Sign me up.. make a DAO.. anything hahaha This guy unlocked everything with 808, Monster set video.. taken me months but becoming coherent what bits mean what.. Pariahs are vowels
In the end when he uses the words " I think my kids" and then explains that he is referring to his students seriously touched me in a way that I never thought possible from a Math teacher. Grant Sanderson please keep inviting these amazing people on your podcast. I am forever grateful to the work you do.
SAME !! I hope he sees my donation and the do not sell until 2027 hahaha.. they even have a better Metcalfe's Law ! nag him for me, cheers.. always so busy being a generalist lately as things just keep falling into my mind out of other things
There’s a quote from the Little Prince that is apt: “If you want to build a ship, don’t drum up the men to gather wood, divide the work, and give orders. Instead, teach them to yearn for the vast and endless sea.” ~Antoine de Saint-Exupéry
Great conversation! I've really loved Strogatz' Infinite Powers book. I've incorporated a few calculus tidbits into my own courses, but mostly it was a masterclass on math exposition that was super helpful
@@theswelldudesfishtanks7461 it means you must enjoy it. the more you enjoy something the more you'll do it, and the more you do somrthing, the better you'll get at it. so the amount of you enjoying something is directly proportional to how good you'll be at it.
This was one of the best 2 hours I've ever spent. words fail to express how grateful I am for both of you for what you do, especially the content you publish for free. Mr. grant, I'm just a 17 y/o student and I really had to sell some old books to buy myself a bright lamp but if I ever gotten rich, I promise you you're getting your share out of my fortune.
This is such insanely good content. I found myself smiling at things that I never thought I'd even enjoy a discussion of so much..This is probably one of the best things I've ever seen in my life.
I remember studying graph theory and seeing the Watts-Strogatz model (how to construct a random social network). And I remember being struck by how visual it was and quite understandable. I was struck by that because to this day I still struggle a lot with math (e.g. simple proof by induction? Not me). My fundamental basis of math had holes, which I'm fixing 10 years later. But that part of graph theory stood with me as a more understandable form of math. And now seeing one of the men behind this model it makes total sense why that part of graph theory did not have the same effect on me as the rest of the course did. This podcast was intense. The love itself (for math) that Steven Strogatz has was so palpable, I could feel it, I almost cried. To hold such emotions within one human being, holy hell, even Grant made a comment among similar lines. It's unreal and so cool to see in someone. On top of that, the content was excellent. I've heard from time to time that math takes a different shape when you get to higher levels, and I feel that I've gotten a very nuanced glimpse at that.
Absolutely brilliant conversation. Steven's enthusiasm for mathematics is so palpable and inspiring. I was grinning the whole way through and at a few points I even had goosebumps. P.S. Grant has always been a huge inspiraion too and a great interviewer to boot. Full of really creative and intelligent questions.
Me too, clapping, posting like a bot so much GOOGLE OI YES YOU keeps fking me over 🤣 thinks im those GPT3 text bots so posts are disappearing.. looks like all they did was add a rate limiter :( ignoring my posts about the impending scammer explosion for almost 2yrs.. priceless We can't fail.. humanity is too powerful, our intention to be good just.. its all so obvious when you grow enough you see it, intuitively or practically.. evolution is relentless its just painful and slow, but fear is at the core of our ego. Fear of the unknown.. after all, what do we do, sense data ? most of it gets discarded. Samurai knew that when you focus on one thing, you unfocus on others - your FOV of your eyes narrows, throwing away more and more data every Planck tick. The amounts are astronomical but we get more meaning from the intuitive stuff anyway, body language.. detail/fidelity is NOT important, and if it was, we would be able to see fundamental quanta.. of course we can't, we perceive them to be physical. So we go optical. The most intriguing bits for me are.. how touch, taste and smell can be combined. And how memory ACTUALLY works. There are a few cases where there is a human who doesn't get brain formation only stem and the inside surface of the skull. In those cases, sometimes the person is not diagnosed during pregnancy as an anomaly ! if you are, then you are actualized then as having that constraint - they are mentally challenged. Meanwhile, one who didn't even know until his 20's, has about avg IQ and works a maths related job HIS MEMORY IS FINE. Memory is stored in "awareness" why else do we need to think about things in a structured way - and why else would CHAOS, entropy be growing in someone who is more and more fearful - old people today cannot interpret these technological advances, you get stuck in your ways interpreting data patterns so when something new comes along,... sometimes it is just too abstract ! voila we have the need to have more than 1 database. You don't grow up all in one lifetime just like maths isn't one very long class.
Grant, two years ago I was aimlessly wandering through my undergraduate. Though I had loved the mathematics that I had encountered, I had not seriously considered majoring in it. Your elegant and magnetising presentations changed that. I shoved my degree into two years, and as of three weeks ago I am embarking on postgraduate work in applied mathematics. Thank you for the inspiration. P.S As an amusing note I watched Strogatz series on Nonlinear Dynamics and Chaos rather than my courses lectures for my dynamical systems unit. Strogatz is just a cut above the rest.
Had me wishing that this conversation just keep going on and on. Being a sophomore, i can relate go to every single thing in the discourse and kind of feel relieved that ppl like grant or steven exist.
It is amazing that Steven Strogatz started with the Stiener-Lehmus problem...the equal bisector problem. There is a lovely proof at the notes section at the end of the Coxeter classic Introduction to Geometry.
A great and interesting discussion between two of my favorite mathematics educators! I hadn't heard about Steven Strogatz' undergrad experience. I too absolutely loved math until I went to college. So many of the math professors talked about "the beauty" of this concept or that. But for us mere neophytes, it was much like hearing master chefs talk about "the beauty" of fresh basil without having never tasted it or eaten pesto or enjoying a great caprese salad. For me, traditional math education methods were very much like discussing recipes without ever tasting the final product or being able to employ improversation to create something new. For example, I never actually grasped differential equations when taught by traditional mathematicians, despite having taken the course twice. I desperately wanted to understand it. I finally had a greater grasp after learning solutions as taught by electrical engineers. Applying the mathematics to an actual problem solidified the concepts. Regardless, great podcast, and both of you please continue doing what you're doing!
I still remember the first time I started reading through the complex system and chaos book he wrote, and was amazed at how easy and delightful reading a course book could actually be compared to all the previous ones I've had gone through. A modern day hero in my eyes!
13:01 this is me every-time I hear Steven talk. I don't know what it is but he is SO damn engaging and interesting to listen to. I can imagine how well his students learn in his classes. Good man.
One of the most inspiring podcast ever. I'm a mathematical physicist working as a professor in a university and I'll send the link to this podcast to all my students! I hope they will be so happy to ear what you talked about as I was :)
Wow a math podcast just made me cry. So much humbleness and passion shines through the whole podcast in general, but especially at 46:26. Thank you Professor Strogatz for the story, and for showing people like me who love math but may not be so great at it that it’s fine, it can still be so fun. And of course, all the questions and everything about this podcast series is amazing. Thanks Grant!
All these talks are so enjoyable. I wish they were 2 times longer. It looks like everyone is having so much fun, everyone is really passionate and they all have great stories.
Grant, your story @1:47:20 of skipping the lecture to read the textbook is exactly the same thing that I did for a statistics class. It was awkward when I had a scheduling conflict with an exam and had to sit in the professor's office taking a final for a class I had pretty much stopped attending 1/3 of the way through. I did something similar for linear algebra, where I supplemented what I was learning with Gilbert Strang's course on MIT Opencourseware.
This was an amazing conversation. That anecdote about clapping after the lecturer's proof (and doing it despite nobody else doing it) is brilliant. Love it.
Hey Grant the question Steven was descrbing in 9:50 is a very popular physics problem in Indian physics books, question of Kinematics, it's thrilling to see Steven Strogatz being so excited about problem solving.
This is so inspiring. As an undergrad, I was amazed by the Nonlinear Dynamics book by Steven Strogatz. And the book was one of the biggest inspirations to pursue research along similar lines. I have come a fairly long way. And in the journey of solving problems, writing a thesis, thinking of jobs, applying, etc. I had somehow lost much of the passion that I had in my undergrad. This conversation just fueled me with a lot of enthusiasm. Thank you so much for sharing this. Getting a glimpse of such an original mind thinking about problems is a privilege. Great work!
I've always felt that the college had a tendency to kill away my natural interest towards any subject through their formal curriculum. Exploring the subject on my own will is always satisfying and I do think channels like yours help a lot. Thanks Grant!
22:53 The vanishing integral around triangles was called _"Goursat's Integration Lemma"_ in our Complex Analysis course. I'll never forget its proof via nested triangles -- it's purely constructive and soooo cute visually!
Grant, I think it's a true skill to make an interview appear as a conversation between friends, yet allow the subject time to fully express their thoughts. Awsum job Cheers Pete
I have a memory of writing down the multiples in a grid; columns and rows. Either I was already aware that multiplication is commutative, or the grid representation showed me so, by way of being diagonally symmetric. The process of memorizing the multiples flowed from that grid representation, each multiple occupying a cell of that grid. It was a single picture representation, natural to mentally visualize, and I would visualize walking down a column.
@@cleon_teunissen I knew it was commutative through memorization as well, but then I understood *why*, by thinking about the area of a rotated rectangle :)
@@DitDede Just to be clear. I did not memorize until after I had the mental representation of all the multiples in a grid. Walking down the columns of the grid (mentally) was slow at first, as I was re-doing the arithmatic each time. In subsequent stages doing-the-arithmatic and producing-from-memory blurred into each other.
Another great episode! I had a thought during "An undermotivated culture" (1:43:20). Would it be valuable for university mathematics departments to start a weekly "Expository Seminar"? Instead of being organized around a particular topic (e.g. Number Theory Seminar), this seminar would be for professors/postdocs to give a talk about their area of research in a way that is meant to explain the topic to other students/professors who are unfamiliar with the topic. In grad school, I had a much more informal version of this experience that was extremely valuable to me. I always went to department tea on Friday afternoons. I'd chat with grad students and professors researching topics much different from my own. Everyone (myself included ) got very good at explaining their research through analogy and/or by relating it to more familiar topics. This led to some useful cross-pollination of mathematical ideas.
You‘re a very good interviewer Grant. I really enjoyed this conversation. Prof. Strogatz is such an interesting and inspiring person. Looking forward to the next podcast!
With regards to giving talks on material that is not your own: in a lot of science PhD programs they require you to give a colloquium on a topic outside of your subfield. In my chemistry PhD program we even had to come up with and present a new research project outside of our subfield. Doing so was helpful to me as a student, but also led to talks that were much easier for everyone to understand. I was skeptical of the value of doing this at first, but I actually came to love the project for showing me the beauty of chemistry in a new way and pushing me to think much deeper into the field as a whole.
I really love hearing about stuff that is completely unrelated to what I am doing but still has so much relevance. So much that has been said in this conversation is in my opinion directly appicable to art. Of course the aspect of practicing but also the idea that you get the deepest understand by learning about stuff yourself and by being frustrated and spending time. No youtube tutorial or teacher can replace that
The timing of me watching this podcast is strange yet amazing. Heres Steven talking about the Cauchy theroem whilst that was the last lecture I had in my complex analysis module. I love maths
Awesome podcast! I have read Strogatz's nonlinear dynamics book and watched his course, but now I just realised I "need" to read all his books! Very inspiring!
Bressoud measure theory book is one of my favorites! His historical approach is very appealing and now I got really interested on that one on real analysis.
I'm a graduate student right now, and have been struggling with the question, "is it me? Are the maths and I not a good fit after all?" Enough so, that I moved to the MS track in the 3rd year of my Ph.D. You just made me realize/remember, that it’s the classes, not the subject. Because I truly love this subject, inside and out, and I have since I was a kid. So, thank you for helping me to stay the course. And for reminding me, it’s university that doesn’t agree with me not maths. On a side note, for my dissertation, I wanted(I still do, but I need to find a new program) to tackle something with a tinge(by which I mean a shovel full) of dynamics to it, and the protein folding problem was one that I had my eyes on. In particular, I thought it would be fun to explore applications of knot theory to PDEs(wave equations in particular) and maybe spice things up with variational methods(like Euler-Lagrange). I was between that or playing with graphs embedded on surfaces and path minimizations. Another “good” fable is the one about why there’s no Nobel for maths. The story is, that it was because Sir Nobel caught his wife having an affair. This got brought up in my Harmonic Analysis course and our professor pointed out… Sir Nobel was never married.
I’ve listened to this in full 3x. Such a great podcast. Really learned a lot, especially about different perspectives/philosophies in teaching mathematics.
On the "morality" remark: Research mathematicians, at least in my experience, 100% think in terms of intuitions, examples, and "morally", this is clear. But I don't think that research papers contain "polished" rigour that removes these intuitions. I understand the point being made, and this is definitely true for many proofs one finds in textbooks. Researcher papers are often written with intuition at the forefront. If a result is published with an abstract, very formal proof, then it will be very likely that researchers in the field will search for a new proof, one that is much more enlightening.
That’s a really interesting perspective. I believe the same thing happens in CS. The textbooks weren’t that fun because they merely presented the final results. Papers are much more inspiring because you can see the growth and development of an idea.
The word "moral" is a bizarre choice, because it's really referring to an aesthetic judgment, but it seems to refer back to the idea that virtues like Truth and Beauty converge. This really plays out in philosophy, where the Anglo-Analytics (rigor) sneer at the Continentals (moral) because they're too fuzzy.
@@ricobarth It's bizarre if you view mathematics only as something which exists outside in the world. Bill Thurston used to emphasise, however, that it's not computers that do mathematics, humans do mathematics. Therefore, mathematics is both something that exists in the outside world, but also in our consciousness -- there's an important interplay.
@@kiran10110 The other thing to bear in mind is the following: Textbooks contain results that have been around for a long time. When you know something for a long time, the curse of knowledge kicks in dramatically, and you assume everyone knows this. In fact, you have evidence for this: All your colleagues know this too, and so, it is as clear as day what is going on. So new results are much easier to exposit than old results that have been known for decades (or centuries).
i work on arrays of josephson junction and was so excited to see him at a conference on the subject before i even started to study physics.. It was a great little surprise as i was so impressed by his book on chaos
I was listening to this on podcast app. Even though I don't see their faces, I knew this would be a good talk! I sometimes find myself laughing insouciantly on some parts of the conversation while commuting or in public listening to this. This is better than my music playlist :D I wished this was longer! Thank you, Grant for this wonderful experience! I'll definitely read Steven's books! With love from Philippines.
That puzzle with the four dogs really is fun, I was trying to get to sleep but just couldn't stop thinking about it, now I have three different solutions
Strogatz spoke about the most important aspect of mathematics pedagogy, where intuition should be the first step and rigour comes afterwards. It is this sequence of steps that makes a maths problem lovable, which would otherwise be so boring and senseless. The intuition part and the visualisation aspects of mathematics pedagogy is what I like the most about Strogatz lectures and Grant's maths videos.
I had that same experience in high school with playing with a problem. I spent a year and a half on 2 great problems, the probabilities of winning a Risk dice roll and the probability of a certain pin being thrown onto a striped pattern and landing in just one stripe. They are slightly weird questions, but they were mine and I had to dig deep and learn new stuff for all of it. I had to come up with a "new" class of numbers for the risk problem and I had to learn calculus for the other, but I was so entranced that I dug in and did it. Now I'm hooked and never going back.
Just looking at that Risk problem, I can see two very different types of "solution". One is the boring, brute-force approach, possibly computer assisted, where you just directly count/calculate the various cases and end up with hard numbers with minimal context. The other is to somehow derive the answers more theoretically, which will give more insight into the problem, but isn't guaranteed to be possible.
@@rmsgrey Yeah, the challenge I gave myself was to figure it out using a derivation instead of brute force. I haven't really completed it so far. I have found a solution for expected losses on a single battle that I checked against a brute force, but I want to derive it further for expected losses over multiple battles. That extended battles problem I have found an algorithm for, but the algorithm is O(3^x).
@@scottbigbrain3944 If you can do expected losses for a single battle starting from an arbitrary (non necessarily integer) troop count can you not just chain battles together to get a campaign? Or do the end effects of low troop counts mess things up too much?
@@rmsgrey You can chain them together, and low troop counts don't really mess with that. The stringing battles together is the O(3^x) algorithm I was talking about. The challenge I am working on with it now is to see if there is a more efficient way to compute further battles.
As someone who loves math and gets excited by teaching this is easily one of the best videos I have seen on UA-cam so far. Thank you for recording this discussion 😁🙏🏻
I have always been driven by the fascination and beauty of science but lately, I have realized that people can do good science for all sorts of reasons. At 31:35, he explains it beautifully. For example, a lot of kids in India are almost forced into science and math because it can take them out of poverty or it opens a path to a better future. And that becomes their motivation. I have observed that these people can work in any field whether they adore the field or not. And also after working in science, math, or engineering for several years, people inadvertently develop some sort of liking for the subject, if not the fascination. But mainly the habit of hard work really works in their favor and they are able to do great work without being absolutely fascinated by the field. A general liking is enough to propel toward success.
Thank you for giving us a podcast like this. Your videos and these podcasts have truly inspired me to move forward towards a graduate studies in mathematics
I love these so much, it's such entertaining and thought provoking conversation. As a side note / suggestion, you should get professor Leonard on here sometime. In case you don't know who he is, he's an undergrad professor who posts his lectures on UA-cam and has gained a pretty substantial following. Great personality and very friendly and helpful for lots of people. I know he helped me get through my first couple calculus classes when I didn't understand my professors, haha!
I caught this via podcast, very nice and far ranging discussion. I believe I'll be sharing this with my students as a way to kick off the semester. And congrats on the new endeavor. One thing that I might be able to share regarding the topic of Newton and decimals. A good place to start is Leo Corry's A Brief History of Numbers. It's not perfect, and leaves out the Indian contribution, but it's very readable and a good intro to the reality that decimals as we understand them were very newly "practically completed" in his time. The roots of these numbers are old, reaching sometime before 800 CE in India, but their development is surprising (at least to people who have only ever seen the finished thing) in its slowness. Islamic mathematicians made progress and began the combination of arithmetic with geometry (in the vein of development that led to Newton-there are other branches) in the Middle Ages, but these were not fully baked, even until Stevin. He seems to usually get credit for the nail in the coffin, but even his version is pretty stunted. It's not until logs, Napier and Briggs, that decimals become a tool for serious calculation, mostly just Astronomy as far as I can tell. Maybe more than the actual technology of decimal calculation, it was also at this time that the vast distinction between magnitude and number finally melts away. For Euclid, 1 was not a number and it's only when we get to Stevin that this changes. Anyway, Corry's book is a good place to start.
57:07 "It's a lot easier to find proofs when you know what the answer's supposed to be." I see a great analogy with coding here: it is a lot easier to write (or rewrite) clean, elegant and readable code once your program already works.
I finished the podcast and started the movie right after, just finished it too. Great combo! Thanks for the suggestion Prof. Strogatz, Jaime Escalante turns out to be a great inspiration for me.
Feynman on the "myth" aspect of the history he tells, from QED p.6: By the way, what I have just outlined is what I call a "physicist's history of physics," which is never correct. What I am telling you is a sort of conventionalized myth-story that the physicists tell to their students, and those students tell to their students, and is not necessarily related to the actual historical development, which I do not really know!
I just loved the entire conversation but the bit towards the end on math expositions in pure maths takes the icing on the cake. For someone pursuing mathematical logic and having to delve with terse proofs with zero exposition, the struggle is real especially if you are stuck in a country where you aren't surrounded by the stalwarts who figured out the perfect proofs to those puszling problems.
Such a FINE conversation! Thank you, Steven, for your on-going awareness! I am sending this interview to my granddaughter, hoping to open some doors. 6th grade, loves math. Thank you!!
Great episode! I remember reading "infinite powers" in high school and it would be interesting to reread it now going into my junior year in university.
Such an illuminating math podcast. I wanted to make a video on commutation of multiplication with the exact same method you described. I taught a kid who was just learning multiplication and fully grasped the idea. I introduce a concept before though - principle of counting. It doesn’t matter which order you count you get the same count. First part would be strongly instilling that idea and how that helps. Which may be obvious to us now, but that is where it starts before you talk about rows and columns.
I can spend an inordinate amount of time trying to really understand what others have done before to my satisfaction. Spurred on my the fact that I know many, many others have understood it before. Perhaps hundreds of years before. I really cannot imagine going off on something that nobody has understood before.
Enjoyed every minute of this. Such an important reflection on how people actually think to understand math. Having done a math bachelor I still always solve everything based on intuition, and only then *put on the iron shackles* to figure out how to translate that to a proof. I actually never memorised any equations either - I haven't even memorised the time tables! =/ Grant is actually younger than me, but I find him such an inspiration - especially to set aside the time to revisit some content to find those motivations or deeper insights.
Unlike Professor Strogatz, I was exposed the connection between abstract and applied math very early, became fascinated by it and grew to love opportunities to take an applied problem, delve into deeper math underlying it, and emerge with a simple solution. At the time, the market for people with math degrees wasn't as obvious as today, and I had a new crisis of confidence; that I would never find my place. Fortunately, things worked out.
Looking at content in timestamp this looks like discussion going to be more philosophical and historical .......and who doesn't love this cultural math. dilemma arguments. looks like Mr. Grant is more into larger aspects of life and rather into healing smaller reality and that's wisdom for sure .
One of the main discussion points coming up several times in this interview - the question whether it's wrong to carry over stories if there's reason to believe they might be just made up - is the argument for American Pragmatism.
Speaking of the problem of being able to google answers; I happened upon the impossible chess board problem through 3b1b and never finished that video. As soon as the problem was stated, I wanted to work on it by myself, with no extra resources. It's been an incredibly rewarding experience, and I thank you so much for it! ... and yea, I'm not done yet. I feel close though :-)
I feel like the discussion on group theory really makes me want an "essence of group theory" 3b1b series
ua-cam.com/play/PLDcSwjT2BF_VuNbn8HiHZKKy59SgnIAeO.html
Perhaps not an entire playlist, but Grant announced recentlly he has sth on Galois Theory in the oven
Sign me up.. make a DAO.. anything hahaha
This guy unlocked everything with 808, Monster set video.. taken me months but becoming coherent what bits mean what..
Pariahs are vowels
@@saulberardo5826 where did he say that???? omg i sooo want grant to cover galois theory
If you truly want to learn group theory, then take a course. A simplified discussion will help, but It won't be enough for deep understanding.
In the end when he uses the words " I think my kids" and then explains that he is referring to his students seriously touched me in a way that I never thought possible from a Math teacher. Grant Sanderson please keep inviting these amazing people on your podcast. I am forever grateful to the work you do.
SAME !! I hope he sees my donation and the do not sell until 2027 hahaha.. they even have a better Metcalfe's Law ! nag him for me, cheers.. always so busy being a generalist lately as things just keep falling into my mind out of other things
"Make the reader love fall in love with the question." -- That's incredible writing advice!
There’s a quote from the Little Prince that is apt:
“If you want to build a ship, don’t drum up the men to gather wood, divide the work, and give orders. Instead, teach them to yearn for the vast and endless sea.”
~Antoine de Saint-Exupéry
Great conversation! I've really loved Strogatz' Infinite Powers book. I've incorporated a few calculus tidbits into my own courses, but mostly it was a masterclass on math exposition that was super helpful
Hay love your lectures as well
I would love to see you do an interview with Grant!!
Sir, you are too good and video which you created helped me so much in graduation, please do as long you can.
hi trefor
Professor I m learning letax watching your videos and now I m writing my thesis in letax because of you thank you so much
Possibly the most encouraging line: 18:20~ "...I was lousy at real analysis."
@DSUM curious, where are you now?
Tbf real analysis will always be difficult unless you're willing to work at it...
continuously.
@@jb31842 Please explain what it means to work continuously, and what it means for the rate at which you work to exist.
@@theswelldudesfishtanks7461 it means you must enjoy it.
the more you enjoy something the more you'll do it, and the more you do somrthing, the better you'll get at it.
so the amount of you enjoying something is directly proportional to how good you'll be at it.
@@jb31842 The effort required to learn it is greater than epsilon.
Damn these maths podcasts are so good...
Ikr!
Yeah, and this is weird who thought a podcast about math would work.
YES
Mind-blowing!
This was one of the best 2 hours I've ever spent. words fail to express how grateful I am for both of you for what you do, especially the content you publish for free. Mr. grant, I'm just a 17 y/o student and I really had to sell some old books to buy myself a bright lamp but if I ever gotten rich, I promise you you're getting your share out of my fortune.
👍
This is such insanely good content. I found myself smiling at things that I never thought I'd even enjoy a discussion of so much..This is probably one of the best things I've ever seen in my life.
Sincerely one of the most delightful conversations I've ever heard with transformative potential
I remember studying graph theory and seeing the Watts-Strogatz model (how to construct a random social network). And I remember being struck by how visual it was and quite understandable. I was struck by that because to this day I still struggle a lot with math (e.g. simple proof by induction? Not me). My fundamental basis of math had holes, which I'm fixing 10 years later. But that part of graph theory stood with me as a more understandable form of math.
And now seeing one of the men behind this model it makes total sense why that part of graph theory did not have the same effect on me as the rest of the course did. This podcast was intense. The love itself (for math) that Steven Strogatz has was so palpable, I could feel it, I almost cried. To hold such emotions within one human being, holy hell, even Grant made a comment among similar lines. It's unreal and so cool to see in someone.
On top of that, the content was excellent. I've heard from time to time that math takes a different shape when you get to higher levels, and I feel that I've gotten a very nuanced glimpse at that.
Absolutely brilliant conversation. Steven's enthusiasm for mathematics is so palpable and inspiring. I was grinning the whole way through and at a few points I even had goosebumps.
P.S. Grant has always been a huge inspiraion too and a great interviewer to boot. Full of really creative and intelligent questions.
Me too, clapping, posting like a bot so much GOOGLE OI YES YOU keeps fking me over 🤣 thinks im those GPT3 text bots so posts are disappearing.. looks like all they did was add a rate limiter :( ignoring my posts about the impending scammer explosion for almost 2yrs.. priceless
We can't fail.. humanity is too powerful, our intention to be good just.. its all so obvious when you grow enough you see it, intuitively or practically.. evolution is relentless its just painful and slow, but fear is at the core of our ego. Fear of the unknown.. after all, what do we do, sense data ? most of it gets discarded. Samurai knew that when you focus on one thing, you unfocus on others - your FOV of your eyes narrows, throwing away more and more data every Planck tick. The amounts are astronomical but we get more meaning from the intuitive stuff anyway, body language.. detail/fidelity is NOT important, and if it was, we would be able to see fundamental quanta.. of course we can't, we perceive them to be physical. So we go optical.
The most intriguing bits for me are.. how touch, taste and smell can be combined. And how memory ACTUALLY works. There are a few cases where there is a human who doesn't get brain formation only stem and the inside surface of the skull. In those cases, sometimes the person is not diagnosed during pregnancy as an anomaly ! if you are, then you are actualized then as having that constraint - they are mentally challenged. Meanwhile, one who didn't even know until his 20's, has about avg IQ and works a maths related job
HIS MEMORY IS FINE. Memory is stored in "awareness" why else do we need to think about things in a structured way - and why else would CHAOS, entropy be growing in someone who is more and more fearful - old people today cannot interpret these technological advances, you get stuck in your ways interpreting data patterns so when something new comes along,... sometimes it is just too abstract ! voila we have the need to have more than 1 database. You don't grow up all in one lifetime just like maths isn't one very long class.
My two favorite math people. This was such a great podcast and both of you are inspirations to this aspiring math teacher.
Agreed but this is a video not a podcast. 👍🤷🏼♀️
36:57 "The knowledge is free."
Totally agree.
Grant, two years ago I was aimlessly wandering through my undergraduate. Though I had loved the mathematics that I had encountered, I had not seriously considered majoring in it.
Your elegant and magnetising presentations changed that.
I shoved my degree into two years, and as of three weeks ago I am embarking on postgraduate work in applied mathematics.
Thank you for the inspiration.
P.S As an amusing note I watched Strogatz series on Nonlinear Dynamics and Chaos rather than my courses lectures for my dynamical systems unit. Strogatz is just a cut above the rest.
That's outstanding, congratulations!
Had me wishing that this conversation just keep going on and on. Being a sophomore, i can relate go to every single thing in the discourse and kind of feel relieved that ppl like grant or steven exist.
It is amazing that Steven Strogatz started with the Stiener-Lehmus problem...the equal bisector problem. There is a lovely proof at the notes section at the end of the Coxeter classic Introduction to Geometry.
Two hours of pure goodness! Thank you!
What a fantastically inspiring conversation
A great and interesting discussion between two of my favorite mathematics educators!
I hadn't heard about Steven Strogatz' undergrad experience. I too absolutely loved math until I went to college. So many of the math professors talked about "the beauty" of this concept or that. But for us mere neophytes, it was much like hearing master chefs talk about "the beauty" of fresh basil without having never tasted it or eaten pesto or enjoying a great caprese salad. For me, traditional math education methods were very much like discussing recipes without ever tasting the final product or being able to employ improversation to create something new.
For example, I never actually grasped differential equations when taught by traditional mathematicians, despite having taken the course twice. I desperately wanted to understand it. I finally had a greater grasp after learning solutions as taught by electrical engineers. Applying the mathematics to an actual problem solidified the concepts.
Regardless, great podcast, and both of you please continue doing what you're doing!
I still remember the first time I started reading through the complex system and chaos book he wrote, and was amazed at how easy and delightful reading a course book could actually be compared to all the previous ones I've had gone through. A modern day hero in my eyes!
Steven's enthusiasm for math, combined with his great ability to explain ideas simply and intuitively is second to none!
13:01 this is me every-time I hear Steven talk. I don't know what it is but he is SO damn engaging and interesting to listen to. I can imagine how well his students learn in his classes. Good man.
One of the most inspiring podcast ever. I'm a mathematical physicist working as a professor in a university and I'll send the link to this podcast to all my students! I hope they will be so happy to ear what you talked about as I was :)
Wow a math podcast just made me cry. So much humbleness and passion shines through the whole podcast in general, but especially at 46:26. Thank you Professor Strogatz for the story, and for showing people like me who love math but may not be so great at it that it’s fine, it can still be so fun.
And of course, all the questions and everything about this podcast series is amazing. Thanks Grant!
I cannot recommend this conversation enough. I felt like I fell in love in math all over again. Thank you for both of you. Much respect.
Listening to this podcast while doing my thesis work is such a pleasure.
All these talks are so enjoyable. I wish they were 2 times longer. It looks like everyone is having so much fun, everyone is really passionate and they all have great stories.
Grant, your story @1:47:20 of skipping the lecture to read the textbook is exactly the same thing that I did for a statistics class. It was awkward when I had a scheduling conflict with an exam and had to sit in the professor's office taking a final for a class I had pretty much stopped attending 1/3 of the way through. I did something similar for linear algebra, where I supplemented what I was learning with Gilbert Strang's course on MIT Opencourseware.
This was an amazing conversation. That anecdote about clapping after the lecturer's proof (and doing it despite nobody else doing it) is brilliant. Love it.
5:27 "That's what she said..."
Btw love your lectures on dynamical systems!!!
Hey Grant the question Steven was descrbing in 9:50 is a very popular physics problem in Indian physics books, question of Kinematics, it's thrilling to see Steven Strogatz being so excited about problem solving.
A great conversation, so motivating especially about the aspect of loving the problem.
This is so inspiring. As an undergrad, I was amazed by the Nonlinear Dynamics book by Steven Strogatz. And the book was one of the biggest inspirations to pursue research along similar lines. I have come a fairly long way. And in the journey of solving problems, writing a thesis, thinking of jobs, applying, etc. I had somehow lost much of the passion that I had in my undergrad. This conversation just fueled me with a lot of enthusiasm. Thank you so much for sharing this. Getting a glimpse of such an original mind thinking about problems is a privilege. Great work!
It's always refreshing to listen to Strogatz. That was a fantastic conversation!
I've always felt that the college had a tendency to kill away my natural interest towards any subject through their formal curriculum. Exploring the subject on my own will is always satisfying and I do think channels like yours help a lot. Thanks Grant!
"Classes will dull your mind; destroy the potential for authentic creativity."
lol
Just like school
This is such an enjoyable talk! I definitely would listen to this for hours and wouldn't be bored!
22:53 The vanishing integral around triangles was called _"Goursat's Integration Lemma"_ in our Complex Analysis course. I'll never forget its proof via nested triangles -- it's purely constructive and soooo cute visually!
Grant, I think it's a true skill to make an interview appear as a conversation between friends, yet allow the subject time to fully express their thoughts. Awsum job
Cheers Pete
two of my most favorite math people...this is 🔥
Lovely!
I vividly remember my own thrill when I suddenly visually-realized multiplication is commutative (sometime during elementary school)
I have a memory of writing down the multiples in a grid; columns and rows. Either I was already aware that multiplication is commutative, or the grid representation showed me so, by way of being diagonally symmetric. The process of memorizing the multiples flowed from that grid representation, each multiple occupying a cell of that grid. It was a single picture representation, natural to mentally visualize, and I would visualize walking down a column.
@@cleon_teunissen I knew it was commutative through memorization as well, but then I understood *why*, by thinking about the area of a rotated rectangle :)
@@DitDede Just to be clear. I did not memorize until after I had the mental representation of all the multiples in a grid. Walking down the columns of the grid (mentally) was slow at first, as I was re-doing the arithmatic each time. In subsequent stages doing-the-arithmatic and producing-from-memory blurred into each other.
Another great episode! I had a thought during "An undermotivated culture" (1:43:20). Would it be valuable for university mathematics departments to start a weekly "Expository Seminar"? Instead of being organized around a particular topic (e.g. Number Theory Seminar), this seminar would be for professors/postdocs to give a talk about their area of research in a way that is meant to explain the topic to other students/professors who are unfamiliar with the topic.
In grad school, I had a much more informal version of this experience that was extremely valuable to me. I always went to department tea on Friday afternoons. I'd chat with grad students and professors researching topics much different from my own. Everyone (myself included ) got very good at explaining their research through analogy and/or by relating it to more familiar topics. This led to some useful cross-pollination of mathematical ideas.
You‘re a very good interviewer Grant. I really enjoyed this conversation. Prof. Strogatz is such an interesting and inspiring person. Looking forward to the next podcast!
This is edifying my soul.
Appropiate way of speaking given your username
1:00:02 no way, I just realized that he actually has the right eye as the logo of 3b1b 🤯🤯
With regards to giving talks on material that is not your own: in a lot of science PhD programs they require you to give a colloquium on a topic outside of your subfield. In my chemistry PhD program we even had to come up with and present a new research project outside of our subfield. Doing so was helpful to me as a student, but also led to talks that were much easier for everyone to understand. I was skeptical of the value of doing this at first, but I actually came to love the project for showing me the beauty of chemistry in a new way and pushing me to think much deeper into the field as a whole.
Finally someone talking about Christiaan Huygens! He is so underrated. 🙌
Such an amazing guest, this was a lovely listen
I really love hearing about stuff that is completely unrelated to what I am doing but still has so much relevance. So much that has been said in this conversation is in my opinion directly appicable to art. Of course the aspect of practicing but also the idea that you get the deepest understand by learning about stuff yourself and by being frustrated and spending time. No youtube tutorial or teacher can replace that
The timing of me watching this podcast is strange yet amazing.
Heres Steven talking about the Cauchy theroem whilst that was the last lecture I had in my complex analysis module. I love maths
Awesome podcast! I have read Strogatz's nonlinear dynamics book and watched his course, but now I just realised I "need" to read all his books! Very inspiring!
Bressoud measure theory book is one of my favorites! His historical approach is very appealing and now I got really interested on that one on real analysis.
I'm a graduate student right now, and have been struggling with the question, "is it me? Are the maths and I not a good fit after all?" Enough so, that I moved to the MS track in the 3rd year of my Ph.D. You just made me realize/remember, that it’s the classes, not the subject. Because I truly love this subject, inside and out, and I have since I was a kid. So, thank you for helping me to stay the course. And for reminding me, it’s university that doesn’t agree with me not maths.
On a side note, for my dissertation, I wanted(I still do, but I need to find a new program) to tackle something with a tinge(by which I mean a shovel full) of dynamics to it, and the protein folding problem was one that I had my eyes on. In particular, I thought it would be fun to explore applications of knot theory to PDEs(wave equations in particular) and maybe spice things up with variational methods(like Euler-Lagrange). I was between that or playing with graphs embedded on surfaces and path minimizations.
Another “good” fable is the one about why there’s no Nobel for maths. The story is, that it was because Sir Nobel caught his wife having an affair. This got brought up in my Harmonic Analysis course and our professor pointed out… Sir Nobel was never married.
I’ve listened to this in full 3x. Such a great podcast. Really learned a lot, especially about different perspectives/philosophies in teaching mathematics.
On the "morality" remark: Research mathematicians, at least in my experience, 100% think in terms of intuitions, examples, and "morally", this is clear. But I don't think that research papers contain "polished" rigour that removes these intuitions. I understand the point being made, and this is definitely true for many proofs one finds in textbooks. Researcher papers are often written with intuition at the forefront. If a result is published with an abstract, very formal proof, then it will be very likely that researchers in the field will search for a new proof, one that is much more enlightening.
That’s a really interesting perspective. I believe the same thing happens in CS. The textbooks weren’t that fun because they merely presented the final results. Papers are much more inspiring because you can see the growth and development of an idea.
The word "moral" is a bizarre choice, because it's really referring to an aesthetic judgment, but it seems to refer back to the idea that virtues like Truth and Beauty converge. This really plays out in philosophy, where the Anglo-Analytics (rigor) sneer at the Continentals (moral) because they're too fuzzy.
@@ricobarth It's bizarre if you view mathematics only as something which exists outside in the world. Bill Thurston used to emphasise, however, that it's not computers that do mathematics, humans do mathematics. Therefore, mathematics is both something that exists in the outside world, but also in our consciousness -- there's an important interplay.
@@kiran10110 The other thing to bear in mind is the following: Textbooks contain results that have been around for a long time. When you know something for a long time, the curse of knowledge kicks in dramatically, and you assume everyone knows this. In fact, you have evidence for this: All your colleagues know this too, and so, it is as clear as day what is going on. So new results are much easier to exposit than old results that have been known for decades (or centuries).
@@KyleBroder Do you know what an aesthetic judgment is? I assure you it's not just internal.
i work on arrays of josephson junction and was so excited to see him at a conference on the subject before i even started to study physics.. It was a great little surprise as i was so impressed by his book on chaos
I could listen to you two talk all day. Greatly hope for a round 2 on a future episode.
I was listening to this on podcast app. Even though I don't see their faces, I knew this would be a good talk! I sometimes find myself laughing insouciantly on some parts of the conversation while commuting or in public listening to this. This is better than my music playlist :D I wished this was longer! Thank you, Grant for this wonderful experience! I'll definitely read Steven's books! With love from Philippines.
I discovered this channel only now (I've been following 3blue for a while) and it's just wonderful. Thank you Grant!
That puzzle with the four dogs really is fun, I was trying to get to sleep but just couldn't stop thinking about it, now I have three different solutions
Strogatz spoke about the most important aspect of mathematics pedagogy, where intuition should be the first step and rigour comes afterwards. It is this sequence of steps that makes a maths problem lovable, which would otherwise be so boring and senseless. The intuition part and the visualisation aspects of mathematics pedagogy is what I like the most about Strogatz lectures and Grant's maths videos.
Mr Strogatz is such an entertaining teacher and the way he isn't shy about telling stories.
Very interesting, informative and worthwhile video. Many thanks for the links to the papers.
Such a lovely professor. Genuine and kind....
I love how attentively Grant is always listening.
Also, insanely interesting podcast! Thank you so much for your work
👍
This is not a podcast, it’s a video🤷🏼♀️
in my functional analysis class, i fell in love with the distance between functions :)
Wow, such an amazing discussion! Grant please return to marking these truly great podcast episodes!!!
I had that same experience in high school with playing with a problem. I spent a year and a half on 2 great problems, the probabilities of winning a Risk dice roll and the probability of a certain pin being thrown onto a striped pattern and landing in just one stripe. They are slightly weird questions, but they were mine and I had to dig deep and learn new stuff for all of it. I had to come up with a "new" class of numbers for the risk problem and I had to learn calculus for the other, but I was so entranced that I dug in and did it. Now I'm hooked and never going back.
Just looking at that Risk problem, I can see two very different types of "solution". One is the boring, brute-force approach, possibly computer assisted, where you just directly count/calculate the various cases and end up with hard numbers with minimal context. The other is to somehow derive the answers more theoretically, which will give more insight into the problem, but isn't guaranteed to be possible.
@@rmsgrey Yeah, the challenge I gave myself was to figure it out using a derivation instead of brute force. I haven't really completed it so far. I have found a solution for expected losses on a single battle that I checked against a brute force, but I want to derive it further for expected losses over multiple battles. That extended battles problem I have found an algorithm for, but the algorithm is O(3^x).
@@scottbigbrain3944 If you can do expected losses for a single battle starting from an arbitrary (non necessarily integer) troop count can you not just chain battles together to get a campaign? Or do the end effects of low troop counts mess things up too much?
@@rmsgrey You can chain them together, and low troop counts don't really mess with that. The stringing battles together is the O(3^x) algorithm I was talking about. The challenge I am working on with it now is to see if there is a more efficient way to compute further battles.
The lecture of nonlinear dynamics system and chaos is really interesting.
As someone who loves math and gets excited by teaching this is easily one of the best videos I have seen on UA-cam so far. Thank you for recording this discussion 😁🙏🏻
W teacher
I have always been driven by the fascination and beauty of science but lately, I have realized that people can do good science for all sorts of reasons. At 31:35, he explains it beautifully.
For example, a lot of kids in India are almost forced into science and math because it can take them out of poverty or it opens a path to a better future. And that becomes their motivation. I have observed that these people can work in any field whether they adore the field or not. And also after working in science, math, or engineering for several years, people inadvertently develop some sort of liking for the subject, if not the fascination. But mainly the habit of hard work really works in their favor and they are able to do great work without being absolutely fascinated by the field. A general liking is enough to propel toward success.
This is the best podcast i've ever listened to. Thanks for all the work!
Thank you for giving us a podcast like this. Your videos and these podcasts have truly inspired me to move forward towards a graduate studies in mathematics
Wow!! I enjoyed every minute of this fascinating podcast!
I can't remember the last time i saw you this excited talking about math! Dr Strogatz is the professor we all wish we had in undergrad
As an undergrad going through analysis this interview chills me out so much
I love these so much, it's such entertaining and thought provoking conversation.
As a side note / suggestion, you should get professor Leonard on here sometime. In case you don't know who he is, he's an undergrad professor who posts his lectures on UA-cam and has gained a pretty substantial following. Great personality and very friendly and helpful for lots of people. I know he helped me get through my first couple calculus classes when I didn't understand my professors, haha!
I was literally watching his videos like an hour ago
So much this, absolutely loved Professor Leonard's videos! Was an absolute carry for me through calc
I honestly thought he was a high school teacher when I found his videos. He just makes the class feel a whole lot less stressful no matter the topic.
Also with professor dave explains UA-cam channel
Leonard is one of the 🐐’s
Woah the Quanta math guy and the UA-cam math guy? That’s a hell of a crossover! ❤️
I also learn dynamical system by watching prof Steven strogatz lecture I found him as good and best teacher
These are the best podcasts available on youtube! I got goosebumps in all 3 episodes I've seen yet.
This was wonderful. Every teacher should listen to this.
I caught this via podcast, very nice and far ranging discussion. I believe I'll be sharing this with my students as a way to kick off the semester. And congrats on the new endeavor. One thing that I might be able to share regarding the topic of Newton and decimals. A good place to start is Leo Corry's A Brief History of Numbers. It's not perfect, and leaves out the Indian contribution, but it's very readable and a good intro to the reality that decimals as we understand them were very newly "practically completed" in his time. The roots of these numbers are old, reaching sometime before 800 CE in India, but their development is surprising (at least to people who have only ever seen the finished thing) in its slowness. Islamic mathematicians made progress and began the combination of arithmetic with geometry (in the vein of development that led to Newton-there are other branches) in the Middle Ages, but these were not fully baked, even until Stevin. He seems to usually get credit for the nail in the coffin, but even his version is pretty stunted. It's not until logs, Napier and Briggs, that decimals become a tool for serious calculation, mostly just Astronomy as far as I can tell. Maybe more than the actual technology of decimal calculation, it was also at this time that the vast distinction between magnitude and number finally melts away. For Euclid, 1 was not a number and it's only when we get to Stevin that this changes. Anyway, Corry's book is a good place to start.
57:07 "It's a lot easier to find proofs when you know what the answer's supposed to be." I see a great analogy with coding here: it is a lot easier to write (or rewrite) clean, elegant and readable code once your program already works.
Wonderful! Both of you are irreplaceable treasures and inspirations to so many of us! Thank you!
I finished the podcast and started the movie right after, just finished it too. Great combo! Thanks for the suggestion Prof. Strogatz, Jaime Escalante turns out to be a great inspiration for me.
Honey, wake up! Grant uploaded a new podcast episode!
Feynman on the "myth" aspect of the history he tells, from QED p.6:
By the way, what I have just outlined is what I call a "physicist's history of physics," which is never correct. What I am telling you is a sort of conventionalized myth-story that the physicists tell to their students, and those students tell to their students, and is not necessarily related to the actual historical development, which I do not really know!
what a wonderful conversation
Excellent conversation!! Waiting for people like Looking Glass Universe and LeiosOS
I just loved the entire conversation but the bit towards the end on math expositions in pure maths takes the icing on the cake. For someone pursuing mathematical logic and having to delve with terse proofs with zero exposition, the struggle is real especially if you are stuck in a country where you aren't surrounded by the stalwarts who figured out the perfect proofs to those puszling problems.
This is pure gold for math lovers
Such a FINE conversation! Thank you, Steven, for your on-going awareness! I am sending this interview to my granddaughter, hoping to open some doors. 6th grade, loves math. Thank you!!
This is a treasure for teachers like us ❤️
Great episode! I remember reading "infinite powers" in high school and it would be interesting to reread it now going into my junior year in university.
Such an illuminating math podcast. I wanted to make a video on commutation of multiplication with the exact same method you described. I taught a kid who was just learning multiplication and fully grasped the idea. I introduce a concept before though - principle of counting. It doesn’t matter which order you count you get the same count. First part would be strongly instilling that idea and how that helps. Which may be obvious to us now, but that is where it starts before you talk about rows and columns.
I can spend an inordinate amount of time trying to really understand what others have done before to my satisfaction. Spurred on my the fact that I know many, many others have understood it before. Perhaps hundreds of years before. I really cannot imagine going off on something that nobody has understood before.
Consuming and Enjoying... Thanks for making this Grant.
Enjoyed every minute of this. Such an important reflection on how people actually think to understand math. Having done a math bachelor I still always solve everything based on intuition, and only then *put on the iron shackles* to figure out how to translate that to a proof. I actually never memorised any equations either - I haven't even memorised the time tables! =/
Grant is actually younger than me, but I find him such an inspiration - especially to set aside the time to revisit some content to find those motivations or deeper insights.
Unlike Professor Strogatz, I was exposed the connection between abstract and applied math very early, became fascinated by it and grew to love opportunities to take an applied problem, delve into deeper math underlying it, and emerge with a simple solution. At the time, the market for people with math degrees wasn't as obvious as today, and I had a new crisis of confidence; that I would never find my place. Fortunately, things worked out.
Looking at content in timestamp this looks like discussion going to be more philosophical and historical .......and who doesn't love this cultural math. dilemma arguments. looks like Mr. Grant is more into larger aspects of life and rather into healing smaller reality and that's wisdom for sure .
One of the main discussion points coming up several times in this interview - the question whether it's wrong to carry over stories if there's reason to believe they might be just made up - is the argument for American Pragmatism.
Speaking of the problem of being able to google answers; I happened upon the impossible chess board problem through 3b1b and never finished that video. As soon as the problem was stated, I wanted to work on it by myself, with no extra resources. It's been an incredibly rewarding experience, and I thank you so much for it!
... and yea, I'm not done yet. I feel close though :-)