| Readings | Assignments |
| Instructor | Kim Plofker |
| Department | History of Mathematics |
| Kim_Plofker@Brown.edu | |
| Office | Wilbour Hall, Room 001 |
| Office phone | 863-1489 |
| Office hours | W 2:30--3:30, Th 11:15--12:15, 3:30--5:00 |
Course Overview: Currently offered as a first-year seminar, "Calculus and its History" is intended for students (whether or not they have already studied calculus) who would like to investigate questions like the following:
The course will meet in C hour (MWF 10:00--10:50) in Sayles 204 according to the University Calendar from 4 September to 9 December 2002. Assigned texts will consist of books, handouts, and electronic texts containing excerpts from primary sources in English translation, including (but not limited to) the following:
| Class participation and mini-assignments | 20% |
| 9 graded homework assignments (lowest two of 11 grades dropped) | 30% |
| Midterm essay | 20% |
| Final paper | 30% |
| Date | Topics |
|---|---|
| W 9/4 | Introduction. (What is calculus? How do we approach it?) Discussion of pre-calculus mathematics, students' mathematical backgrounds. Reading original mathematics: Babylonian tablet. Explanation of course requirements, course objectives. Focus on original sources, historical development, and mathematical understanding. |
| F 9/6 | Before Calculus: Practical mathematics in ancient civilizations. (We explore the basic mathematical tools that ancient cultures developed for problem-solving, and link them to some of our modern tools such as notation.) "Original urban-civilization" mathematics: Egyptian/Babylonian arithmetic. Babylonian area calculations. (Explicit discussion of algebraic "translation" of pre-algebraic works.) |
| M 9/9 | Philosophical limitations to mathematical knowledge. (We discuss the evolving definition of mathematical knowledge and its shocking consequence: the discovery that we can know exactly that there are things we cannot know exactly.) Babylonian into Greek mathematics. The role of mathematics in Greek philosophy. What is number? The Pythagoreans: numerical cosmology, ratios, means. Commensurability and irrationality. |
| W 9/11 | Developing mathematics within the limitations, I. (The ancient Greeks ask: If we concede that there are limits to what mathematical (The ancient Greeks ask: If we concede that there are limits to what mathematical knowledge can tell us, what can we know within those limits?) How do we separate what is mathematics from what isn't? The axiomatic deductive method and the geometrical approach: strict definitions of number, separation of number from magnitude. Geometric proofs and their "plots". |
| F 9/13 | Developing mathematics, II. (What kind of problems can Greek mathematics solve and still be confident that it produces exact knowledge?) Difficulties with intuitive notions of infinity and continuity. Zeno and the paradoxes of motion. |
| M 9/16 | Areas and volumes. (In the days before coordinates and graphs, how do we talk about curved lines? Can we discuss any of them mathematically? Which ones? How?) Quadratures and cubatures. Squaring rectinilinear figures; squaring the lune. |
| W 9/18 | Archimedes; curves other than the circle. (The greatest mathematician of antiquity comes up with some clever answers.) The method of exhaustion and reduction to absurdity; "principle of Eudoxus." The quadrature of the parabola. |
| F 9/20 | The "Method" for finding solutions. Mechanical arguments for computing with "infinitesimal" areas and volumes. Indivisibles and the quadrature of the parabola. "Cleaning up" the results with rigorous proof. |
| M 9/23 | The end of classical mathematics; the "other" classical mathematics. (What ever happened to Greek mathematics? What did the rest of it look like?) The declining impact of classical geometry in late antiquity. Efforts to systematize and classify it; ideas of analysis and synthesis. The other side of Greek mathematics: the need for computation. |
| W 9/25 | The "other" classical mathematics, cont'd. Astronomical/geographical problems: trigonometry, computing chord values. Classical systems for representing and calculating with numbers. |
| F 9/27 | Mathematics in motion, I; new tools from other cultures. (Drawing pictures of change: late medieval mathematicians try to quantify change and motion.) Discussion of the latitude of forms. Qualities and quantities; rates of change; the mean speed theorem. Calculation techniques and the "Arabic" numbers. |
| M 9/30 | New tools, II: the emergence of early modern math. Islamic algebra techniques. Picking up the separate strands of "original-urban-civilization" math, classical Greek geometry, the "other" classical mathematics, Indian numerals, Islamic developments, the European university curriculum: unification into the Latin mathematics of the early modern period, with its "analytical art". |
| W 10/2 | New people and new ideas in mathematics. Innovative applications and developments attract popular interest in mathematics. "A very large number of very small pieces:" new approaches to calculating areas and volumes. Methods of indivisibles, Cavalieri's principle. Philosophical objections. The integral power law. |
| F 10/4 | New tools, III; computational requirements spurring mathematical developments. Medieval development of trigonometric functions from ancient Greek chords. Trigonometric tables and problems, and their computational burden. Number-crunching with Logarithms: replacing multiplication by addition. |
| M 10/7 | Mathematics in motion, II. (Mathematicians return to the discussion of motion and change, and draw more pictures.) Classical and medieval notions of motion: quantifying celestial motions. Quantifying terrestrial motion: velocity and acceleration, projectiles, mathematical and graphical descriptions of motion in Galileo. |
| W 10/9 | The beginning of general solutions, I. (The amazing discovery that a lot of problems that look different are really the same.) Descartes: analytic geometry and coordinate systems. Introduction of equations for curves. The principle of nonhomogeneity. Fermat's e and the "adequality" approach for maxima and minima. |
| F 10/11 | Beginning of general solutions, II. Review of conic sections in ancient and early modern guises. Representing the curve and finding the tangent. Barrow: finding tangents can be related to finding areas. |
| M 10/14 | NO CLASS |
| W 10/16 | (Mid-semester) Systematizing the math of arbitrarily small variations. Review/preview of course topics week by week. Identifying the two basic types of problems for these methods: tangents/maxmins, areas/volumes. | /TR>
| F 10/18 | General solutions, III: the techniques for attacking the two basic types of problems. Sample methods for tangents/maxmins: Fermat's dividing by not-quite-zero. Sample methods for areas/volumes: unexpected application to finding line-lengths. |
| M 10/21 | Historical context of the calculus, I.
(Why is all of this mathematics happening now? How are these concepts justified?)
Historical and philosophical background of the calculus in the early seventeenth
century: education, technology, motivation.
(The "Scientific Revolution" as a historical concept, 1450--1750. Movable type and the fall of Constantinople; implications for revival of ancient learning. Factors fostering the redevelopment of research mathematics: intermediate social class(es); educational institutions and resources; employment opportunities; source of research problems; collegial communication. Comparison of medieval universities' curricula with actual influences on 16th--17th c. mathematicians' training and employment. Element of chance in mathematical careers; absence of real professionalization of the field at this point. Philosophical/mystical acceptance of paradox permits more speculation on notions such as the infinitely large and small.) |
| W 10/23 | The differential calculus, I: Leibniz. (At last, a truly general algorithm for all (?) curves.) Pascal's sines and the "tiny tangent triangle." Leibniz' New Method and basic rules of differentiation. |
| F 10/25 | Class visit to Lownes Collection of John Hay Library. View early editions of Euclid, Fermat, Descartes, Leibniz, Newton, etc., as well as some of Brown's cuneiform text collection. Discussion of the physical sources of mathematical texts, their dissemination and use. |
| M 10/28 | The differential calculus, II: Newton. (Simultaneously (?), another truly general algorithm.) Completing Leibniz's "calculus of differentials" with the equivalent of the chain rule. Comparing this approach to Newton's o-method and fluxions: motivation from physical problems of motion. |
| W 10/30 | Historical context, II: the great dispute.
(Who invented the calculus? Why are we fighting about it?)
Reception of the two calculus techniques and the argument over priority
and plagiarism. National loyalties (the "scurvy English" vs. the "Leipzig
rogues"); the Royal Society's inquiry.
("Protocalculus" vs. "calculus" concepts: justification of attribution of calculus to Newton and Leibniz rather than Fermat or Barrow. Power and flexibility of the new techniques; proliferation of calculus textbooks; fame and prestige of the inventors. Chronology of the discoveries and the accusations in the dispute; challenges and counter-challenges. Extracurricular motivations: political issues concerning England and Hanover; Newtonian vs. Continental physics.) |
| F 11/1 | The differential calculus, III: further testing. (Does the new "calculus of differentials" work for all the curves we can think of?) Interpretation of Bernoulli and L'Hopital; systematization into early calculus books. Profusion of applications to physical problems. Extrema, concavity, higher derivatives. |
| M 11/4 | The integral calculus, I: Leibniz and Newton again. (The two inventors present their versions of the other half of the calculus.) Leibniz and Newton on integration; using the fundamental theorem to find areas. |
| W 11/6 | The integral calculus, II. (Systematizing the techniques of integration.) Finding the area under various curves; tables of integrals. Rectification. |
| F 11/8 | Historical context, III.
Making an algorithm into a discipline:
notation, textbooks, inclusion in the curriculum, new problems.
(Late 17th--late 18th c.: calculus changes from research mathematics to instruction for schoolchildren and even suggested recreation for ladies. How has it been "mainstreamed"? Development of standardized mathematical notation; Sudden development of mathematical "alphabet" in 17th c. to express expanding range of concepts and procedures. Mathematization at present affects only the physical sciences; example of a contemporary calculus textbook and its "physico-mathematical" problems.) |
| M 11/11 | The integral calculus, III. Further techniques of integration and other applications. Tying integration in with differentiation. |
| W 11/13 | Historical context, IV: philosophical objections to the
calculus.
("The emperor has no clothes, and he's dividing by zero:" complaints that
the "new analysis" has abandoned mathematical certainty.)
Newton and Berkeley. Fluxions and the "ghosts of departed quantities."
(The Academy of Lagado in Gulliver's Travels: satirical assessment of the Royal Society and "progressive" attitudes toward science. "Responsible opposing viewpoint" re the Scientific Revolution, shared by Berkeley. Developments in sciences and scholarship color Enlightenment ideas of literal truth in religion. The Analyst as a response to freethinking abetted by mathematical rationalism. Details of criticisms: higher fluxions and differences inconceivable, manipulation of infinitesimals illogical.) |
| F 11/15 | Responses to the critiques. (Discussion of final paper: study a renowned problem solved (or controversy caused) by calculus and write a 13--15-page paper explaining the mathematics and discussing its historical development. A list of sample topics and a description of desired goals and methods for the paper will be provided.) Counterarguments provided by the reasoning of Leibniz, Nieuwentijdt, Bernoulli, Grandi, Euler. Is the calculus philosophically defensible? Need we care? |
| M 11/18 | The calculus explosion, I: Euler. (Is there anything calculus can't solve? Eighteenth-century mathematics takes off.) The differential equation; power series. Leonhard Euler and the new organization of mathematics. |
| W 11/20 | The calculus explosion, II. The exponential and the logarithm. |
| F 11/22 | The calculus explosion, III. (Is there any physical problem calculus doesn't apply to?) Euler and the world of "mixed mathematics". |
| M 11/25 | Rigorization of the calculus, I. (Can we restore mathematical certainty to the calculus? Sure; just change the definitions.) Educational necessities lead to a re-examination of philosophical complaints about the calculus. D'Alembert, Lagrange, Cauchy and the limit; definition of the derivative. |
| W 11/27 | NO CLASS |
| F 11/29 | NO CLASS |
| M 12/2 | Rigorization, II. (More changing of definitions.) Definite integrals: Cauchy and Riemann. Is this just Archimedes all over again? |
| W 12/4 | Rigorization, III. (The dreaded epsilon-delta limit finally appears.) Foundations of the limit: Heine and Weierstrass, epsilon-delta definition. The "fundamental theorem." |
| F 12/6 | Conceptual changes after the 19th century. (How did all this become the calculus we know today? Is it going to stay that way?) Standard approach to the rigorous calculus. Nonstandard analysis; Robinson and infinitesimals. |
| M 12/9 | Conclusion. |
| Back to the History of Math homepage |
|---|