. From the introduction of the English translation of de l’Hôpital’s Analyse des Infiniment Petits by E. Stone.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Optimization and Gradient Descent INFO-4604, Applied Machine Learning University of Colorado Boulder
    Optimization and Gradient Descent INFO-4604, Applied Machine Learning University of Colorado Boulder September 11, 2018 Prof. Michael Paul Prediction Functions Remember: a prediction function is the function that predicts what the output should be, given the input. Prediction Functions Linear regression: f(x) = wTx + b Linear classification (perceptron): f(x) = 1, wTx + b ≥ 0 -1, wTx + b < 0 Need to learn what w should be! Learning Parameters Goal is to learn to minimize error • Ideally: true error • Instead: training error The loss function gives the training error when using parameters w, denoted L(w). • Also called cost function • More general: objective function (in general objective could be to minimize or maximize; with loss/cost functions, we want to minimize) Learning Parameters Goal is to minimize loss function. How do we minimize a function? Let’s review some math. Rate of Change The slope of a line is also called the rate of change of the line. y = ½x + 1 “rise” “run” Rate of Change For nonlinear functions, the “rise over run” formula gives you the average rate of change between two points Average slope from x=-1 to x=0 is: 2 f(x) = x -1 Rate of Change There is also a concept of rate of change at individual points (rather than two points) Slope at x=-1 is: f(x) = x2 -2 Rate of Change The slope at a point is called the derivative at that point Intuition: f(x) = x2 Measure the slope between two points that are really close together Rate of Change The slope at a point is called the derivative at that point Intuition: Measure the
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Berkeley's Case Against Realism About Dynamics
    Lisa Downing [Published in Berkeley’s Metaphysics, ed. Muehlmann, Penn State Press 1995, 197-214. Turbayne Essay Prize winner, 1992.] Berkeley's case against realism about dynamics While De Motu, Berkeley's treatise on the philosophical foundations of mechanics, has frequently been cited for the surprisingly modern ring of certain of its passages, it has not often been taken as seriously as Berkeley hoped it would be. Even A.A. Luce, in his editor's introduction to De Motu, describes it as a modest work, of limited scope. Luce writes: The De Motu is written in good, correct Latin, but in construction and balance the workmanship falls below Berkeley's usual standards. The title is ambitious for so brief a tract, and may lead the reader to expect a more sustained argument than he will find. A more modest title, say Motion without Matter, would fitly describe its scope and content. Regarded as a treatise on motion in general, it is a slight and disappointing work; but viewed from a narrower angle, it is of absorbing interest and high importance. It is the application of immaterialism to contemporary problems of motion, and should be read as such. ...apart from the Principles the De Motu would be nonsense.1 1The Works of George Berkeley, Bishop of Cloyne, ed. A.A. Luce and T.E. Jessop (London: Thomas Nelson and Sons, 1948-57), 4: 3-4. In this paper, all references to Berkeley are to the Luce-Jessop edition. Quotations from De Motu are taken from Luce's translation. I use the following abbreviations for Berkeley’s works: PC Philosophical Commentaries PHK-I Introduction to The Principles of Human Knowledge PHK The Principles of Human Knowledge DM De Motu A Alciphron TVV The Theory of Vision Vindicated and Explained S Siris 1 There are good general reasons to think, however, that Berkeley's aims in writing the book were as ambitious as the title he chose.
    [Show full text]
  • Newton: 'Opticks'
    University of Dayton eCommons Imprints and Impressions: Milestones in Human Commentaries on the Exhibit’s Works Progress November 2014 Newton: ‘Opticks’ Follow this and additional works at: http://ecommons.udayton.edu/rosebk_commentary Recommended Citation "Newton: ‘Opticks’" (2014). Commentaries on the Exhibit’s Works. 30. http://ecommons.udayton.edu/rosebk_commentary/30 This Article is brought to you for free and open access by the Imprints and Impressions: Milestones in Human Progress at eCommons. It has been accepted for inclusion in Commentaries on the Exhibit’s Works by an authorized administrator of eCommons. For more information, please contact [email protected], [email protected]. Reflections on the various works in the exhibit Imprints and Impressions: Milestones in Human Progress Highlights from the Rose Rare Book Collection, Sept. 29-Nov. 9, 2014 Roesch Library, University of Dayton Isaac Newton Opticks: Or, a Treatise of the Reflexions, Refractions, Inflexions and Colours of Light London, 1704 First edition Reflection 1 Perhaps the key point of Newton’s observations was that white light is made up ow is a rainbow formed? Why of a superposition of many different “rays” do I see colors like magenta, of light with different colors. Furthermore, H yellow, and turquoise blue on an each individual color of light, in a oily puddle or when my kids blow soap continuous spectrum, has its own property bubbles? Why does a prism project the of “refrangibility,” or refraction. When a ray colors of the rainbow on the wall? And why of red light (such as a laser pointer) hits a are all the really big telescopes based on glass surface at an angle, it will be bent as it mirrors and not lenses, which was the passes into the new medium.
    [Show full text]
  • Maxima and Minima
    Basic Mathematics Maxima and Minima R Horan & M Lavelle The aim of this document is to provide a short, self assessment programme for students who wish to be able to use differentiation to find maxima and mininima of functions. Copyright c 2004 [email protected] , [email protected] Last Revision Date: May 5, 2005 Version 1.0 Table of Contents 1. Rules of Differentiation 2. Derivatives of order 2 (and higher) 3. Maxima and Minima 4. Quiz on Max and Min Solutions to Exercises Solutions to Quizzes Section 1: Rules of Differentiation 3 1. Rules of Differentiation Throughout this package the following rules of differentiation will be assumed. (In the table of derivatives below, a is an arbitrary, non-zero constant.) y axn sin(ax) cos(ax) eax ln(ax) dy 1 naxn−1 a cos(ax) −a sin(ax) aeax dx x If a is any constant and u, v are two functions of x, then d du dv (u + v) = + dx dx dx d du (au) = a dx dx Section 1: Rules of Differentiation 4 The stationary points of a function are those points where the gradient of the tangent (the derivative of the function) is zero. Example 1 Find the stationary points of the functions (a) f(x) = 3x2 + 2x − 9 , (b) f(x) = x3 − 6x2 + 9x − 2 . Solution dy (a) If y = 3x2 + 2x − 9 then = 3 × 2x2−1 + 2 = 6x + 2. dx The stationary points are found by solving the equation dy = 6x + 2 = 0 . dx In this case there is only one solution, x = −1/3.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Calculus 120, Section 7.3 Maxima & Minima of Multivariable Functions
    Calculus 120, section 7.3 Maxima & Minima of Multivariable Functions notes by Tim Pilachowski A quick note to start: If you’re at all unsure about the material from 7.1 and 7.2, now is the time to go back to review it, and get some help if necessary. You’ll need all of it for the next three sections. Just like we had a first-derivative test and a second-derivative test to maxima and minima of functions of one variable, we’ll use versions of first partial and second partial derivatives to determine maxima and minima of functions of more than one variable. The first derivative test for functions of more than one variable looks very much like the first derivative test we have already used: If f(x, y, z) has a relative maximum or minimum at a point ( a, b, c) then all partial derivatives will equal 0 at that point. That is ∂f ∂f ∂f ()a b,, c = 0 ()a b,, c = 0 ()a b,, c = 0 ∂x ∂y ∂z Example A: Find the possible values of x, y, and z at which (xf y,, z) = x2 + 2y3 + 3z 2 + 4x − 6y + 9 has a relative maximum or minimum. Answers : (–2, –1, 0); (–2, 1, 0) Example B: Find the possible values of x and y at which fxy(),= x2 + 4 xyy + 2 − 12 y has a relative maximum or minimum. Answer : (4, –2); z = 12 Notice that the example above asked for possible values. The first derivative test by itself is inconclusive. The second derivative test for functions of more than one variable is a good bit more complicated than the one used for functions of one variable.
    [Show full text]
  • Mean Value, Taylor, and All That
    Mean Value, Taylor, and all that Ambar N. Sengupta Louisiana State University November 2009 Careful: Not proofread! Derivative Recall the definition of the derivative of a function f at a point p: f (w) − f (p) f 0(p) = lim (1) w!p w − p Derivative Thus, to say that f 0(p) = 3 means that if we take any neighborhood U of 3, say the interval (1; 5), then the ratio f (w) − f (p) w − p falls inside U when w is close enough to p, i.e. in some neighborhood of p. (Of course, we can’t let w be equal to p, because of the w − p in the denominator.) In particular, f (w) − f (p) > 0 if w is close enough to p, but 6= p. w − p Derivative So if f 0(p) = 3 then the ratio f (w) − f (p) w − p lies in (1; 5) when w is close enough to p, i.e. in some neighborhood of p, but not equal to p. Derivative So if f 0(p) = 3 then the ratio f (w) − f (p) w − p lies in (1; 5) when w is close enough to p, i.e. in some neighborhood of p, but not equal to p. In particular, f (w) − f (p) > 0 if w is close enough to p, but 6= p. w − p • when w > p, but near p, the value f (w) is > f (p). • when w < p, but near p, the value f (w) is < f (p). Derivative From f 0(p) = 3 we found that f (w) − f (p) > 0 if w is close enough to p, but 6= p.
    [Show full text]