Foundations of Calculus in the 1700'S

Total Page:16

File Type:pdf, Size:1020Kb

Foundations of Calculus in the 1700'S Foundations of Calculus in the 1700’s “Ghosts of departed quantities” Nagging Doubts • Calculus worked, and the practitioners, including Newton, Leibniz, the Bernoullis, Euler, and others, rarely made mistakes or got into trouble in using it. • Still, the explanation of why it all worked was not logically satisfactory –it was not as logically put together as Euclid, for example. • Of particular concern were these pesky infinitesimals that were sometimes ignored as being essentially 0, and sometimes not – the “abominable little zeros,” as your book says. Bishop Berkeley • The most important criticism of the foundations of early calculus came from George Berkeley, an Irish philosopher who was appointed a Bishop in the church. • His tract, The Analyst: A DISCOURSE Addressed to an Infidel Mathematician, was a response to recent attempts of the part of some philosophers and scientists to argue that Newton’s physics obviated the need for a personal God who was involved in the daily workings of the universe. Bishop Berkeley • Thus he attempted to show that the mathematics upon which Newtonian physics was based was not at all certain. • It was probably addressed to the English astronomer Edmond Halley, who expressed the opinion that religious arguments were mysterious and illogical. • So he was returning the favor. Bishop Berkeley • He didn’t deny the usefulness or validity of the results of calculus, but wanted to show that mathematicians had no valid justification for the methods they used (and thus their arguments were no more logical or valid than those of religion). From The Analyst • Berkeley pointed out “…the fallacious way of proceeding to a certain Point on the Supposition of an Increment, and then at once shifting your Supposition to that of no Increment . Since if this second Supposition had been made before the common Division by 0, all had vanished at once, and you must have got nothing by your Supposition.” From The Analyst • “Whereas by this Artifice of first dividing, and then changing your Supposition, you retain 1 and nxn‐1. But, notwithstanding all this address to cover it, the fallacy is still the same.” • “The minutest errors are not to be neglected in mathematics.” ‐‐ Berkeley used a quote from Newton. From The Analyst • “And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?” From The Analyst • “… he who can digest a second or third fluxion, a second or third difference, need not, methinks, be squeamish about any point in divinity.” • So There. Judith Grabiner on Berkeley • “Berkeley’s criticisms of the rigor of the calculus were witty, unkind, and—with respect to the mathematical practices he was criticizing—essentially correct” From Grabiner, Judith (1997), "Was Newton's Calculus a Dead End? The Continental Influence of Maclaurin's Treatise of Fluxions", The American Mathematical Monthly (Mathematical Association of America) 104 (5): 393–410 Responses to Berkeley • MacLaurin’s response was perhaps the best conceived in that it did answer Berkeley’s objections and showed a real understanding of the calculus. • However, it relied heavily on classical geometry, was 754 pages, and was rather difficult to read. MacLaurin’s Response • The processes of Newton’s calculus could be made rigorous by proving that, for example, if the rate of change of y is R, R cannot be less than 2x and cannot be more than 2x, so R must be exactly 2x. This method is similar to that used by Archimedes prove that the area of the circle couldn’t be less than 2π, and could not be greater than 2π; in other words, the good old method of exhaustion, coupled with the double reductio. MacLaurin’s Response • The problem was that, although this might work as a proof method, still the only method of finding the rates of change involved the suspect methods of infinitesimals, fluxions, etc. In addition, the proofs in general cases were quite difficult. MacLaurin’s Response • He also introduced the notion of instantaneous velocity as part of his explanatory framework. • He did not recognize that these could be measured by using limits, but his work did leave that door open. Responses to Berkeley • d’Alembert used the idea of finding the limit of the ratio as particular magnitudes vanished. In fact, d’Alembert went farther by giving a definition of limit that sounds somewhat like ours: • “One magnitude is said to be the limit of another magnitude when the second may approach the first within any given magnitude, however small, though the second magnitude may never exceed the magnitude it approaches.” Responses to Berkeley • In summary, the need for some idea of limit began to be obvious, but had its own conceptual problems. • The final appearance of a rigorous definition of limit would have to wait until Cauchy in the next century. And the next chapter..
Recommended publications
  • Nonstandard Analysis (Math 649K) Spring 2008
    Nonstandard Analysis (Math 649k) Spring 2008 David Ross, Department of Mathematics February 29, 2008 1 1 Introduction 1.1 Outline of course: 1. Introduction: motivation, history, and propoganda 2. Nonstandard models: definition, properties, some unavoidable logic 3. Very basic Calculus/Analysis 4. Applications of saturation 5. General Topology 6. Measure Theory 7. Functional Analysis 8. Probability 1.2 Some References: 1. Abraham Robinson (1966) Nonstandard Analysis North Holland, Amster- dam 2. Martin Davis and Reuben Hersh Nonstandard Analysis Scientific Ameri- can, June 1972 3. Davis, M. (1977) Applied Nonstandard Analysis Wiley, New York. 4. Sergio Albeverio, Jens Erik Fenstad, Raphael Høegh-Krohn, and Tom Lindstrøm (1986) Nonstandard Methods in Stochastic Analysis and Math- ematical Physics. Academic Press, New York. 5. Al Hurd and Peter Loeb (1985) An introduction to Nonstandard Real Anal- ysis Academic Press, New York. 6. Keith Stroyan and Wilhelminus Luxemburg (1976) Introduction to the Theory of Infinitesimals Academic Press, New York. 7. Keith Stroyan and Jose Bayod (1986) Foundations of Infinitesimal Stochas- tic Analysis North Holland, Amsterdam 8. Leif Arkeryd, Nigel Cutland, C. Ward Henson (eds) (1997) Nonstandard Analysis: Theory and Applications, Kluwer 9. Rob Goldblatt (1998) Lectures on the Hyperreals, Springer 2 1.3 Some history: • (xxxx) Archimedes • (1615) Kepler, Nova stereometria dolorium vinariorium • (1635) Cavalieri, Geometria indivisibilus • (1635) Excercitationes geometricae (”Rigor is the affair of philosophy rather
    [Show full text]
  • Casting out Beams: Berkeley's Criticism of the Calculus
    Casting Out Beams: Berkeley's Criticism of the Calculus Eugene C. Boman Penn State - Harrisburg First cast out the beam out of thine own eye; and thou shalt see clearly to cast out the mote out of thy brother's eye. King James Bible, Matthew 7:5 Calculus burst upon the scientific community in the late 17th century with unparalleled success. Nevertheless it took another 200 years and the combined efforts of many first rate mathematicians to finally "get it right." In particular the fundamental concept of a limit did not take on its final form until the 1800's with the work of Karl Weierstrass. The Method of Fluxions as it was known in Britain, was invented by Isaac Newton in the 1660's. For reasons known only to him, he did not publish his invention until much later. In the meantime Gottfried Wilhelm Leibniz published his own formulation of Calculus which he called Calculus Differentialis in 1684. Newton's approach was kinematic. Quantities were said to "flow" in time and the fluxion (derivative) of a flowing quantity was interpreted as a velocity. Leibniz's approach was based on the idea of infinitely small quantities: infinitesimals. Their public priority brawl is both legendary [11] and a stain on both of their reputations. A slightly less well known public argument, but mathematically far more important, began in 1734 when Bishop George Berkeley published a treatise entitled The Analyst. Berkeley derided the lack of rigor in the fundamentals of both renderings of the topic. To be sure, neither Newton's nor Leibniz's formulation of the Calculus would stand up to the modern standard of rigor.
    [Show full text]
  • Arxiv:0811.0164V8 [Math.HO] 24 Feb 2009 N H S Gat2006393)
    A STRICT NON-STANDARD INEQUALITY .999 ...< 1 KARIN USADI KATZ AND MIKHAIL G. KATZ∗ Abstract. Is .999 ... equal to 1? A. Lightstone’s decimal expan- sions yield an infinity of numbers in [0, 1] whose expansion starts with an unbounded number of repeated digits “9”. We present some non-standard thoughts on the ambiguity of the ellipsis, mod- eling the cognitive concept of generic limit of B. Cornu and D. Tall. A choice of a non-standard hyperinteger H specifies an H-infinite extended decimal string of 9s, corresponding to an infinitesimally diminished hyperreal value (11.5). In our model, the student re- sistance to the unital evaluation of .999 ... is directed against an unspoken and unacknowledged application of the standard part function, namely the stripping away of a ghost of an infinitesimal, to echo George Berkeley. So long as the number system has not been specified, the students’ hunch that .999 ... can fall infinites- imally short of 1, can be justified in a mathematically rigorous fashion. Contents 1. The problem of unital evaluation 2 2. A geometric sum 3 3.Arguingby“Itoldyouso” 4 4. Coming clean 4 5. Squaring .999 ...< 1 with reality 5 6. Hyperreals under magnifying glass 7 7. Zooming in on slope of tangent line 8 arXiv:0811.0164v8 [math.HO] 24 Feb 2009 8. Hypercalculator returns .999 ... 8 9. Generic limit and precise meaning of infinity 10 10. Limits, generic limits, and Flatland 11 11. Anon-standardglossary 12 Date: October 22, 2018. 2000 Mathematics Subject Classification. Primary 26E35; Secondary 97A20, 97C30 . Key words and phrases.
    [Show full text]
  • Slopes, Rates of Change, and Derivatives
    2.2 Slopes, Rates of Change, and Derivatives APPLICATION PREVIEW The Confused Creation of Calculus* Calculus evolved from consideration of four major problems that were current in 17th century Europe: how to define instantaneous velocity, how to define tangent lines to curves, how to find maxi- mum and minimum values of functions, and how to calculate areas and volumes, all of which will be discussed in this and the follow- ing chapters. While calculus is now presented as a logical mathe- matical system, it began instead as a collection of self-contradictory ideas and poorly understood techniques, accompanied by much controversy and criticism. It was developed by two people of dia- metrically opposed temperaments, Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716), each while in his twenties (Newton at age 22 and Leibniz at about 29). Newton attended mediocre schools in England, entered Cam- bridge University (with a deficiency in Euclidean geometry), im- mersed himself in solitary studies, and graduated with no particular distinction. He then returned to his family’s small farm where, during a 2-year period, he singlehandedly developed calcu- lus, in addition to the science of optics and the theory of universal gravitation, all of which he kept to himself. He then returned to Cambridge for a master’s degree and secured an appointment as a professor, after which he became even more solitary and intro- verted. A contemporary described him as never taking “any recre- ation or pastime either in riding out to take the air, walking, bowling, or any other exercise whatever, thinking all hours lost that was not spent in his studies.” He was often seen “with shoes down at heels, stockings untied,..., and his head scarcely combed.” He was not a popular teacher, sometimes lecturing to empty class- rooms, and his mathematical writings were very difficult to under- stand (he told one friend that he wrote this way intentionally “to avoid being bated by little smatterers in mathematics”).
    [Show full text]
  • The Early Criticisms of the Calculus of Newton and Leibniz
    Ghosts of Departed Errors: The Early Criticisms of the Calculus of Newton and Leibniz Eugene Boman Mathematics Seminar, February 15, 2017 Eugene Boman is Associate Professor of Mathematics at Penn State Harrisburg. alculus is often taught as if it is a pristine thing, emerging Athena-like, complete and whole, from C the head of Zeus. It is not. In the early days the logical foundations of Calculus were highly suspect. It seemed to work well and effortlessly, but why it did so was mysterious to say the least and until this foundational question was answered the entire enterprise was suspect, despite its remarkable successes. It took nearly two hundred years to develop a coherent foundational theory and, when completed, the Calculus that emerged was fundamentally different from the Calculus of the founders. To illustrate this difference I would like to compare two proofs of an elementary result from Calculus: That the derivative of sin⁡(푥) is cos(푥). The first is a modern proof both in content and style. The second is in the style of the Seventeenth and Eighteenth centuries. A MODERN PROOF THAT THE DERIVATIVE OF 푠푛(푥) IS 푐표푠(푥) 푑(sin(푥)) A modern proof that = cos(푥) is not a simple thing. It depends on the theory of limits, which we 푑푥 will simply assume, and in particular on the following lemma, which we will prove. sin⁡(푥) Lemma: lim = 1. 푥→0⁡ 푥 Proof of Lemma: Since in Figure 1 the radius of the circle is one we see that radian measure of the angle and the length of the subtended arc are equal.
    [Show full text]
  • Arxiv:1210.7750V1 [Math.HO] 29 Oct 2012 Tsml Ehdo Aiaadmnm;Rfato;Selslaw
    ALMOST EQUAL: THE METHOD OF ADEQUALITY FROM DIOPHANTUS TO FERMAT AND BEYOND MIKHAIL G. KATZ, DAVID M. SCHAPS, AND STEVEN SHNIDER Abstract. We analyze some of the main approaches in the liter- ature to the method of ‘adequality’ with which Fermat approached the problems of the calculus, as well as its source in the παρισoτης´ of Diophantus, and propose a novel reading thereof. Adequality is a crucial step in Fermat’s method of finding max- ima, minima, tangents, and solving other problems that a mod- ern mathematician would solve using infinitesimal calculus. The method is presented in a series of short articles in Fermat’s col- lected works [66, p. 133-172]. We show that at least some of the manifestations of adequality amount to variational techniques ex- ploiting a small, or infinitesimal, variation e. Fermat’s treatment of geometric and physical applications sug- gests that an aspect of approximation is inherent in adequality, as well as an aspect of smallness on the part of e. We question the rel- evance to understanding Fermat of 19th century dictionary defini- tions of παρισoτης´ and adaequare, cited by Breger, and take issue with his interpretation of adequality, including his novel reading of Diophantus, and his hypothesis concerning alleged tampering with Fermat’s texts by Carcavy. We argue that Fermat relied on Bachet’s reading of Diophantus. Diophantus coined the term παρισoτης´ for mathematical pur- poses and used it to refer to the way in which 1321/711 is ap- proximately equal to 11/6. Bachet performed a semantic calque in passing from pariso¯o to adaequo.
    [Show full text]
  • Gottfried Wilhelm Leibniz (1646 – 1716)
    Gottfried Wilhelm Leibniz (1646 – 1716) From Wikipedia, the free encyclopedia, http://en.wikipedia.org/wiki/Gottfried_Leibniz Leibniz occupies a prominent place in the history of mathematics and the history of philosophy. He developed the infinitesimal calculus independently of Isaac Newton, and Leibniz's mathematical notation has been widely used ever since it was published. He became one of the most prolific inventors in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal's calculator, he was the first to describe a pinwheel calculator in 1685 and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. He also refined the binary number system, which is at the foundation of virtually all digital computers. In philosophy, Leibniz is mostly noted for his optimism, e.g., his conclusion that our Universe is, in a restricted sense, the best possible one that God could have created. Leibniz, along with René Descartes and Baruch Spinoza, was one of the three great 17th century advocates of rationalism. The work of Leibniz anticipated modern logic and analytic philosophy, but his philosophy also looks back to the scholastic tradition, in which conclusions are produced by applying reason to first principles or prior definitions rather than to empirical evidence. Leibniz made major contributions to physics and technology, and anticipated notions that surfaced much later in biology, medicine, geology, probability theory, psychology, linguistics, and information science. He wrote works on politics, law, ethics, theology, history, philosophy, and philology. Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters, and in unpublished manuscripts.
    [Show full text]
  • Archiv Für Mathema- Tische Logik Und Grundlagenforschung, 6, 7–29
    INFINITESIMALPROBABILITIES Sylvia Wenmackers 5 Suppose that a dart is thrown, using the unit interval as a target; then what is the probability of hitting a point? Clearly this probability cannot be a positive real number, yet to say that it is zero violates the intuitive feeling that, after all, there is some chance of hitting the point. —Bernstein and Wattenberg (1969, p. 171) It has been said that to assume that 0 + 0 + 0 + ... + 0 + ... = 1 is absurd, whereas, if at all, this would be true if ‘actual infinitesimal’ were substituted in place of zero. —de Finetti (1974, p. 347) Infinitesimals played an important role in the seventeenth century devel- opment of the calculus by Leibniz and—to a lesser extent—by Newton. In the twentieth century, calculus was applied to probability theory. By this time, however, Leibnizian infinitesimals had lost their prominence in mainstream calculus, such that “infinitesimal probability” did not become a central concept in mainstream probability theory either. Meanwhile, non- standard analysis (NSA) has been developed by Abraham Robinson, an alternative approach to the calculus, in which infinitesimals (in the sense of Equation 1 below) are given mathematically consistent foundations. This provides us with an interesting framework to investigate the notion of infinitesimal probabilities, as we will do in this chapter. Even taken separately, both infinitesimals and probabilities constitute major topics in philosophy and related fields. Infinitesimals are numbers that are infinitely small or extremely minute. The history of non-zero in- finitesimals is a troubled one: despite their crucial role in the development of the calculus, they were long believed to be based on an inconsistent concept.
    [Show full text]
  • Math in 18Th Century.Pdf
    MATHEMATICS IN THE 18th. CENTURY-SELECTIVE CHRONOLOGY 1707- Leonhard Euler born near Basel, Switzerland- his father was a pastor. 1713-Posthumous publication of Ars Conjectandi, by Jakob Bernouilli (1654-1705), a treatise on probability. 1714- Brooke Taylor (1685-1731), of `Taylor’s formula’ fame (published 1715) publishes a paper dealing with the motion of a taut vibrating string. 1721- Euler studies at the University of Basel, and is advised in his studies by Johann Bernouilli (1667- 1748). He earns a master’s degree in philosophy and enters divinity school, but quits in favor of math. 1725-Daniel Bernouilli (1700-1782, Johann’s son) assumes the chair in mathematics at the new Academy of Sciences in St. Petersburg (Russia). 1727-Euler arrives in St. Petersburg, initially to occupy the chair in physiology/medicine. In 1733 Daniel Bernouilli gets a position in Switzerland, and Euler is named to the mathematics chair. 1730- Euler’s paper on the factorial function 1733.-Euler marries Katharina Gesell, daughter of a Swiss painter living in Russia. They would be together until her death in 1773, and have 13 children (only 5 survived to adolescence, 3 survived their parents.) 1733- Abraham de Moivre (1667-1754) publishes a version of the `central limit theorem’, as well as an approximation to the factorial equivalent to `Stirling’s formula’, published by James Stirling (1693-1770) in 1730. 1734- Bishop George Berkeley publishes The Analyst, which includes an attack on the foundations of calculus (infinitesimals.) 1735- Euler solves the `Basel Problem’ (sum of reciprocals of squares.) 1736- Euler’s textbook Mechanica; analytical development of Newton’s dynamics of a point mass.
    [Show full text]
  • Studies on Da Costa's Paraconsistent
    South American Journal of Logic Vol. 2, n. 2, pp. 319{332, 2016 ISSN: 2446-6719 Studies on da Costa's Paraconsistent Differential Calculus: Hypermetric and Hyperfunctions Itala M. Loffredo D'Ottaviano and Marcelo Reicher Soares We dedicate this paper to our friend Chico Miraglia, as a testament of our affection and respect for this exemplary scholar and tireless battler for social causes. Abstract In this paper we present a new proposal for the development of da Costa's paraconsistent differential calculus, from the introduction of a new concept of ∗ metric, a hypermetric. After presenting the hyperring A and the quasi-ring A , that extend the set R of the real numbers, we introduce a hypermetric on A and generalized concepts of limit, continuity and derivability of hyperfunctions. Keywords: Paraconsistent differential calculus, infinitesimals, infinite numbers, hyper- real numbers, hypermetric, hyperfunctions, limit of hyperfunctions, continuity, deriva- bility. 1 Introduction Ever since the beginning of the 18th century, the criticisms by various philosophers and mathematicians of the differential and integral calculus, created independently by Leibniz and Newton at the end of the 17th century, have been well known. The apparent intrinsic inconsistency of the concept of the infinitesimal, which was not clarified by either Leibniz or Newton, is at the center of these discussions, as seen, for instance, in George Berkeley's well-known The Analyst (see Berkeley (1774)). Independently of such critiques, during the next two centuries several mathematici- ans contributed to the development of the calculus, in special Jean le Rond d'Alembert and Augustin-Louis Cauchy, both having proposed that the concept of the limit of a function should give the foundation of the calculus.
    [Show full text]
  • DE MOTU and the ANALYST the New Synthese Historical Library Texts and Studies in the History of Philosophy
    DE MOTU AND THE ANALYST The New Synthese Historical Library Texts and Studies in the History of Philosophy VOLUME 41 Series Editor: NORMAN KRETZMANN, Cornell University Associate Editors: DANIEL ELLIOT GARBER, University of Chicago SIMO KNUUTTILA, University of Helsinki RICHARD SORABJI, University of London Editorial Consultants: JAN A. AERTSEN, Free University, Amsterdam ROGER ARIEW, Virginia Polytechnic Institute E. JENNIFER ASHWORTH, University of Waterloo MICHAEL AYERS, Wadham College, Oxford GAiL FINE, Cornell University R. J. HANKINSON, University of Texas JAAKKO HINTIKKA, Boston University, Finnish Academy PAUL HOFFMAN, Massachusetts Institute of Technology DAVID KONSTAN, Brown University RICHARD H. KRAUT, University of Illinois, Chicago ALAIN DE LmERA, Ecole Pratique des Hautes Etudes, Sorbonne DAVID FATE NORTON, McGill University LUCA OBERTELLO, Universita degli Studi di Genova ELEONORE STUMP, Virginia Polytechnic Institute ALLEN WOOD, Cornell University The titles published in this series are listed at the end of this volume. GEORGE BERKELEY DE MOTU AND THEANALYST A Modem Edition, with Introductions and Commentary Edited and translated by DOUGLAS M. JESSEPH North Carolina State University, Raleigh, U.S.A. SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. Library of Congress Cataloging-in-Publication Data Berkeley, George, 1685-1753. [De motul De motu, and, the analyst I by George Berkeley; a modern edition with introductions and commentary by Douglas M. Josseph. p. cm. -- (The New synthese historical library ; v. 41) Inc 1udes index. ISBN 978-94-010-5144-6 ISBN 978-94-011-2592-5 (eBook) DOI 10.1007/978-94-011-2592-5 1. Science--Philosophy. 2. Mathematics--Philosophy. 1. Jesseph, Douglas M. II. Berkeley, George, 1685-1753.
    [Show full text]
  • Differentials and Differential Coefficients in the Eulerian
    Historia Mathematica 31 (2004) 34–61 www.elsevier.com/locate/hm Differentials and differential coefficients in the Eulerian foundations of the calculus Giovanni Ferraro via Nazionale, 38, I-80021 Afragola, Naples, Italy Dedicated to P.L. Butzer, L. Carbone, F. Jongmans, F. Palladino, and the ethics of scientific research Abstract In the 18th-century calculus the classical notion of quantity was understood as general quantity, which was expressed analytically and was subject to formal manipulation. Number was understood as the measure of quantity; however, only fractions and natural numbers were considered numbers in the true sense of term. The other types of numbers were fictitious entities, namely ideal entities firmly founded in the real world which could be operated upon as if they were numbers. In this context Eulerian infinitesimals should also be considered as fictitious numbers. They were symbols that represented a primordial and intuitive idea of limit, although they were manipulated in the same way as numbers. This conception allowed Euler to consider calculus as a calculus of functions (intended as analytical expressions of quantities) and, at the same time, to handle differentials formally. 2003 Elsevier Inc. All rights reserved. Sommario Nel diciottesimo secolo la classica nozione di quantità fu sviluppata fino ad essere intesa come quantità generale, la quale, analiticamente espressa, era soggetta a manipolazioni formali. Il numero era inteso come una misura della quantità; tuttavia solo le frazioni e i naturali erano considerati numeri nel vero senso del termine. Le altre specie di numeri erano entità fittizie, cioè entità ideali ben fondate nel reale che potevano essere manipolate come numeri.
    [Show full text]