MTH 381/URI
History of Mathematics

Course Information and Syllabus, Fall 2001

Text: History of Mathematics, David M. Burton, Fourth Edition 

Prerequisites: MTH 142 or 132

Exams and Grading: There will be two exams, some homework assignments, and the
presentations.

TWO HOUR TESTS: 40 percent 

PRESENTATIONS AND HOMEWORK: 40 percent

FINAL EXAM: 20 percent

Aims and Objectives-Short Version
 

The course aims to illustrate the following:

1. How mathematics has been, and still is, a developing subject.

2. How advances in mathematics are driven by problem solving and how initial formulations
often lacked rigour.

3. How good mathematical notation is vital to the development of the subject.

4. How mathematical ideas that are considered "elementary" today have great level of sophistication.

5. To teach you how to use the library and technology, especially the internet.

6. To improve your oral and written communication skills in a technical setting.

By the end of the course students are expected to be able to:

1. understand, describe, compare and contrast the main ideas and methods studied in the
course.

2. apply the methods to  given examples.

3. develop a broad historical appreciation of the development of mathematics.

4. understand that even very abstract results of pure mathematics affect everyday lives.

5. have effective presentation style in a technical setting.
 

Objectives-Detailed Version

The main aim of this course is to introduce the study of the history of mathematics. This means both telling the story of the development of mathematics in the past, and practicing the historical judgements and methods that enable the story to be told. The course should also deepen your understanding of the role the mathematics has played in society.

Topics

The course is intended for interested people from a variety of backgrounds: students of mathematics who want more understanding of its historical development, teachers of mathematics at all levels, who will find such material enriching to their students' learning, and people who have a general interest in social and cultural history.

Our approach is based on texts and the materials that can be found on the Internet.

 The major topics that will be covered are:

Mathematics in the ancient world moves from the earliest evidence for mathematical activity, before the time of the Egyptians and Babylonians, through the achievements of classical Greece to Euclid's Elements and the great geometers Archimedes and Apollonius.

Through the Middle Ages to the seventeenth century . We follow the development of the algebraic approach through Muslim culture and then the rediscovery in Europe of classical Greek texts at the end of the sixteenth century, which helped lead to a flowering of mathematics in the next century. We look at the time of Napier (logarithms) in Scotland; Descartes (algebraic geometry) in France; Kepler in Germany and Galileo in Italy applying mathematics to the world; and the invention of the calculus.

The seventeenth and eighteenth centuries. The calculus was invented, independently and in rather different ways, by Newton and Leibniz (building on the work of many earlier mathematicians). What were the consequences of this? We trace some developments through the eighteenth century, and examine how algebraic concerns reached almost their modern form in the work of the great Swiss mathematician Leonhard Euler.

Topics in nineteenth-century mathematics. Is Euclid's 'parallel postulate' necessarily true, or can other logically consistent geometries be devised? Can a formula be found for solving equations of the fifth degree or, if not, why not?  Were the foundations of the calculus secure - if not, what to do about it?  Can calculation be mechanized, and at what cost? Can you 'prove' a theorem by using a computer? These are some of the questions discussed in this survey of characteristic topics of nineteenth-century mathematics that are the basis for many of the concerns and approaches of mathematics in the twentieth century.

Topics in twentieth century mathematics. This part is based on John L. Casti book, Five More Golden Rules, Wiley, 2000 which brings a brilliant collection of 20th-century mathematical theories, leading the reader on a fascinating journey of discovery and insight. The topics discussed will range from the knot theory to the Hopf bifurcation theorem and the chaos. This part will be presented through the students projects that may involve the computer presentation.

Instructor: Dr. M. Kulenovic, Tyler 216, X44436,

e-mail:  kulenm@math.uri.edu

Online information: www.math.uri.edu/courses or www.math.uri.edu/~kulenm
Office hours: MWF: 10-11 

Time: MWF: 12-1                              Room: