Undefined terms: What are they?

Recall that we have discussed about the meaning of “undefined” in different contexts.  In this post we are going  to discuss the meaning of undefined terms. What do we really mean when we say undefined terms in mathematics?

The time we started to learn mathematics, we also started to learn its vocabulary. We have learned about the definitions of terms such as triangles, fractions, polynomials, axioms, etc.  In our years of studies, we also have acquired the skill of being concise and precise in what we say or write.  Like in other fields of endeavors, we need to understand each other and make sure that we mean the same thing when we say a particular word. That is why we need to define them.  For example, when we say p0lygon (see figure above), we mean (loosely) that it is a plane figure bounded  by a finite number of line segments. Notice that from the definition itself, we must also define the other terms used. What do we mean by plane figure or line segment?

Now, suppose we define line segment as a part of a line that is bounded by two endpoints, again, we have to define the word line and endpoints.  If we, however, define these terms, we will have to use other terms which means that we also need to define them.  And, this will go on forever. That is why we have to stop somewhere. We have to have some undefined terms — terms that do not require a definition. These terms are used as a “base” to define other terms. They serve as building blocks of other mathematical terms.

In Geometry, we have three undefined terms: point, line, and plane.  Of course, even though we call them undefined terms, it does not really mean that we are forbidden to describe them.  In fact, many books still attempt to do so. For instance, a point is usually described and represented as a dot on a page, some even describe it as a star in the sky, but authors would always caution that a point has no dimensions (length and width). A line is described as like the edge of a ruler that never ends and a plane is like a football field that extends indefinitely. Of course, as teachers, we should be careful with these descriptions since they can cause misconceptions.

Another example undefined term in mathematics is set. A set is described as a collection of objects where we can determine if a particular object belongs or does not belong to that set.