Announcements

Grove City College

Monday and Tuesday, September 14-15, 2009

5:00 – 6:00 pm "The Axiomatic Method"

Since 1900, the axiomatic method has become the standard way to develop and present mathematical results. Students of mathematics will have a much improved overview of mathematics if they make the effort to understand what is meant by the axiomatic method. The standard was set by the Greek mathematician Euclid around 300 B.C. when he wrote the Elements as an organized presentation of more than 400 theorems, along with their logically correct proofs. This book contains the knowledge about geometry and number from Euclid’s day. Here are some of the topics that we may consider.

• The meaning of an interpretation and a model of an axiom system
• The consistency of an axiom system
• The dependence or independence of an axiom system
• The meaning and importance of non-Euclidean geometry
• The use of the axiomatic method in non-mathematical settings
• The completeness of an axiom system and Godel’s theorem

This talk does not assume extensive mathematical knowledge.

7:00 – 8:00 pm "Cantor’s Theory of the Infinite"

The results about infinite sets presented by Georg Cantor during the closing decades of the 19^th century provide some of the most useful, interesting, and controversial results in all of mathematics. The concept of the infinite is connected to our understanding of God as well as our philosophical views about the world. It also serves an essential role in the development of mathematics in such subjects as calculus. This talk will present the basic results that Cantor discovered about what have since been called transfinite numbers. Their proofs require little mathematical content. We will also talk about the concept of infinity as we see it in the Bible, and try to relate it to the many different infinities that are part of Cantor’s theory. Questions are welcomed during each talk.

9:00 – 4:00 pm Free time to meet with students and/or faculty over coffee, during lunch, or other informal settings. Let me know what works best.

5:00 – 6:00 pm "Three Main Themes in Algebra"

Algebra is one of the major branches of mathematics, and most students in junior and senior high school study this subject for many years. It is usually viewed as a collection of techniques to be applied without much understanding or enjoyment. This talk will show one way to increase understanding of this important subject by considering three of its major themes – those of solving equations, finding properties of numbers, and developing algebraic structures. Specific examples of each will be presented. It would be helpful to have seen in advance one example of an algebraic structure, such as a vector space from linear algebra or a group from abstract algebra. In all talks, questions will be welcomed at any time.

7:00 – 8:00 pm "Bringing Rigor into Calculus"

Many college students have taken a course of calculus and know the basic results about derivatives and integrals. What makes calculus so useful is the fundamental relationship between these two concepts that was discovered by Newton in the 1660s. For the next 150 years, calculus was successfully applied to solve a wide range of problems from the physical world. But this amazing structure of results had no solid foundation, and paradoxes began to appear. During the years from 1820 to 1860, most of these problems were resolved, and calculus was put into its modern form with all the concepts based on the limit definition. A significant portion of the historical development of calculus and analysis is covered in this talk.