17 February 2016

"Why doesn't that work?" "That's it?" "Why is that the solution?" These are questions that I've heard many times from people struggling with a math problem. They always bothered me and I didn't really know what to say. It felt like something was just missing.

In so many cases, the immediate reason for the struggle is simple: The solver doesn't understand the problem. But that doesn't answer the bigger question. Why didn't they understand? I still don't think I have the whole picture, but I think I have a piece.

A function $$f$$ is continuous if for every $$x$$ and every $$\varepsilon > 0$$, there exists a number $$\delta > 0$$ such that $$|f(x) - f(y)| < \varepsilon$$ for every $$y$$ with $$|x - y| < \delta$$.

## A Linguistic Diversion

What does this sentence mean?

A guard stood in front of every house.

Did you imagine one guard, perhaps going to each house in turn and standing in front of it, or did you imagine one guard per house, and all the standing being simultaneous? Both meanings are possible. There are two quantifiers in the sentence: an existential quantifier ("a guard"), and a universal quantifier ("every house"). In English, those quantifiers can be read in either order, producing the two meanings. This situation is called a quantifier scope ambiguity.

Generally, when you come across these ambiguous sentences, context will make one reading stand out over the other. There's no point in reading the sentence closer to determine which reading is "right" because English simply doesn't make a distinction.

How would you interpret this exchange?

Q: Did everybody pass the exam?
A: No, everybody did not pass the exam.

Does it mean:

1. There is somebody who did not pass the exam.
2. Nobody passed the exam.

In my experience, the first meaning is almost always the one intended. This seems to be true despite the fact that there is an alternate construction "Not everybody passed the exam." that would be unambiguous and mean exactly that.

## Back To Math

What do these ambiguities have to do with having trouble with the $$\varepsilon-\delta$$ definition? The existence of multiple quantifiers. If I were to write out the above continuity condition without words, it would look like this:

$$\forall x. \forall \varepsilon > 0. \exists \delta > 0. \forall y. (|x - y| < \delta \Rightarrow |f(x) - f(y)| < \varepsilon)$$

But what happens if someone encounters the definition and reads the quantifiers in the other order? In English that's totally fine; the sentence is ambiguous. In this case however, you'd get a much less sensical condition:

$$\forall x. \exists \delta > 0. \forall \varepsilon > 0. \forall y. (|x - y| < \delta \Rightarrow |f(x) - f(y)| < \varepsilon)$$

An astute reader might notice that this condition is only true when $$f$$ is a constant function. So you can imagine that a student encountering this definition for the first time might be extremely confused if they read it this way.

## What To Do?

Unfortunately, I don't have any students to try teaching calculus to in order to see if any of these help, but here are a few things that I could see helping.

### Emphasize the dependence of $$\delta$$ on $$x$$ and $$\varepsilon$$.

The simplest change might be to write $$\delta_{x, \varepsilon}$$ instead of just $$\delta$$. The fact that $$\delta$$ depends on $$\varepsilon$$ should emphasize that $$\varepsilon$$ is quantified first.

It might be even better to write this dependence as a function itself, so instead of $$\delta$$, write $$g(x, \varepsilon)$$. It's essentially the same idea, but could help with separating the variables' roles in students' minds.

### Do work with simpler statements involving quantifiers before introducing the definition.

I can't think of any topic earlier in the standard math curriculum that involves so many quantifiers. Of course people are going to be confused, since they have no practice. I'd suggest making sure that students are comfortable with at least the following ideas (with example applications, of course):

• $$\lnot (\forall x. P(x))$$ is equivalent to $$\exists x. \lnot P(x)$$
• $$\lnot (\exists x. P(x))$$ is equivalent to $$\forall x. \lnot P(x)$$
• $$\forall x. (P(x) \land Q(x))$$ is equivalent to $$(\forall x. P(x)) \land (\forall x. Q(x))$$
• $$\exists x. (P(x) \land Q(x))$$ implies, but is not equivalent to $$(\exists x. P(x)) \land (\exists x. Q(x))$$
• $$\exists x. (P(x) \lor Q(x))$$ is equivalent to $$(\exists x. P(x)) \lor (\exists x. Q(x))$$
• $$\forall x. (P(x) \lor Q(x))$$ is implied by, but is not equivalent to $$(\forall x. P(x)) \lor (\forall x. Q(x))$$
• $$\forall x. \forall y. P(x, y)$$ is equivalent to $$\forall y. \forall x. P(x, y)$$
• $$\exists x. \exists y. P(x, y)$$ is equivalent to $$\exists y. \exists x. P(x, y)$$
• $$\forall x. \exists y. P(x, y)$$ is very different from $$\exists y. \forall x. P(x, y)$$

### Introduce a topological view of continuity.

Teaching an introduction to point-set topology seems like it would have some advantages over a traditional calculus course. While topology is more abstract, the basic concepts are all relatively simple. And, despite the abstraction, there are tons of illustrative examples that can be given.

Topology serves as a good practice ground for set theory, and the ideas there are essentially the same as the logical rules from before (replacing intersection for and, union for or, and subset for implication), so it serves to reinforce those ideas. Then once you get to continuity, the definition is much simpler:

A function $$f$$ is continuous if the preimage of any open set is open.