Question

When introducing students to highly technical definitions for seemingly intuitive concepts (e.g., homotopy, continuity), how do you motivate the necessity of the definition? On the one hand, you would hope that the students are mathematically mature enough to appreciate a rigorous definition such as the epsilon-delta formulation of continuity. But if they are not (for example, epsilon-delta arises in their first truly proof-based class), are there standard cautionary tales that are especially convincing in conveying the worth of the technical definition?

(Context: I am teaching a course for students many of whom have not taken classes beyond linear algebra. The course serves as an introduction to proofs, and one part of the curriculum is continuity; for some of the students, this is the only place they will ever encounter epsilon-delta.)

Answer 1

I once asked my honours real analysis class to define the concept of an integer to a hypothetical bright young kid who was already perfectly familiar with the natural numbers and the operations one could perform on them, but had not yet been exposed to negative numbers. The response was both enthusiastic and chaotic; I remember one student, for instance, giving a heuristic to explain why the product of two negative numbers was positive, which was interesting but not directly useful for the problem at hand.

Nevertheless, the question served its purpose; when I did then introduce a rigorous definition of the integers (as formal differences of natural numbers, quotiented by equivalence), the need for such a formal definition was made much clearer by the lack of an "obvious" way to do it by other means. And I think it also had a residual effect in motivating the fancier epsilon-delta definitions that arose later in the course.

Another example I have seen, at the early high school level, is to challenge students to come up with a watertight definition of a rectangle. This is remarkably difficult to do for students without training in higher mathematics; not only does one have to deal with degenerate cases (e.g. line segments), but often crucial properties (e.g. that the four sides of a rectangle have to be connected at the vertices) are omitted. One can also get into interesting debates, such as whether a square should be considered a rectangle.

Answer 2

I think an excellent way of motivating precise definitions is to walk through a sequence of candidate weaker/less precise definitions, and for each of them construct an explicit counter example that leads to a failure of whatever property you'll often be interested in deducing in the class.

Perhaps also as a supplementary/extra credit exercise in homeworks, have similar work (albeit with some sort of mention of how if the counter example can be found on wikipedia, the explanation of why it fails must be especially rigorous), and then after each such homework is collected, walk through some of the counter examples for these questions.

Answer 3

Cauchy published a "proof" that a convergent sequence of continuous functions converges to a continuous function, relying on a not-completely-rigorous idea of continuity. This is particularly notable given Cauchy's role in giving precise definitions here, and also given how easy it is to think of counterexamples.

The discussion in the following link may be relevant: http://www.math.usma.edu/people/Rickey/hm/CalcNotes/CauchyWrgPr.pdf

Answer 4

I think it's darned near impossible, unless you are actually going to use the definition in a proof of some non-obvious fact. A semi-obvious fact, such as the intermediate value theorem, won't quite do, for you now face the problem of explaining why this requires a proof at all. (Given a notion of continuity as “the function graph is connected”, it is not at all clear that this theorem is not obvious.)

Sorry to be such a downer – but you say it's a course on proofs, so maybe it's not so bad after all. I mean, it should be clear that the very idea of a proof requires definitions to work with? So you can tell them that if the need for this definition is not clear at present, it will become clear once you start trying to prove stuff. Mike's answer is of course excellent, if you can dig up the actual mistake Cauchy made and if it isn't going to be too advanced. All sorts of pathological examples, like space filling curves or functions that are everywhere continuous but nowhere differentiable, are good to drive home the point, but again there is a risk they might be too demanding.

Answer 5

I like explaining that without definitions, they don't even know what they think they already know. For example: What is the definition of $2^3$? $2$ times $2$ times $2$. Right. What is $2^{-3}$? $1/2^3$, right. What is $2^{1/3}$? $3^{rd}$ root of $2$, right. What is $2^{\sqrt{2}}$? Nobody? What does it even mean?

Answer 6

Sipser's Introduction to the Theory of Computation spends some time motivating the need for precise definitions, since it's aimed towards a computer science audience who may not have experience with proofs. Here's what he has to say before the introduction of the first really formal definition in the book on page 35:

In the preceding section we used state diagrams to introduce finite automata. Now we define finite automata formally. Although state diagrams are easier to grasp intuitively, we need the formal definition, too, for two specific reasons.

First, a formal definition is precise [emphasis added]. It resolves any uncertainties about what is allowed in a finite automaton. If you were uncertain about whether finite automata were allowed to have 0 accept states or whether they must have exactly one transition exiting every state for each possible input symbol, you could consult the formal definition and verify that the answer is yes in both cases. Second, a formal definition provides notation [emphasis added]. Good notation helps you think and express your thoughts clearly.

The language of a formal definition is somewhat arcane, having some similarity to the language of a legal document. Both need to be precise, and every detail must be spelled out.

For a more mathematical example, you might want to spend some time talking about how hard it is to come up with a formal definition of polyhedron such that Euler's formula is actually true.

Answer 7

I usually like to tell students that math isn't an experimental science, and proofs is the only way we can be sure something is true. Topology offers many opportunities to illustrate this point. As the course progresses, and students see some non-obvious facts derived, they get better appreciations of proving as the method to validate mathematical truths. At some point I like to discuss that nobody knows for sure whether ZFC is consistent, so we should take this method with a tiny grain of salt.

Answer 8

For the particular case of continuity, it seems to me that functions like $x \mapsto \begin{cases} x\sin\frac{1}{x} & x \neq 0 \\ 0 & x = 0\end{cases}$ and $x \mapsto \begin{cases} \sin\frac{1}{x} & x \neq 0 \\ 0 & x = 0\end{cases}$ are good motivators -- specifically, they both show that the naive definition doesn't always allow us to distinguish continuous from discontinuous functions. (I might hold an in-class vote about whether those two functions are continuous, for example -- I suspect that opinions would differ, and this would provide motivation for a definition that could be unambiguously tested.)

Answer 9

This won't be a very precise answer, but might still be useful. I have occasionally been able to convince someone that a precise definition is a useful thing because you can know for sure when you've checked it. For example, it's surprisingly involved to define whether a graph is connected, under people's usual intuition: for all pairs of vertices, there exists a finite number $n$, such that there exists a sequence of vertices, such that for all vertices $v_i$ in that sequence, $(v_i,v_{i+1})$ is in your edge set. $\bf But$: once you've gone to the bother of making that precise, it's often pretty easy to show that one or another reasonably defined graph is connected. (Then there's the exercise to show that this connectedness is iff there doesn't exist a separating function onto {0,1}.)

Answer 10

If you make people see enough mathematics, eventually they'll catch on. I don't think a first exposure course for proofs is bound to fail, even if $\epsilon,\delta$ would be people's first exposure to rigorous mathematics. It's just the sort of thing you have to get exposed to a lot if you truly want to get to grips with it. If there is somebody at university level who still ponders if there's any use for the $\epsilon,\delta$-formulation of continuity after three weeks of contemplation, mathematics is just not their trade.

EDIT: As my post surely suggests, I'm a fan of 'staring at it until it's obvious'. I'm still a newbie, but this sort of thing have helped me a great deal in my courses. I wouldn't suspect everyone in class to be familiar with this sort of approach, as I suspect it only makes sense fields like mathematics and physics.

A good motivation to get thinking about this sort of thing is having some sort of non-trivial project to work with. What works for me is if it's intrinsically motivated. This is most often the case I made it up myself, but not everyone will be interested in doing that. What I'd consider doing was requiring a hand-in where the student tries to motivate their own definition of continuity. Then providing an example or two of it working. If it doesn't work, have them say a few words of what might be the problem, and what might fix it. If they end up at $\epsilon,\delta$, make them spill their mind on how and why.

In evaluation, I wouldn't pay too much attention to whether or not they actually came up with a sensible definition of continuity. Just make sure the points are well reasoned, and that they seem aware of any problems that might be there. The students might hate you for this, but I think it'll pass for most/EDIT