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What's the Use? Interview with Professor Ian Stewart
4.3k Plays6 months ago
Welcome to another engaging episode of the Breaking Math Podcast! Today's episode, titled "What is the Use?," features a fascinating conversation with the renowned mathematician and author, Professor Ian Stewart. As Professor Stewart discusses his latest book "What's the Use? How Mathematics Shapes Everyday Life," we dive deep into the real-world applications of mathematics that often go unnoticed in our daily technologies, like smartphones, and their unpredictable implications in various fields.
We'll explore the history of quaternions, invented by William Rowan Hamilton, which now play a critical role in computer graphics, gaming, and particle physics. Professor Stewart will also shed light on the non-commutative nature of quaternions, mirroring the complexities of spatial rotations, and how these mathematical principles find their correspondence in the natural world.
Furthermore, our discussion will encompass the interconnectivity within mathematics, touching upon how algebra, geometry, and trigonometry converge to paint a broader picture of this unified field. We also discuss the intriguing concept of "Fearful Symmetry" and how symmetrical and asymmetrical patterns govern everything from tiger stripes to sand dunes.
With references to his other works, including "Professor Stewart's Cabinet of Mathematical Curiosities" and "The Science of Discworld," Professor Stewart brings an element of surprise and entertainment to the profound impact of mathematics on our understanding of the world.
So stay tuned as we unlock the mysteries and the omnipresent nature of math in this thought-provoking episode with Professor Ian Stewart!
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Transcript
Introduction and Ian Stewart's Background
00:00:05
Speaker
Hello, Breaking Math listeners. This is your host, Gabriel Hash. I am thrilled to announce that today I'll be doing an interview with the author of a popular mathematics book called What's the Use? with the subtitle, How Mathematics Shapes Everyday Life. Now, the book is authored by Ian Stewart, and some of you may be wondering, who is Ian Stewart? Prior to this interview, I went ahead and pulled up the Amazon page for Ian Stewart, and there's quite a lot to say. Dr. Stewart, can you tell us a little bit about your background?
00:00:35
Speaker
Well, I'm a retired math professor. I was and still have some activities at the University of Warwick in England, which is particularly well known in the mathematics community because it's one of the best math departments in the UK. But I retired about 12 years ago.
Math Research and Podcast Audience
00:00:51
Speaker
And I do a lot of popular math writing. I also write science fiction as a sort of sideline. And I'm still doing math research. I have a very good friend in the States. He lives in Columbus, Ohio, and we've been working together for about 40 years now.
00:01:12
Speaker
Math actually research level is is a social activity, but it's a very small social group. And yeah, I think you're absolutely right. What's exciting about this podcast in particular is I think it's found people who are passionate about mathematics, the world over, you know, and that way, like it narrows down your audience, so you don't have to cater your audience as much. So wow, yeah, that's that's very cool.
00:01:35
Speaker
You had mentioned you've authored some science fiction. I have now interviewed four different, either mathematicians or physicists, who have gotten into science fiction. What got you into science fiction?
Science Fiction Passion and Book Collection
00:01:46
Speaker
I got interested in science fiction when I was about 13, which is a fairly standard sort of entry route. And I just enjoyed the books. And I now have a collection of 8,500 books and magazines in the house, which my wife kindly tolerates.
00:02:03
Speaker
I have to ask you this question real quick. So when I ask myself that question about my own interest in mathematics and science and science fiction, I think about elements of stories themselves and fantasy, and then what separates fantasy from science fiction. There's a book called Harry Potter and the methods of rationality. Have you heard of that one yet? I've heard of it. I've not read it, but a few years ago now, have you heard of Terry Pratchett? Yes.
Collaboration on 'The Science of Discworld'
00:02:30
Speaker
Okay, because I've done four books with Terry Pratchett and with a biologist friend of mine who died a few years ago. We did a series of books called The Science of Discworld, and of course what people always assume that's about is it's like these sort of science of the X-Files and so on books, Science of Star Trek, which uses the
00:02:51
Speaker
the TV programme or the movie or the book, as a vehicle for explaining, you know, how does the walk drive in Star Trek work? That kind of thing. Well, we didn't do it that way, because when we had the idea, we talked to Terry Pratchett and said, Terry, why don't we do a Science of Discworld book, a bit like these Physics of Star Trek and so forth? And he said, I'd love to do that, but there's no science in Discworld, it's fantasy.
00:03:16
Speaker
It's based on the power of story. It's based on narrative imperative. And it's based on magic. And magic is what happens when people want and science isn't like that. But we found a way to do it, which is in the fantasy world. And so basically, Terry wrote a short story for readers who don't know about this world. It's a fantasy world with witches and wizards. It's flat. It rides on the back of four elephants on the back of a giant turtle who swims through space. And he uses it as a vehicle for satire, basically.
00:03:46
Speaker
But we found we could use it to talk about science, provided
Fantasy and Science in Discworld
00:03:50
Speaker
that the wizards of Unseen University on Discworld brought into existence by magical means Roundworld, which is our universe and our planet. And from the outside, they observe what's going on in Roundworld, and it makes absolutely no sense to them because it's not it's not even a flat world like any sensible world ought to be.
00:04:10
Speaker
So we then interspersed chapters commenting on what is puzzling the wizards. So everything that we write about is actually going on in Round World. It's in our universe. But the story kind of sets it into the Discworld setting. And this just turned out to be a very enjoyable and fruitful way of talking about any bit of science we want. It doesn't have to be something that has any connection with Discworld as such.
00:04:39
Speaker
It just has to be something that the Wizards are observing going on and thinking, what on earth are these people doing? So we had a series of four books. We were only going to do one, but it was all the fans liked it. So they insisted we did the second one and it kind of carried on from there.
00:04:56
Speaker
Fantastic. So for our listeners right now who won't want to read that book or any of you other previous books, where would you suggest some math in that? Which of my books should people start with?
Engaging Math Books to Start With
00:05:08
Speaker
One that I has been very successful and that I really like and really enjoyed doing don't let the title put you off. It's called 17 equations that change the world or in the American title is
00:05:25
Speaker
is less off-putting because it doesn't mention equations. That was so scary, aren't they? For everybody. In pursuit of the unknown is the American title. So the unknown is what the equations are about, of course. But what it is, it takes about 18 or 20 really important mathematical equations, explains their history, explains where they came from, and what
00:05:53
Speaker
in our lives now depends upon mathematicians and physicists and engineers and other such people having come up with these equations and understood them. So for example, radio and television depends on the wave equation for electromagnetic waves, and so on. So in pursuit of the unknown, that's a good one to start with. And it tackles equations head on. Nice. But you don't have to actually understand the equation.
00:06:18
Speaker
Yes, you know what I saw a really good show on Stephen Hawking's information paradox with black holes and it was actually produced in Britain Oh gosh, what was it called? It may have just been called the information paradox. It was released probably 18 years ago But what was really cool is it took that approach it actually took
00:06:34
Speaker
Stephen Hawking's approach on information entropy in black holes, and it took every single letter, every single one, and it had
Approachable Math and Creative Engagement
00:06:42
Speaker
a neat diagram and picture and story. Now, not heavy math at all, just like stories and things that people can relate to. This started out, it was actually a Dutch publisher who asked my English publisher whether they had anything on
00:06:58
Speaker
of this kind. And basically the response was, no, but we can get one written. But the idea, yes, I think in show business there is a saying which says, if you have a wooden leg, wave it. If there's something that you really would quite like to conceal, don't.
00:07:17
Speaker
Put it right up front. So yeah, every chapter starts with a page which has the equation in very big letters with lots of arrows and things saying what the symbols are and what they mean. You know, E equals mc squared, E is the energy, m is the mass. I don't know to use that equation because everyone knows that one.
00:07:34
Speaker
Yeah, maybe the only one they don't. And it tries to make the equation seem friendly, because it's equations are telling you something, they contain a lot of information. This is why mathematicians like them. They're a very compact way of representing key ideas in mathematics. So if you can start to treat equations as something that isn't off putting, but is kind of not exactly warm and fuzzy, but at least friendly, it's trying to help you.
00:08:02
Speaker
Yes, no, no, I hear you, I hear you. You know, early in our Breaking Math podcast, our idea phase, we did a lot of promotion on social media, and one of the things that we wanted to make equations relatable is we actually took a whole series of famous photographs and famous paintings through all history, tons and tons of them, and we would superimpose a mathematical equation on the top or the bottom to get people to think. Now, the first one, you're gonna laugh.
00:08:30
Speaker
You know that famous photograph of Merlin Monroe where her dress is blowing up over the tent? Oh yes. We did that one and beneath that we put the fluid dynamics equation.
00:08:40
Speaker
I was thinking you might. Yeah, exactly. Aerodynamics. Yeah, yeah, yeah, yeah. We did. Absolutely. There's a couple other great ones. And these are nothing more than conversation pieces. Like, okay, what does that equation have to do with what's happening here? There's another one we did where there's a famous photograph that I recently heard was actually fraudulent. In other words, it didn't actually happen. It shows a bunch of construction workers eating their lunch on a beam.
00:09:07
Speaker
Do you know what I'm talking about? Oh, I know the one. Yeah, I think I know the one you're talking about. It's possible it was fake. I mean, whatever. Oh yeah, you can fake almost anything. Oh yeah, yeah. So that one we did, potential gravitational energy, because that's pretty obvious. There's one that I never did, that I've been wanting to do. I want to superimpose an equation, I just haven't found the right one, on what's the painting, is it called, the scream, where it shows a person just slapping their face and screaming?
00:09:34
Speaker
So what would be the most horrible equation ever that drives physicists, you know, to madness that we could put on there, maybe turbulence or something? No, I mean, the obvious one would be the wave equation for sound. Okay, okay, yeah. Yeah, but I mean, it's what's going on inside his head.
00:09:52
Speaker
that's really the point here. It's what's driving the screen, isn't it? So you want some sort of, there's a thing called the Hodgkin-Huxley equation for the nerve impulse, which is one of the things mathematical biologists use to understand how the nervous system works and in particular the brain. Yeah. So at least you'd be getting
00:10:12
Speaker
towards sort of brain mathematics rather than sound. But it's the interpreter is the mind, it's what it means to the person who is screaming. And that's much harder to turn into an equation, I think.
00:10:27
Speaker
Yeah, yeah, totally, totally. So yeah, that'd be fun. Now, I would like to real quickly redirect toward the book that you are selling right now. Again, the book is called What's the Use? And it's available right now on Amazon or anywhere folks buy books. I saw it at Barnes and Noble on their website and all kinds of places.
00:10:46
Speaker
So, what's the use, of course, the inspiration of this is kind of what we're doing right now.
Math's Role in Technology
00:10:52
Speaker
It's explaining what's the significance of a given math and how to use it every day. That's essentially what you'd say the purpose of the book is. Yeah. It's a kind of, what is math doing for us? Why do we have mathematicians at all? Yes. Okay. From the pointed view of somebody who's not in the academic world, who is not necessarily into anything scientific, they're just living their lives the way we do nowadays.
00:11:15
Speaker
One of the points that the book starts with making is you don't see the math that is being used. You pick up your phone. I mean, a phone is an amazingly complex mathematical machine. Yes. Smartphone, a cell phone.
00:11:31
Speaker
The amount of math that goes into a cell phone without which it would not work is enormous. And the user does not see this, probably doesn't know this. Why should they? You don't need to. I mean, when I use a phone, I don't actually think very hard about math. If you had to, then nobody would ever buy one. But without the math, those things won't work. So we need to have enough people and it turns out to be rather a lot by the time you put all this stuff together.
00:11:57
Speaker
Yeah, so one of the ideas was to try and explain to people that you may not see it in action, but it is happening all the time in stuff that you're using. And the other aspect of it was to try and make the math interesting in its own right and surprising. Yes. With perhaps the biggest surprise of all being that that kind of math
00:12:19
Speaker
is getting used for this kind of problem. Because nearly all of the math that's in the book was not invented with the current use in mind. Some of it wasn't invented with any use in mind. It turned out a few hundred years later, gosh, well, we can actually make use of this stuff.
00:12:37
Speaker
Isn't that wild? And then it turned out to be really quite important. Yeah, so I was trying to collect stories which are a little bit surprising, because the trying to put over the message that you can't judge a piece of math purely by its content. Maths that seem to be about something that's totally unrelated to the real world can overnight turn into something that is very closely related to the real world, because it suddenly turns out to be useful in solving some kind of real world problem. Yeah.
00:13:03
Speaker
Yeah, that's one
Unexpected Uses of Math and Philosophy
00:13:04
Speaker
of my favorite parts of mathematics. In fact, one of our first episodes, we actually talk about useless or quote unquote useless math that eventually had a purpose and the greater, I guess, philosophical musing of how does human knowledge work. It's almost like mathematics, pure math, and then like what's the opposite of mathematics, I guess, just like intuition. I don't know.
00:13:26
Speaker
We were talking about like two strands of DNA about how we understand our world. You know what I mean? And yeah, pure mathematics is definitely part of it and you don't know what you don't know. So I'm trying to make this sound deep here, but I think I'm stumbling. Right. Well, no, but there is a philosophical point here and it's one that I don't think anyone still understands. There was a physicist called Eugene Wigner who gave a lecture about the unreasonable effectiveness of mathematics.
00:13:52
Speaker
Yes. And the important word there is unreasonable. Yes. If mathematicians study and physicists and so on study the flow of air around objects and you end up with aerodynamics and it helps you design aeroplanes that fly, that's not a surprise. Yes.
00:14:07
Speaker
Yeah, it's very worthy. It's very important, but it's not it's a reasonable use of mathematics. It was invented for that purpose. Yes. But if you take something that somebody came up with off the top of their head just because it looked like fun about 300 years ago. Yeah. And suddenly it turns out to have medical applications or technological applications to something that they could never have the inventor of this math could never have imagined was possible.
00:14:33
Speaker
then that's an unreasonable use. So the book is about the unreasonable uses. Oh yeah. Because they're more fun and they're more surprising. Yes. Oh yeah. For sure. For sure. One other thing we talked about prior to this interview, while we were just getting set up, is who the target audience of this is. And I found that perhaps I have a bias.
00:14:56
Speaker
I was going to say, you know, even like a high school student who's interested in mathematics or, you know, I don't know, even down to eighth grade. But I thought, no, wait a minute. I say this as somebody who already has a master's degree in electrical engineering. So I've already been familiarized with a lot of this stuff.
00:15:12
Speaker
So it's kind of hard to tell exactly who it's for. However, you and I both said, if there's an unfamiliar term or an unfamiliar concept, it's so easy this day and age to go to YouTube or even Wikipedia is fairly reliable a lot of the time. It's pretty good old math actually.
00:15:33
Speaker
Yeah, yeah. So really, I would say anybody who is passionate about math, you won't be solving any equations in this. I mean, you could, but like you won't be doing a lot of math yourself. So you can read the book and, you know. It's not trying to teach people things. Correct. Correct. It's just to inform people who are interested, you need to have some sort of sympathy towards the idea of reading something about math, I think, as far as it goes.
00:15:58
Speaker
You don't have to be an enthusiast, you don't have to be a fan. I try to explain the technical ideas in non-technical terms. There are lots of pictures which I always find helpful to illustrate ideas. The point of it is to try and make it as accessible to as many people as possible.
00:16:19
Speaker
I never have a specific target audience in mind for books because I think books tend to find their own audience, in fact. I once had a long argument with a television producer about this who reckoned that if you're doing TV, you know exactly who your audience is going to be. And I said, well, it's really not like that with books. Publishers get big surprises.
00:16:42
Speaker
They think they're onto a winner and it flops. They do a book that I think no one's ever going to buy and it ends up on the best seller list. Isn't that amazing? And isn't it nice that it's like that? Yes, yes. If anything was predictable, it would be so boring. That itself, I think, would be a fascinating story, a fascinating podcast. It's just to sit with some of these seasoned book publishers and talk about their surprises and their misses and all that kind of stuff, you know?
00:17:10
Speaker
It's one of the things I quite enjoy about writing the book because to some extent I write it because I'm interested in it and I think other people will be interested if I can tell them about it in a sufficiently accessible way. And then we get to see whether in fact that's true and what happens.
00:17:30
Speaker
had a big success some years ago with a book was just a collection of about 200 very short pieces of miscellaneous interesting math. It was called Professor Stewart's Cabinet of Mathematical Curiosities. Nice.
00:17:47
Speaker
Yeah, it went to number six on Amazon in the UK. Fabulous. Just after Christmas some years ago. And nobody expected that including me, my agent and the publisher. It was the right book at the right time. I think people were looking for Christmas presents and
00:18:05
Speaker
so on if it's only two paragraphs long then you're not going to get bogged down in anything that you can't bail out of and move to a different page. Yeah, yeah, for sure, for sure. The entire list here of books, there's a lot I'd love to read here and I'd love to talk about like,
00:18:22
Speaker
I don't know. You'd mentioned earlier in the interview a few recommended spots to start with. But yeah, if you've got any other suggestions among your publications, let me know, because I'd like to fill up my basket while we're talking. One of the things that's very interesting about math for people who are that way inclined is the history. Yes. Because it has a very, very long history. Yes.
00:18:44
Speaker
Yeah, it goes back at least 4000 years to the ancient Babylonians and some of the things that the Babylonians understood are still in use and worth knowing today. Yes. They were pretty smart mathematicians, in fact. So there's one book I did called Significant Figures, which is essentially it takes 25 important mathematicians, not the most important,
00:19:05
Speaker
for all sorts of reasons. But 25 who did something of considerable significance mathematically, it might or might not have applications. And roughly speaking, I tried to arrange them in approximately historical order so that there was a kind of mathematical thread that runs through the whole thing. So I think people who are interested in the history, it perhaps differs slightly from some of the other history of math books, because it's really driven by the people and their story.
00:19:35
Speaker
the math is just part of what they do. Wow. Very cool. And you said this and I didn't catch it. The name of that book, oh, Significant Figures. Oh, Significant Figures. I did catch it. Okay. I thought for a minute that I didn't catch it. You did. Okay. What's interesting is as I'm talking to other folks who are academic in nature and who have say a podcast or a TikTok or whatever about an academic topic, a lot of the folks who make things on say history or philosophy, they say, oh,
Math History's Appeal
00:20:02
Speaker
mathematics. That's terrifying. So if there's a book that's more focused on the history of mathematics, that might give them something to chew on and to relate to. So that's really cool, actually. Now, as we're talking, I have this whole list here on Amazon of your other publications, and there's some interesting ones here. There's a book titled Fearful Symmetry. What is that about?
00:20:25
Speaker
That's about pattern formation. It's about the math that helps
Math in Natural Patterns and Physics
00:20:32
Speaker
us understand a lot of striking patterns in the natural world. So if you throw a stone into a pond, you see circular ripples which spread out from the center. Well, that's a mathematical pattern, the circle. It's a very simple object, but it's also a very symmetric object.
00:20:53
Speaker
Yes. Mathematicians are very keen on symmetries because there's some beautiful mathematical theory behind it called group theory, which kind of formalizes the idea of a symmetry transformation. A symmetry isn't a thing, it's a transformation. The symmetries of circles are, if you rotate a circle through some angle, it still looks like a circle.
00:21:15
Speaker
If you've got a square you have to rotate it through 90 degrees, otherwise you'll notice that it's moved. But if you have just a blank square and turn your back and someone rotates it through 90 degrees and you turn around and have a look, it will look exactly the same. Actually it's moved. So the symmetries are the ways to move objects which leave them looking just like they did beforehand. And there's a whole
00:21:39
Speaker
beautiful mathematics of pattern formation, which covers a huge number of different types of application from biology through to engineering, using essentially the same basic mathematical toolkit.
00:21:52
Speaker
which is a mixture of this group theory, which is the kind of algebra of symmetries, and dynamics, which is the mathematics of how things change with time. So fearful symmetry, you probably recognize it as a William Blake poem, tiger, tiger burning bright in the forest of the night, what immortal hand or eye, dare frame thy fearful symmetry. So grubbing around for a title, he ended up with
00:22:19
Speaker
with fearful symmetry as the title. Oh, okay. Very cool. Very cool. I like that. Interesting. Yeah. We, um, on a tiger are an example of this kind of pattern formation. Yes. Yeah. There's a mathematical equation which governs how the stripes form in the embryo before the tiger's born. Oh, wow.
00:22:37
Speaker
crazy. So there's a pattern which is not visible when the tiger is first born but there's a chemical pattern on the skin which at a future stage triggers the formation of the proteins in the hair which give it color and that's where the stripes come from. So there's a pre-pattern which is laid down in the embryo following mathematical rules which then as the tiger develops some bits go black, some go brown, some bits go white and you get a stripey pattern.
00:23:07
Speaker
I like tigers. Nice. Yeah, that's cool. So now when you talk about symmetry, I was curious if this topic was brought up in the book, Fearful Symmetry, and that is in physics and in the history of the universe, breaking symmetry. Absolutely. In fact, breaking symmetry is
00:23:27
Speaker
the pattern forming mechanism. So the idea is that, in fact, there's two different ways to break symmetry. One is the mathematical equations remain symmetric with the solutions stop being symmetric. That's called spontaneous symmetry breaking. And then there's another one, which is you actually change the equation so that they're not quite symmetric anymore. So you kind of
00:23:51
Speaker
give them a bit of a nudge. So mostly what the book is about is the spontaneous symmetry breaking, which in physics, for example, the Higgs boson, which was this very famous particle discovered fairly recently, which gives all the other particles mass.
00:24:07
Speaker
Now, the Higgs boson is an example of spontaneous symmetry breaking. The mathematical equation for it has circular symmetry, but the state of the particle is not circularly symmetric. But I can give an example that's perhaps a little... One that I'm fascinated by is the patterns of sand dunes.
00:24:29
Speaker
If you look at Google Earth from high above at a desert, you will see amazing patterns in the sand dunes in the Namib Desert or indeed on Mars. You have to go to NASA's website for that. But there are enormous fields of sand dunes on Mars. And they have patterns. They have quite interesting, often quite regular patterns. In fact, the simplest one is like the tiger. It stripes long parallel dunes.
00:24:56
Speaker
Now, those form the kind of equation that governs the formation of these dunes. The desert starts out completely flat. It's like a blank sheet of paper. It's very, very symmetric, boringly symmetric, because you can move a sheet of paper any way you like, and it still looks like if you just focus on what's in front of you, it looks like a bit of a sheet of paper.
00:25:18
Speaker
You can't really tell if it's blank. As soon as you rule some straight lines on it, then those move around. But if you have parallel lines and you move them at right angles so that they just move one line across and then they overlap again, they end up looking exactly the same. So stripes are a series of equally spaced parallel lines. That's a symmetry. And that's a broken symmetry from the flat desert. And all of the sand dune patterns
00:25:46
Speaker
can be explained and understood in terms of breaking symmetry from a flat desert, there is also the possibility that the wind is blowing. And if the wind is blowing in irregular ways, which it can do, now you're breaking the symmetry of the equations as well. So you've got the other kind of symmetry breaking. But it would be possible to write an entire book about the mathematics of sand dunes.
00:26:11
Speaker
Wow. That's a topic where I hear what you're saying, but I'm trying to comprehend symmetry breaking. There's a feeling when I really understand something and I definitely am not there yet. Obviously, I'm grasping it like, well, why? You know what I mean? That's quite a topic.
00:26:32
Speaker
Yeah, I mean, there are some simpler examples than sand dunes. But those are very visual. Yeah. The mathematics is just that the if you have a symmetric system of equations, then there is always a there's a solution of the equations, which is has as many symmetries as the equations do. It has all of the symmetries to the equations. But physically, that can become unstable. That is, you won't see that solution happening. It's
00:27:00
Speaker
It's like a pencil balanced on its point. In theory, there is a perfect balance point, which is completely symmetric under rotations if it's a round pencil. But you can't balance it there. If you let go, it will fall over. It's broken the symmetry. But in fact, there's a whole circle's worth of directions in which it could have fallen over, just depending which way the tiny breath of wind was blowing. So a pencil balance on its tip and what happens when you let go is actually a very good mathematical example.
00:27:30
Speaker
Okay. Very cool. Very cool. But not very exciting. Well, so sort of. Now again, I'm a little obsessed with math because I don't know. I think it was a show I was watching from Neil deGrasse Tyson on the Big Bang. And again, it's pressing that question of, well, how do you know this? You say all these things, but how do you know? And it's for that reason that even if something might even seem uninteresting, I'm still obsessed with it and I still file it away like, okay, so this will explain something down the road or something.
00:27:57
Speaker
You know what I mean? So even if something is not really super intuitive or clear, I don't know. It's certainly worthy of discussion. One of the things that I want to mention, and again, you'll have to forgive me for my jumping around style. I just think of so many things that I just have to do this. So in your current book, there's one chapter that I read several times, and I realized there's a reason why I read it so many times.
00:28:21
Speaker
The chapter was called, Papa, Can You Multiply Triplets? The reason why I read it so many times was because there was a rather recent video from the brilliant YouTube channel, 3Blue1Brown, where he illustrates graphically exactly what quaternions are. He does many examples and you see beautiful graphs in 3D.
00:28:46
Speaker
Those two tied together so well. So for all of our listeners, I want to suggest that they read that book, but also watch that YouTube clip on quaternions. It'll help. Now, I'm sorry, I feel like I interrupted you. You were going to tell us a little bit about that chapter. Yeah, that's a fascinating story because it's about William Rowan Hamilton, who was an Irish mathematician back in the 1800s, who was essentially trying to invent a new number system. And it was a number system that would let you do three-dimensional geometry.
00:29:16
Speaker
The precursor of this is something that for mathematicians is perhaps one of the biggest discoveries, most important ever is what are called complex numbers. This is where you allow minus one to have a square root. But the natural geometry of complex numbers is the two dimensional plane. And enormous amounts of physics in two dimensions can be done using complex numbers. So Hamilton wanted to do the same for 3D. And he spent years trying to invent this new number system.
00:29:45
Speaker
And eventually he figured out you couldn't.
00:29:48
Speaker
do it in 3D. In fact, nowadays, we can prove you can't do it in 3D. But you can do it in four dimensions. And this is why they're called quaternions. The quatter part is the Latin for four. There's basically four different basic numbers which you combine together by adding them to get all of this system of quaternions. So inside the four-dimensional space is a three-dimensional space where you can use them to do various things in physics.
00:30:19
Speaker
Hamilton invented these, and he was quite obsessed with them. I think a lot of people at the time thought he was a bit of a crackpot about it. He was a very good mathematician, but he got to be in his bonnet about these strange quaternion things. But they did turn out to be very useful, not for the kind of physics he had in mind, but in mathematics itself.
00:30:42
Speaker
in the computer graphics industry for games and movies because what you want to do in those is
00:30:49
Speaker
move objects around in three dimensions and project them onto your two-dimensional computer screen or movie screen, and you want them to look realistic. And so you need very efficient, fast ways of representing motion in 3D, moving sideways, rotating, spinning, all of this kind of thing. Hamilton's quaternions are absolutely perfect for that. Essentially, they eliminate errors that might
00:31:17
Speaker
accumulate if you tried other methods. And the other thing is they're now turning out to be actually quite important in fundamental physics, particle physics, areas like that.
00:31:28
Speaker
which were not the sort of classical physics that Hamilton had in mind. Yeah, in fact, you know, I have a colleague that we start after breaking math. We began a YouTube show called touring rabbit holes named after Alan Turing. My co-host was Dr. Alex Alanis. He was obsessed with quaternions and their use in particle physics. We never really did a show on it. And but I'm curious, what what can you tell us about their use in particle physics?
00:31:55
Speaker
Okay, I mean, basically, the key to particle physics is symmetry.
00:32:01
Speaker
But fundamental particles are very, very weird things, even to physicists. Now, there's some sort of bunch of waves in a quantum world, and they obey quantum rules. But there are symmetries in particle physics, which are not just the physical ones. If you can take a particle and move it, or you can rotate it, you can actually transform it into a different particle.
00:32:27
Speaker
You can apply a mathematical transformation to the equations, which solutions that will be one kind of particle are then turned into a different kind of particle. And the symmetries that crop up here are, among other things, these quaternions and objects like them are fundamental to the mathematics of understanding those symmetries.
00:32:52
Speaker
So the standard model of particle physics is a rather complicated bunch of equations, which have quite a lot of symmetry. But the symmetries are what mathematicians call continuous groups. They are not just you move something in a small number of different ways, you move them in infinitely many different ways.
00:33:15
Speaker
in the way that I could take an object, hold it in my hand, and move it sideways any distance I like. So that would be potentially an infinite number of possible distances I could move it. Whereas with a square, if you want symmetries of a square, there are only eight symmetries. You can rotate it.
00:33:31
Speaker
through four different angles, and you can flip it over about four different axes. And that's it. There's eight. The quaternions are not greatly useful for these finite groups, but for the continuous ones, they often show up. They're very basic to the whole area. So physicists working in those areas
Quaternions and Their Modern Applications
00:33:51
Speaker
have to have some understanding of quaternions and some related mathematical gadgets. I mean, it's a very well developed, sophisticated area. And some of them really like thinking about quaternions and get, they're fascinating things because they're
00:34:12
Speaker
a bit of a twist on the usual number systems we have. They don't obey all of the rules of algebra. That was the other thing that Hamilton didn't understand at first. He wanted all of the rules of algebra to apply. And in particular, the order in which you multiply two numbers together shouldn't make any difference to the answer. So two times three is six, three times two is six. And he thought that should happen for his number system. You can't do it that way. Yeah. Quaternions
00:34:39
Speaker
you multiply them in a different order you get a different result usually oh yeah kind of like um and again i'm i'm gonna expose my ignorance here sort of like in matrices how you can multiply you can yeah well you can represent quaternions as four by four matrices okay okay of a particular kind um and yeah it's all all of this stuff is related but with rotations um
00:35:04
Speaker
if you rotate an object through two different angles about two different axes in space, depending on which order you do those rotations, you can get different result. So, so it's not surprising, given the link between caternions and rotations, it's not surprising that they don't obey this commutative law of algebra, as we call it. Yes, because the thing you're trying to model, which is movements in space, that doesn't obey that law.
00:35:32
Speaker
Yeah. It obeys almost everything else. Interesting. Yeah. Very cool. What I also I love about this interview itself, and I'm sure you've experienced this many, many times in your research and in your discussions, are how wind topics overlap each other. Because, you know, we were just talking about symmetry, what was it 10 minutes ago? And then we talk about quaternions and how symmetry is a huge aspect of that with respect to, you know, particle physics and things like that. So that's one part of
00:35:57
Speaker
of this that I love, talking about math, just, you know, everything overlaps. So that's pretty cool. Yeah, it's unified. It's something which I think when we first do it in high school or earlier, it comes in, there's arithmetic, there's algebra, geometry, trigonometry, and so on. And you don't see many connections between them, at least not to begin with.
00:36:26
Speaker
Yes. But really, they're all part of one much bigger picture. The problem is the entire picture is so huge, that no living mathematician would actually understand all of it. Yes, yeah. There are some who come close, and they're very impressive. But yeah, no, most most of us find an area within the subject that appeals to us. And we kind of specialize in that. We've got any sense we keep an eye on what's going on everywhere else.
00:36:54
Speaker
in general terms. Because sometimes it's important for what we're interested in. Yes. Yeah. And, you know, I'm not saying that that's, that's the test of whether it's worth doing, but it's the test of whether we should pay attention to it. Yes. And, you know, but it's a huge subject. I mean, my research colleague in the States and I are putting together a book on our
00:37:18
Speaker
part of research mathematics, which is about networks and dynamics on networks. And we've pretty much finished writing it. And it's 900 pages long. Oh, goodness. Wow. It's going to be a big fat volume. But in order to tell the story and the applications as well as the theory, we eventually decided we just had to make it that big. You know, there's just a big story to tell. So
00:37:47
Speaker
No, and that's our tiny little corner of mathematics. Yes. Dang, that's fascinating. Man, I'll tell you what, when you're ready to publish it or just let us know. We would love to have you and any of your co-authors back on the show to talk a little bit about that. That sounds like a fascinating project.
00:38:03
Speaker
The nice thing about it is everyone knows what a network is. It's also called a graph by mathematicians, but it's a lot of dots joined by lines. It looks like a spider's web built by a very drunken spider. But huge numbers of things in the natural world have that structure. Airports connected by airlines is an obvious one.
00:38:24
Speaker
nerve cells in the brain connected by synapses, genes connected by this one reacts with that one, and so on. So the network math has lots and lots of interesting applications. And you can find out a lot about it and kind of understand what's going on without getting tangled up too much in the theory. So it could be quite a nice topic.
00:38:45
Speaker
I had one more question that I wanted to ask you. This probably motivated a little bit selfishly. One of our obsessions on breaking mathematics or on breaking math is the topic of mathematical modeling and trying to describe human language.
Math in Machine Intelligence
00:39:02
Speaker
We always try to go about it from Shannon's information theory as well as themes from the book Godel Escher Bach. So I'm curious, is this an area that you have read much about or do you have any thoughts on that field?
00:39:17
Speaker
I've read Gerda Lesher Bach is one of my favorite books of all time. I think it's absolutely brilliant. Yes. I have read Shannon and Weaver's book on information theory. Yes. One of the equations in my in pursuit of the Unknown Book is in fact the information equation from Shannon.
00:39:45
Speaker
But I'm not an expert on any of that. Yes. I'm aware of it. That's how I feel. I think language is very interesting mathematically, but there's a big gulf between language has actually spoken by people and sort of theoretical grammatical structures and some of the things that people were trying to study.
00:40:09
Speaker
But of course, where I think we're where math and language do come into contact now is machine intelligence and yes. You know, it is now possible to talk to your computer and it will understand you. I mean, it will talk back to you.
00:40:24
Speaker
And these things are getting surprisingly good compared to what anyone thought was possible even a few years ago. And behind this is a lot of math, but it's not so much information theory math. It's matrices and graph theory and stuff like that, I believe. Deep learning algorithms, that sort of stuff. This is clearly going to be, well, it's already very important. It's going to be more important. It's another one. It's not in the book because
00:40:54
Speaker
Well, I had to save some in case there was a sequel. You can't put all your best material into into volume one just in case. Oh, yeah. Yeah. I think all authors, the back of their minds, you're thinking, well, you know, if you put all the good stuff into the first book, then you don't have many material left for the next one.
00:41:12
Speaker
No, I hear you. Fascinating. It's a topic where here's how I'll describe it. My intuition is strongly pushing me in that direction where there's a lot of great ideas and things that have not been published yet. It's rich ground for publication. I'm not there yet. It's still cooking.
00:41:38
Speaker
Yeah, I mean, one of the things that interests me, and I think Gerd Lechebach goes into this to some extent, there's a big difference between information and meaning. Yes. Yes. Between syntax and semantics. Yes. The statement 2 plus 2 equals 5 contains the same amount of information as 2 plus 2 equals 4. Yes.
00:42:00
Speaker
Right, but one of them's correct and the other one's not. Yeah. And that's important as well. Absolutely, absolutely. And yeah, and like part of what I've thought about is language and how it's used for or how I'm not even speaking correct grammar here. I should do an entire and entire podcast of incorrect grammar that is perfectly coherent and talk about how you can have meaning without necessarily having grammar and you know, how those rules work. But I've thought a lot about song lyrics that are
00:42:27
Speaker
complete nonsense, yet evoke a strong sentiment. Like, oh, things by the Beatles, or by the Red Hot Chili Peppers. I'll have to think of examples here. Or even things that people can say that sound super deep and are hypnotic, but are complete nonsense. You know what I mean? Yeah, there's a lot of things. If you look too closely, a lot of lyrics have popped. When I was an undergraduate, I was lead guitarist in a
00:42:53
Speaker
rock band for two years and yeah we were singing this stuff and then you think about it after what did that mean exactly yeah but you know but it evoked a response in the audience and
00:43:11
Speaker
Yeah, well, we mostly played the dances and it was the response was they danced to it. But yes, you know, they liked it. Yes. And if some things worked and some things didn't, when we rapidly found out which ones didn't, we stopped playing those. Indeed. It's almost like talking about, you know, there's there's physics for poets. We also need poetry for physicists. So yes. Yeah.
00:43:35
Speaker
Thank you so much for coming on the show. Come on on again. We'd love to hear your thoughts on anything, any of your future publications, or if you want to just come and talk about whatever. We'd love to have you. So thanks again. Thank you, Gabriel. That was great fun. And yeah, I'll take you up on that offer. Wonderful, wonderful.