Center for Lifetime Study

The Center for Lifetime Study provides opportunities for intellectual and cultural exploration and development for people of retirement age. It is connected to Marist College in Poughkeepsie, NY. This page represents the plans being made for an eight week course on the history of mathematics to be conducted beginning April 7 and concluding May 26. What follows is early planning for these lectures. Suggestions would be most welcome.

Title of the course: Marvels of Mathematics History

Time:      8 weeks

Abstract:   Mathematics has a long and fascinating history. We will look at several individuals and events in mathematics that have positively influenced our modern world. We will consider Archimedes the Master, the contributions of the Arab world, Fermat's Theorem, the magnificent Euler, mathematics on the plain of West Point, Lewis Carroll's logic, women in mathematics,  and Alan Turing's provocative question of whether computers can think. This course is designed for the inquisitive lay person who is willing to do a bit of advanced preparation and who wants to actively participate in an intellectual dialogue.  Further information about the plans for this course is available at www.dean.usma.edu/math/people/rickey/talks/cls/ (which is this page).

Class Manager: Stan Schmidt

Presenter: V. Frederick Rickey, United States Military Academy, West Point, NY.

For high level biographical information about the individuals (and their work) who will be mentioned in these presentations, see the Dictionary of Scientific Biography, edited by Charles Coulson Gillispie. This is available in the Marist Library (Q141 D5 1981) and the Vassar Library (Q141 D5). If you would prefer to look on the web, the St Andrews site in Scotland usually has reliable information on individuals and their contributions.

Lecture 1: Archimedes, A Genius from Antiquity.

Archimedes was the greatest mathematician of antiquity, and one of the greatest of all time, but his writings were difficult and so survive in very few copies, some in unique copies. The most fascinating of these, which contains his wonderful work that we call "the Method," was sold at auction by Christie's in 1998. This manuscript was copied in the tenth century, palimpsested in the twelfth, rediscovered and published early in the twentieth, and sold at auction in 1998. It is presently under restoration and extensive scholarly study at the Walters Art Gallery in Baltimore. Scholars have great hope for what the manuscript will reveal. All of this makes for a fascinating story. To begin, we will discuss what little is known of the life of Archimedes, discuss a few of his contributions to mathematics and then describe his Method and debate whether it was a precursor of the calculus. Our main focus will be on the history of this manuscript, its restoration by the Walter's staff and its study by scholars. 

Before the lecture:

• Look at the web site of the Walters Art Gallery that deals with Archimedes.
• PBS Nova had a show about the Palimpsest. Their web page contains a chronology and other information.

Further reading:

• Sherman Stein, Archimedes: What Did He Do Besides Cry Eureka?
• Roger Easton, a member of the imaging team, has a page about the palimpsest together with a power point presentation.
• T. L. Heath, The Works of Archimedes. Briefly reviewed by Fernando Q. Gouvêa.
• You may enjoy looking at models of the Archimedian solids. Be sure to use your mouse to turn them.

Show and Tell:

• The Archimedes Palimpsest. Christie's catalog 9058 for the Auction of 29 October 1998.
• Christopher De Hamel, Medieval Craftsmen: Scribes and Illuminators, University of Toronto Press, 1992.
• Reviel Netz, Ken Saito and Natalie Tchernetska, "A new reading of Method Proposition 14: Preliminary evidence from the Archimedes palimpsest," SCIAMVS: Sources and Commentaries in Exact Sciences, from volumes 2 (19) and 3 (2002).

Thoughts about things to include.

Lecture 2: The Arab World Preserves and Expands Mathematics.

The Arabic speaking world played a vital role in the development of mathematics. They valued learning and so preserved the Greek texts from the ancient world. They translated them into Arabic, developed those ideas, and created entirely new areas of mathematics such as algebra. We will discuss the work of a number of important mathematicians, including al-Khwarizmi (whose work gave us the word 'algebra'), Al-Haytham (who is pictured on the new Iraqi 10,000 dinar note), and Omar Khayyam (the poet who worked on cubic equations).

Before the lecture:

• Iraq: the history of mathematics and the aftermath of war, by Eleanor Robson

Further reading:

• J. Len Berggren, Episodes in the Mathematics of Medieval Islam, New York, NY: Springer-Verlag, 1986.
• George Joseph, Crest of the Peacock, I. B. Taurus, 1991. (Also Pengiun Books, 1993, and Princeton)

http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Arabic_mathematics.html

http://www.arabicnews.com/ansub/Daily/Day/980422/1998042208.html

www.math.tamu.edu/~dallen/masters/islamic/arab.pdf

Lecture X: Fermat and his Last Theorem.

In June of 1993, Andrew Wiles announced his proof of one of the most long standing problems in mathematics, Fermat's Last Theorem. It is a seemingly easy generalization of the Pythagorean Theorem, which says that a triangle is right if an only if the sum of the squares of the lengths of the sides of the triangle is equal to the square of the length of the hypotenuse. But unlike the Pythagorean Theorem, FLT was not a theorem of geometry, but of Number Theory. It states that the equation xⁿ + yⁿ  = xⁿ  has no solution in positive integers when n is bigger than 2. This conjecture has fascinated the mathematical community every since Fermat's son first published it in1670. We shall explain the background of the problem, describe some of the partial solutions of the problem, and then discuss some of the events surrounding the proof of Wiles and its reception by the mathematical community. The mathematics that Wiles did is too complicated even to give a summary of, but the story is very interesting.

Before the lecture:

• Read the biography of Fermat at St. Andrews.

Further Reading:

• Amir D. Aczel, Fermat's Last Theorem. Unlocking the Secret of an Ancient Mathematical Problem, New York and London: Four Walls and Eight Windows, 1996.
• Simon Singh, Fermat' Enigma. The Epic Quest to Solve the World's Greatest Mathematical Problem, New York: Walker and Company, 1997.
• Michael S. Mahoney, The Mathematical Career of Pierre de Fermat, 1601-1665, Princeton University Press, 1973.
• Marilyn vos Savant, The World's Most Famous Math Problem, New York: St. Martin's Press. This is a terrible book for the author knows nothing of which she writes. Because of that, it is a wonderful example to help people become skeptical of what they read.

Video:

• The Proof. A Nova show.
• Fermat's Last Theorem: an exploration of issues and ideas. Presented at the Fermat Fest held in San Francisco, California, July 28, 1993. Includes an interview with Andrew Wiles.

Lecture 4: Leonhard Euler.

Leonhard Euler (1705 - 1783) was the most prolific mathematician of all time. We will discuss his life, some of his works, and his influence on mathematics and society.

Before the lecture:

• Reading of original sources: The Bridges of Konigsburg

Further reading:

• William Dunham, Euler: The Master Of Us All.
• N. L. Biggs, E. K. Lloyd, R. J. Wilson, Graph Theory, 1736-1936. Oxford: Clarendon Press, 1976.
• H. Martyn Cundy and A. P. Rollett, Mathematical Models. Oxford: Clarendon Press, 1961.
• Leonhard Euler, Introduction to Analysis of the Infinite, Springer Verlag, 1988.
   

Lecture 5: How West Point Influenced American Mathematics.

Before the lecture:

Further Reading:

• Here is a paper I wrote about the history of the Department: 201 Years of Mathematics at West Point.
• Chris Arney, West Point's Scientific 200: Celebration of the Bicentennial. Biographies of 200 of West Point's Most Successful and Influential Mathematicians, Scientists, Engineers, and Technologists, Lexington SC: Palmetto Bookworks, 2002. Review by CPT Ray Eason and Prof. Fred Rickey, USMA.
• Theodore J. Crackel, West Point: a Bicentennial History, Lawrence, Kansas: University Press of Kansas, c2002. This is the best one volume history of the United States Military Academy.
• James L. Morrison, Jr., The Best School in the World. West Point, The Pre-Civil War Years, 1833-1866. The Kent State University Press, 1986.

Something extra: Field Trip to West Point.

This field trip will have two aspects. We will go on a tour of Math on the Plain which will discuss the contributions of some of West Point's scientific graduates. Then we will Visit the Rare Book Room and look at some of the books which have come up in these lectures. Items for the visit to the Rare Book Room will be especially chosen for this group. The link given is just a sample. You will need to provide your own transportation.

Before the lecture:

Read the text of the Math on the Plain tour. Make suggestions about what you would like to see on the library tour. Here is my running list of things to show.

Lecture 6: Lewis Carroll and His Friend Alice.

Read some short portion, say the Jabberwocke. Discuss various points where mathematics come up in Alice. Give a general discussion of Oxford in his day and what sort of people were there and what they did.

Before the lecture:

• There is a nice biography of Carroll at the St. Andrews web site.
• You might want to look at the Lewis Carroll Home Page or at some of the photographs that Carroll took.
• Martin Gardner, The Annotated Alice. Alice's Adventures in Wonderland & Through the Looking Glass by Lewis Carroll, Illustrated by John Tenniel, New York: Clarkson N. Potter, Inc, 1960. There is a new edition, The Annotated Alice: The Definitive Edition, that has been very favorably reviewed.

Further reading:

• John Fauvel, Raymond Flood, and Robin Wilson, Oxford Figures. 800 Years of the Mathematical Sciences, Oxford University Press, 2000. While this is superb on background information about Oxford, there is no separate chapter on Carroll. Reviewed  by Eisso Atzema.
• Ivars Peterson, Hankies, Snarks, and Triangles, discusses a new book by Martin Gardner on Carroll.

Video:

• Alice in Wonderland, The Learning Channel, Great Books.

Show and tell:

• Alice's Adventures in Wonderland: A Pop-up Adaption. Written by Lewis Carroll, designed by Robert Sabuda.
• Reflections in a Looking Glass. A Centennial Celebration of Lewis Carroll, Photographer, by Morton N. Cohen, Aperture, 1998.
• The Mathematical Pamphlets of Charles Ludwidge Dodgson and Related Pieces, edited by Francine F. Abeles, University of Virginia Press, 1994.

Lecture 7: Women in Mathematics.

Hypatia, du Chatelet, Agnesi (play some music by her sister), Kovalevskaya, Noether,

Julia Robinson. We will discuss the social issues and prejudices that prevented women from entering mathematics and argue that things have changed and that women are now welcomed into the mathematical ranks.

Before the lecture:

• Look at the web site of the Association for Women in Mathematics and some the biographies on this page.
• The students of Larry Riddle at Agnes Scott College have written nice biographies of many women mathematicians.
• There is a nice site for Black women mathematicians. Note the effect that Lee Lorch has had on the community.

Further reading:

• Notable Women In Mathematics: A Biographical Dictionary edited by Charlene Morrow and Teri Perl. Reviewed by Libby Krussel.

Lecture 8: Alan MathisonTuring: An Original Math Nut

His life is a very interesting one. His work on computers. See if his paper on distinguishing between a computer and a person could be read by the group.

Keith Devlin, Reflections on Deep Blue. This deals with the question of whether computers can think.

Further Reading:

• Andrew Hodges, Alan Turing:The Enigma, New York: Simon and Schuster, 1983. Hodges maintains an extensive web site on Turing.

• Christos H. Papadimitriou, Turing (A Novel about Computation).  Reviewed by Darren Glass.

   

Video:

• Alan Turing, BBC Horizon.

Return to Fred Rickey's Home Page.