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Department of Mathematics, LSU Baton Rouge
Math 4005: Geometry

I.E. Addendum: What are Definitions?

I am always surprised by how difficult it seems to be for students to begin to use definitions in the way a mathematician does--as perfectly solid and fully adequate foundations for reasoning. They commonly assume more than what the definitions state or feel the need to prove more. For example, a parallelogram is by definition a quadrilateral with opposite sides parallel. But asked what a parallelogram is, many students give more information, e.g., it's opposite sides are congruent. And asked to prove that a figure is a parallelogram, they often prove more than just that its opposite sides are parallel.

Natural concepts are hazy, flexible and redundant. There are good reasons for this, for in many natural situations these properties are desirable. When people communicate, they adapt their cognitive frames to the frames of those with whom they're communicating. When faced with novel problems, people seek solutions by modifying existing cognitive patterns. And as any engineer knows, redundancy can protect a system from faults and errors.

In mathematics, we we set up artificial conceptual systems that are clear, rigid, and sparse. Such systems give us the power to find truths that natural systems cannot see. There is a cost. To use the artificial systems, we must be far more observant of detail, far more more sensitive to fine distinctions far more literal than most people are naturally inclined. Sometimes, we need to suppress the thoughts that spring from older, natural habits.

Yet, when we learn to use mathematical concepts, we do not cease to reason in the natural way. Our natural reasoning powers provide guidance and intuition as we craft demonstrations using the formal concepts. Both kinds of thinking are important, but they are separate. The formal system is the final arbiter of truth, and what mathematicians share with their mathematical peers is framed in this system.

Readings on the idea of a definition: Here are three readings that I think provide useful insights into these issues, particularly in relation to mathematics education.