The Exchanges

Q That's a fascinating story. Like, it strikes me that math is something that we develop this mindset about for whatever reason, that We're just not mathematical, or math isn't my strong, and you know, this sort of like happens around grade four or five or six, and it sort of like follows us. Is that a teaching thing? Like, why do most of us find math so boring and dry?

A You're right about that, aren't you? I mean, that is, if you just look around and ask your friends, the average person did not have a good experience in math class, although it tends to be a little more complicated than that. Most people will say, I did like math until And then they'll tell you until we got to fractions. Those were really confusing with the common denominators. Or, or I liked it until we got to algebra, and then it was all those x's and y's instead of numbers, and I didn't really know what was going on. Or I even liked algebra, but then I lost it in geometry, or I hit the wall in calculus. So different people have these different places, but it seems it sort of happens to everybody at some point, or almost everybody. And so, why? Um, there's a few explanations. You know, one is that the subject, the way it's traditionally taught, is, is made to seem like a tower. You know, each thing builds on the, the topic before it, and if you fall off at any stage, it's a long way down off the tower. You know, that, that is, it, it's a very linear architecture, whereas the reality is that math is a web, not a tower. That you can jump in at different points in the network of ideas, And then find your way around from there. And so it's not really true. I mean, it's complicated. In some ways, it is sort of like a tower. The idea of algebra and variables and functions, and I kn…

AI assessment note: “one is that the subject, the way it's traditionally taught, is, is made to seem like a tower”

Q Let's see if I understood this correctly. So I plug my phone into my car, turn on Apple CarPlay, get directions. It's triangulating between three or four different satellites, and the time that the satellites sync up their atomic clocks, and then it's triangulating between how long it gets from each of those satellites to my phone, and then based on that, it can triangulate where I am. Did I?

A Yep. You've got it perfectly, because those, those clocks on those satellites are very accurately maintained, and the military keeps extremely care, careful measurements of where the satellites are. So they have to know exactly where those satellites are, and also, you know, what time it is on board the satellites. And so, as you say, you, now, they're all at somewhat different distances from your car, from your receiver, and, ah, so yes, you get to measure, as you say, triangulate, you get to measure several distances, three or four distances. To all those different satellites. And knowing those four distances, knowing where those satellites are, that places you uniquely on the Earth. Not only that, it also places your velocity. I mean, so GPS can tell you how fast you're moving as well as where you are in three dimensions.

AI assessment note: “Yep. You've got it perfectly, because those, those clocks on those satellites”

Q And our effectiveness at that. And it, it also played a role in GPS. Can you explain that one? Or does, it's the fundamental role in GPS, I think.

A Well, that's right. Sure, anytime we use our GPS gadgets to find our way to a strange destination, or sometimes even to find our way home when we're lost after going far away, Um, GPS is a wonder of calculus. It's got so many different aspects of calculus built into it in the, the way that it acquires signals from the satellites overhead, um, the way that the satellites and the whole system estimates distances, um, by doing a complicated mathematical calculation. You know, it has to look at distances to three or four different satellites overhead. Um, there's also Really what the GPS system does is it doesn't directly measure distances, it measures time and converts those into distances. So the time is the time it takes for a signal to go from the satellite to your GPS receiver, given that it's traveling at the speed of light, because it's an electromagnetic wave, this signal, um, it's going to move at the speed of light. And so by, you have to time very precisely how long it takes for the signal to get to your receiver from the time it was emitted By the, the satellite overhead. Now, what's really tricky about all that Is that the, um, the satellites need to keep extremely accurate time. They have onboard atomic clocks in them, the most accurate time pieces we know of, that are based on principles of quantum theory, which itself is built on calculus as its math, its infrastruc…

AI assessment note: “GPS is a wonder of calculus. It's got so many different aspects of calculus”

Q What are the other terms that we would use around rates of change?

A Well, in economics, people are always using the word marginal. So if they say the marginal utility, you know, how much extra pleasure or utility do you get from something for the next dollar that you spend? Or what's your marginal return on an investment? You invest one more dollar, how much bang do you get for that buck? Um, so those are rates of change, the rate of change of return with respect to investment. That would be an example, and, and that's a common sense thing. You know, you don't want to At, at some point, the marginal returns start going down, so it's not worth it to put in that extra dollar of investment. Um, so we use that idea a lot there, but certainly in physics, we talk about speed. That's a rate of change of position with respect to time. We have acceleration. That's the rate of change of velocity. But even something as simple as a paycheck, you know, when I say I'm making six dollars an hour, well, let's not say that. That talk about minimum wage. Let's say we're making, uh, ok, 15 dollars an hour is what they're talking about now. Like, everyone should be able to make that. If, you know, if you had a decent wage, that's still a rate of change, right? It's dollars per hour. Anything where you say per is a rate. Exchange rates, you know, how many marks per dollar or how many pounds per dollar. Those are rates.

AI assessment note: “Well, in economics, people are always using the word marginal.”

Q You, you didn't have an easy path in math, dude. Was it first year university where you sort of Uh, got the worst grade you've ever gotten, and you switched to physics, or was that, when was that?

A That is true. That was my first year. Well, that was that class, so I was just describing how our freshmen, the first year students, can do astonishing things. I was, I was, um, supposedly one of those students when I went to college, so they grouped a bunch of us together in, like, whiz kid linear algebra, and, uh, for, this was for all the kids who had done extremely well in calculus in high school, so we, and it's sort of like, You had a little, a mark on your head. This is a future professional mathematician, or at least potential to be. And the other thing, though, is that, that professors know that kids who are good in math in high school, a lot of them are going to get, so to speak, weeded out. They don't really have the right stuff to be mathematicians of the future. At least that was the thinking at the time. It's a pretty nasty pedagogical philosophy, I must say, but, but they would throw all of us in With one of the worst teachers in the school, I don't know why they assigned him to us. I mean, this was a really pathetically terrible person for teaching. I mean, the first day of class, he came in and sort of slithered along the wall, didn't make eye contact with the students, very shy person, who didn't say welcome to Princeton, welcome to the, you know, this is going to be exciting, nothing. He just, just started with the definition of the key ideas at the beginning…

AI assessment note: “That is true. That was my first year.”

Q Can you give me some examples of, like, the stuff that you cover?

A Oh, yeah, sure. So here's one. You know, we're used to a standard kind of geometry, but there are other geometries. Here's an example. Suppose we're in New York City or some other city that has a grid of streets. So if I say how far is my location from some other location, you could say it in terms of miles or kilometers as the crow flies, but that's not very relevant if you're driving on the grid. You might be more likely to say you have to go three blocks north and, you know, eight blocks west or something like that. So you would give units of blocks as measured on the grid. Now, that's an interesting kind of geometry. If you, if you define the distance between two points in a plane as being, um, the number of units north-south plus the number of units east-west, right? You can define a distance like that. It's not the Euclidean distance. It's not the standard one, because you're not allowed to take diagonal paths in this geometry. You can only go on the grid. Now, if you ask someone, A question like, what does a circle look like in this geometry? Um, that's an interesting question. You know, what, in other words, what's the set of all points three blocks away from a given point? So try to picture that in your head. You could go three blocks north, or you could go two blocks north and one over, either east or west, or you could go one up and two over. And if, if you can pictu…

AI assessment note: “So here's one. You know, we're used to a standard kind of geometry”

Q we'll explore how math helps us better understand the world and make better decisions, and as you'll see, Steve makes learning things fun and practical. Let's get started. It's time to listen and learn. How did you get interested in math? Was it a story that pulled you in? Like did this, were you innately interested in it? Was it something like a teacher? Like how, how did this happen?

A I, well, I was interested in all of school, and I still am. I like all the different subjects, but, so math didn't stand out for me, um, I'd say really pretty much for most of my childhood. It was just one among many, but then I do remember one or two moments where it started to take special significance. One was I had two friends when I was about, well, let's see, we used to call it junior high school, so it was seventh and eighth grade. I must have been about 12 years old. Two other boys, and, um, They were both good at math and, and, ah, vocabulary and geography and everything else. I used to like to compete with them. We would play chess against each other and other things. And at that age, just being better than your friend meant a lot to me. I don't know. Or, or having them feel better than me inflicted a kind of pain. So I do remember one time getting a little higher score than, than this other kid on a math exam, and it just felt good. So I'm not saying this is a noble reason to be interested in math, but it was just an early memory that it was very objective. You know, you could say, I'm better than you, and it actually meant something, which to a thirteen-year-old boy, that was important. But, but on a more elevated level, um, I'd say the real turning point was in my, uh, would have been my sophomore year of high school, so by then I was about 14 or 15, and A teacher …

AI assessment note: “I do remember one or two moments where it started to take special significance.”

Q You mentioned economics. I'm curious as to why Physics math is so accurate, and economics math is so, uh, wildly inaccurate. Is that because the, the mathematical models don't apply as well, or, I mean, to what extent does biology give us a better model for economics than physics?

A Lots of great ideas in that question. Well, ok, I, the short answer would be that physics is a lot simpler than economics. You know, when you, when you measure the moon, it doesn't mind. It doesn't react. You know, it's just an inanimate thing. Um, the laws governing inanimate bodies are just a lot simpler and more easily quantified than the laws governing populations or, or individual people. So, the, the task of the social sciences is extremely hard. Um, they get to feed back on the system that's measuring them. It's also their ethical issues. You can't, Do experiments as easily, or, or sometimes you can't do them at all on people or on populations of people. Um, it's much harder to do controlled experiments. You try to pin down some variables and they just pop up somewhere else. So, if you think about the history of science, which sciences were solved first, or where did we make progress? The first science to really make good progress was astronomy, which I always think is kind of surprising at first, given that the moon and the stars and The planets are very far from us. They're very remote, and you might think, like, why shouldn't biology be the subject where we make the most progress? It's very important, medicine, and, and helping sick people, and prolonging life, and all that. You would have thought we'd devote so much attention to that, but it's not, we did, you know, …

AI assessment note: “the short answer would be that physics is a lot simpler than economics.”

Q And correct me if I'm wrong, you're, you're more interested just, uh, intellectually in sort of the orderly side of nature than the chaotic side, correct?

A I would say that's true, yeah. My, my interest has always been in how does order emerge from chaos? So I do find chaos interesting as a starting point, but, um, whereas some people revel in the chaos, I, I like self-organization. I like systems that spontaneously show astonishing feats of, um, collective behavior, where they somehow Get their act together on their own with no, no commander, no outside force telling them how to behave. Because I think that's, you know, we see that all around us. We see, I, I assume life evolved spontaneously, you know, in the consistent with this position I'm giving that it's all materialism. I, I think that's one of the great mysteries. How do you get life from non-life without a creator? You know, how, it must somehow be in my world view that the laws of nature And the right conditions will lead to life emerging from chemistry, but understanding the origin of life is one of our great scientific mysteries people are working on. It's not hopeless, but it's, we don't have the answer just yet.

AI assessment note: “I would say that's true, yeah. My, my interest has always been in”

Q Can you tell us the story of how calculus, or math, I guess in this case, and calculus influenced, uh, our treatment options for HIV?

A Well, ok, yeah. Um, so take your mind back to, I'd say, you know, like the mid-nineteen-eighties, where in the West, here in the U.S. or Canada, the, the HIV epidemic was really starting to hit. It was this very mysterious disease. Um, wasn't so clear what was causing it. Uh, and, but the symptoms, Were very predictable, that a person who got infected would at first show flu-like symptoms. They'd feel kind of sick for two weeks, but then they'd get better. You know, they might have a fever. Um, they'd have, you could see their T cells, the important components of their immune system had a measurable change. Um, could detect some virus in the blood, but, but they'd get over it, you know. And so, after two weeks, It seems like the person was sort of better, and then years could go by without any particular symptoms except for this strange low level of T cells, these crucial components in the immune system. Um, it seemed to be that the T cells were being depleted somehow by the, by the presence of the virus, but otherwise people weren't that, that sick. But then after maybe 10 years, suddenly A tremendous crash would seem to happen, and they would become terribly sick, and that's when HIV would become AIDS, at which point all kinds of nasty opportunistic infections would set in, weird kinds of pneumonia that you wouldn't normally see, weird cancers that were very uncommon, and the…

AI assessment note: “take your mind back to, I'd say, you know, like the mid-nineteen-eighties”

Q How should parents Engage their kids in math. Like, rather than give them the answer, or how do we get kids excited? And how do we find that in ourselves again, as we're helping our kids?

A Yeah, because I think the parents are critical to this in that a parent who says, well, I never used math in my life, so it doesn't matter, that's not going to help. You know, we have to try not to pass on our own anxieties to our kids. What I try to do and what I would recommend other people do is to not be afraid to admit that you don't know something. That's, that's a big, strong reaction to say, I don't know. Let's figure it out. You know, either we can figure it out by thinking about it, or we can look it up on the web. I mean, that's maybe a second best choice, but sometimes that's the best you can do. Um, but, but the key being, it's okay to not know everything. Now, some, I suppose in some models of parenting, the parent is the authority figure, and to relinquish authority is a, is a big concession, and so these are maybe in more traditional homes. The parent will never do that. You're not supposed to be the equal or even the lesser of your kid. But, um, in intellectual matters, I happen to like honesty, and if you don't know something, why are you trying to pretend? You're gonna get found out anyway.

AI assessment note: “What I try to do and what I would recommend other people do is”

Q Does anything else come to mind when you're asking that question of your students or yourself?

A Yes, a big thing is personality. I mean, in Duncan Watt's case, since I was just talking about him with the cricket problem that became the small world problem, and maybe I should just say, I mean, it's gross, it's kind of, kind of come off like bragging, but just to finish that thought, it's, this paper is now among the hundred most highly cited papers of all time in any scientific discipline. I mean, it's the biggest home run I'll ever hit in my life, for sure, and long after I'm dead, it's the one thing that Duncan and I will be remembered for. This was a monumental Game-changing bit of work that started from, you know, from just a crazy question.

AI assessment note: “Yes, a big thing is personality. I mean, in Duncan Watt's case”