To epsilon-delta or not to epsilon-delta?

Hi! Have you started making the real analysis essence videos? I was interested in contributing to those discussions as a patreon. Do you know when you'll start working on that?

I think that, if it's possible, you should make a sort of "optional" or "aside" video that talks about where epsilons and deltas fit in. Or, when you do make the real analysis video, make sure that you include an explanation of limits that's reasonably accessible to newcomers. I really like the explanation of epsilon-delta continuity that you gave 10 minutes into your Hilbert's curve video. If you could provide a visualization like that for limits, I'm sure it would be **immensely** valuable to those starting out with the idea.

Benjamin Grossmann

While I think it's not important to teach people the complex mechanics of how to do delta-epsilon proofs, I'd still include a basic description of the idea with an example, because from that description all the limit properties used in Calculus make more sense and at least seem like they come from somewhere one can imagine about. Otherwise, it would just seem completely unfounded and seemingly random to say the limit as x approaches a of f(x)/g(x) is equal to the limit of f(x) as it approaches a divide by the limit of g(x) as it approaches a, only except when g(x) approaches zero, which is in essence core to the idea of Calculus.

I remember when learning calculus in high school, my teacher did epsilon-delta proofs, and at least one person broke down into tears because they were so confused. I was confused too, but when I did understand it I felt my overall grasp of calculus improved a lot. Precisely because its a difficult topic I think it would be great for you to explain it.

It's helpful to get some guidance. I would never just blindly do what a poll says, but this comment thread and the relatively even split above did help quite a bit to clarify for me what the right approach/goal might be here.

3blue1brown

More to the point: we love your videos because your approach is always unique, and informed by a much richer mathematical background than most of us have; and so I would be very confused if polls helped you make better videos.

Jason Orendorff

It's weird that the poll is roughly tied yet most of the comments are from yes voters. …I think the answer is in the question: topics that are "largely beside the point" don't belong in "the essence of calculus". The high-school calc class I took started with the epsilon-delta definition of limits. None of us understood why they were relevant; and when they disappeared in week 2, never to be seen again, they were not missed. Now I understand. But it seems a really weird pedagogical choice, like teaching set theory before addition.

Jason Orendorff

I remember you had a visualization of continuity a while back; it seems like that could be adapted to explain epsilon-delta visually and with relative ease. Then again, I'd be surprised if you asked this question without a visualization already in mind. After all, the brute force approach to that triply-quantified monstrosity didn't make sense to me until after I had several years of experience with formal logic. All that just to explain why I'm voting yes (even though my first thought was "infinitesimals!").

Okuno Zankoku

In my limited experience, epsilon-delta cleared up the vagueness of limits to many of my fellow classmates. We were in an honors class but I think saving it for the end of the video and making it clear how limits are well defined can help. Epsilon-delta is something I struggled with a lot, and including it may make the video more confusing, or it may save many budding mathematicians who are super confused and find your explanation valuable (I have no doubt you'll do a great job). Also if it were in a real analysis series, a high school student or college student learning calculus may not find your intuitive explanation of epsilon-delta. My vote is to put it in and make a longer video.

I'm with Aidiakapi in that I'm one of those oddballs for whom the epsilon-delta definition actually helped solidify and clarify my understanding of limits and derivatives; nonetheless, I would love to hear any insights you have to give (and you usually have plenty) about epsilon-delta that I haven't yet keyed on, and I think it would probably help a lot of students who are encountering the concept in a Calc I course. All that being said, I still am very interested in hearing about whatever is stewing up in that head of yours regarding the relationship between formalization and real analysis, so if you think that the epsilon-delta definition of a limit is a particularly good example to illustrate that point (and intuitively I feel that I can understand why you would) then I would encourage you to include it also in that later series (which I am freaking PUMPED for - real analysis is one of those topics that I'm inevitably steered toward essentially every time I search for answers regarding some mathematical query/curiosity.

Strange as it may be, epsilon-delta is what actually made limits make sense to me. Without it, at this point, I wouldn't even have a basic understanding of what limits are actually about. My knowledge of calculus however is very limited!

Thanks for the perspective, and I quite like the metaphor!

3blue1brown

So I learned something today: That you can define limits with epsilon-delta. God I'm happy that our prof just used the definition using sequences. So obviously, even though I'm normally always for more formality, I'm biased against an epsilon delta version and hope for an sequence based version in your real analysis series ;)

Zairaner

I had a horrible time understanding limits in Calc 1,2,3. I could manipulate limits in certain contexts, but what I had been taught in class was borderline useless when I found myself in a new situation. I knew that the instructor could derive things, but I didn't know how to convince myself of the validity of my own derivations. For me, the key to writing convincing derivations relies on having a strong standard of evidence. The epsilon-delta definition of a limit is the *gold standard* of evidence. If you are unsure of some limit, if you have a hunch, but at the same time you are skeptical of your hunch, where do you turn if not the epsilon-delta definition of a limit? In general, I think that the true beauty of mathematics lay in how it can be *discovered*, not just learned from others. Providing students exposure to strong standards of evidence gives them the opportunity to discover math for themselves. And what a thrill that is!

Riley Murray

i didn't learn it this way when i was learning calc 1 (or i guess teaching it to myself for the most part lol), but when i was helping a friend with their hw and their prof was teaching it to them. I think it'd be helpful to at least introduce the idea, not only because of the additional rigor but also because when someone's not looking from a more rigorous perspective, they might not even question what "gets close to" means, and i think especially around series and stuff it might be confusing why some (like the harmonic series) don't actually converge.

V

To me, the Epsilon Delta definition felt weird because you start knowing what the limit is and then proving that it should exist in the first place. I could see you do a "let's zoom in really tight and nudge the input by 0.001, 0.0001, etc." and show how there are still values to the function. You never have to mention epsilon or delta, but it still conveys the key idea.

Colin Williams

From the perspective of someone taking calc for the first time, I don't think you should ignore it completely, this is mostly because day one my teacher basically went on a whole thing about epsilon-delta and then dropped it completely the next day. I think mentioning it in the context of limits, even just an overview, and at some later point in time really getting into the nitty-gritty for those who want to know more would be a good compromise. Dip your toe in the water today and full on belly-flop at some point in the future.

I could see this working as a small, mostly geometric, interpretation for visualizing how limits work. You have already hinted at this in your discussion of derivatives. I feel like something along those lines would work.

Peter Su

The core value of your videos to me so far is developing better intuitions about the concepts covered. Formalism aside, if it is essential in building an understanding put it in otherwise leave it out or as suggested create a bonus video if you have capacity. Your work is much appreciated!

I think this is an important concept. When I first learned calculus I was really uncomfortable with exactly what a limit was until I saw it spelled out formally.

I generally would prefer epsilon-delta be avoided in introductory calculus. But that being said, it is still required where I teach and is one of the more challenging things for students to grasp. So I would like to see it included in such a video.

Given that I don't know about any plans to include other formalism topics I chose to vote for leaving it out.

Lionel Pöffel

Of course the answer to whether a mention of the epsilon-delta definition fits in depends a bit on how much formalism is planned overall for the calculus series. There is, indeed, the slight risk that this stands out in what is otherwise a mostly intuition-driven video series.

Lionel Pöffel

I think a short mention to acknowledge its existence and say that it is not necessary for an initial approach to Calculus is the best thing to do, since it is sooo common to have it as a topic in early courses in Calculus that you're initially led to think that they are fundamental for making calculus. I never was taught this in Calculus, but some friends were and I felt like I was missing out something very important (In the end, my friends ended up with a very mechanical view of the concept, so it wasn't like they understood its relevance anyway)

Ever Salazar

I also vote for a bonus video or at least some small mention here. I like to imagine calc students watching this series before/during the course, and I'm not sure many would recreationally venture into a real analysis video series during the school year and might be left thinking the formalism is superfluous if their authority on it (you) skips it. But if you have a different vision for how you want this series to be used, then let that guide you.

Evan Miyazono

Bonus video tieing into your fractal dimension stuff. Epsilon-delta was invented by Cauchy (?) to deal with messy function. It would be good to know that things like Weirstrauss exist, where the limit is everywhere defined (continuous), the derivative is nowhere to be found, and yet the integral defined across the whole domain. Seeing that before RA makes for a lot less lost brain cells later.

武明帥

Can you clarify what type theory has to do with limits?

Jake Palmer

Bonus video sounds spot-on. It's not core to an "essence of" understanding. But it would be useful to a lot of people studying calculus, because it's going to be in their calculus courses whether it should be or not.

The Delectable Electro The CyberPuppy

A bonus video will be a good idea. If you drop the epsilon-delta concept completely, that will not complete the discussion.

When I was in my first year at university, I would have greatly appreciated a video that clarified the confusing epsilon-delta-definitions.

Andreas Blatter

I think the first semi-rigorous calculus course I took used a story about a pedantic demon who can challenge the mathematician with any value of epsilon he desires.

Jacob Rus

I like the idea of a bonus video. When I've taught this, I've found it useful to ground the explanation using social cognition. It's analogous to how people are terrible at figuring out how to check a rule like, "If there is an even number on one side of the card, there has to be a vowel on the other side," (they typically check the card with a vowel on one side) but good at figuring out how to check an entirely isomorphic rule like "If you're having beer, you have to be over 21," (they always check the person drinking beer and the person who's 19.) So if you frame it as a sort of competition between you, trying to prove that the derivative is some value, and another person who's trying to cast doubt by getting you to prove that the value is within an increasingly tiny window centered on that point, people find that easier than the "for all / there exists" formulation.

john kraemer

Skip it. Epsilons and deltas are just one choice of ways to retroactively justify the clearly correct results we get from calculus in the case of smooth functions. The rigor is useful if you’re trying to push definitions to their limits and figure out how to consistently handle every edge case, or if you’re trying to satisfy every pedant, or if you want to prepare students for the standards of research mathematics, but is not really necessary for an “essence of” series of short videos.

Jacob Rus

One short sentence that conveys meaning is better than a chapter that conveys the same meaning. Break it up, Have clear borders, you can always refer to earlier works to complete pictures later on.

0123456asdfb

First calculus courses tend to either mention it and gloss over it (which can be very confusing to students) or not mention it at all, which honestly is the better choice. In general, not enough time is given for sufficient discussion, examples, and exercises required to really process it. It takes a long time to get properly comfortable with, and that time just isn't there in a first course in calculus. When I teach I feel no pressure to mention it. Leave it for later, and really do it justice then. If you're going to include it, that's a fine decision as long as you really make use of it over the rest of the course (in this case you end up with a calculus-with-analysis course, as in Spivak's book, which is an awesome book, but definitely for a prepared and motivated group).

I'd be ok with the non-standard definition, but without going too much into type theory.

Alexey Badalov

I voted to include, but I want to qualify my vote by explicitly stating that it should be a bonus video. Much of the power of your series comes from making the viewer feel that she is smart; you accomplish this with your insanely intuitive approach. You may lose that relationship if you sacrifice intuition for lines of equations. Although, a very good counterargument could be made that you are also very good at prefacing your videos in such a way that the viewer orients her brain to receive the information through the appropriate lens such that she protects herself from feeling dumb if she doesn't understand the first time. In any case, making it a bonus video seems to resolve all of these concerns.

Kathryn Schmiedicke

I'd love the idea of a bonus video

Javier Almeida

Bonus video?

Seems pretty clear which one you want to do. Happy to validate your leanings. :)

jason black

I say include it. In contrast to something like "compactness" it seems to me that concepts like limit and continuity are fairly accessible to beginner students once translated into plain English. I'm thinking of something like "you can choose any distance in the output space, as small as you like. If I can always find a distance in the input space that's small enough..." Maybe my understanding is naive, but anyway I know that you'll do a great job of giving a simple and intuitive explanation of what the formal version is saying and we may as well get it down in video format now rather than later!

Dan Davison

After all the hand-wringing about epsilon-delta in calc class, I read the *tiny* Dover book 'Infinitesimal Calculus' by James Henle, and all that stuff disappears. Simple, intuitive, fully mathematically rigorous approach to limits, using the Nonstandard Analysis developed by Robinson in the 60's.

Calculus and Analysis are separate. I changed my vote to save epsilon and delta for Analysis.

Jacob Mirra

I'd say leave it out. I think continuity is one area of calculus where people already have a fine intuitive understanding in terms of graphs, and epsilon-delta discussions just muddy it up. Save it for analysis where it belongs.

It's all nu to me

Don Sanderson

I, too, think you should make the presentation both now and later with real analysis

John C. Vesey

BetterExplained compares epsilon-delta as zooming in here: <a href="https://betterexplained.com/articles/an-intuitive-introduction-to-limits/" rel="nofollow noopener" target="_blank">https://betterexplained.com/articles/an-intuitive-introduction-to-limits/</a> . On one hand that perspective is very beautiful, on the other I needed to do several exercises to fully understand it. Since your videos already include zooming in explanations, I say save it for another video

Albert Martinez

I was highly ambivalent, but voted to include. Limits are intuitive, and there are lots of different types of limits in Calculus. Namely, three come to mind: the limit as x approaches a of a function f(x); the limit as n approaches infinity of a sequence a-sub-n (for series); and the limit as mesh-size approaches zero of a Riemann sum. By themselves, these operations wouldn't warrant a rigorous discussion. Together, they pose a problem that shouldn't be brushed under the rug: what is it we're doing exactly, when we say that a quantity is a limit of quantities? The idea of a limit isn't just for "formalizing" the notion of derivative or integral or infinite sum. It's a central idea in math (or, at least analysis): it's often impossible or unrealistic to get our hands on the exact answer to a problem, so we have to instead get our hands on an approximation of it. If you want your YouTube channel to just be a place where people go to seek comforting, simple, and, yes, beautiful, explanations, then leave limits out. But if you want to do something more - break students out of the Styrofoam box of "exact answers" - then invest the time in finding a meaningful, beautiful explanation of epsilon and delta, in the way you always do.

Jacob Mirra

I say keep it, but present an intuitive interpretation, much like this: <a href="https://mathwithbaddrawings.files.wordpress.com/2015/12/20151214072308_00004.jpg?w=584" rel="nofollow noopener" target="_blank">https://mathwithbaddrawings.files.wordpress.com/2015/12/20151214072308_00004.jpg?w=584</a>

Jeffrey Sun

I'd vote for a bonus video just about this topic, if that was an option. While I agree it's better suited for the real analysis series, it is a topic many calculus students struggle with a lot.

André Mello

I would love to get a visual intuition for epsilon-delta as I feel it would give me a deeper insight into what continuity really means. The discussion in the video on Hilbert curves was great and I'd like more of it. I suspect many of the people watching your videos already know some calculus and are looking for that kind of deeper insight.

Richard Futrell

The rigor would be nice, but it is a bit beside the basic essentials of calculus. However -- it would still be awesome to hear about it.

PseudsPie

I spent a semester fifty years ago beating epsilon delta into my head.

Doug Fort