Intimations of Infinity, vol. 51, number 7

Overture The comments above (see the next page for who made them and when) represent one type of thinking about infinity. There are other types, as we will see, and they all create difficulties for students, philosophers, and even mathematicians. The purpose of this article is to show how a particular theory about how people come to understand mathematics, APOS Theory, can be helpful in understanding the thinking of both novices and practitioners as they grapple with the notion of infinity. In APOS theory, which will be more fully explained later, an individual develops an understanding of a concept by employing certain mechanisms called interiorization, encapsulation, and thematization. These mechanisms are used to build and connect mental structures called actions, processes, objects, and schemas. To get a feeling for the complexity of how people grapple with infinity, see how you and perhaps some of your colleagues would answer the following questions. How do you think your answers compare with what has been said by mathematicians and philosophers over the last 3,000 years or by students today? • If the slow tortoise starts a little ahead of the swift Achilles, how can this demi-god ever catch up? For Achilles must first advance to where the tortoise started, by which time the plodder has moved on a little, so Achilles must then advance to that spot, and so on, forever. • How can the quantity dx be treated both as a positive quantity with which calculations can be made, and something that can be ignored as if it were 0? • Is 0.999 · · · = 1? • Suppose you put two tennis balls numbered 1 and 2 in Bin A and then move ball 1 to Bin B, Kirk Weller is associate professor of mathematics at the University of North Texas. His email address is [email protected].