THE 2001 MAA NORTH CENTRAL SECTION SUMMER SEMINAR

THE HISTORY OF MATHEMATICS

Abstracts of Talks and Suggested Readings

Nota Bene: Biographical information on almost all of the individuals mentioned below and detailed information about all of their work, with references to both primary and secondary literature can be found in the Dictionary of scientific biography/ Charles Coulston Gillispie, editor-in-chief. The articles about mathematicians have been pulled out and published as the Biographical dictionary of mathematicians: reference biographies from the Dictionary of scientific biography, New York: Scribner, 1991. This source is highly recommended for information on deceased mathematicians. 

  1.                  Early examples of integration and differentiation.

  The Greeks knew how to find tangents to circles, parabolas, and the other curves that they knew about. They also knew how to find the area of a number of curvilinear figures. In the seventeenth-century new curves arose and mathematicians learned to find their tangents, areas, and volumes. This work involved every important mathematician of the century, especially, Kepler, Cavalieri, and Wallis. We will consider some of these examples such as cycloid and Torricelli’s infinitely long solid with finite volume. To illustrate what a great advance Leibniz made, we shall consider Barrow’s Theorem that the sum of the subnormals to a curve is half the final ordinate squared. We also need to mention the tangent methods of Descartes, Fermat, Hudde, and Sluse, as well as von Heureat rectification of the semi-cubical parabola. It should be clear that the twin problems of the calculus, tangents and areas, were well developed by mid-seventeenth century but that they lacked --- if you will excuse the pun --- integration.

This material is nicely laid out by Kirsti Pedersen in “Techniques of the calculus, 1630-1660,” which is Chapter 1 in Ivor Grattan-Guinness’s From the Calculus to Set Theory, 1630-1910, which was reprinted in 2000 in paperback by Princeton University Press. The same material is discussed by C. H. Edwards, Jr., in his The Historical Development of the Calculus, in Chapter 4 on “Early indivisibles and infinitesimal techniques” and Chapter 5 on “Early tangent constructions.” Barrow’s Theorem is in Struik’s Source Book of Mathematics, 1200-1800, pp. 260-262. This will be a difficult read, but seeing how things were done before Newton and Leibniz will impress you with what they did. 

For another source see Carl B. Boyer and Uta C. Merzbach, A History of Mathematics (2nd edition, 1989), sections 16.18, 16.19, 16.22, 16.23, chapters 17 and 18. Or see Victor J. Katz, A History of Mathematics: An Introduction (1993), the first part of chapter 12 on "The Beginnings of the Calculus."

2.                  What Newton and Leibniz did.

The seminal work of Newton (1642-1727) and Leibniz (1646--1716) will be discussed so that we can see precisely why history has placed such great value on their work. This necessitated the previous lecture, which showed what their predecessors did, and the succeeding ones, which will show how their successors developed and extended their ideas. Thus we will be in a position to frame a definition of the calculus of Newton and Leibniz (it is instructive to think about this in advance) and to contrast it with the calculus we do today. In 1684 Gottfried Wilhelm Leibniz presented his "Nova methodus," his new method of maxima, minima, and tangents to the scientific community. This was the first paper on the differential calculus and demands our attention. Our goal is to show that the work of Leibniz (and Newton too) was a watershed; mathematics in succeeding generations was considerably different than mathematics before them. This will show why Leibniz and Newton are justly cited as the creators of the calculus.

Never at Rest (Cambridge 1980) by Richard S. Westfall is a superb scientific biography of Newton, but unfortunately there are not a lot of mathematical details. On the other hand Leibniz in Paris, 1672-1676; his growth to mathematical maturity by Joseph E. Hofmann (Cambridge 1974) is loaded with mathematical details. A nice exposition of the ideas of Leibniz are in "Some fundamental concepts of the Leibnizian Calculus," pp. 83-97 of Lectures in the history of mathematics (AMS, 1993) by  Henk J. M. Bos. Details on the work of Newton are found in The Mathematical Papers of Isaac Newton, edited by D. T. Whiteside (Cambridge, 8 volumes); the introductions to these volumes present a detailed introduction to Newton’s work. On an expository level, I immodestly recommend, "Isaac Newton: Man, Myth, and Mathematics," The College Mathematics Journal, 18(1987), 362-389. For the philosophical side of the Newton-Leibniz controversy see A. Rupert Hall, Philosophers at War: the Quarrel Between Newton and Leibniz (Cambridge 1980).

One could also read Boyer and Merzbach, chapter 19, or Katz, the end of chapter 12.

3.                  The Bernoullis and Euler.

When Johann Bernoulli (1667-1748) was in Paris in 1691-92, he wrote manuscripts, for the use of L'Hospital (1661-1704), on the differential and integral calculus. The first of these was the model for L'Hospital's Analyse des infiniment petits, the first calculus textbook (1696). Another student of Bernoulli, Leonard Euler (1707-1783), was the most prolific mathematician of all time (or are there recent counterexamples?). We shall ignore his hundreds of research papers and concentrate on the three “textbooks” which he wrote on the calculus, so that it will be clear that Euler’s calculus is not the calculus of Newton and Leibniz, but is really the calculus we teach today.

Euler: The Master of Us All by William Dunham (MAA 1999) is excellent; chapters 2 (logarithms), 3 (infinite series), and 5 (complex variables) are the most relevant. See the review by Ed Sandifer. Hopefully this book will encourage you to read Euler himself; consider: Introduction to Analysis of the Infinite, translated by John D. Blanton (Springer-Verlag, 1988-1990). The November 1983 (vol. 56, no. 5) of Mathematics Magazine is "A Tribute to Leonhard Euler, 1707-1783" although none of the articles pertains exclusively to the calculus. 

Chapters 21 and 22 of Boyer and Merzbach deal with the Bernoullis and Euler as does chapter 13 of Katz.

On the web see: http://www.euler2007.com/

4.     Cauchy and Weierstrass develop foundations.

Augustin-Louis Cauchy (1789-1857) is usually credited with providing a rigorous foundation to the calculus, but commonplace claims like this are never so simple when given a careful historical look. Karl Weierstrass (1815-1897) really deserves much of the credit. We will look at the lives of both of these individuals and then look at the basic definitions given by Cauchy to see how rigorous they really are. As a special example we will look at Cauchy's famous wrong proof that the limit of a convergent sequence of continuous functions is continuous.

Judith V. Grabiner, The origins of Cauchy's rigorous calculus (MIT Press, 1981) is a wonderful treatment of Cauchy's ideas on the calculus, including his debt to Lagrange. There is also a full biography by Bruno Belhoste, Augustin-Louis Cauchy: a Biography (Springer Verlag 1991). Unfortunately, information about Weierstrass is not so easy to give. For Cauchy's famous wrong proof see the appendix to Proofs and Refutations: the Logic of Mathematical Discovery (Cambridge 1976) by Imre Lakatos; edited by John Worrall and Elie Zahar.

See Boyer and Merzbach, chapter 23 and 25 and Katz, chapter 16 on "Rigor in analysis."

5.     Cantor’s leap from calculus to set theory.

After Georg Cantor (1845-1918) received his PhD in number theory and took a position at Halle, his senior colleague Eduard Heine (1821-1881) got him interested in the question of representing a function in a unique way as a Fourier series. He proved several theorems of the form “If the Fourier series converges except on such and such exceptional points, then the representation is unique.” Soon however, the sets of exceptional points became of interest in themselves, and this led to such concepts as the derived set, and thus to set theory. We will discuss Cantor’s first (and almost unknown) proof of the uncountability of the reals, and also his proof that there are as many points on a line as in the plane. Some glimpses of later work will be discussed.

Georg Cantor, His Mathematics and Philosophy of the Infinite (1979) by Joseph W. Dauben is a full scientific biography that has details about much of Cantor’s work. Many popular works contain an exposition of Cantor’s work on set theory. The most recent (which I have not read) is Amir D. Aczel, The mystery of the aleph: mathematics, the kabbalah, and the search for infinity (New York: Four Walls Eight Windows, c2000). My thoughts about the equinumerosity of the plane and the line are in "The necessity of history in teaching mathematics," pp; 251-256 in Vita Mathematica (MAA 1996). For Cantor in his own words see Contributions to the Founding of the Theory of Transfinite Numbers (1915, reprinted by Dover) with a fine introduction by P. E. B. Jourdain (1879-1919).

6.     Robinson and non-standard analysis.

Abraham Robinson (1918-1974) developed Nonstandard Analysis in the fall of 1960 thereby providing a rigorous foundation for infinitesimals, a tool that had been used throughout the entire development of the calculus. After presenting a biographical sketch of this most interesting individual two proofs that infinitesimals exist will be given. The first uses the compactness theorem of Gödel, the second an ultrapower construction (do you remember what an ultrafilter is?), which is somewhat akin to how the integers are constructed from the natural numbers. The concepts of continuity, uniform continuity, differentiability, and integrability will be defined and illustrated with simple examples.

For biographical information see George B. Seligman, “Biography of Abraham Robinson,” pp. xi-xxxix in Selected Papers of Abraham Robinson, volume 2, Nonstandard Analysis and Philosophy, North-Holland, 1979 and A. D. Young, et alia, “Abraham Robinson,” Bulletin of the London Mathematical Society, 8 (1976), 307-323. For a full biography see Abraham Robinson. The Creation of Nonstandard Analysis. A Personal and Mathematical Odyssey, by Joseph Dauben, Princeton University Press, 1995. A popular presentation of the mathematics is “Nonstandard Analysis,” by Martin Davis and Reuben Hersh, Scientific American, vol. 226 (June 72), pp. 78-84 and 86. A simple presentation of the details of Nonstandard Analysis can be found in Alexander Abian, “Nonstandard models for arithmetic and analysis,” Studia Logica, 33 (1974), #1, pp. 11-22. There are calculus textbooks based on Nonstandard Analysis by H. Jerome Keisler, Elementary calculus: an infinitesimal approach (2^nd edition, 1986) and by James M. Henle and Eugene M. Kleinberg, Infinitesimal Calculus (MIT, 1980).

Prepared by V. Frederick Rickey, July 2001. Send comments to fred-rickey@usma.edu .