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The Beauty of Calculus [video] (yale.edu)
320 points by okket on May 10, 2019 | hide | past | favorite | 54 comments

Very nice lecture that connects calculus to the laws of nature. I've always thought applied math has a leg up on abstract math because of the connections to real-world ideas.

I encourage anyone who liked the talk to try out some of the calculations on their own using https://live.sympy.org/ which is an online REPL with build in calculus function like integrate, diff, limit, summation, etc. Here is an example:

   >>> summation((1/2)**(2*n), [n,0,oo])
via https://live.sympy.org/?evaluate=summation((1%2F2)**(2*n)%2C... note the answer is an exact rational number (class 'sympy.core.numbers.Rational) and not a float approximation 1.33333...

For a quick tutorial on how to use SymPy, check "Taming math and physics using SymPy" which is avail in printable format https://minireference.com/static/tutorials/sympy_tutorial.pd... or notebook https://nbviewer.jupyter.org/github/minireference/sympytut_n...

>I've always thought applied math has a leg up on abstract math because of the connections to real-world ideas.

I don't particularly like this dichotomy. Abstract math is connected to the real world because if it wasn't it wouldn't exist. Overall people who say this stuff are just referring to math that is only derived to work from an experimental standpoint and not from an axiomatic standpoint.

Machine learning is one example of a mathematical field with no axiomatic basis and the the pythagorean theorem is an example of one that is derived from the axioms of euclid. Both exist in the real world and are therefore applicable.

> Abstract math is connected to the real world because if it wasn't it wouldn't exist.

I'm not a mathematician by any means, but seeing that we know very few things about the foundations of the real world I think that trying to strongly link mathematics to it (meaning to the real world) also means "dragging" maths down to not really having any clear foundations.

Yes, I do know of the opposite road taken by some very smart people, i.e. tying maths to reality, finding the foundations of maths and how maths work => we now have a pretty good out idea of the foundations of reality (for lack of a better expression) and how reality "works" via its connections with maths.

It's just that even though we've been quite successful up until step 2 (meaning in postulating the maths <-> reality connection and in finding out how maths really works) it is my understanding that until now we have failed quite miserably at step number 3, as we haven't got the slightest idea of what this "reality" is made up of (again, for a lack of a better expression).

seeing that we know very few things about the foundations of the real world

we haven't got the slightest idea of what this "reality" is made up of

What in the world are you talking about? Setting aside unanswerable nonsense philosophical questions, we have a damn good idea what reality is made of.

In math, we actually know and can prove beyond all doubt something to be true.

The same can't be said about the reality as we can never be sure that we've accounted for all variables and filled them correctly. This applies to everything in physics, even something as basic as "how long will this Apple take to fall to the ground"... You can make an estimate and then let reality take it's course. And when you're finally looking at the numbers it might've been correct down to milliseconds... But there is going to be some drift

in math, it's gonna be one exact point, as everything is accounted for. In reality, the Apple is not going to land at exactly the same time, as you forgot to include something like the the current air pressure or a comet vaporizing the planet in flight!

This is true. Not only is it impossible to prove anything in science and therefore reality, but we also view things through a blurry lens. Results arrive with a limited amount of significant figures or in the form of a statistic. Technically, although things cannot be proven in science, things can actually be disproven in science, but due to the blurry lens we are unable to fully do so. Instead science is largely about establishing correlations and to an even harder extent trying to give a blurry statistic to causation.

Do note that for math, you can prove things but two things must be assumed. The first assumption is that all axioms related to the proof are true. The second assumption you make is that logic as we know it is true and always consistent regardless of context.

There is one more interesting thing regarding science. Because nothing can be proven in science and because all science is, is establishing correlations, one thing that we assume is true in the real world is probability. In math the axioms of probability are basically arbitrary ratios assigned to sets with the theorems blossoming outward from different compositions of these sets and ratios. The theory itself has nothing to do with random events. So literally the theory of probability is just about sets and an associated rational number representing a portion of that set.

If we had a 6 sided dice and we rolled it billions of times the reason why the number 5 appears close to a 1/6 portion of all the results is a mystery. We literally assume this is the case and that the axioms of probability which are essentially just sets and ratios actually applies to random events and happenings. All other science is derived from this assumption.

See y4mi’s reply. Also, we cannot account for about 70% or 80% of the material Universe (the one we presume has weight), we even found a fancy name for it, “dark matter”. This is pure mysticism, this is definitely not knowing what reality is made up of.

I don't think it's worthwhile to engage in applied versus pure "math war" either, but I don't really agree with the point you've made. All applied mathematics has an axiomatic basis. When certain mathematical theories can be "applied" to real world problems, it simply means the relevant axioms and definitions are a robust approximation of reality.

To speak to your example directly: machine learning absolutely has an axiomatic basis. You can conduct legitimate research in implementations and software or hardware optimizations thereof; however, fundamentally every experimental result in machine learning is an application of a variety of theorems in linear algebra, probability theory or calculus.

Imho this is not the right way to think of it. Experimental results aren't applications of any theorem, they are just measurements. And the fact that these measurements may or may not come from machines we know how to "perfectly" measure (eg digital computers) doesn't give them any axiomatic basis. I'm not sure anyone is doing machine learning using symbolic computation, afaik it's mostly about low-precision floats--which do have some foundations themselves, but quite far away from the reals-based vector spaces upon which optimization theory is based. As addition to this point: most convergence results are much better in practice than in theory, ie we do not yet have satisfying predictive theories for experimental optimization.

On this pure vs applied math thing, imho it is indeed a false dichotomy: there are symbolic objects of study and there are empiric or otherwise "preexisting" objects of study. We may use symbolic objects to approximate empiric objects and provide "applied theories" with predictive power, but we may also abstract empiric objects into new symbolic theories (both usually being done together). There is a funny phenomenon in the more abstract domains of math where at some point everyone is routinely talking about "intuition", "seeing things" and "morality" of facts. For me, building up intuition in a domain is about taking an axiomatic theory and transforming your view of it into something of the more empiric kind, one where you believe in an external understanding of how things behave. A symbolic fact being "moral" when it's consistent with the empiric counterpart you developed in your mind. My conclusion: some things are axiomatic/symbolic and some are not, but we mostly treat them the same way, ie by building empiric mental models. How else would we have proof ideas?

> I've always thought applied math has a leg up on abstract math because of the connections to real-world ideas.

You may enjoy this conversation from 1972 (https://youtu.be/avSHHi9QCjA) with Cornelius Lanczos and others on his dissatisfaction with the applied/pure math dichotomy.

> I've always thought applied math has a leg up on abstract math because of the connections to real-world ideas.

godel settled this, so it's ultimately moot.

does sympy render latex in the repl...? also haha oo as an alias for float("inf") or whatever is clever.

What I feel about calculus, esp school calculus, is that it seems to comprise largely of specific tricks of symbol manipulation, rather than general approaches that work in all cases. Any ways, analytic integration/differentiation seems to be full of such tricks, of which how the core intuition was first obtained remains a mystery most of the time(rather than it being a result of scientific/mathematical method or process).

When I was learning it, I always felt like calculus (and honestly algebra and trigonometry too) struggled to “justify” itself: whenever they presented real-world applications, they always seemed so artificial that I ended up feeling even more confident that higher math was just complex for the sake of being complex. Although I understand that there are professions that “directly” apply calculus (i.e. solving integrals or derivatives) I went into computer programming where direct applications of calculus are few and far between, and ever the places where it’s applied like gradient descent and elliptic curve cryptography can be used without any understanding or appreciation of the base theory anyway. Still, I’ve come to appreciate that the real-world applications were deliberately simplified because the real real-world applications are mind-meltingly complex and that the presentation of calculus teaches you more to solve problems methodically than to apply specific processes and procedures.

Putting aside the common issues with pedagogy, the reason you learn a "bunch of tricks" for calculus is because that's more or less all you have in many cases.

If you take integration as an example, there is no single approach to solving every integral. More importantly, it's extremely common to encounter integrals for which there exists no closed form antiderivative. In fact it's technically exceptional to find an integral which can be neatly solved in the space of all possible integrals.

As a direct result, solving an integral becomes an (often frustrating) exercise in transforming it into something equivalent integrals up to a negligible constant. Nonlinear optimization problems and differential equations are similar in this regard.

There is something to be said for the depth of analysis, which does provide a deeper meaning and rigor to the "bag of tricks" in calculus. Outside the US it's somewhat common to skip calculus entirely and begin straight away with analysis, and I think there's merit to that. But the profundity and power of analysis doesn't provide you with any fundamentally more complete methods of solving calculus problems except insofar as they become more advanced and rigorous. Ultimately analytic mathematical work (as opposed to algebraic) is characterized by this kind of pattern-matching; this frequently results in seemingly inspired, bizarre looking proofs compared to how neat everything is in algebra.

Perhaps for some of the students,a course that skips indefinite integrals, and focusing on definite integrals and numerical integration may be sufficient.

I thought so.... until I took a real analysis course. Most calculus tricks build upon on a "small" set of big ideas. The big ideas (such as compactness, convergence, continuity, diff/integration..) are limited in numbers to convince oneself on, yet are generalizable tools to think about A LOT of complicated mathematical phenomena in a concise, clear way. Should they fall short to evaluate a certain math phenomenon, they should do so unambiguously rather than opaquely. Real analysis is a start to developing such tools.

Look into automatic differentiation. It's magic.

Makes me think about Paulos. I read various books of him. He has an interesting view on the world as a mathematician.


Follow Steven on twitter for more great content. Amazing person too!


As a non mathematician I enjoyed David Berlinksy's "A Tour of the Calculus" when I first read it back in the late 90's. It's similar to this lecture in that it strives to provide an understanding to non-mathematical types by exploring the principles without tossing lots of notation at the reader early on.

By dumb luck I scrolled to around 26 minutes, where he gives a summary.

Basically, the laws of nature are written in calculus, which is more than a language, and we can use this to manipulate nature.

>Basically, the laws of nature are written in calculus

Calculus is a tool that powers certain specific useful predictive models of our observations of nature in certain specific domains.

Having a mental model in which the laws of nature are related to calculus in some deeper fashion (or where calculus is always an appropriate tool to describe observations of nature) is likely to lead to some erroneous intuitions.

Incredible, that's where the Wadsworth Constant would put you in the lecture as well.

I think whilst the video is great, the guy missed the deep question raised by the audience.

Using the language in the talk, the question is may be God speak Calculus but is it the only language it/he/she speak?

You have that concern whilst just like a predecessor in post Pythagoras, post Newton, post Einstein, post QM, post AI, we got a strong and strong "understanding" and manipulation of the universe. And those understanding provide us with a great physical well being and more wealth. But is the world a better place?

The tradition question is still why we can send people to the moon but never able to solve the problem of ghetto.

Minor issue: Even he has to admit the assumption of continuity (which mathematically different from differential but mostly do), there are many are discrete and non-differential. Hence God if it/he/she has to cover everything, he must have to speak another language. QED

BTW I like the T-shirt and go to study more calculus. Great talk.

Do they not teach why Newton invented calculus anymore? They did when I was taught physics and it all made sense. I don't recall if I considered it beautiful at that time but I do recall how impressed I was with Newton's feat; moving himself farther along with his studies in physics.

For anyone interested, here's the BBC looking into Newton and Leibniz notebooks as they cook it up for the first time - https://www.youtube.com/watch?v=ObPg3ki9GOI

Also Steven Strogatz is an excellent writer. His other books Sync and Joy of X can be read by anyone.

He wanted to calculate and predict the movement of celestial bodies. The mathematical knowledge he had wasn't capable of expressioning the relations, but he saw patterns. Then he worked out the mathematical language to express these patterns.

Not sure what's in the curriculum today, but I was taught calculus (20 years ago) heavy on physics applications. I don't think I would have appreciated it as much without that.

A while ago I saw someone posting a book about Calculus in HN forum that was written decades ago, probably 40s or 50s. I liked it and could not find it back. Does anyone have link to it?

I have Thomas' Calculus from the 1960s, the same book that Knuth promotes in interviews as being responsible for him choosing a math degree. Very high signal to noise ratio without being as terse as Apostol https://www.amazon.com/Calculus-Analytic-Geometry-Supplement...

However I really enjoy algebraic calculus and wish somebody had taught me this in highschool https://www.youtube.com/playlist?list=PLIljB45xT85CSlgGh3681...

This was my text at MIT in the 80s and I still have it on my shelf!

Was it Calculus Made Easy?


It seems easy to find a pdf on the web, but I didn't want to post that.

The original (not the Martin Gardner version) is public domain at this point. Project Gutenberg has a PDF of it.


Ok, thank you. I did not realize this version was PD.

yes, I was looking for this book. Thank you. I wish I had read this when I was in school. I liked the book for explaining complex topic in a very simple, easily understandable way. This would be a great book to introduce calculus to your kids. I read chapter 2 (on different degrees of smallness) and found it enlightening.

You're welcome. Happy reading!

There's also a website if you don't like PDFs: http://calculusmadeeasy.org/

Probably not it, but this looks interesting: https://archive.org/details/calculus00marciala/page/n5

Maybe Apostol's Calculus, or Spivak's Calculus. Both 1967 though

The beauty of Calculus can only be revealed to young recruits by Herb Gross' videos and Spivak's Calculus. If those don't tickle your brain, nothing will.

"Spivak's Calculus" is a reference to the book "Calculus" by Michael Spivak. https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/091409...

Many consider the book's presentation of the topic utterly beautiful, bordering even on the spiritual.

I also found 3Blue1Brown's (Grant Sanderson) YouTube series "Essence of Calculus" quite enjoyable.

link? not sure what you're referring to


There are many, many ways to study calculus, so take this as a preference.

For example, this wouldn't be my first choice.

In general, I find the "Definition. Axiom. Theorem." approach very dry and just doesn't fit reality. Nobody ever discovered mathematics like this.

One of the best books I've read is Gausses "Disquisitiones Arithmeticae" rather than any modern Theory of Numbers book.

I've had more insight in sums manipulations (and various little tricks, some of them not even justified in modern mathematics) from Euler's letters than any other book on this subject.

I don't know what happened in the meantime. It sure doesn't look like 18/19 century books were so dry like the ones today.

Thanks for sharing both those things. We've added these to our Calculus topic here: https://github.com/learn-awesome/learn-awesome/blob/master/c...

Steve strogatz is a pretty excellent speaker even for people who don’t like maths

Really nice!

BTW, LearnAwesome's calculus topic is relatively barren: https://github.com/learn-awesome/learn-awesome/blob/master/c... I'd appreciate if HNers here can point us to best learning resources for Calculus. Even better if you can send a pull request.

Just created a pull-request; I added links to some articles and sections hosted on Paul's Online Notes page (http://tutorial.math.lamar.edu). This site helped me quite a lot during my studies, and I'm sure the same holds for a lot of other students and professionals :)

The section about Usian Bolt running the 100m dash was very interesting. Usian needs 41 strides to reach 100m while other runners need 44.

Steven Strogatz is amazing. you can get lost in his lecture like Hollywood movie. it is so engaging

From the video 22:10

"... must have been one of the greatest aha moments in history ..."


Two statements/quotes that I personally found interesting through prism of current times.

"... Algebra is quite sterile, Algebra is not about anything …."

(at about 1:17:28 )


One thing I feel a little bad about, and I feel I cannot properly correct it but I just want to show you that I am woke.

I mean, I really am. I am aware of this. Women are part of the story. And so are people in India, China, and Japan. And then the Mayan civilization. There are a lot of calculus being done around the world.

With woman, honestly, it is only sort of , around 1800s that women were allowed to go to universities and hear lectures and stuff. …"

Tired: Calculus

Wired: automatic differentiation

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