American Mathematical Monthly - January 2000

Articles

Mathematics at the Turn of the Millennium
by Phillip A. Griffiths
pg@ias.edu
The last century has been a golden age for mathematics. Many important, longstanding problems have been resolved, in large part because of the growing understanding of the complex interactions among the subfields of mathematics. As those relationships continue to expand and deepen, mathematics is beginning to reach out to explore interactions with other areas of science. These interactions, both within diverse areas of mathematics and between mathematics and other fields of science, have led to some great insights and to the broadening and deepening of the field of mathematics. I discuss some of these interactions and insights, describe a few mathematical achievements of the 20th century, and pose some challenges and opportunities that await us in the 21st century.

The Commandant's Corollary
by C. W. Groetsch
groetsch@email.uc.edu
Galilieo's parabolic model of the trajectory of a point particle in a resistance-less medium is considered by many to be the genesis of the science of dynamics. While discussions of the Galiean model are widespread in the undergraduate curriculum, elementary results on resisted trajectories seldom find their way into calculus courses. One such result is a barely known, and quite pretty, theorem of a fin de siècle French artillery commander that relates the general resisted trajectory to two enveloping Galilean trajectories. This note is an exposition of that theorem based on results from first year calculus.

Using Statistics to Determine Causal Relationships
by Jerome Reiter
jerome.p.reiter@williams.edu
Increasingly, causal questions are being answered with statistics. For both scientists and consumers, it has become important to understand how valid causal studies can be designed and how suspicious studies can be identified. This paper aims to further these understandings by explaining the statistical principles and techniques that underlie valid studies of causal relationships. It explains how randomized experiments yield reliable estimates of causal effects and describes techniques used to estimate causal effects in non-randomized studies.

Paradoxes Involving Conflicts of Interest
by James Case
jcase66777@aol.com
We argue that the boundary separating genuine paradox from mere subtlety advanced significantly in 1944, with the publication of The Theory of Games and Economic Behavior, by J. von Neumann and O. Morgenstern. Much of what had previously seemed paradoxical suddenly seemed less so. The thesis is developed with the aid of specific examples, including the original "Holmes-Moriarty paradox", the so-called "backward induction paradox", and the "surprise examination" or "hanging paradox".

Start Where They Are: Geometry as an Introduction to Proof
by J. E. McClure
mcclure@math.purdue.edu
All of us who teach mathematics majors would like our students to be able to do proofs, but it is a common experience that students have great difficulty making the transition from their early mathematical training to an environment in which proof is emphasized. In response to this difficulty, many mathematics departments have created "bridge" courses, usually focusing on a set theory or the elements of analysis. Another common approach is to use linear algebra as the students' first encounter with formal proof. Both of these methods can be used with reasonable success to train students to write elementary proofs that obey the rules, but I argue that neither of them is the best foundation for further progress because neither really addresses the root causes of our students' difficulties. Students who are learning about axiomatic formal proof for the first time have underlying, often unarticulated, confusions that are not easily perceived by professional mathematicians to whom formal proof has become second nature, and my contention is that the only truly satisfactory way to meet and answer these confusions is to begin with Euclidean geometry. Since there has been a trend for many years to de-emphasize or even eliminate formal proof at the high school level, I suggest that for most students the natural role of a bridge course is to provide a thorough study of the use of proofs in Euclidean geometry. I describe one effective way to organize such a course, and I also argue that Euclid's Elements should be one of the tools we use to teach axiomatic methods.