# The Trouble with Pedagogy involving the Infinite

2012/11/04 2 Comments

There’s a problem I have with teaching certain mathematical concepts, especially when they involve the word “infinity.” In algebra, one of the ways this pops up is during interval notation, where if you have an unbounded interval, you use the symbols or to denote that it is not bounded on whichever side. One of the problems with this is that now it seems as though we’re treating “infinity” as though it’s a number. People have some conceptual notion of infinity prior to any formal knowledge of how mathematicians use the idea Childhood number topping games trumped by crying out “Infinity,” only to be refuted with a “Infinity + 1” come to mind. Encouraging it seems like a bad idea as if one every gets to talking about when such concerns about infinite sets or real numbers arise the naive notion inclusion of infinity as a number. The usual way of explaining it away is the unsatisfying answer that “Infinity is a concept, not a number.” But that’s strange, what makes something a “concept” and what for that matter is a “number”?

Then again, I have tried the opposite approach of introducing the extended real line in a calculus course. Of course, this often fails, because of the subtleties involving not having all arithmetic operations available. So I ask non-rhetorically, is it better for people to treat as part of an “extended” real number line, or to say “Infinity is a concept, but not a number”? If the latter, what are we trying to say with that?

As one final thought, yesterday, I talked about infinite sums. What is probably obvious to everyone who has dealt with analysis is that we do not in fact, add infinitely many number together. The entire point is that we can find finite sums that are within any margin of error to our value and therefore the infinite sum *is* that value. This is difficult to understand at the first exposure, since it’s a loaded concept with some much behind it. But the intuitive idea of adding infinitely many things together is probably how most calculus students think of the idea. Isn’t this why people have so much trouble with the equation ?

The best approach for relatively low-level classes is some version of “concept, not number”. The “not number” part means infinity is not located on the real number line, arithmetic operations are not defined for infinity, inequalities are not defined for infinity, etc. The “concept” part means we are, admittedly, using a naive childhood conception of infinity, but only as a notational mnemonic device. The interval (4, infinity) is defined to contain all numbers greater than 4. An infinite summation is defined by the limit of partial sums. And so forth. I think this is illuminated by the fact that we could overhaul our notation system in such a way that we never need to use the infinity symbol, or anything that references it, in college algebra and calculus. If we did so we wouldn’t need to have this discussion at all.

Thank you for your comment. I guess part of this post was recalling the feeling that infinity was almost a number back in my high school days. It doesn’t help that complex numbers also had a similar status of “not really existing” in high school despite their status as number (at least in my work). Having notation that suggests that infinity is a number validated in my mind that naive thought, even though I knew it was heuristic then. I would like notation that avoids this issue, especially since it’s not necessary as you say.