One of the better episodes! Lots of meat here and clear explanations. It's refreshing to see institutions like Hopkins still value gifted educators like Emily. Too many pure researchers and grant-writing bureaucrats in higher education nowadays.
so basically, category theory is like the Facebook of mathematics, where to find out more about what a mathematical object is like, you look how it interacts with accounts of other people, instead of the original account itself 😂
No, not really. Category theory is an abstraction of many/most/all of the constructs used by mathematics and in particular logic(and hence thinking). Category theory is more like UML or design patterns, in some sense. But these abstract structures apply to anything.
Sean is as busy as an one-legged person in an ass kicking competition with delivering us good information through his different communication channels. Love it. Never stop kicking.
My answer to the question "what is the point of maths ?" is "what is the point of comparative literature ?", "what is the point of musicology ?", "why do everybody asks this question about theoretical scientific subject and never about other subject ?" When asked "Why did you want to climb Mount Everest?", George Mallory is replied "Because it's there". And the same goes for maths and theoretical physics.
I was an undergraduate physics major And this had taken a number of advanced mathematics and physics courses. I had a friend who was a mathematics major. I remember looking through her topology textbook one day and thinking “wow this is some pretty complicated, esoteric stuff. “
What a wonderful conversation! I miss a few extra selling points of category theory. Specifically, being able to study the equivalence between categories makes it possible to re-use theorems and, in particular, proof techniques from one subject of mathematics to another. I remember in my undergraduate studies, when I had just learned about linear algebra the previous semester, and was now taking a course on linear differential equations, specifically systems with constant coefficients. The professor was working through an example on the black board when at the end I "saw the light" and realized that systems of ordinary differential equations with constant coefficients was essentially the same as studying systems of linear equations, i.e. basically just linear algebra in a new disguise. This was a jaw-dropping moment for me, and that's when I really saw the beauty of mathematics - the interconnectedness of everything. For the computer programmers / software developers out there, it might help to consider category theory as similar to "Design Patterns", where structures/patterns are re-used from one context to another. Another relevance for software developers is "Type Theory", e.g. used in compilers. Also the programming language Haskell can be considered an application of category theory.
i love topology! well, at least from what i imagine it to be about as a first year undergrad. im trying so desperately to try to fit it into my schedule for my 3rd and 4th year cuz i know i wanna take it.
This is exactly the sort of thing that I want more light shined on. During the pandemic I've spent a lot of time reviewing old course notes and finding Wikipedia articles that generalize things I know from linear algebra or the basic sort of group theory people invoke in physics to this weird thing called category theory. I think we can joke about category theory being too absurdly abstract to really matter but it's everywhere. And I don't entirely understand what it even is it but I want to.
knots and holes have amazing properties, literally mind bending. 44:00 lol. fields. it has another application in tv, i used to work in computer graphics and frames per second in animation is different to frames per second in film and different again from frames per second in television. animation by hand is 25 frames per second, as is animation by computer (although these days it can be 50 fps or 60fps - it depends) film runs at 24 frames per second and computer graphics is either 25 or 50 fps, it depends. there is also a thing we call interlacing, with hand animation you get one image per frame of film (if you imagine old camera film) but with television you get lines of the image building up from bottom to top, one line at a time every 1/50th of a second. progressive scanning means you get one image built up from the bottom one line at a time, but with interlaced scanning you get line 1 of frame 1 but line 2 is line two of frame 2, effecftively you get two frames at once, which makes for very smooth movement. but if you "hold" on one of these interlaced frames, they flicker between the two ACTUAL frames in the image, this is two "fields" interlaced refreshing at 50 fps. we once did a title sequence that involved a combine harvester doing it's thing, and when one of the frames that flickered on fields came up our apprentice asked "why is it flickering" and we all fell about with the answer "because it's on fields". never mind.
I've seena lot of science on podcast with an audience. Kind of like a real-to-life robin hood. Learned the benefits of spending time alone and only for the First and Second laws,. God bless you and may you continue to use that God given talent for his Glory!!!
Regarding rings capturing geometric properties... Aren't knot invariants, like the alexander or the jones polynomial, a little like that? At least the geometric meaning of the first one is well understood if i remember correctly...
@7:10 Structuralism is fine, but as a complete ontology (if you like) always fails, as does any attempt at ontology that only describes internal relations but which is not "the entire universe" so-to-speak. Linguists had the same problem in their version of structuralism.It's always nice to here from a platonist, they tend to be prepared to give the best explanations and motivations for mathematical research. Thanks for this one.
sometimes n=0 is defined as the "first" natural number, sometimes n=1 There are advantages and disadvantages for either choice. IMO n=1 is a better choice, but it is difficult to give the reason(s) why in a few sentences
If you are a patron of the podcast, you get to ask a question every month, but recently not every questions are answered, Sean picks the ones, to which he has something interesting to say. Also as a patron you can write an direct email, he most likely will answer it, hovewer IT may take a while.
My only complaint about this interview is that it isn't 5 hours long
One of the better episodes! Lots of meat here and clear explanations. It's refreshing to see institutions like Hopkins still value gifted educators like Emily. Too many pure researchers and grant-writing bureaucrats in higher education nowadays.
Not making this episode 10 times as long was an absolute CRIME.
so basically, category theory is like the Facebook of mathematics, where to find out more about what a mathematical object is like, you look how it interacts with accounts of other people, instead of the original account itself 😂
No, not really. Category theory is an abstraction of many/most/all of the constructs used by mathematics and in particular logic(and hence thinking). Category theory is more like UML or design patterns, in some sense. But these abstract structures apply to anything.
Sean is as busy as an one-legged person in an ass kicking competition with delivering us good information through his different communication channels.
Love it. Never stop kicking.
non banger
My answer to the question "what is the point of maths ?" is "what is the point of comparative literature ?", "what is the point of musicology ?", "why do everybody asks this question about theoretical scientific subject and never about other subject ?"
When asked "Why did you want to climb Mount Everest?", George Mallory is replied "Because it's there". And the same goes for maths and theoretical physics.
What is the point of musicologymusic?
I was an undergraduate physics major And this had taken a number of advanced mathematics and physics courses. I had a friend who was a mathematics major. I remember looking through her topology textbook one day and thinking “wow this is some pretty complicated, esoteric stuff. “
What a wonderful conversation! I miss a few extra selling points of category theory. Specifically, being able to study the equivalence between categories makes it possible to re-use theorems and, in particular, proof techniques from one subject of mathematics to another.
I remember in my undergraduate studies, when I had just learned about linear algebra the previous semester, and was now taking a course on linear differential equations, specifically systems with constant coefficients. The professor was working through an example on the black board when at the end I "saw the light" and realized that systems of ordinary differential equations with constant coefficients was essentially the same as studying systems of linear equations, i.e. basically just linear algebra in a new disguise. This was a jaw-dropping moment for me, and that's when I really saw the beauty of mathematics - the interconnectedness of everything.
For the computer programmers / software developers out there, it might help to consider category theory as similar to "Design Patterns", where structures/patterns are re-used from one context to another.
Another relevance for software developers is "Type Theory", e.g. used in compilers. Also the programming language Haskell can be considered an application of category theory.
Excellent interview! It s an interesting point of view that one should talk about mathematics the same way as about art, literature or history.
i love topology! well, at least from what i imagine it to be about as a first year undergrad. im trying so desperately to try to fit it into my schedule for my 3rd and 4th year cuz i know i wanna take it.
Make sure you are well-prepared. I rushed into my first topology course under-prepared and I got almost nothing out of it.
Category theory is absolutely fascinating. Emily is great.
There.....is..an... ENTIRE theory on categories. That is both hilarious and awesome at the same time hahaha
Instant 'like', this should be interesting!
Thank you for the new episode, Prof. C!
I had no idea I'd be so interested in this!
*_Nice experience, keep on, my friend. Greeting from HONG KONG. SUBSCRIBED already_*
Podcasts grow up so fast, don't they.
This is exactly the sort of thing that I want more light shined on. During the pandemic I've spent a lot of time reviewing old course notes and finding Wikipedia articles that generalize things I know from linear algebra or the basic sort of group theory people invoke in physics to this weird thing called category theory. I think we can joke about category theory being too absurdly abstract to really matter but it's everywhere. And I don't entirely understand what it even is it but I want to.
Don't be intimidated, category theory isn't actually that hard to learn.
Functional programmers love them some category theory
"To topologist a pair of pants and a thong are the same." Damn, that's deep. Gotta remember that line.
knots and holes have amazing properties, literally mind bending.
44:00 lol. fields. it has another application in tv, i used to work in computer graphics and frames per second in animation is different to frames per second in film and different again from frames per second in television. animation by hand is 25 frames per second, as is animation by computer (although these days it can be 50 fps or 60fps - it depends) film runs at 24 frames per second and computer graphics is either 25 or 50 fps, it depends. there is also a thing we call interlacing, with hand animation you get one image per frame of film (if you imagine old camera film) but with television you get lines of the image building up from bottom to top, one line at a time every 1/50th of a second. progressive scanning means you get one image built up from the bottom one line at a time, but with interlaced scanning you get line 1 of frame 1 but line 2 is line two of frame 2, effecftively you get two frames at once, which makes for very smooth movement. but if you "hold" on one of these interlaced frames, they flicker between the two ACTUAL frames in the image, this is two "fields" interlaced refreshing at 50 fps.
we once did a title sequence that involved a combine harvester doing it's thing, and when one of the frames that flickered on fields came up our apprentice asked "why is it flickering" and we all fell about with the answer "because it's on fields".
never mind.
Great guest and interview!
43:26 - The overt and subtle DnD references. ^.^
I've seena lot of science on podcast with an audience. Kind of like a real-to-life robin hood. Learned the benefits of spending time alone and only for the First and Second laws,. God bless you and may you continue to use that God given talent for his Glory!!!
@seancaroll I love this podcast!! need more of these plz.....
You have fitted perfectly with this podcast. I am freshly after David Foster Wallace History of Infinity and I have ∞s on my mind ;)
Rate the greatness of various physicists ! I am interested mainly on that !
Regarding rings capturing geometric properties... Aren't knot invariants, like the alexander or the jones polynomial, a little like that? At least the geometric meaning of the first one is well understood if i remember correctly...
Discovered I was good at mathematics when I was nearly 50 and convinced two university lecturers so, but hopeless at school, end of
This really needs video :(
@7:10 Structuralism is fine, but as a complete ontology (if you like) always fails, as does any attempt at ontology that only describes internal relations but which is not "the entire universe" so-to-speak. Linguists had the same problem in their version of structuralism.It's always nice to here from a platonist, they tend to be prepared to give the best explanations and motivations for mathematical research. Thanks for this one.
Yep. Too difficult to follow for gym music
Thanks this is amazing
Appearance of infinity
C of X C of Z C of Y 🤔
cohomoligy is a topological invariant that is a ring why did she forget :| @48:30
Sean Carroll calculus
Oh man, we need to get Prof. John Baez on this podcast.
Guest is afraid Sean knows more than her. Why are yourasking about infinite category? That just makes definition more difficult
"or somewhere like that" cuh. the.center.of.the.universe. england.
Mind bender
Is '0' a natural number? I thought natural number set is {1,2,3,...}.🙏
sometimes n=0 is defined as the "first" natural number, sometimes n=1
There are advantages and disadvantages for either choice.
IMO n=1 is a better choice, but it is difficult to give the reason(s) why in a few sentences
How can I contact you sir Sean Carroll
He has a patreon I do believe where you can ask questions
@@things_leftunsaid but I tried sending many times but he didn't reply
I would guess he gets wayyyy more mail than he can respond to. Eminem wrote a song about this called Stan
If you are a patron of the podcast, you get to ask a question every month, but recently not every questions are answered, Sean picks the ones, to which he has something interesting to say. Also as a patron you can write an direct email, he most likely will answer it, hovewer IT may take a while.
Did you develop a theory of everything?
higher brain level
Would like to hear Sean interview an infinity denier.
That would be fun. Especially Wildberger.
What is the use of a new born baby?
@@things_leftunsaid it was a rhetorical question.
Didn't find the speaker engaging, sorry, plus she sounds kinda nervous in a "coffeinated" way. Maybe I'm just not intelligent enough.
Maybe you should try listening to the content.
Sir Caroll could u see me ?
First to comment
An achievement comparable to Einstein's GR.