
The Beauty of Calculus [video] - okket
https://frankeprogram.yale.edu/event/steven-strogatz-lecture-april-26-2019
======
ivan_ah
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/](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])
       4/3
    

via
[https://live.sympy.org/?evaluate=summation((1%2F2)**(2*n)%2C...](https://live.sympy.org/?evaluate=summation\(\(1%2F2\)**\(2*n\)%2C%20%5Bn%2C0%2Coo%5D\)%0A%23--%0A)
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...](https://minireference.com/static/tutorials/sympy_tutorial.pdf#page=5)
or notebook
[https://nbviewer.jupyter.org/github/minireference/sympytut_n...](https://nbviewer.jupyter.org/github/minireference/sympytut_notebooks/blob/master/notebooks/Calculus.ipynb)

~~~
crimsonalucard
>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.

~~~
paganel
> 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).

~~~
solipsism
_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.

~~~
y4mi
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!

~~~
crimsonalucard
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.

------
billfruit
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).

~~~
throwawaymath
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.

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

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

[http://johnallenpaulos.com/booksandreviews.html](http://johnallenpaulos.com/booksandreviews.html)

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

[https://twitter.com/stevenstrogatz](https://twitter.com/stevenstrogatz)

------
markbnj
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.

------
lordnacho
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.

~~~
trevyn
> _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.

------
kwccoin
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.

------
0x445442
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.

~~~
kodz4
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](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.

------
guru4consulting
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?

~~~
throwaway55554
Was it Calculus Made Easy?

[https://www.amazon.com/Calculus-Made-Easy-Silvanus-
Thompson/...](https://www.amazon.com/Calculus-Made-Easy-Silvanus-
Thompson/dp/0312185480/ref=sr_1_1_sspa?keywords=calculus+made+easy&qid=1557496980&s=gateway&sr=8-1-spons&psc=1)

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

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

[http://www.gutenberg.org/files/33283/33283-pdf.pdf](http://www.gutenberg.org/files/33283/33283-pdf.pdf)

~~~
throwaway55554
Ok, thank you. I did not realize this version was PD.

------
molteanu
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.

~~~
edge17
link? not sure what you're referring to

~~~
rofo1
[https://www.youtube.com/watch?v=MFRWDuduuSw&list=PL3B08AE665...](https://www.youtube.com/watch?v=MFRWDuduuSw&list=PL3B08AE665AB9002A)

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.

~~~
learn_awesome
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...](https://github.com/learn-awesome/learn-
awesome/blob/master/calculus.md)

------
bigred100
Steve strogatz is a pretty excellent speaker even for people who don’t like
maths

------
mathnmusic
Really nice!

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

~~~
giu
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](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 :)

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

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

------
sjcsjc
From the video 22:10

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

[https://youtu.be/1r6893ga_So?t=1330](https://youtu.be/1r6893ga_So?t=1330)

------
75dvtwin
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. …"

------
adamnemecek
Tired: Calculus

Wired: automatic differentiation

