This Is the Calculus They Won't Teach You

"Infinity is mind numbingly weird. How is it even legal to use it in calculus?"

"After sitting through two years of AP Calculus, I still feel like I don’t know anything about it."

"What do you mean that historically, integrals came before derivatives? Seriously, what’s the deal with calculus?"

There is an inspiring story hidden behind the formulas and word problems of single-variable calculus–a story that should be told in every introductory calculus course, but usually isn’t. Regardless of if you’ve taken a few calculus courses before or are just beginning your study of calculus, this video aims to give a taste of the insights that can be gained from learning about the history and philosophy behind the key ideas of this field of math… brought to you by a dog and some rough animations!

This video was part of #SoME2, a math video making contest created by @3blue1brown in the summer of 2022.

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Chapters:
0:00 Chapter 1: Infinity
1:49 Chapter 2: The history of calculus (is actually really interesting I promise)
2:19 Chapter 2.1: Ancient Greek philosophers hated infinity but still did integration
7:43 Chapter 2.2: Algebra was actually kind of revolutionary
11:14 Chapter 2.3: I now pronounce you derivative and integral. You may kiss the bride!
19:53 Chapter 2.4: Yeah that’s cool and all but isn’t infinity like, evil or something
24:14 Chapter 3: Reflections: What if they teach calculus like this?

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Thank you very much to Dr. David Bressoud for letting me interview him!

If you have any questions, thoughts, recommendations or you notice any mistakes, please do share!

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CORRECTIONS LIST (periodically being updated):

3 : 47 I start a description of Archimedes' circle proof here, but I want to note that the one I present is a modified version of it; it does contain a few "hand-wavy" mistakes near the end and may still be difficult to follow; that's on me. My goal with showing this proof wasn't to give a proper explanation of it, but to give a taste of what I thought is the spirit of rigorous reasoning. If you would like to see something that is closer to Archimedes' argument, there's a link in the further reading you can check out--or you could also just google "Archimedes circle proof".

8 : 45 I made a typo... Viete lived from 1540-1603, not 1540-1693

14 : 35 This is embarrassing. Kepler's 2nd law is true because angular momentum is constant, not because velocity is constant. Whoops!

A lot of comments have brought up Indian contributions to calculus that I left out, particularly the Kerala School and Madhava of Sangamagrama, where the first instances of derivation came about. A few reasons for why I missed this in my research could be because attributing the beginnings of calculus with Indian mathematicians seems like a relatively new historical narrative, and many sources in English likely have a bit of Eurocentric bias. Regardless, this goes to show how the narrative I shared does have its flaws, and similarly to the misleading impression of calculus I criticized modern introductory courses of spreading, this narrative is not the whole picture. If you have any sources you'd like me to add to the further reading, please let me know!

I may have suggested that Euler discovered the number e, but he did not--some attribute it to Napier, and many others before Euler used it. On that note, I probably misleadingly attributed a lot of other ideas I brought up to one or a few people, when they are much more nuanced; so take the briefly brought up associations in this video with a grain of salt. This video is intended to share a broad narrative instead of establish definitive history of calculus. Hopefully this can just act as an intro to the subject!

[AWellRestedDog]

#SoME2 is finally over. What a journey it was! Thank you all for the attention and constructive comments. I'm incredibly honored to have been chosen as a 'winner'... and I'm very excited to be getting that pi plush lol

That being said, if you're interested in learning more about the history of calculus, there's a link to the further reading (and a list of corrections) in the description.

Have a good one!

[kushalbadal]

I strongly believe that every field should have historical context in its introductory phase, and this is a great example of it. Very well put together...

[cemmy]

I really like how you paired the music with the time periods, starting with ancient and moving through the baroque, classical, and romantic periods. That's a very nice touch

[ren]

As a first year engineer and past experiences with calculus i find what David Bressoud said so relatable "students only start paying attention when the professor actually starts doing the examples"

[evanedgeworth]

I thought I hated math but when you combine it with the historical context it’s almost a thing of beauty 🤷🏻‍♂️

[mattdamron]

This was honestly an emotional experience for me. It put into perspective how far as a culture we have come, that we can now learn things as teenagers what it used to take entire lifetimes to touch. I cannot tell you how much I appreciate the obvious effort and love you put into making this video. Thank you

[A_Random_Rat]

Just found out this was your only upload and now I’m devastated.

[michelemassa]

The amount of effort you put into this masterpiece is unimaginable from the animations to going through years and years of math history. Seeing a single video on this channel saddened me a little I wont lie, because this is extremely well made and after taking calculus 1 through 3 made me appreciate what I have learnt more, you gave it value. Very few times do I get out of a 30 minute YT video feeling as good as this video made me feel, this deserves to be put as summer prep for every calculus course out there. Simply splendid.

[LagAttacktoSlay]

I'm a private Maths tutor. I've always wanted my students to understand WHERE and how the concepts we study together came to be, but, sadly, it's extremely inefficient, especially at a college level... They DO engage, though, so I usually give them extra hours for free even though it REALLY hurts my wallet, so I stopped doing that for the time being. Reading through the comments, and seeing how grateful people are for this video makes me genuinely happy! Beyond measure! Enough to make my eyes tear a little!! I have saved this video, and who knows? Perhaps a future student of mine will voluntarily ask me those questions, in which case, I have the perfect video to refer them to! My appreciation can not be expressed in words! Thank you so much for putting this together, and for showing me that students all over the world really do want to know the history behind Mathematics!!!!

[georgehammond]

Great job. I am an engineer who graduated in 1972. The math that we were taught was mostly memory work. I learned more about calculus from my engineering professors than I ever did in math class. Math could definitely be taught differently and could be easier to understand if taught like this.

[notjackschannel]

I’m an engineer who exactly fits the stereotype your guest mentions. I struggled to get through calculus in school and only barely made it. Adding the humanity of how calculus was developed over generations, as this video does, is exactly what was missing from my courses. I wish this concept (or YouTube for that matter) existed 25 years ago.

[chris-cukl]

This video reminded me greatly of my algebra 2 teacher in high school who was notoriously behind on the pace that the curriculum demanded, but for the sake of having students scratch their heads at the ideas with each other. The kinds of ideas behind math feel like such an incredible system that’s been arranged to work with itself in so many different ways, but often the class is framed in pure stress and only in how to do the executional requirements of math. Her class really illuminated the things about math that I enjoy, but it is very difficult to keep this enthusiasm up taking in more math classes with professors who just see it as I’m giving you the tools to pass the test. This video really brings to light how much there is to improve with schools teach subjects in a way that can be more informative and natural to understand.

[scottaseigel]

You are very gifted at explaining calculus. I was a math teacher and I definitely would have used your video if it existed back then! A lot of history and also notoriously hard concepts to teach are now well presented on YouTube. That’s the idea behind classroom flipping. Calculus always seemed best taught by exploration such as doing labs and puzzles, or in an applied context like physics. Perhaps it’s like the proverbial elephant and we’re all among the blind men saying rope, fan, tree, snake, wall, etc.. This suggests that math is discovered and maybe that’s the real key take away (because I’m pretty sure it’s not L’Hôpitals rule).

[thattimestampguy]

1:52 Where it began?
2:39 Geometry
3:30 Zeno’s Paradox
4:03 Rectangle smaller than Circle. Or not?
7:43 800s Arabic Numerals
9:00 Coordinate Geometry
10:28 Sin(X) Cos (X)
12:06 Derivative, Integral
12:50 Galileo & Kepler
14:57 Newton
16:08 Leibnitz
17:05 d - differential, infinitesimal
Fluxions
18:29 Diplomat Leibnitz brought ideas to more smart people
19:32 They contributed huge

20:05 Infinite Series
21:17 Limits
21:56 Function (X, F(X))
23:20 Over and over again
23:43 Infinity Itself ♾

24:17 Reflections
• Built around a story
25:38 Jousting with Infinity
26:24 I want to understand important math
27:53
• Professor tries to explain what’s going on
• Once he starts doing examples, I start paying attention

Infinitesimals - philosophers wrestled with it

Expect students to stumble

29:27 Calculus Reform Movement 1990s

[real_michael]

This is what I've been saying to myself all this time. We need to incorporate the history of the math we're learning otherwise it just feels like I'm learning it to pass a class. Thank you for this video. SoME is truly a godsend.

[robroy]

It doesn't matter what the subject is, finance, calculus, databases, cell biology, integrated circuit boards, inorganic chemistry, accounting, psychology, etc. I discovered the best way to learn is to go back from where it started historically. It's not necessary to study the individuals who contributed to the subject-matter but what they contributed and how that led to the next contribution. It's the only way I care to learn and I don't understand why it's not being taught this way. I went through College calculus never understanding how I could ever use the equations I was solving.

[AayushPatel-gcfw]

I feel sad that these western mathmeticians published their theories and discoveries and that are still available, but Indian textbooks where burnt off in attacks on oldest Universities of the world, what knowledge could be there !

[thejackbancroft]

This would probably have to be the best made video I've seen from someone with under 10k subscribers.
This is the quality you'd get from professional full-time channels with teams.

Very well done.

[alekdaniels]

I'm a nerd and I already finished school but I never got any education on calculus. My business course does not have this subject as our maths only revolve around business stuff (accounting, depreciation, taxation, interest rates, etc.). It was all very mathematically simple. So I hopped on to YouTube and tried to get a head start on learning more math. And I gotta say, THIS IS THE MOST IMPACTFUL video I've ever watched re: Calculus. I actually feel more interested to move forward now. I tried jumping into equations and solutions but I always thought, "why are people doing this? What's the point?" And this video gave me the answers. Thank

[jakeharms]

A historically motivated teaching method would be fantastic; I do think that going down a level to the problem-solving motivation would be even better, especially for us engineers