January 4, 2010

Students Learn the Philosophy Behind Mathematics

Interdisciplinary course was taught by Joe Shieber, assistant professor of philosophy, and Lorenzo Traldi, Metzgar Professor of Mathematics

A new team-taught course offered during the fall semester provided advanced philosophy and math majors with an introduction to the set theory, the logical foundation of mathematics, and the philosophical discussions of mathematics. The course was supported in part by a grant from the Andrew W. Mellon Foundation.

Infinite Sets, taught by Joe Shieber, assistant professor of philosophy, and Lorenzo Traldi, Marshall R. Metzgar Professor of Mathematics, was designed to appeal to math or philosophy majors, particularly those interested in attending graduate school.

“In the current state of the discipline, philosophers of mathematics cannot be taken seriously unless they have a very solid grounding in mathematics itself,” says Shieber. “For this reason, students interested in even gaining a basic familiarity with the philosophy of mathematics need an exposure to the mathematical techniques and results in set theory and other areas of interest to philosophers. An added source of interest in set theory for philosophers is that many of the prominent set theorists at the end of the 19th century and the beginning of the 20th century were also philosophers.”

Set theory is the branch of mathematics that studies sets, or any collection of objects. Students in the high-level seminar course studied the theory’s implications for philosophical questions concerning the meaning of mathematical expressions, the nature of mathematical objects, and the extent of mathematical knowledge.

Traldi says that the interdisciplinary approach was beneficial to students in several ways.

“Philosophers seem to be more critical than mathematicians — they wonder why certain techniques are considered valid, rather than simply learning how to use them. Mathematicians tend to be less critical perhaps, but we are very careful about using techniques accurately. Both attitudes are valuable for advanced students,” Traldi says.

The professors also benefited from teaching the course as a team.

“For me as a teacher, I have benefited greatly from the chance to observe Professor Traldi in action; he is truly a consummate master-teacher and an excellent explicator of what are, quite often, very difficult mathematical ideas. It has also been useful for me to have to explain the philosophy of mathematics to the prospective mathematicians among our students, many of whom have never considered the sorts of questions that philosophers pose about mathematics. For me as a researcher, given my interest in the nature and sources of knowledge, the examples that I encounter in mathematics offer me excellent case studies for reflection and productive fodder for articles,” Shieber says.

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