Q&A with Grant, windy walk edition

Who's your favorite mathematician? I always find favorite questions kind of silly, but I will tell you about two different mathematicians that have been on my mind lately. So, a month or two ago I watched this documentary about Claude Shannon called The Bit Player, and then that prompted me to read a little bit more about Shannon in books like The Idea Factory or James Glick's The Information. And this guy was super interesting. I almost think of him as kind of a mixture between Donald Knuth and Adam Savage. So on the one hand, you know, he's the father of information theory. He wrote this absolutely seminal, super important paper about transmitting information and storing it and encoding it and things like that. It really gave birth to the information age and a lot of the foundations for computer science. To give you some indication of just how important this work was, there's this one science fiction novel where all of the years, instead of being measured B. C. and A. D. with respect to the birth of Christ, everything is before Shannon or after Shannon. It's viewed where the foundation of information theory is so pivotal in any kind of civilization that it's worth setting your entire timeline around that point. So that's Shannon. Super influential on that one hand. But on the other hand, he was incredibly playful and very willing to kind of pour his life into totally useless toys. So he spent a lot of time building different kinds of unicycles. He built this flaming trumpet for his son. He, you know, built this automatic juggling machine, this mechanical calculator that would do computations in Roman numerals. Just wholly, completely useless things done for the purpose of play and nothing else. What I think you have to wonder is if this playfulness is wholly incidental and unrelated to the important work that he did. Or if there's something that's necessary about it, right? If it's required that the kind of personality who's coming up with something totally new, like original enough that it is a foundational moment for a new kind of science, almost has to be the kind of personality that's also willing to make gizmos and toys for his own pleasure that other people look at and see as useless, if somewhat fun. Another mathematician I've been thinking about for kind of related reasons is Edward Lorenz, who is maybe not the father of chaos theory, but certainly one of the fathers. And he wasn't even a mathematician, actually. He was a meteorologist. But really, he was a mathematician to his core, but wearing the external veneer of a meteorologist. And he's maybe most famous for this system of equations. It's just three variables, three unknowns. It's one of the earliest examples of something called a strange attractor. But it came about when he was studying weather patterns, right? And weather is famously hard to understand. You know, at the extreme, there's a huge number of unknown variables. If you go all the way to one extreme, you could say that every atom in the atmosphere has six degrees of freedom. So you could have this truly ungodly monstrous system of equations that no one could ever hope to analyze. So inevitably, you have to do something to simplify things down for pragmatism's sake. And a lot of people, I think, thought that the reason weather is hard to predict is simply because of the number of variables at play. And one of the important contributions from Lorenz was to be able to simplify down the unpredictability of it into a surprisingly simple set of equations to say, hey, some of the facts here that are hard to predict when you make a, you know, when you make some kind of measurement with a little bit of error around it, and that error kind of propagates, that's not just because of the number of variables at play. You can have that kind of chaos arising from surprisingly simple circumstances. So in order to do this, right, you have to have the strange mixture, again, of a kind of pragmatism with a kind of more pure mathematician instinct. And I wonder if this is ever something that a pure mathematician could have done. If he wasn't grounded in a problem like weather, where he was doing all of these computational models, and really just in the weeds of, you know, convection or whatever else it might be. So in the same way that Shannon kind of represents this contrast between playfulness and pragmatism, in my mind, Lorenz sort of represents this contrast between, like, applied science and then pure math. And just as Shannon is the father of information theory, it doesn't seem like a coincidence that we find in the father of another very instrumental science to our modern age, chaos theory, this sort of middle ground between those two. Now obviously there's a lot of selection bias at play here, right, like most people that find themselves somewhere between pure and applied or somewhere between pragmatic and playful don't give rise to completely new fields of study. But I do have to wonder if the novelty required to father a new field necessitates breaking the norm and not falling into one clear-cut path. Some of you might be wondering if every question in this Q& A will have me pontificating for many minutes, but I do have a broader point here, which is that I think a number of people watching this channel, especially those on the younger end, are clearly into pure math, right, they're curious about it. I wouldn't be surprised if a lot of them are contemplating becoming mathematicians. And for that set of people, I kind of want to put out the question of where do you think there's more value? Do you think it's if all of these people with this inclination towards pure math go into that field, and that's where they get to collaborate with a lot of folks who are like-minded, who think like them, they resonate on the same wavelength, maybe they amplify each other's strengths? Or is there more value if each one of them kind of gets dispersed into some completely different field, and each one is not so much a mathematician as a mathematician plus X, right, they're a mathematician plus a builder plus a meteorologist plus whatever have you. And they take those instincts of playfulness towards puzzles or desire to abstract the simplest form of a specific kind of hardness and basically bring that mathematician instinct to something totally different, right, which of those worlds has more value?

A teenage kid walks up to you and says they hate math. What do you tell or show them? I get the impression that the spirit of this question is for me to answer with some piece of math that's so enticing that even the most ardent of math haters would have to bring about some kind of affection for the subject. But the thing is, if someone comes to you and they admit that they don't like math, or that they hate math, I don't think you should show them a piece of math, even if you do want to convert them. It's a little bit like if, let's say you really love coffee, right, and someone comes to you and they say, I just hate coffee, but you like it a lot and you're kind of this snob and you really want to turn them on to it. The way to do it is not to try to find the world's best cup of coffee and then give it to them, because no matter what, it's going to taste like dirt. They don't like it. They're not addicted to it at this point. Instead, if you want to turn someone on to it, we have a couple options. One of them can be to first breed necessity, and then from there maybe get an addiction. So let's say it's a student and they need to stay up all night to finish some kind of paper, and so they begrudgingly have to take some caffeine, and after that a kind of culinary Stockholm syndrome kicks in, and through the addiction they come to kind of like the substance. That's not the greatest, but that's a way to do it. The analog with the world of math might be trying to find something where math is the drug necessary for someone to accomplish what they want to. You know, maybe they're really into video games, so they want to make their own video game, and often in writing the software for that you have to use geometry, trigonometry, maybe bits of calculus. It kind of depends on the game that you're building. So you breed the necessity. But in either case, I think the key is to just try to build up familiarity, just be exposed to math in a lot of different contexts, and importantly for it to not ever be traumatic when that happens. I think the reason a lot of people say that they hate math is because their only exposure to it is tightly linked with a sense of failure, right? It's a hard subject, but importantly it's a very cumulative subject, so if you miss one little part, it looks like you're failing in all of the later parts, even if you would have been really good at those other parts without that missing link. So instead if the math comes about in a less judgmental context, something where there's no grading, you know, maybe it's just playing with puzzles with friends, or it's writing that bit of software for the video game, or whatever it is where you're gaining exposure without that trauma, I think after enough time, you're just going to like the subject. Because I think if we're all really honest with ourselves, and we look back on why we like the things that we like, it's often because someone in our life liked it. The time we were spending with them brought us to spending time with that thing, and then it just stuck with us. What advice would you give to a math enthusiast suffering from anxiety disorder

clinical depression, and ADHD? I'm not entirely sure how the word math enthusiast in the question changes the answer, but I will say that for anything health related, and this goes double from mental health, definitely seek professional help earlier than you think you need to. So don't be shy about finding a therapist or asking your doctor about these kinds of things. One thing I will say, as a professional YouTuber, I do think it's probably healthy for people to spend less time on the internet. You know, I always get this uneasy feeling if I hear about someone binge watching all of my videos or something like that. Because on the one hand, I often do measure success in terms of how much time people are watching the videos. It's some indication of how much it's reaching the world, how deeply people want to engage with it, and all of that. But on the other hand, if I think of someone kind of staying up all night to watch YouTube, or even just sitting in their house all day to watch YouTube, no matter what that video is, no matter how enlightening or how educational it seems to be, that's gotta be bad for you and unhealthy in comparison to occasionally, you know, going outside or spending time with real people in the world, or if you want to engage with math, doing it in a more physical, social kind of way.

Is Ben and Blue still a thing? Ah, yes, the podcast with the Bens. No, I don't think it'll continue. I don't think that's a bad thing. I think we had some really good conversations about education in there. I'm not sure how much we necessarily had to say. And I don't think all projects should live forever. I think there's something kind of nice about doing something a little different, a little experimental, and then being willing to just walk away from it if it seems better to spend time on, say, animated math videos or whatever else your main occupation might be. Favorite podcasts?

Alright, so there's three different podcasts where I get actively excited if I see something new in the feed. And it's unfortunate because each one doesn't upload super regularly. First one is The Anthropocene Reviewed by John Green. So, as many of you probably know, John Green is an excellent writer. The podcast itself, he kind of reviews everyday things or aspects of human existence that range from deeply philosophical ones to very mundane, like the Taco Bell breakfast menu. It's maybe 70% personal memoir, and you get this view into his own very thoughtful but also very twisted and tortured mind. So, highly recommended. The second, which I think is super popular, so certainly don't need me saying this, is Hardcore History by Dan Carlin. He often likes to go deep into the human experience of certain wars or certain very violent times in human history that shaped the direction of things. And he does a good job just giving a super abundant amount of context, maybe way too much context, which is why these podcasts sometimes last as long as six hours and will be many, many part series. But sometimes you like that in a podcast. And then maybe foremost, that I like a lot, and this will just come as no surprise, is the Numberphile podcast. I'm slightly biased because I was on it, but I actually think I'm the least interesting guest there. Super interesting to hear from different mathematicians what their story is, how they relate to the subject, what motivates them, and of course Brady Haran is a phenomenal interviewer, so it's just perfect for anyone who's into math, even in the slightest of ways. If you had both the responsibility and opportunity to best introduce the world of

mathematics to curious and intelligent minds before they are shaped by the antiquated, disempowering, and demotivational education system of today, what would you do? Asking because I will soon be a father. First of all, let's just acknowledge that it's very weird for me to be asked a question that has anything with the flavor of parenting advice, because, well, look at me, like I'm 27, what the hell do I know? But one interesting place to look here, if we're thinking, how do you make a child entering the world love math, or things related to math, is Richard Feynman's dad. So, evidently, Feynman's dad was incredibly interested in making him into, like, a physicist or an engineer or something like that, and even when young Richard was just a newborn baby, he would paint these interesting patterns that were meant to instill young Richard's mind with a sense of mathematical patterns through the raw exposure. And as he grew up a little bit later, he would give these very deep, thoughtful answers to questions about how the world worked, or why when you, like, tug at a wagon that has a ball in it, the ball doesn't move. Feynman would tell all of these stories, they were some of his favorites to tell. You know, if I look back at my own childhood, there's definitely a lot of influence from a very attentive, thoughtful father in that respect. I think I said this in a previous Q& A, but I'll just say it again here. I remember these games where he would stack these sugar cubes in interesting geometric arrangements, and I would be asked to count how many there are. And you couldn't just straight up count because it was, some of them were hidden in certain ways, right, so you are effectively cubing numbers or something like that. And, of course, if I got it right, then I would get one of the sugar cubes as this Pavlovian reward. And if you look at the success of someone like Feynman when it comes to problem solving, and I definitely don't view myself as, like, a great problem solver, but I do love the subject. I have this, like, deep-seated affection for it that probably is not unassociated to the kind of games my dad would play with me. It's not so much that I think, oh, the painting of those patterns really did instill young Richard's mind with a sense of math, or that the answers that his father gave to certain questions were the ones that made him deep and thoughtful later. I think it's just that when you're a parent and you're showing a lot of attention towards something, you're signaling to the kid that something is important and it's worth thinking about. So all of the signaling that probably came from young Richard Feynman's dad showing this deep attentiveness to questions about the physical world, about mathematical patterns, probably made it such that young Richard would spend a lot of his own time thinking about those things, because they just pattern match off of their parents. Another thing I might say is try to draw a distinction between school math and math, right? They can just be very separate things, and kind of separating the brand of those two can't hurt.

What's something you think could have been discovered long before it was actually discovered? Well, just last week, all of the internet has been very abuzz about a certain result that came from these three physicists studying neutrinos about eigenvectors and eigenvalues, which is a crazy fundamental thing. You kind of wouldn't imagine that there are any new things to be discovered about computing eigenvectors or computing eigenvalues, because it's so, well, it's kind of old and it's very, I don't know, routine at this point. But they found what they thought was a result and they sent it to Terry Tao, who actually responded and his initial thought was, this can't be true, it would be in every textbook if it was true. And then within two hours, I think he found three independent proofs of the thing, and yeah, it's just a different way to compute eigenvectors that was discovered in 2019, even though that totally could have been discovered hundreds of years ago.

Can we fix math on Wikipedia? Really serious here. I constantly go there after your vids for a bit of a deeper dive and learn nothing more, ever. Compared to almost any other topic in the natural sciences or physics, where at least I get an outline of where to go next, it's such a shame. So this question kind of reminds me of that classic trope where you've got a girl and she's dating a boy and, you know, he's kind of a bad boy, he does a couple things that are wrong that adds to his allure, he's kind of sexy in that way, but she's thinking, oh, I can change him, right? He's flawed, but I can fix him. And everyone in her life is looking and saying, oh, honey, like, he's not going to change. People don't change. You have to find someone else. In the same way, if I see someone trying to learn math from Wikipedia and not use it as a reference, it's like, it's not going to change. Don't try to change it. You've got to find a different source. There's lots of really great blogs that you can go to, or Math Exchange and Quora are great in terms of people trying to explain things in approachable ways, and don't forget about just good old textbooks. In math, more so than a lot of fields, I think there's a strong contrast between what makes good reference material and what makes good pedagogical material. And a general rule of thumb, this is not universal, but things that are single-authored, I believe, are better pedagogically. And I suspect the reason for this is that when you want to explain a topic, often the best route to making it understandable is to start off by being a little bit wrong. You explain kind of a simplified version of something that isn't entirely accurate, but it's easier to get a foothold in. Then once you have that foothold, you slowly carve away what's wrong about it until you end up at what is entirely accurate, but more complicated. But you've taken this path through incorrectness. Now when you have multiple authors, I think the tendency is that you sort of wipe away and edit away the things that are incorrect. That's like the stable equilibrium that you reach. So what you're left with is a source that's entirely factually correct, but it's harder to get a foothold into for that reason. And Wikipedia just represents the extreme of this, but I also think you see it if you look at a textbook that has, you know, three or four authors. Again, there are exceptions, but I like that as a rule of thumb. Real quick, I want to tell you about two new items that have been added to the 3Blue1Brown store for any math enthusiasts. The first one, in the spirit of upping the level of formality on things, is this knot theory themed tie. So as you can see, the pattern includes a lot of different simple mathematical knots. So almost any knot that you are likely to tie with your tie anyway is going to be topologically equivalent to one of these, unless you just go totally crazy. In sourcing this, we wanted to make sure that it was, you know, a legitimately high quality tie, and I'm really happy with what we found. Then as a supplement to the ties, I also got these vector field socks produced, and what they represent is the phase space of a pendulum, which some of you may know is most naturally represented on a cylinder, hence printing it on a sock. So the whole item is just sort of a subtle nod to that fact. I believe DFTBA is going to do some kind of sale on Black Friday and Cyber Monday, so if you're watching this before then, definitely check it out. And with that, I will see you all in the next, probably much more typical, video. Thanks for watching!