"What is a Gröbner basis?"
If that is your intent for what this Q&A episode is going to be,
as far as technicality and deep explanation is concerned,
you're gonna be grossly disappointed.
Same goes to whoever asked about what the Fourier transform has to do with quantum computing.
I can say at a high level it's because the Fourier transform gives you a
unitary operation,
and quantum computing is very fast
when it comes to anything that can be expressed as a unitary matrix,
but those words won't make sense
If you don't already know what it means
and this is not at all
meant to be a video that's going to go into some deep math explanation,
but when I do cover quantum computing,
which I will at some point.
What would you do professionally if it weren't for Youtube/what are you doing professionally?
So a lot of you might know I used to work for Khan Academy and I think if I wasn't doing this
I would definitely seek out some other way of doing math outreach online.
The intent of the question is maybe more what would I do that has nothing to do with math outreach.
I spent a lot of time
doing like random software engineering things through college
Like my summer internships were often spent at a tech company,
rather than doing something explicitly math related
But if I really, you know, turn on that parallel universe machine,
I think going into data science was a very real possibility.
You know, one of the internships that I was doing,
at the end of it they asked if I wanted to,
instead of going to college again,
just stick around and maybe have a full-time job
and just see what unfolded there
And I seriously considered it, you know, it was pretty compelling.
Ultimately the love for pure math won out,
so I did go back to school as you're supposed to,
but I kind of do wonder what would have been if instead
I went that professional data science route.
How arbitrary do you think our mathematical perspective is, as humans on earth?
If an alien civilization developed math from scratch,
do you think we would see clearer similarities and their development of the fields,
like number theory, trigonometry, and calculus?
So this is an interesting question because it cuts right to the heart of, is math invented or discovered?
And I like the phrasing,
where you kind of imagine an alien civilization coming and comparing your math to theirs
It's hard to speculate on this, right?
Like I have no idea. If some aliens came,
we have no way of knowing whether their math would look completely different from ours.
One thing you can be pretty sure of,
and this might seem superficia,l
is that the notation would be entirely different, right?
There's a lot of arbitrary choices and how we write things down.
Newton's notation for calculus versus Leibniz's notation for calculus
You know a lot of the really silly things we have, like
which side of the variable the function goes on.
Writing out the letters s-i-n for sine and cosine
I had the whole triangle of power video about, you know,
notations for radicals and exponential's
and on the one hand that might not feel substantive,
but I think it's really interesting to contemplate other ways where
the notation shapes the way that we think about it
and shapes the axioms and theorems we even choose
more so than we give it credit for.
So a project I'm actively working on right now is about quaternions
and I was a little bit surprised to learn how up in the air the potential notations and conventions
for teaching students about vectors was
like a lot of the actual
notation and terminology we have for vectors and cross products, dot products, the way we think of them in 3d
ultimately stems from quaternions,
you know, even the fact that we use i, j, and k as letters to represent the x, y, and z directions and
if Hamilton had had his way
we would still teach engineering students primarily about quaternions
and then things like the dot product and cross product would be viewed as
subsets of what the quaternions do and what quaternion multiplication is
and I think there's a compelling case to be made for the fact that
we would use that if we could visualize four dimensions better
But the reason that quaternions never really won out as the notation de jure
is because they're confusing, because no one really understood them.
There's all sorts of hilarious quotes from Lord Kelvin and the like about how
quaternions are just needlessly confuddling when you're trying to phrase some fact about the universe
Like Maxwell's equations were originally written much more quaternionically
than we teach them to students now
And arguably, they're much more elegant that way
but it's confusing because we can't visualize it
So I think if you had some alien civilization that came,
but they had a very good spatial conception for four dimensions
they would look at our vector notation and think that it was
not capturing the deeper realities of math.
Arguably. Who knows?
What do you think is the main thing that drives people away from math?
Always hard to answer on these kind of things,
but I really suspect that as soon as you
wrap something at a certain kind of judgment,
of there's a notion of being correct or incorrect,
or an implicit statement,
there's a notion of being good at math—some people are math people,
some people aren't math people—as soon as you get someone identifying that they're not a math person,
First, you know insinuating that that even makes sense and then insinuating that they fall into that
Like, of course you're not gonna like it
Of course your natural mind turnings
aren't going to go in the direction of some puzzle
because you'd much rather to think about things that you're good at and that
make you feel happy.
All of the latest stuff about growth mindsets and
Carol Dweck and Jo Boaler,
really behind that,
You know the idea that if you're trying to tell a student something about how they're doing with math,
rather than framing it around,
"Oh, you must be so smart," right?
Frame it around, "Oh, you must have worked very hard."
"You must have put a lot of...a lot of time into that."
There's a lot of much less judgmental things that we have out there like,
reading. Even though there's some notions of reading comprehension tests for students in school,
and you know, "you're reading at an eighth grade level,"
people usually aren't like, "Oh, I'm not a reading person," right?
Like I—those words like some people that just make sense to them, but for me those letters, I don't know how they come together
When it comes to contest map like the AMC, I think those can be really good for high schoolers as a bank of problems
I think they can be really bad for high schoolers as an insinuation of there's some like top tier math folk and they can do
These problems really quickly
but if you give the same questions to a student and you say rather than being forced to go through all of these in
75 minutes you say let's let's spend 30 minutes on just one of them by really doughy to it
they're really solid problems that kind of engage in the spirit of problem solving and
You know removing that judgemental aspect removing that time aspect. I think, you know, I can help out a lot lot of people ask about
Certain things that I've made promises for but haven't necessarily delivered on in a recent video, you know
I did one on divergence and curl and I mentioned at the end an example of using complex numbers to model fluid flow
and a certain model for flow around a wing and
You might notice I have yet to actually put out a video on that and I've certainly seen a number of commenters
You know hampering on me for that fact
If there's ever a thing that I promise them that I don't make a video on it
It's probably because I spent a good amount of time trying to write a script for it that I just didn't feel was compelling for
Whatever reason and I think maybe the granddaddy here is the probability series which at the moment
I have five videos that I've made that are, you know released to patrons. I
Don't I just don't feel great about them and I kind of want the stuff that I put out to you guys to be something
I feel is you know, if not original something that wouldn't be out there
Otherwise from other creators and there's a lot of good probability material online
I will probably do something to release the material that I have either just as it is but on some second channel with the acknowledgement
Hey, this isn't the greatest work. I think I've done or trying to rework them and make them stand Alone's
but as far as you know, essence of blank content
I feel much clearer about how I would want to extend the linear algebra series rather than spinning my wheels on certain scripts and animations that
I ultimately don't think are gonna deliver something to you guys that I would feel proud of do you have any questions - Brill?
I'm just reading from reading from some reddit ones here
But we do something live how much compromise if any do you have to give with?
Like what you can animate versus like what your script is trying to convey
Usually if I can't animate a thing and it's a mathematical thing. It's not like a frivolous cartoonish type thing
I change the tools so that it can animate that thing right and then that might take more time and it's possible that
Subconsciously that means I resist topics that I know would be more difficult to animate. I don't think that happens
I like to use that to encourage creation of new things right like on the divergence curl didn't have good fluid flow stuff
But it was fun to play around with that for
quaternions right now
I think there's a lot of 3d related things that I wanted to sort of upgrade because
the previous way I was doing a lot of 3d animations was clunky and
Not as extensible as I wanted it to be so usually use that as a good excuse to just improve the graphics tool
We have somewhere a question on here. What sort of music do you listen to?
Which I mostly wanted to answer to
Like mention my renewed love for the Punch Brothers. I don't know if any of you know about them. They're actually super weird
they're like a
Avant-garde
Bluegrass band. It's just five geniuses who get together and put out phenomenal art, so can't complain about that
how do you compare making your videos to making videos for Khan Academy, so
very different
processes right like Khan Academy you imagine sitting next to someone and tutoring them and just explaining it you're writing everything by hand for the
Most part you do it live
On this channel, I obviously script things
I put a lot of time into creating the visuals for it sometimes in a way that makes me feel you know
If it kind of Quetta me I could sit down and make like three videos in an afternoon and here it's taking me like three
Weeks to do one video, you know, which of these actually carries more of an impact?
I think there's a proper balance for both of them
And I think there's a lot of people out there who do the Const Isle stuff to include Khan Academy
But also many others the way I like to think about things is what wouldn't happen if I wasn't doing it
but there is that little part of me that thinks maybe I should start some sort of second channel of the
Super cheap just like me and a notebook and a pencil like scrapping through some sort of explanation super quickly
Who makes the awesome music playing in your videos Vince Rubin Eddy link in the description link in all of the descriptions?
Actually, he does really good work and just go
You know download some of the music and leave him a little tip if you feel like it's something that you enjoy
What is your favorite polyomino? Polyamino? Polymino?
This one, I'll figure it out later and insert it on the screen. Alright folks
Thanks for watching, stick around for whenever the next upload is it's going to be on quaternions and I hope you like it
This is your close-up, this is your wife, all right
Does that probably mean I should be looking at the wide one?
Or do I do a dramatic like camera number two?