Foundations of Calculus in the 1700 s. Ghosts of departed quantities
|
|
- Collin Stevens
- 6 years ago
- Views:
Transcription
1 Foundations of Calculus in the 1700 s Ghosts of departed quantities
2 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 not as logically put together as Euclid, for examples. Of particular concern were these pesky infinitesimals that were sometimes ignored as being essentially 0, and sometimes not.
3 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.
4 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.
5 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).
6 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 o, all had vanished at once, and you must have got nothing by your Supposition.
7 From The Analyst Whereas by this Artifice of first dividing, and then changing your Supposition, you retain 1 and nx n 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. A quote from Newton.
8 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?
9 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.
10 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):
11 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.
12 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.
13 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.
14 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.
15 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.
16 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.
Rigorization of Calculus. 18 th Century Approaches,Cauchy, Weirstrass,
Rigorization of Calculus 18 th Century Approaches,Cauchy, Weirstrass, Basic Problem In finding derivatives, pretty much everyone did something equivalent to finding the difference ratio and letting. Of
More informationInfinity. Newton, Leibniz & the Calculus
Infinity Newton, Leibniz & the Calculus Aristotle: Past time can t be infinite because there can t be an endless chain of causes (movements) preceding the present. Descartes: Space as extension; the res
More informationChapter 10. Definition of the Derivative Velocity and Tangents
Chapter 10 Definition of the Derivative 10.1 Velocity and Tangents 10.1 Notation. If E 1 (x, y) and E 2 (x, y) denote equations or inequalities in x and y, we will use the notation {E 1 (x, y)} = {(x,
More informationHow Euler Did It. by Ed Sandifer. Foundations of Calculus. September 2006
How Euler Did It Foundations of Calculus September 2006 by Ed Sandifer As we begin a new academic year, many of us are introducing another generation of students to the magic of calculus. As always, those
More informationAn Introduction to a Rigorous Definition of Derivative
Ursinus College Digital Commons @ Ursinus College Analysis Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) 017 An Introduction to a Rigorous Definition of
More informationPaper read at History of Science Society 2014 Annual Meeting, Chicago, Nov. 9,
Euler s Mechanics as Opposition to Leibnizian Dynamics 1 Nobumichi ARIGA 2 1. Introduction Leonhard Euler, the notable mathematician in the eighteenth century, is also famous for his contributions to mechanics.
More informationName Class Date. Ptolemy alchemy Scientific Revolution
Name Class Date The Scientific Revolution Vocabulary Builder Section 1 DIRECTIONS Look up the vocabulary terms in the word bank in a dictionary. Write the dictionary definition of the word that is closest
More informationIn today s world, people with basic calculus knowledge take the subject for granted. As
Ziya Chen Math 4388 Shanyu Ji Calculus In today s world, people with basic calculus knowledge take the subject for granted. As long as they plug in numbers into the right formula and do the calculation
More informationBeyond Newton and Leibniz: The Making of Modern Calculus. Anthony V. Piccolino, Ed. D. Palm Beach State College Palm Beach Gardens, Florida
Beyond Newton and Leibniz: The Making of Modern Calculus Anthony V. Piccolino, Ed. D. Palm Beach State College Palm Beach Gardens, Florida Calculus Before Newton & Leibniz Four Major Scientific Problems
More informationMath 4388 Amber Pham 1. The Birth of Calculus. for counting. There are two major interrelated topics in calculus known as differential and
Math 4388 Amber Pham 1 The Birth of Calculus The literal meaning of calculus originated from Latin, which means a small stone used for counting. There are two major interrelated topics in calculus known
More information1). To introduce and define the subject of mechanics. 2). To introduce Newton's Laws, and to understand the significance of these laws.
2 INTRODUCTION Learning Objectives 1). To introduce and define the subject of mechanics. 2). To introduce Newton's Laws, and to understand the significance of these laws. 3). The review modeling, dimensional
More informationMthEd/Math 300 Williams Fall 2011 Midterm Exam 3
Name: MthEd/Math 300 Williams Fall 2011 Midterm Exam 3 Closed Book / Closed Note. Answer all problems. You may use a calculator for numerical computations. Section 1: For each event listed in the first
More information9.2 Taylor Polynomials
9.2 Taylor Polynomials Taylor Polynomials and Approimations Polynomial functions can be used to approimate other elementary functions such as sin, Eample 1: f at 0 Find the equation of the tangent line
More informationContents. Preface. Chapter 3. Sequences of Partial Sums Series in the 17th century Taylor series Euler s influence 96.
Contents Preface ix Chapter 1. Accumulation 1 1.1. Archimedes and the volume of the sphere 1 1.2. The area of the circle and the Archimedean Principle 6 1.3. Islamic contributions 9 1.4. The binomial theorem
More informationDate: Tuesday, 25 September :00PM. Location: Museum of London
Ghosts of Departed Quantities: Calculus and its Limits Transcript Date: Tuesday, 25 September 2012-1:00PM Location: Museum of London 25 September 2012 Ghosts of Departed Quantities: Calculus and its Limits
More informationLecture 8. Eudoxus and the Avoidance of a Fundamental Conflict
Lecture 8. Eudoxus and the Avoidance of a Fundamental Conflict Eudoxus of Cnidus Eudoxus, 480 BC - 355 BC, was a Greek philosopher, mathematician and astronomer who contributed to Euclid s Elements. His
More information8. Reductio ad absurdum
8. Reductio ad absurdum 8.1 A historical example In his book, The Two New Sciences, Galileo Galilea (1564-1642) gives several arguments meant to demonstrate that there can be no such thing as actual infinities
More informationTHE MATHEMATICS OF EULER. Introduction: The Master of Us All. (Dunham, Euler the Master xv). This quote by twentieth-century mathematician André Weil
THE MATHEMATICS OF EULER Introduction: The Master of Us All All his life he seems to have carried in his head the whole of the mathematics of his day (Dunham, Euler the Master xv). This quote by twentieth-century
More information2500 years of very small numbers
2500 years of very small numbers (The myth of infinitesimals?) David A. Ross Department of Mathematics University of Hawai i Definition: An infinitesimal is a quantity which is smaller than any finite
More information= 0. Theorem (Maximum Principle) If a holomorphic function f defined on a domain D C takes the maximum max z D f(z), then f is constant.
38 CHAPTER 3. HOLOMORPHIC FUNCTIONS Maximum Principle Maximum Principle was first proved for harmonic functions (i.e., the real part and the imaginary part of holomorphic functions) in Burkhardt s textbook
More informationTo Infinity and Beyond
To Infinity and Beyond 25 January 2012 To Infinity and Beyond 25 January 2012 1/24 The concept of infinity has both fascinated and frustrated people for millenia. We will discuss some historical problems
More informationThe Limit of Humanly Knowable Mathematical Truth
The Limit of Humanly Knowable Mathematical Truth Gödel s Incompleteness Theorems, and Artificial Intelligence Santa Rosa Junior College December 12, 2015 Another title for this talk could be... An Argument
More informationTOOLING UP MATHEMATICS FOR ENGINEERING*
TOOLING UP MATHEMATICS FOR ENGINEERING* BY THEODORE von KARMAN California Institute of Technology It has often been said that one of the primary objectives of Mathematics is to furnish tools to physicists
More informationLecture 2: What is Proof?
Lecture 2: What is Proof? Math 295 08/26/16 Webster Proof and Its History 8/2016 1 / 1 Evolution of Proof Proof, a relatively new idea Modern mathematics could not be supported at its foundation, nor construct
More informationChapter 0. Introduction. An Overview of the Course
Chapter 0 Introduction An Overview of the Course In the first part of these notes we consider the problem of calculating the areas of various plane figures. The technique we use for finding the area of
More informationLeibniz and the Discovery of Calculus. The introduction of calculus to the world in the seventeenth century is often associated
Leibniz and the Discovery of Calculus The introduction of calculus to the world in the seventeenth century is often associated with Isaac Newton, however on the main continent of Europe calculus would
More informationGhosts of Departed Errors: 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
More informationMATH1014 Calculus II. A historical review on Calculus
MATH1014 Calculus II A historical review on Calculus Edmund Y. M. Chiang Department of Mathematics Hong Kong University of Science & Technology September 4, 2015 Instantaneous Velocities Newton s paradox
More information8. Reductio ad absurdum
8. Reductio ad absurdum 8.1 A historical example In his book, The Two New Sciences, 10 Galileo Galilea (1564-1642) gives several arguments meant to demonstrate that there can be no such thing as actual
More informationSession 7: Pseudoproofs and Puzzles - Handout
Session 7: Pseudoproofs and Puzzles - Handout Many who have had an opportunity of knowing any more about mathematics confuse it with arithmetic, and consider it an arid science. In reality, however, it
More informationO1 History of Mathematics Lecture VIII Establishing rigorous thinking in analysis. Monday 30th October 2017 (Week 4)
O1 History of Mathematics Lecture VIII Establishing rigorous thinking in analysis Monday 30th October 2017 (Week 4) Summary French institutions Fourier series Early-19th-century rigour Limits, continuity,
More informationSam Butler Hunziker (Department of Mathematics)
Sam Butler Hunziker (Department of Mathematics) George Berkeley s Mathematical Philosophy And The Calculus a Berkeley s Life As noted by Ronald Calinger (Calinger 1982), George Berkeley was born (1685)
More informationIntroduction to Real Analysis MATH 2001
Introduction to Real Analysis MATH 2001 Juris Steprāns York University September 13, 2010 Instructor Instructor: Juris Steprāns My office is N530 in the Ross building. Office hours are Mondays from 4:00
More informationHow Euler Did It. Today we are fairly comfortable with the idea that some series just don t add up. For example, the series
Divergent series June 2006 How Euler Did It by Ed Sandifer Today we are fairly comfortable with the idea that some series just don t add up. For example, the series + + + has nicely bounded partial sums,
More informationL1. Determine the limit of a function at a point both graphically and analytically
L1. Determine the limit of a function at a point both graphically and analytically The concept of a limit is essential to the development and understanding of Calculus. Limits are used in the definition
More informationDoes Really Equal 1?
The Mathematics Educator 2011/2012 Vol. 21, No. 2, 58 67 Does 0. Really Equal 1? Anderson Norton and Michael Baldwin This article confronts the issue of why secondary and postsecondary students resist
More informationO1 History of Mathematics Lecture VI Successes of and difficulties with the calculus: the 18th-century beginnings of rigour
O1 History of Mathematics Lecture VI Successes of and difficulties with the calculus: the 18th-century beginnings of rigour Monday 22nd October 2018 (Week 3) Summary Publication and acceptance of the calculus
More informationEuler s Galilean Philosophy of Science
Euler s Galilean Philosophy of Science Brian Hepburn Wichita State University Nov 5, 2017 Presentation time: 20 mins Abstract Here is a phrase never uttered before: Euler s philosophy of science. Known
More informationEquivalent Forms of the Axiom of Infinity
Equivalent Forms of the Axiom of Infinity Axiom of Infinity 1. There is a set that contains each finite ordinal as an element. The Axiom of Infinity is the axiom of Set Theory that explicitly asserts that
More informationSymbolic Analytic Geometry and the Origins of Symbolic Algebra
Viète and Descartes Symbolic Analytic Geometry and the Origins of Symbolic Algebra Waseda University, SILS, History of Mathematics Outline Introduction François Viète Life and work Life and work Descartes
More informationNewton s Work on Infinite Series. Kelly Regan, Nayana Thimmiah, & Arnold Joseph Math 475: History of Mathematics
Newton s Work on Infinite Series Kelly Regan, Nayana Thimmiah, & Arnold Joseph Math 475: History of Mathematics What is an infinite series? The sum of terms that follow some rule. The series is the sum
More informationNumbers, proof and all that jazz.
CHAPTER 1 Numbers, proof and all that jazz. There is a fundamental difference between mathematics and other sciences. In most sciences, one does experiments to determine laws. A law will remain a law,
More informationZeno s Paradox #1. The Achilles
Zeno s Paradox #1. The Achilles Achilles, who is the fastest runner of antiquity, is racing to catch the tortoise that is slowly crawling away from him. Both are moving along a linear path at constant
More informationFinal Exam Extra Credit Opportunity
Final Exam Extra Credit Opportunity For extra credit, counted toward your final exam grade, you can write a 3-5 page paper on (i) Chapter II, Conceptions in Antiquity, (ii) Chapter V, Newton and Leibniz,
More informationfrom Euclid to Einstein
WorkBook 2. Space from Euclid to Einstein Roy McWeeny Professore Emerito di Chimica Teorica, Università di Pisa, Pisa (Italy) A Pari New Learning Publication Book 2 in the Series WorkBooks in Science (Last
More informationHonors Day 2015 Archimedes Balancing Calculus
Honors Day 2015 Department of Mathematics Florida State University Honors Day Florida State University, Tallahassee, FL April 10, 2015 Historical Time Line 1906 Palimpsest discovered Rigorous development
More informationINTRODUCTION TO THE QUADRATURE OF CURVES. Isaac Newton. Translated into English by. John Harris
INTRODUCTION TO THE QUADRATURE OF CURVES By Isaac Newton Translated into English by John Harris Edited by David R. Wilkins 00 NOTE ON THE TEXT This translation of Isaac Newton s Introductio ad Quadraturam
More informationLecture 5. Zeno s Four Paradoxes of Motion
Lecture 5. Zeno s Four Paradoxes of Motion Science of infinity In Lecture 4, we mentioned that a conflict arose from the discovery of irrationals. The Greeks rejection of irrational numbers was essentially
More information3 The Semantics of the Propositional Calculus
3 The Semantics of the Propositional Calculus 1. Interpretations Formulas of the propositional calculus express statement forms. In chapter two, we gave informal descriptions of the meanings of the logical
More informationMathematical Misnomers: Hey, who really discovered that theorem!
Mathematical Misnomers: Hey, who really discovered that theorem! Mike Raugh mikeraugh.org LACC Math Contest 24th March 2007 Who was buried in Grant s tomb? Ulysss S. Grant, of course! These are true too:
More informationIs mathematics discovery or invention?
Is mathematics discovery or invention? From the problems of everyday life to the mystery of existence Part One By Marco Dal Prà Venice - Italy May 2013 1 Foreword This document is intended as a starting
More informationPoincaré, Heisenberg, Gödel. Some limits of scientific knowledge. Fernando Sols Universidad Complutense de Madrid
Poincaré, Heisenberg, Gödel. Some limits of scientific knowledge. Fernando Sols Universidad Complutense de Madrid Henry Poincaré (1854-1912) nonlinear dynamics Werner Heisenberg (1901-1976) uncertainty
More informationOn divergent series. Leonhard Euler
On divergent series Leonhard Euler Because convergent series are defined in that manner, that they consist of continuously decreasing terms, that finally, if the series continues to infinity, vanish completely;
More informationCarrying On with Infinite Decimals
Carrying On with Infinite Decimals by Jim Propp (University of Massachusetts, Lowell), with the members of the MIT Chips and Rotors Research Initiative (Giuliano Giacaglia, Holden Lee, Delong Meng, Henrique
More informationEuclidean Geometry. The Elements of Mathematics
Euclidean Geometry The Elements of Mathematics Euclid, We Hardly Knew Ye Born around 300 BCE in Alexandria, Egypt We really know almost nothing else about his personal life Taught students in mathematics
More informationInfinity: is the universe infinite? Peter J. Cameron School of Mathematics and Statistics University of St Andrews
Infinity: is the universe infinite? Peter J Cameron School of Mathematics and Statistics University of St Andrews The Library of Babel The influential Argentine writer Jorge Luis Borges was fascinated
More informationReal Analysis. a short presentation on what and why
Real Analysis a short presentation on what and why I. Fourier Analysis Fourier analysis is about taking functions and realizing them or approximating them in terms of periodic (trig) functions. Many, many
More informationA Conundrum concerning the area of a sphere
return to updates A Conundrum concerning the area of a sphere by Miles Mathis Since the surface area of a sphere and the surface area of an open cylinder of equal height are both 4πr 2, let us look at
More informationO1 History of Mathematics Lecture XIII Complex analysis. Monday 21st November 2016 (Week 7)
O1 History of Mathematics Lecture XIII Complex analysis Monday 21st November 2016 (Week 7) Summary Complex numbers Functions of a complex variable The Cauchy Riemann equations Contour integration Cauchy
More informationThe Copernican Revolution
The Copernican Revolution The Earth moves and Science is no longer common sense 1 Nicholas Copernicus 1473-1543 Studied medicine at University of Crakow Discovered math and astronomy. Continued studies
More information(Ill) The square of the period of the planet's revolution around the sun is proportional to the cube of the semi-major axis of the elliptic orbit.
CALCULUS 11 1 Dr. M. K. Siu Translation by Dr. P. Y. H. Panl In the last episode, we mentioned the work of the father and son mathematician team Zu Chongzhi and Zu Geng during the North-South Period (AD
More informationSyllabus for MTH U201: History of Mathematics
Syllabus for MTH U201: History of Mathematics Instructor: Professor Mark Bridger Office: 527 Nightingale; ext. 2450 Hours: M,W,Th, 1:00-1:30, and by appointment e-mail: bridger@neu.edu Text: The History
More informationTHE RISE OF MODERN SCIENCE CHAPTER 20, SECTION 2
THE RISE OF MODERN SCIENCE CHAPTER 20, SECTION 2 ORIGINS OF THE SCIENTIFIC REVOLUTION 335 BCE-1687 CE A New View of the Universe Scientists of the 1500s asked same questions as Greeks: What is the universe
More informationAN INTRODUCTION TO LAGRANGE EQUATIONS. Professor J. Kim Vandiver October 28, 2016
AN INTRODUCTION TO LAGRANGE EQUATIONS Professor J. Kim Vandiver October 28, 2016 kimv@mit.edu 1.0 INTRODUCTION This paper is intended as a minimal introduction to the application of Lagrange equations
More informationTHE CATECHISM OF THE AUTHOR OF THE MINUTE PHILOSOPHER FULLY ANSWERED. Jacob Walton. Edited by David R. Wilkins
THE CATECHISM OF THE AUTHOR OF THE MINUTE PHILOSOPHER FULLY ANSWERED By Jacob Walton Edited by David R. Wilkins 2002 NOTE ON THE TEXT The following spellings, differing from modern British English, are
More informationThe paradox of knowability, the knower, and the believer
The paradox of knowability, the knower, and the believer Last time, when discussing the surprise exam paradox, we discussed the possibility that some claims could be true, but not knowable by certain individuals
More informationIt s a Small World After All Calculus without s and s
It s a Small World After All Calculus without s and s Dan Sloughter Department of Mathematics Furman University November 18, 2004 Smallworld p1/39 L Hôpital s axiom Guillaume François Antoine Marquis de
More informationGalilean Spacetime (Neo-Newtonian Spacetime) P t = 2
(Neo-Newtonian Spacetime) Figure V t t t = t* vt x P t = 2 t = 1 t = 0 O x x vt x I used figure V above to show you how to change the coordinates of point P in the coordinate system S to coordinates in
More informationStories from the Development of Real Analysis
Stories from the Development of Real Analysis David Bressoud Macalester College St. Paul, MN PowerPoint available at www.macalester.edu/~bressoud/talks Texas Sec)on University of Texas Tyler Tyler, TX
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES In section 11.9, we were able to find power series representations for a certain restricted class of functions. INFINITE SEQUENCES AND SERIES
More informationCISC-102 Fall 2018 Week 11
page! 1 of! 26 CISC-102 Fall 2018 Pascal s Triangle ( ) ( ) An easy ( ) ( way ) to calculate ( ) a table of binomial coefficients was recognized centuries ago by mathematicians in India, ) ( ) China, Iran
More informationThe Analyst: a Discourse addressed to an Infidel Mathematicia. George Berkeley
The Analyst: a Discourse addressed to an Infidel Mathematicia George Berkeley Table of Contents The Analyst: a Discourse addressed to an Infidel Mathematician...1 George Berkeley...1 THE ANALYST: A Discourse
More informationWave-Particle Duality & the Two-Slit Experiment: Analysis
PHYS419 Lecture 17: Wave-Particle Duality & the Two-Slit Experiment 1 Wave-Particle Duality & the Two-Slit Experiment: Analysis In the last lecture, we saw that in a two-slit experiment electrons seem
More informationBefore we work on deriving the Lorentz transformations, let's first look at the classical Galilean transformation.
Background The curious "failure" of the Michelson-Morley experiment in 1887 to determine the motion of the earth through the aether prompted a lot of physicists to try and figure out why. The first attempt
More informationContents Part A Number Theory Highlights in the History of Number Theory: 1700 BC 2008
Contents Part A Number Theory 1 Highlights in the History of Number Theory: 1700 BC 2008... 3 1.1 Early Roots to Fermat... 3 1.2 Fermat... 6 1.2.1 Fermat s Little Theorem... 7 1.2.2 Sums of Two Squares...
More informationMthEd/Math 300 Williams Fall 2011 Midterm Exam 2
Name: MthEd/Math 300 Williams Fall 2011 Midterm Exam 2 Closed Book / Closed Note. Answer all problems. You may use a calculator for numerical computations. Section 1: For each event listed in the first
More informationSCIENTIFIC REVOLUTION
SCIENTIFIC REVOLUTION VOCABULARY: SCIENTIFIC REVOLUTION Revolution a sweeping change Geocentric earth-centered universe Astronomer scientist who studies the motion of stars and planets Heliocentric sun-centered
More informationTo Infinity and Beyond. To Infinity and Beyond 1/43
To Infinity and Beyond To Infinity and Beyond 1/43 Infinity The concept of infinity has both fascinated and frustrated people for millennia. We will discuss some historical problems about infinity, some
More informationEuler s Identity: why and how does e πi = 1?
Euler s Identity: why and how does e πi = 1? Abstract In this dissertation, I want to show how e " = 1, reviewing every element that makes this possible and explore various examples of explaining this
More informationSTATION #1: NICOLAUS COPERNICUS
STATION #1: NICOLAUS COPERNICUS Nicolaus Copernicus was a Polish astronomer who is best known for the astronomical theory that the Sun was near the center of the universe and that the Earth and other planets
More informationA Little History Incompleteness The First Theorem The Second Theorem Implications. Gödel s Theorem. Anders O.F. Hendrickson
Gödel s Theorem Anders O.F. Hendrickson Department of Mathematics and Computer Science Concordia College, Moorhead, MN Math/CS Colloquium, November 15, 2011 Outline 1 A Little History 2 Incompleteness
More informationTwo hours UNIVERSITY OF MANCHESTER. 21 January
Two hours MATH20111 UNIVERSITY OF MANCHESTER REAL ANALYSIS 21 January 2015 09.45 11.45 Answer ALL SIX questions in Section A (50 marks in total). Answer TWO of the THREE questions in Section B (30 marks
More informationSome Thoughts on the Notion of Kinetic Energy.
Some Thoughts on the Notion of Kinetic Energy. Jeremy Dunning-Davies, Institute for Basic Research, Palm Harbor, Florida, U.S.A. and Institute for Theoretical Physics and Advanced Mathematics (IFM) Einstein-Galilei,
More informationZentrum für Technomathematik Fachbereich 3 Mathematik und Informatik. R, dx and ε. Derivatives and Infinitesimal Numbers
R, dx and ε Derivatives and Infinitesimal Numbers 1 We use derivatives all day Looking for extrema: f (x) = 0 Expressing conntection between quantities: y = f(y, x) Calculating norms or constaints: f =
More informationO1 History of Mathematics Lecture VIII Establishing rigorous thinking in analysis. Monday 31st October 2016 (Week 4)
O1 History of Mathematics Lecture VIII Establishing rigorous thinking in analysis Monday 31st October 2016 (Week 4) Summary French institutions Fourier series Early-19th-century rigour Limits, continuity,
More informationGrade 8 Chapter 7: Rational and Irrational Numbers
Grade 8 Chapter 7: Rational and Irrational Numbers In this chapter we first review the real line model for numbers, as discussed in Chapter 2 of seventh grade, by recalling how the integers and then the
More informationAstronomy 1010 Planetary Astronomy Sample Questions for Exam 1
Astronomy 1010 Planetary Astronomy Sample Questions for Exam 1 Chapter 1 1. A scientific hypothesis is a) a wild, baseless guess about how something works. b) a collection of ideas that seems to explain
More informationFormal (natural) deduction in propositional logic
Formal (natural) deduction in propositional logic Lila Kari University of Waterloo Formal (natural) deduction in propositional logic CS245, Logic and Computation 1 / 67 I know what you re thinking about,
More informationthan meets the eye. Without the concept of zero, math as we know it would be far less
History of Math Essay 1 Kimberly Hannusch The Origin of Zero Many people don t think twice about the number zero. It s just nothing, after all. Isn t it? Though the simplest numerical value of zero may
More informationTHE LOGIC OF COMPOUND STATEMENTS
THE LOGIC OF COMPOUND STATEMENTS All dogs have four legs. All tables have four legs. Therefore, all dogs are tables LOGIC Logic is a science of the necessary laws of thought, without which no employment
More informationMathematical Reasoning. The Foundation of Algorithmics
Mathematical Reasoning The Foundation of Algorithmics The Nature of Truth In mathematics, we deal with statements that are True or False This is known as The Law of the Excluded Middle Despite the fact
More informationRigor in Analysis: From Newton to Cauchy
Collingwood 1 Rigor in Analysis: From Newton to Cauchy James Collingwood Drake University Dr. Daniel Alexander Dr. Alexander Kleiner jrc016@drake.edu james.collingwood@gmail.com 2202 240 th St. Williamsburg,
More informationInduction 1 = 1(1+1) = 2(2+1) = 3(3+1) 2
Induction 0-8-08 Induction is used to prove a sequence of statements P(), P(), P(3),... There may be finitely many statements, but often there are infinitely many. For example, consider the statement ++3+
More informationWave-particle duality and the two-slit experiment: Analysis
PHYS419 Lecture 17 Wave-particle duality and two-slit experiment: Analysis 1 Wave-particle duality and the two-slit experiment: Analysis In the last lecture, we saw that in a two-slit experiment electrons
More informationGreece In 700 BC, Greece consisted of a collection of independent city-states covering a large area including modern day Greece, Turkey, and a multitu
Chapter 3 Greeks Greece In 700 BC, Greece consisted of a collection of independent city-states covering a large area including modern day Greece, Turkey, and a multitude of Mediterranean islands. The Greeks
More informationIn Being and Nothingness, Sartre identifies being-for-itself as consisting in the
POINCARÉ, SARTRE, CONTINUITY AND TEMPORALITY JONATHAN GINGERICH Thus temporality is not a universal time containing all beings and in particular human realities. Neither is it a law of development which
More informationMATH10040: Chapter 0 Mathematics, Logic and Reasoning
MATH10040: Chapter 0 Mathematics, Logic and Reasoning 1. What is Mathematics? There is no definitive answer to this question. 1 Indeed, the answer given by a 21st-century mathematician would differ greatly
More informationLearning Challenges and Teaching Strategies for Series in Calculus. Robert Cappetta, Ph.D. Professor of Mathematics College of DuPage
Learning Challenges and Teaching Strategies for Series in Calculus Robert Cappetta, Ph.D. Professor of Mathematics College of DuPage Ian, a second grader working with Mathman Don Cohen. www.mathman.biz
More informationAtomic Theory. The History of Atomic Theory
Atomic Theory The History of Atomic Theory This model of the atom may look familiar to you. This is the Bohr model. In this model, the nucleus is orbited by electrons, which are in different energy levels.
More information226 My God, He Plays Dice! Entanglement. Chapter This chapter on the web informationphilosopher.com/problems/entanglement
226 My God, He Plays Dice! Entanglement Chapter 29 20 This chapter on the web informationphilosopher.com/problems/entanglement Entanglement 227 Entanglement Entanglement is a mysterious quantum phenomenon
More information