# $\epsilon, \delta$…So what?

Over the course of my studies I often encounter phrases in reference material of the type "and this avoids the need for using $\epsilon$, $\delta$ definitions" or "by this we can omit those complicated $\epsilon, \delta$ arguments", etc. In other words performing stunts in order to get around $\epsilon, \delta$. I've seen enough of this to think that it should be categorized as epsilondeltophobia, if you all will permit. Personally, I was thrilled to learn definitions in these terms because it was one of the first rigorous definitions given to me, all in terms of quantifier logic, and it was used for very fundamental things whose real meaning I always wondered about. In the beginning of course I didn't have a clue how to use the language, but I loved it anyways because it was like, "wooow, deep maan". Not to mention that later on, I began to see that all of the higher-order constructions that were built upon $\epsilon, \delta$-objects worked out perfectly, giving me more satisfaction that whoever came up with $\epsilon, \delta$ language knew what they were doing. So I'm not saying that it's not ok to develop an epsilondeltophobia, as we all do naturally in the beginning...but textbooks (some) seem to promote this fear, even some teachers, and this is what I'm not happy about. I think $\epsilon, \delta$ is great.

Question: who thinks likewise? oppositely?

Edit: I don't want this to come off as a pedantic "rigor or death" statement, or as a suggestion that first courses on calculus should always include $\epsilon, \delta$ (although maybe yes in mathematics). I'm just against the predisposal to it in a negative way.

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Examples please? Where have you encountered this? – nbubis Jan 9 '14 at 17:00
at least give the context - a physics book? calculus chapters on derivatives? – nbubis Jan 9 '14 at 17:05
I might suggest some of Terry Tao's blogs about non-standard analysis and "epsilon management" as instances of this, though he hardly has a "phobia" of hard analysis. – Ryan Reich Jan 9 '14 at 19:52
$\varepsilon$ and $\delta$ proofs are like assembly languages. As soon as you have a higher-level command for a loop, you should not program a loop in assembly language because it is both hard to do and both hard to understand. In the same vein, you use the definition to prove properties of limits and continuous functions and then you use these properties instead of the definition. This has nothing whatsoever to do with lack of rigor. It is possible to give more and less rigorous proofs with and without lower-case Greek letters. – Phira Jan 9 '14 at 20:50
@GPerez But it is more than that. The key issue is to choose the "right" properties to prove. If you later need $\varepsilon$ and $\delta$ again, this is a sign that you are missing an appropriate general property. This can give the impression of phobia, but it is an important heuristic. – Phira Jan 9 '14 at 21:39

I think that is a complex issue; we have both pedagocical aspects and "foundational" ones.

First, according to my point of view, and assuming that I'm not prepared to discuss the pedagogical side, I think that we cannot avoid in teaching mathematics (and not only) some amount of "dogmatism". Past failure in the efforts to introduce naive set language in advance to elementary arithmetics was significative.

Try for a moment with this "conceptual experiment" : teaching in secondary school algebra and calculus starting from axiomatized $ZF$ and building all mathematical stuff "from scratch" (the empty set) . Do we really think it feasible ?

A recent book by John Stilwell, The Real Numbers An Introduction to Set Theory and Analysis (2013), start with the following consideration :

any book that revisits the foundations of analysis has to reckon with the formidable precedent of Edmund Landau’s Grundlagen der Analysis (Foundations of Analysis) of 1930. [...] so few books since 1930 have even attempted to include the construction of the real numbers in an introduction to analysis. On the one hand, Landau’s account is virtually the last word in rigor. [...] On the other hand, Landau’s book is almost pathologically reader-unfriendly.

I've tried re-reading Landau : it is very "unfriendly" !

Second : please, don't forget the enormous amount of effort it takes, form Newton and Leibniz until (at least) Cauchy (see the wonderful book of Judith Grabiner, The Origins of Cauchy's Rigorous Calculus - 1981) to "distill" the rigorous $(\epsilon − \delta)$ definition! And also mathematical standards of "rigor" are evolving.

I spoke above about "dogmatism" (suggestion : think how to apply Thomas Khun's considerations in SSR about the "positive" role of dogmatism in "normal science" to mathematics).

My personal feeling is that the best antidote to the (unavoidable) use of dogmatism in teaching is the historical perspective: to learn how we arrived at current ideas (included our current standard of rigor and our current ideas about "foundations") can be very useful.

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Regarding the "conceptual experiment", I think it'd be very positive if the definition of a vector space over a field were somehow introduced in high-school, and elementary proofs like $\lambda v = 0 \Leftrightarrow \lambda = 0 \text{ or } v=0$, because it's relatively simple, yet rigorous at the same time. I don't know... I just like the concept of not talking about something until you define what you're talking about. Logically, a balance has to be found, otherwise we'd all be reading Lanadu before learning to count. – GPerez Jan 10 '14 at 13:13
Also, about the historical perspective, yes yes and yes. Because we always get the format: axioms$\to$definitions$\to$theorems, as if it happened that way chronologically, whereas the real diagram is probably the most complex of directed graphs, if it were simplified. – GPerez Jan 10 '14 at 13:17

I believe that the pushback against $\epsilon,\delta$ definitions (which unfortunately spills over to pushback against $\epsilon,\delta$ techniques) is entirely justified because $\epsilon,\delta$ definitions arise from the (unfortunately widespread) confusion between a statement being formal and a statement being rigorous.

Consider the formal "definition" of continuity of a function $f$ at a point $a$: $$\forall\epsilon\exists\delta\forall x(0<|x-a|<\delta\rightarrow |f(x)-f(a)|<\epsilon)$$ This is just an objuscated way of stating the informal, but rigorous:

For every ball $B_{f(a)}$ centered at $f(a)$, there is a ball $B_a$ centered at $a$ so that $f$ sends every point of $B_a$ into $B_{f(a)}$.

which is logically equivalent to the conceptually clearer, though still informal, though still rigorous:

Whenever the image $f(S)$ of a set $S$ is separated from the image $f(a)$ of a point $a$, the set $S$ was already separated from the point $a$.

which is the contrapositive of the, informal and rigorous, intuitive definition of continuity of $f$ at a point $a$:

Whenever a set $S$ of points are close to a point $a$, the set of images $f(S)$ of those points are close to image point $f(a)$.

I strongly believe that the equivalence of the blocked statements and the IDEA that equivalence expresses, which is that we CAN distill an intuitive notion into a rigorous definition, is much more interesting, important, and memorable, than the formal $\epsilon,\delta$ "definition". Furthermore, I can't even bring myself to calling the formal "definition" a definition, since what it expresses is not a description of what it means for a function to be continuous, but a technique (of $\epsilon,\delta$ proofs) for how to check that a function is continuous.

This, in my opinion, is the reason for the pushback against $\epsilon,\delta$ "definition" and arguments: instead of expressing the rigorous idea or concept of continuity, the $\epsilon,\delta$ "definition" only gives a technique for working with continuity, and, when presented as a definition, only obfuscates the meaning of the concept (in a very efficient way, I might add, since the path from the intuitive and meaningful definition to the $\epsilon,\delta$ definition involves taking a contrapositive...).

Finally, I do think that being aware of how to rigorously translate (as above) from the intuitive definition of continuity to the statement of the $\epsilon,\delta$ technique will certainly not hurt, and I suspect could actually help students in using the ($\epsilon,\delta$) technique, especially with the simple functions that arise in Calculus and basic analysis.

(Someone might criticize the above saying that the notion of a ball is confusing in single-variable Calculus. My perhaps controversial response is that there really isn't any good reason not to teach Calculus using $2$ or $3$ variables from day $1$ and that the narrow viewpoint offered by single-variable Calculus obscures more than it simplifies).

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It's okay though, I mean accepting $\epsilon,\delta$ as a definition doesn't mean prohibition of developing intuitions about what it means. Personally I always made this diagram (thank you geogebra) mentally whenever imagining it. As a side note what is "what something means" if not the definition we have decided for it? – GPerez Jan 9 '14 at 20:14
The intuitions, hence the meaning, come before the definitions. That's just how mathematical practice is, and one's choice of definitions reflects either one's active preference for some intuitions, or one's complacent (unexamined) acceptance of other intuitions... – Vladimir Sotirov Jan 9 '14 at 20:43
Why of course I'm complacent in accepting these intuitions, I happen to agree with them. And maybe we have conceptions a priori in our minds, but until you define what it means, you can't "do things" with them. – GPerez Jan 9 '14 at 21:57
Thanks Vladimir for your response. I could not agree more. $\epsilon, \delta$ are just a shorthand notation to save typing effort in place of their equivalent statements in English (or whatever language the student is being taught in). For me $\lim_{x \to a}f(x) = L$ means that the values of $f$ can be made arbitrarily near to $L$ by choosing values of $x$ sufficiently near to $a$ (but not equal to $a$). And there is nothing non-rigorous in the last statement. To me $\epsilon, \delta$ is a sign of formalism and not rigor. – Paramanand Singh Jan 10 '14 at 4:30
I'm not sure "rigorous" and "formal" are exactly the right words to use (I admit that the word "rigorous" in math has become a pet peeve of mine; I'm no longer sure what it means), but I find the sentiment of this answer dead-on. Calculus textbooks approach the $\epsilon$-$\delta$ definition by emphasizing formal algebraic manipulations of inequalities, which seems especially cruel since anyone who teaches calculus knows that students are absolutely untrained in and bad at this. So the underlying geometric/topological meaning is lost, despite the fact that it could be easier to understand. – Pete L. Clark Jan 13 '14 at 14:34

It turns out that engineers, scientists, and financial folks need to use calculus, but they don't need to understand calculus.

The construction of the typical university education feeds all of those students, plus math students, through the same introductory calculus courses. This is done for cost efficiency, and also because of a potentially mis-placed ideal that career mathematicians should teach mathematics to people for whom mathematics is ultimately really just an annoying means to an end.

So eliding $\epsilon - \delta$ arguments streamlines this process, saving trouble for the students and the instructors, at the expense of the math students. But those math students will encounter it later, anyways.

I'm not saying it's the best approach, but it's a bit more efficient perhaps. Mechanical engineers don't want to learn $\epsilon - \delta$, and math professors don't want to teach $\epsilon - \delta$ to students who will never truncate a Taylor series beyond the linear term.

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As I say to my students: "If you don't understand it, you are not allowed to use it". – Siminore Jan 9 '14 at 17:41
"It turns out that engineers, scientists, and financial folks need to use calculus, but they don't need to understand calculus." Wow, what a clear statement of what seems implicitly to be the dominant position in the contemporary pedagogy of freshman calculus...and which I find to be alarming and pernicious bordering on repugnant. I wish we college math teachers could have an explicit discussion to see whether we really believe this and fully understand its implications. – Pete L. Clark Jan 13 '14 at 13:53
Two opening salvos: (i) Both real world exposure to young people and much research bears out the hypothesis that almost everyone begins their study of (e.g.) mathematics wanting to understand everything. The tragedy of mathematics education is that we beat this desire out of 99% of the students so systematically and thoroughly that by the time they get to calculus they have a Stockholm syndrome level identification with their torturers. (ii) In contemporary society, the name for the thing that does calculations without conceptual understanding is "computer", not "engineer". – Pete L. Clark Jan 13 '14 at 14:00
Let me also say -- if I can say something without wanting to back it up in these comments -- that I have often been less than favorably impressed with the amount of "understanding of calculus" evinced by the way calculus textbooks are written. A lot of times what mathematics textbooks at this level call "rigor" I would be tempted to call "pedantry brought on by a limited, fragile understanding". – Pete L. Clark Jan 13 '14 at 14:13
@Arkamis: Well, the first step is to identify and agree on the problem. And then....can I get back to you? – Pete L. Clark Jan 13 '14 at 22:35

$(\epsilon,\delta)$ techniques are fundamental to developing the foundations of real analysis, but sometimes insight can be gained via alternative techniques that one doesn't see as readily via $(\epsilon,\delta)$. Consider for example the failure of the squaring function to be uniformly continuous. This is quite a tedious exercise to motivate if you are limited to $(\epsilon,\delta)$ techniques. Possibly 90% of undergraduates will be unable to reproduce such an exercise other than in a passive way.

An alternative possibility would be to note that $f(x)=x^2$ fails to be microcontinuous at a single infinite point $H$ and is therefore not uniformly continuous. In this approach uniform continuity is defined by requiring $f$ to be microcontinuous at all points (standard and nonstandard) of its extended hyperreal domain. Thus, if one considers an infinitesimal $\alpha=\frac{1}{H}$, then $f(H+\alpha)=H^2+2H+\alpha^2$ and $f(H+\alpha)-f(H)=2+\alpha^2$ which is not infinitesimal. Thus we see that $f$ is not microcontinuous at $H$.

This definition makes it transparent that uniform continuity in this case has to do with the behavior of the function "at infinity". This remark can be formalized in the context of an infinitesimal-enriched continuum, but cannot be formalized in the context of the real continuum.

Thus the $(\epsilon,\delta)$ approach has its advantages but it also has serious pedagogical shortcomings.

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1) Yes, uniform continuity is a killer. But, characterizations exists and they help. 2) As far as pedagogy goes, I disagree. Saving people from abstracting does not help them. Also, when somebody finally gets it, 100% of the time they feel really good about themselves. This is good. We want people to feel good about the things they are learning. Of course one asks: what if they don't get it? They will, eventually. The only problem is that not doing so on time has dire consequences, and so we turn to question how the whole testing and grading model is set up, but this is a different discussion. – GPerez Jan 9 '14 at 18:24
@GPerez, as to 1): I precisely formulated such a "characterization", as you put it, and it helps. as to 2): I am not sure what you mean by "abstracting". If you are referring to mathematical rigor, then the hyperreals are just as rigorous as the reals. The only "dire consequences" I see here is for people who refuse to consider alternatives. – Mikhail Katz Jan 10 '14 at 8:37
I don't doubt that the hyperreals are valid, or abstract. In fact they are more abstract then the real numbers. But it all depends, do you want to give a mathematical education where everything is constructed and built upon, and proving steps along the way? In this case at some point you'd have to include a proof that the hyperreals are consistent if and only if the reals are. I don't know if it's complicated or not for students, if it weren't though then I'd have no complaints about this being used in mathematics education. If the education is to be given without proofs well, okay. – GPerez Jan 10 '14 at 12:53
Taking the nonstandard analysis argument, and deleting some words and concepts, gives a simpler standard argument proving the same thing in the same way. All the work consists of determining what scale of change to $H$ causes an appreciable change to $H^2$. There is nothing about NSA that would have indicated to choose $\alpha = 1/H$ as the infinitesimal, and the proof does not work with arbitrary $\alpha$. The understanding of which choices work comes from the derivative, not foundations, though of course once you have a working argument it can be written in either language. – zyx Jan 11 '14 at 8:44
I very much like Terry Tao's approach, by the way: he clearly explains both standard and nonstandard analysis, is especially clear about the relations between them, and gives examples of which type of problem one might be preferred to the other. He bears this out in his research, so it is clear that SA and NSA are both very useful for him. I have never seen him say that hyperreals have "more expressive language and utility in research": his position is much more nuanced than that. – Pete L. Clark Jan 13 '14 at 15:13

There's definitely something to @Arkamis's remarks (if they are somewhat more relevant to the American system), but there's also something to be said for the opposite.

$\epsilon-\delta$ language tends to be overly technical; it's simple enough to phrase to 1st year students, and precise enough to practice rigorous mathematics, but all these technicalities can also obscure the point (a la the famous trees-forest analogy). The concepts of open sets and preimages under functions can be somewhat more powerful, and/or point to the crux of the proposition one considers, whereas having to deal with too many quantifiers might be cumbersome.

So, when you see proper math textbooks using seemingly complex constructs to avoid speaking in $\epsilon-\delta$ language, I put forward that most of the time that is done in the name of abstractness, to better phrase underlying concepts, or better deal with new and more general notions the author wishes to present.

I had to return here when I encountered this, an example where OP did a great job solving a problem with $\epsilon-\delta$ techniques, but still seemed to be feeling uncomfortable with the results. For my money, that's exactly because this language hides the crux of the problem, the reason things work the way they do. Having completed the exercise, I believe OP still wouldn't have pinned the underlying property that's present in the cases for which the answer is 'yes', and absent where it's 'no'.

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Oh no, by no means am I criticizing the generalization of these concepts, as it becomes a must when there's no longer a metric function – GPerez Jan 9 '14 at 17:39