Syllabus for History of Mathematics

The Textbook
There will be two required textbooks for the course. One is Journey Through Genius: Great Theorems in Mathematics by William Dunham. The other is a standard history of mathematics text; however, these books now cost about $140, so I am not assigning a specific book. What you can do is locate a new or used book of your choice. I suggest books by one of the following: Carl Boyer, David Burton, Howard Eves or Victor Katz. The book need not be the latest edition.
An Overview
This course is designed to show what mathematics is, how mathematics has developed from man's efforts to understand the world around him, what the mathematical approach to real problems can accomplish, and the extent to which mathematics has molded our civilization and culture. The extent to which civilization and culture have affected mathematical development will also be investigated. Although the course is about mathematics, the intimate relationship of mathematics to science, philosophy, religion, music, painting, and other arts, cannot be overlooked. For here lie many of the motivations for mathematical studies. Since mathematics is what mathematicians do, a look into the lives of mathematicians will be an integral part of the course. On occasion, you will be asked to solve some mathematical problems related to the historical period or person being discussed. A midterm and final exam will be given. In addition, a semester project will be required. The semester project may be a paper or project appropriate to the course. The course can be basically separated into eight parts (not equal in length), as given below. Basically the coverage will be chronological; however, certain topics will be covered out of order. For example, Part II, Numbers and Number Systems, will cover the evolution of numbers and number systems from ancient times to the present. This is done so that the students have a general idea of the problems mathematicians faced with inadequate numbers and number systems at various times in the history of mathematics.
Part I.	INTRODUCTION
	What is mathematics? A look at the difficulties arising from an effort to precisely define the term "mathematics." What is a mathematician? A look at the wide spectrum covered by mathematics from philosophy to technology. When does a philosopher become a mathematician? When does a mathematician become a technician? A chronological listing of events in mathematics and history. Quite often dates important in mathematics have no meaning to a student without reference to familiar historical events. This will be an overview of the time period covered by the course for later reference.
Part II.	NUMBER AND NUMBER SYSTEMS
	Counting and the history of number. Numbers systems. An investigation of number systems from antiquity to the present. Number systems in various bases will be seen, as well as the effectiveness of each system. Idiot Savants. - Human calculators. Here we look at the interesting phenomenon of people who can calculate difficult numerical problems with machine speed and how their views of numbers and number systems aided their mental computations.
Part III.	ANCIENT MATHEMATICS
	Babylonian and Egyptian mathematics. The development of arithmetic and geometry for use in commerce and agriculture. No interest in pure mathematics. Greek mathematics. The arrival of mathematics as a pure discipline. Pythagoras. The life of an ancient mathematician and his school. How his mysticism and worship of the whole number influenced and held back further discoveries in mathematics. Euclid. The man who is credited with formalizing the study of mathematics. The axiom, definition, theorem development of mathematical topics presented in his Elements produced the foundation of "Modern Mathematics." During this period we will meet a variety of mathematicians who did remarkable work with elementary mathematics--one of the most remarkable was Archimedes. India, the Middle East and China. Was mathematical knowledge isolated or did it spread geographically? Who did what first?
Part IV.	THE DARK YEARS
	The decline of mathematics and the sciences. In the first ten centuries or so C.E., advances in mathematics and the sciences seemed to come to a halt. Mathematics and religion. An investigation of the effects of religious thoughts on the development of mathematics. To quote St. Augustine (circa 400 C.E.) "The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell." The rebirth of mathematical thought. This is a lead-in to the "golden age" of mathematics - how the use of mathematics in art rekindled interest in mathematics.
Part V.	THE "GOLDEN AGE" OF MATHEMATICS
	A chronological look at the "golden age" of mathematics. The period of the 14th through 19th centuries were very rich in mathematical development. Many events of mathematical as well as historical importance occurred here, so it is valuable to once again have reference points. Mathematics and the arts. The contribution of mathematics to the arts and architecture will be discussed throughout the course; however, in this period the topic deserves special attention. Mathematics and society. Mathematics was widely studied in this period for general educational purposes. Educated persons took great pride in their knowledge of mathematics and new developments in mathematics. Mathematics enjoyed its greatest popularity during this period. It was not uncommon for newspapers and general periodicals to publish mathematical papers.
Part VI.	BIOGRAPHICAL SKETCHES OF MALE MATHEMATICIANS FROM THE "GOLDEN AGE" OF MATHEMATICS
	A partial list of some of the male mathematicians whose lives we may discuss in class. Not all can be discussed in the time available; however, there are some who must be discussed.      Descartes. (1596-1650) French.      Fermat. (1601-1665) French.      Pascal. (1623-1662) French.      Newton. (1642-1727) English.      Euler. (1707-1783) Swiss.      Laplace. (1749-1827) French.      Gauss. (1777-1855) German.      Abel. (1802-1829) Norwegian.      Galois. (1811-1832) French. The list of mathematicians was chosen from a list of many. The choice is the instructor's. Other people teaching this course may choose a different collection. These were chosen because of the interesting aspects of their lives. Of course, Newton and Gauss have made such prominent contributions their names would probably appear on everyone's list.
Part VII.	BIOGRAPHICAL SKETCHES OF FEMALE MATHEMATICIANS
	A partial list of some of the female mathematicians whose lives we may discuss in class. Not all can be discussed in the time available; however, there are some who must be discussed. For many years female mathematicians were ignored in the history of mathematics writings.      Hypatia (370-415) Greek      Sophie Germain (1776-1831) French      Sonya Corvin-Krukovsky Kovalevsky (1850-1891) Russian      Emmy (Amalie) Noether (1882-1935) German The list of female mathematicians was chosen from a list of many. The choice is the instructor's. Other people teaching this course may choose a different collection. These were chosen because of the interesting aspects of their lives. As well as for their contributions to mathematics.
Part VIII.	THE 20th CENTURY
	A brief history of the development of the computer. The computer as a tool. These two sections will cover the development of the computer and its value in science, mathematics and society. Very few people have a good understanding of what computers really do as a tool. Mathematics and modern society. Have the giant strides of advancements in mathematics in the 19th and 20th centuries left the average person floundering in its wake? What effects have the educational processes of the "new math" of the 1960 had on modern education? What are the new standards for K-12 mathematics education? Where do we go now? Here we will discuss what may be in store for the mathematician in the future. Might the mathematician, in the classical sense of the word, disappear with future advances in computer technology? Or, will there always be a need for people who dream mathematical thoughts?
THE STUDENTS' RESPONSIBILITY
The student is responsible for attending class and for all reading materials and assignments. Take care not to postpone doing work until the last minute. Assignments will consist of written assignments, two exams and a term project.  All written assignments must contain a bibliography.  Not all references should be from the internet and no encyclopedias (Brittanica, Compton, etc.) should be used.  Get to know your library!  Cutting and pasting from web sites is NOT permitted--the punishment will be harsh. The written assignments will consist of: 1. a short paper containing biographies of five mathematicians, who lived before 1100 C.E. 2. a short paper containing biographies of ten mathematicians, who lived after 1100 C.E. (at least four of these should be women) 3. a paper on the history of non-Euclidean geometry 4. a term paper, or project, on a topic of your choice (approved by me).
The Final Examination will be Thursday, Dec. 17, from 8:00 AM to 9:50 AM.
GRADES
Grades will be determined by performance on the assignments. The written assignments, midterm exam, final exam (this will be over the material covered after the midterm exam) and term project will be weighted as follows:  written assignments 40%, exams 20% each, term project 20%. The following scale gives an idea of the worst grade you would receive. 100 - 94 A 93 - 90 A- 89 - 86 B+ 85 - 82 B 82 - 80 B- 79 - 74 C+ 73 - 68 C 67 - 65 C- 64 - 60 D+ 59 - 55 D 54 - 50 D- 49 -> F

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