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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.)

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I see that someone (not me) has downvoted the question. I understand the impulse, for this sort of question doesn't really fall within the charter of MO; the question does not have a definite answer, and the discussion can quickly become argumentative. Still, we all struggle with this, so I think we should give the discussion a chance – if only as an experiment. –  Harald Hanche-Olsen Mar 20 '10 at 3:24
I think that experimental discussions should be community wiki. –  Jonas Meyer Mar 20 '10 at 8:16
If epsilontics is your students' first experience with proofs, then it's bound to fail. Sorry. –  darij grinberg Mar 20 '10 at 12:51
I have hit this with the wiki-hammer, concurring with Jonas. –  Scott Morrison Mar 20 '10 at 16:08
@darij: there is an overwhelming experimental data to contradict your claim: as far as I know most CS / Math / physics in Europe start their first year in the university with epsilon-delta (my personal experience is Israel), and about half of them move on to second year. I'm not saying it's the greatest idea, but it is certainly not bound to fail. –  David Lehavi Mar 20 '10 at 19:02
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9 Answers

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.

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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.

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I like this idea. It can be seen as a type of Monster bashing. This was discussed by the philosopher Imre Lakatos. You think you have a good definition for something where there is a reasonable intuitive notion, then you show how the definition needs to evolve if MONSTERS are to be avoided! Grothendieck maintained that `dragging concepts out of the dark' was a very important activity yet we do not attempt to discuss this with student. –  Tim Porter Mar 20 '10 at 8:19
I like the idea of going through counterexamples, but on simpler definitions in areas where the students' intuitions are valid, not for the technical definition you want to introduce. –  Douglas Zare Mar 20 '10 at 18:32
Amuzingly, this is exactly what Dave Parnas does when teaching the need for precise requirements for software! He used to go one step further and give it as an assignment to the students to code up a program using these less-than-satisfactory specifications; the next assignment would have students 'swap' programs and now they would be responsible for testing someone else's program. They would then find lots of bugs. The level of understanding thus gained is MUCH higher than the usual 'show and tell' at the blackboard. [And by Curry-Howard, specs and definitions are essentially the same] –  Jacques Carette Mar 20 '10 at 21:51
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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

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This is exactly the error that the famous quote about Dirichlet is making reference to. About how when A proves something, there's a good chance it's true, when Cauchy proves something, youv'e got a 50-50 shot, but when Dirichlet proves something, it is a real proof. –  Harry Gindi Mar 20 '10 at 18:47
"A" being Gauss in Jacobi's quote. –  Dror Speiser Mar 20 '10 at 20:02
fqpc, I am accustomed to seeing that wind up with some resolution of the form “when B says that he has proved it, you know that it's false”. I don't remember who B is canonically, but I suspect we can all advance our own candidates. :-) –  L Spice Mar 20 '10 at 23:07
Regarding the role of Cauchy in development of analysis: I always thought the key name, if not the key figure, here was Weierstrass? –  Yemon Choi Mar 21 '10 at 8:49
Yemon, I think that one may view Cauchy more as the father of modern real analysis, and Weierstraß more as the father of modern complex analysis. (Well, that's my simplistic view, anyway—probably someone will set me straight.) –  L Spice Mar 21 '10 at 23:55
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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.

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Unless I've misread the original question: it's not a course on proofs, but rather a course in which it is expected that most students will first encounter proofs. –  Yemon Choi Mar 20 '10 at 3:25
Actually space filling curves might rather lead the students to the conclusion that the epsilon-delta-definition is not good because it includes such pathological examples. –  Peter Arndt Mar 20 '10 at 3:28
@Peter: So grant them the point. Introduce Lipschitz continuity and explore it a bit. It seems very natural, and the “right” definition until the students realize that the square root function is not Lipschitz. At which point you might introduce Hölder continuity, which seems to help, but now you have to keep track of the Hölder exponent instead. Finding the right definition is often a question of balancing the admission of pathological examples versus being too (arbitrarily) restrictive. –  Harald Hanche-Olsen Mar 20 '10 at 13:19
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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?

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Whether they know what it means is a fascinating question, because at some level they do, even if they can't give a rigorous definition. I think if they understand raising to a rational power, they will sort of feel that 2 to the root 2 is going to be well approximated by 2 to the 1.414, for instance. And from there they will see how in principle you could work out 2 to the root 2 to arbitrary accuracy. But of course they may never have explicitly articulated such thoughts. –  gowers Mar 20 '10 at 10:13
This could well lead into all sorts of fascinating philosophical discussions about what it means to know something. What is the usual definition – knowledge is well founded true belief? So the problem becomes one of what “well founded” means, assuming you don't want to lead the students into a discussion about what “true” means. I think I'd recommend against trying the latter. –  Harald Hanche-Olsen Mar 20 '10 at 12:45
Strictly speaking, one must prove that $2^x$ is continuous. –  Lunasaurus Rex Dec 20 '11 at 4:02
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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.

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Qiaochu, I seem to remember that Lakatos's “Proofs and refutations” takes exactly your final idea as its primary motivation. –  L Spice Mar 20 '10 at 23:10
I knew I'd gotten that idea from somewhere, but I couldn't remember where... –  Qiaochu Yuan Mar 21 '10 at 6:41
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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.

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I like to think that maths is an experimental science, and trying to prove the things we suspect or hope are true is the lab work. –  Scott Morrison Mar 20 '10 at 16:11
Aren't we saying the same thing in different words? My point was to explain to students why proofs are essential to what we do. –  Igor Belegradek Mar 20 '10 at 17:54
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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.)

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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}.)

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