The Beauty of Calculus [video]
frankeprogram.yale.edu
frankeprogram.yale.edu
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])
4/3
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 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.
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.
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'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).
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.
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!
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.
godel settled this, so it's ultimately moot.
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.
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.
Basically, the laws of nature are written in calculus, which is more than a language, and we can use this to manipulate nature.
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.
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.
Also Steven Strogatz is an excellent writer. His other books Sync and Joy of X can be read by anyone.
https://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/...
It seems easy to find a pdf on the web, but I didn't want to post that.
However I really enjoy algebraic calculus and wish somebody had taught me this in highschool https://www.youtube.com/playlist?list=PLIljB45xT85CSlgGh3681...
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.
Many consider the book's presentation of the topic utterly beautiful, bordering even on the spiritual.
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.
"... must have been one of the greatest aha moments in history ..."
"... 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. …"
Wired: automatic differentiation