Department of Mathematics, LSU Baton Rouge
Math 4005: Geometry

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. The natural system, on the other hand, is for many people the route to discovery and understanding. But communication and understanding are partners. One has little value witout the other.

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.

Poincaré. This reading consists of excerpts form a classic essay on definitions in education that is included in Poincaré's famous book of essays: Science and Method. Poincaré considers the educational problems that arise from the peculiarities of mathematical definitions.
Mathematics educators David Tall and Schlomo Vinner wrote a famous essay on the distinction between "concept image" and "concept definition". Their ideas seem in accord with the Murphy reading (below). Concept image is what people tend to have in mind when they think of something--a quadrilateral, say. Concept definition is the exact formal rule for the use of a term: "a quadrilateral is a union of four segments, each of which meets two others at its endpoints and only at its endpoints." It may be very different from the concept image, yet it is what counts in formal reasoning. A comparison to the law might be useful. When you think of "trespassing" you probably have specific mental images. The law defines trespassing in a very precise way that may not accord with your mental images. When you go to court, it's the law that counts, not your images. (Here are additional comments by Tall.)
Murphy. Excerpts form Gregory L. Murphy, The Big Book of Concepts. Cambridge: MIT Press 2002. This is a textbook on cognitive science. This chapter demonstrates that the "natural" way in which concepts work is quite different from the way mathematical concepts function. A mathematical definition for a concept gives necessary and sufficient conditions for a thing to be an instance of that concept. ("Parallelogram: a quadrilateral in which the both pairs of opposite sides are parallel." The conditions are necessary: if something is not a quadrilateral or if it is, but it fails to have opposite sides parallel, then it's not a parallelogram. The conditions are sufficient if something has the properties in the definition, then it's a parallelogram. Nothing more is needed.) But natural concepts (like "dog" or "game") cannot be specified by necessary and sufficient conditions. No matter what properties you might list in order to give necessary and sufficient conditions for being a dog, you will likely find some dogs that don't have all the properties on your list, or some things that do, but are not dogs.
Wu. This is a contemporary essay by a mathematician on the use of definitions in mathematics and math education.