WEBVTT - Why is math so important for physics? 0:00:08.600 --> 0:00:11.560 Sometimes I'll bump into a stranger, maybe on an airplane, 0:00:11.640 --> 0:00:14.080 and they'll ask me the inevitable question, what do you 0:00:14.120 --> 0:00:16.760 do for a living? When I say that I'm a physicist, 0:00:16.840 --> 0:00:21.279 I often get the reaction I hate physics so much math, 0:00:21.520 --> 0:00:23.840 And that makes me think, if it's the math you 0:00:23.880 --> 0:00:26.440 didn't like, then hey, hate the math, but you can 0:00:26.440 --> 0:00:29.040 still love the physics. But of course the two are 0:00:29.200 --> 0:00:32.400 closely linked. You can't love Shakespeare if you hate the 0:00:32.440 --> 0:00:36.160 English language, and that of course makes us wonder why 0:00:36.560 --> 0:00:40.320 math and physics are so intertwined. I mean, if people 0:00:40.479 --> 0:00:44.840 can actually enjoy Shakespeare and other languages, then it has 0:00:44.920 --> 0:00:48.479 something about it that's transcending the original English words. Is 0:00:48.520 --> 0:00:51.760 it possible for physics to transcend math or are they 0:00:51.840 --> 0:00:55.280 shackled to each other with math woven deeply into the 0:00:55.280 --> 0:01:14.280 fabric of physics. Hi, I'm Daniel. I'm a particle physicist 0:01:14.360 --> 0:01:17.080 and a professor at U c Irvine, and I'll admit 0:01:17.160 --> 0:01:20.920 that I love math. Some people find it confusing, but 0:01:21.000 --> 0:01:23.039 when I was a kid, I found it to be 0:01:23.200 --> 0:01:25.880 crisp and logical in a way that the rest of 0:01:25.880 --> 0:01:29.319 the world was sort of fuzzy and complicated. Like people 0:01:29.400 --> 0:01:32.880 For example, people are complicated and hard to understand when 0:01:32.880 --> 0:01:34.480 you're a kid. Are they going to be mean to 0:01:34.520 --> 0:01:36.240 you or nice to you if you sit next to 0:01:36.240 --> 0:01:38.640 them at lunch. It's hard to predict from one day 0:01:38.680 --> 0:01:41.360 to the next. But the rules of math were cast 0:01:41.480 --> 0:01:44.400 in iron two plus two equals four every day of 0:01:44.440 --> 0:01:47.480 the week, and if you know the rules, the answer follows. 0:01:47.880 --> 0:01:51.120 Math is reliable, it's predictable, and that's what led me 0:01:51.200 --> 0:01:55.240 to physics, the ability to use math to understand and 0:01:55.360 --> 0:01:59.240 predict the universe. And welcome to the podcast Daniel and 0:01:59.320 --> 0:02:01.920 Jorge ex Playing the Universe, where we do a deep 0:02:02.000 --> 0:02:05.000 dive into the rules of the universe, doing our best 0:02:05.040 --> 0:02:07.960 to reveal what science has uncovered in terms of the 0:02:08.040 --> 0:02:10.840 machinations of the universe, and laying out for you what 0:02:11.040 --> 0:02:14.920 science and scientists are still puzzling over. We tear the 0:02:15.000 --> 0:02:17.480 universe down to its smallest bits and put them back 0:02:17.520 --> 0:02:21.720 together to explain how things work and expose our remaining ignorance. 0:02:21.800 --> 0:02:23.880 And we can do that, thinks, in no small part 0:02:24.040 --> 0:02:27.800 to the power of math. Math underlies all of the 0:02:27.880 --> 0:02:31.359 stories that we tell ourselves about the universe. If you 0:02:31.400 --> 0:02:33.240 want to predict the path of a baseball, and you 0:02:33.280 --> 0:02:36.600 can use math to calculate its trajectory and tell you 0:02:36.840 --> 0:02:39.720 where it will land. Physics can predict the future, but 0:02:39.800 --> 0:02:43.640 the language and the machinery of it are all mathematical. 0:02:43.760 --> 0:02:46.400 There are times when our intuition fails us when the 0:02:46.480 --> 0:02:50.040 universe does things that don't align with our expectations, like 0:02:50.360 --> 0:02:53.560 in the case of quantum mechanics, which tries to describe 0:02:53.680 --> 0:02:57.880 tiny little objects that follow rules that seem alien to us, 0:02:58.440 --> 0:03:02.919 but they do follow rules, and those rules are mathematical 0:03:03.240 --> 0:03:07.440 described by equations. So when we've lost our intuition, we 0:03:07.480 --> 0:03:10.880 can close our intuitive eyes and just follow the math 0:03:10.960 --> 0:03:13.720 and trust that it will guide us to the right 0:03:13.880 --> 0:03:18.120 physical answer. On this podcast, we actually usually are trying 0:03:18.160 --> 0:03:21.839 to do the opposite, to avoid the mathematics, and that's 0:03:21.880 --> 0:03:24.920 partly because it's an audio program not well suited to 0:03:25.040 --> 0:03:28.919 equations or geometric sketches, but also because we are trying 0:03:28.960 --> 0:03:32.239 to feed your intuition about how the universe works, to 0:03:32.720 --> 0:03:35.920 strip away the opaqueness of math and make it all 0:03:36.000 --> 0:03:38.480 makes sense to you. And I'll admit that this is 0:03:38.720 --> 0:03:41.920 sometimes a struggle to accomplish. For some of the deeper 0:03:41.960 --> 0:03:46.120 concepts and physics like gauge invariants. The way I learned 0:03:46.160 --> 0:03:49.360 them is mathematical, and the way that I understand them 0:03:49.480 --> 0:03:53.920 is mathematical, which has become part of my intuition. So 0:03:54.040 --> 0:03:57.280 it's not always easy to know how to translate those 0:03:57.320 --> 0:04:00.800 concepts into pure intuition and talk about the without math, 0:04:00.880 --> 0:04:03.760 but you know, finding ways to talk about them intuitively 0:04:04.160 --> 0:04:08.160 has also led me to a deeper understanding of the ideas. 0:04:08.640 --> 0:04:12.360 That's one of the underappreciated joys of teaching. It forces 0:04:12.400 --> 0:04:15.120 you to strengthen your own knowledge. But to me, it 0:04:15.200 --> 0:04:19.400 raises a really interesting question. Is it possible actually to 0:04:19.640 --> 0:04:24.640 divorce physics from math? Is math truly the language of 0:04:24.680 --> 0:04:28.520 physics or is it just useful like a shorthand notation. 0:04:29.040 --> 0:04:31.839 Is math the language of the universe itself or is 0:04:31.880 --> 0:04:34.719 it just the way the humans like to think about it. 0:04:34.839 --> 0:04:42.520 So on today's episode, we'll be asking the question why 0:04:42.880 --> 0:04:46.479 is math so important for physics? A few weeks ago 0:04:46.520 --> 0:04:49.760 we talked to a philosopher of mathematics, Professor Mark cole Evan, 0:04:50.000 --> 0:04:54.559 about whether the universe was mathematical. Today, I've invited someone 0:04:54.560 --> 0:04:56.880 from the other side of the issue to join us. 0:04:57.120 --> 0:05:00.520 We'll talk today to a working mathematicians some and who 0:05:00.560 --> 0:05:04.039 spends his day's building mathematical tools and using them to 0:05:04.120 --> 0:05:07.800 describe the patterns and structures of the universe. So it's 0:05:07.839 --> 0:05:11.200 my pleasure to welcome Professor Steve Strogatz. He's the Jacob 0:05:11.240 --> 0:05:15.280 gould Sherman Professor of Applied Mathematics at Cornell University, having 0:05:15.279 --> 0:05:17.880 taught previously at M I T and having earned a 0:05:17.880 --> 0:05:20.520 PhD in math from Harvard, So those are some pretty 0:05:20.520 --> 0:05:24.120 impressive credentials. But he's also an expert in applying math 0:05:24.200 --> 0:05:28.120 to the real world, including understanding the math of firefly swarms, 0:05:28.200 --> 0:05:31.440 choruses of chirping crickets, and the wobbling of bridges. He's 0:05:31.480 --> 0:05:34.440 also a well known podcaster, host of the podcasts The 0:05:34.520 --> 0:05:36.600 Joy of X and The Joy of Why, both of 0:05:36.640 --> 0:05:40.159 which I highly recommend, as well as a prolific author. 0:05:40.560 --> 0:05:44.040 One of his recent books is Infinite Powers, How Calculus 0:05:44.040 --> 0:05:46.919 Reveals the Secrets of the Universe, the book I recently 0:05:46.920 --> 0:05:50.000 read and thoroughly enjoyed, and which inspired me to invite 0:05:50.040 --> 0:05:53.240 Steve on the podcast to talk about calculus, infinity, and 0:05:53.279 --> 0:05:57.080 the deep relationship between physics and math. Steve, Welcome to 0:05:57.120 --> 0:05:59.680 the podcast and thank you very much for joining us. 0:05:59.760 --> 0:06:02.359 Thank it's a lot, Daniel. It's a great pleasure to 0:06:02.360 --> 0:06:04.080 be with you. That's going to be fun. It's a 0:06:04.120 --> 0:06:06.160 treat to have you here as I've been very much 0:06:06.200 --> 0:06:08.920 enjoying listening to your podcast series and reading your book. 0:06:08.920 --> 0:06:10.680 And so I'd like to start by asking you a 0:06:10.760 --> 0:06:14.320 question I've heard you ask several of your guests about definitions. 0:06:14.320 --> 0:06:16.920 Your book is about calculus, a word that a lot 0:06:17.000 --> 0:06:19.560 of people have heard but might not really know what 0:06:19.600 --> 0:06:23.359 it means. Can you define for us what is calculus? Sure, 0:06:23.520 --> 0:06:25.919 let's try it in a sequence of definitions, and you 0:06:25.920 --> 0:06:28.000 could stop me when I get too detailed. So if 0:06:28.040 --> 0:06:29.760 I were giving it to you in one word, I 0:06:29.800 --> 0:06:33.680 would say, it's the mathematics of change. That's the keyword change. 0:06:33.760 --> 0:06:36.720 If we want to go a little more into it, 0:06:36.720 --> 0:06:40.800 it's the mathematics of continuous change, and especially things that 0:06:40.839 --> 0:06:43.040 are changing at a changing rate. So you say it's 0:06:43.040 --> 0:06:46.400 the mathematics of change. What exactly is changing there? Like 0:06:46.480 --> 0:06:48.599 if I just want to describe how a ball is 0:06:48.640 --> 0:06:51.440 moving through the air, what exactly is changing about the 0:06:51.440 --> 0:06:53.880 ball's motion? So in that case, what's changing is the 0:06:53.920 --> 0:06:56.919 position of the ball, or also possibly the speed of 0:06:56.960 --> 0:07:00.000 the ball. So your listeners will remember from high schoo 0:07:00.000 --> 0:07:03.640 Well algebra, we do problems about change and motion. That 0:07:04.040 --> 0:07:07.120 gets summed up in the mantra. Distance equals rate times time, 0:07:07.240 --> 0:07:10.000 and so that's motion at a steady speed or at 0:07:10.000 --> 0:07:14.160 a steady velocity. And you can handle that with algebra. 0:07:14.320 --> 0:07:17.520 It's just a matter of multiplication. Distance equals rate times time. 0:07:17.560 --> 0:07:20.520 The rate is the speed, and you're driving sixty miles 0:07:20.520 --> 0:07:23.880 an hour. For an hour, you're gonna go sixty miles. Okay, 0:07:23.920 --> 0:07:27.360 So in that case the distance is changing, position of 0:07:27.400 --> 0:07:29.640 the car on the highway is changing, but the speed 0:07:29.680 --> 0:07:31.480 of the car is not changing. We said it was 0:07:31.680 --> 0:07:34.600 a steady sixty miles an hour. And so at the 0:07:34.680 --> 0:07:38.680 time of Isaac Newton or even Johannes Kepler or Galileo, 0:07:38.840 --> 0:07:42.760 scientists started to become very interested in motion that was 0:07:42.840 --> 0:07:46.080 not just simple motion at a constant speed. You know, 0:07:46.160 --> 0:07:48.480 in connection with the things you mentioned dropping a ball, 0:07:48.760 --> 0:07:52.000 the apocryphal or maybe true dropping the cannon ball off 0:07:52.080 --> 0:07:55.240 the leaning tower of pizza from Galileo, certainly Kepler with 0:07:55.320 --> 0:07:57.680 thinking about the motions of the planets. In all of 0:07:57.680 --> 0:08:00.440 those cases there were things that were changing. I mean 0:08:00.480 --> 0:08:02.760 we should also keep in mind with the planets. Another 0:08:02.800 --> 0:08:06.120 thing that can change is direction. So instead of motion 0:08:06.200 --> 0:08:08.560 in a straight line, if you have an orbit, then 0:08:08.800 --> 0:08:11.800 the direction of the planet as it's moving is changing. 0:08:11.840 --> 0:08:14.120 It's curving as it's going around the Sun. And so 0:08:14.240 --> 0:08:17.520 geometry is a big part of calculus too, when we 0:08:17.560 --> 0:08:20.440 start to deal with curved shapes as opposed to shapes 0:08:20.680 --> 0:08:23.880 made of straight lines or or planes. So in this case, 0:08:23.920 --> 0:08:27.040 do you feel historically like physics was in the lead 0:08:27.240 --> 0:08:30.400 or mathematics. I mean, people have been thinking about things 0:08:30.440 --> 0:08:34.120 that were moving and changing for thousands of years. But 0:08:34.160 --> 0:08:36.600 calculus is just a few hundred years old. Was it 0:08:36.640 --> 0:08:40.480 invented to solve a particularly difficult problem or did it 0:08:40.559 --> 0:08:43.640 appear in the minds of intelligent people and then allow 0:08:43.760 --> 0:08:46.400 us to solve problems that had been standing for thousands 0:08:46.440 --> 0:08:48.880 of years. That's interesting that you say it's only a 0:08:48.880 --> 0:08:53.760 few hundred years old. Most historians and certainly most scientists 0:08:53.800 --> 0:08:56.880 would say, yeah, calculus is from the middle sixteen hundreds, 0:08:56.920 --> 0:09:00.680 from Isaac Newton and Gottfried Wilhelm Leibnitz. But I don't 0:09:00.720 --> 0:09:05.400 personally I want to endorse that position because I think 0:09:05.440 --> 0:09:07.440 we can see if you want. I mean, I don't 0:09:07.480 --> 0:09:11.040 equibble about definitions, but there are definitely ideas of calculus. 0:09:11.480 --> 0:09:15.319 Almost two thousand years earlier in the work of Archimedes 0:09:15.600 --> 0:09:18.320 in Syracuse, in what was at the time the Greek 0:09:18.520 --> 0:09:23.839 Empire Um to fifty BC. Archimedes is calculating volumes of 0:09:23.960 --> 0:09:27.760 solids with curved faces, or also areas under a parabola. 0:09:28.000 --> 0:09:29.840 Or he's the one that gives us the volume of 0:09:29.840 --> 0:09:32.280 a sphere or the surface area of a sphere. Those 0:09:32.320 --> 0:09:35.720 are all calculus problems. We teach those today in calculus 0:09:35.720 --> 0:09:38.880 when we're teaching students about integrals, which is a generalization 0:09:38.920 --> 0:09:41.360 of the idea of area and volume. And so he's 0:09:41.440 --> 0:09:44.600 totally doing calculus in two fifty BC. In fact, he's 0:09:44.640 --> 0:09:47.840 doing the harder part of calculus, integral calculus. But we 0:09:47.880 --> 0:09:52.000 don't usually call it calculus because of I don't know why. Actually, 0:09:52.040 --> 0:09:54.640 I mean, I think it is calculus. So back to 0:09:54.720 --> 0:09:57.600 your question, what is calculus, I mean, another way of 0:09:57.640 --> 0:10:01.120 talking about it is it's the systematic use of infinity 0:10:01.160 --> 0:10:06.480 and infinitesimals to solve problems about curved shapes, about motion 0:10:06.600 --> 0:10:10.200 at a non constant speed, and about anything else that's 0:10:10.280 --> 0:10:13.240 changing in a non constant way. It could be amount 0:10:13.280 --> 0:10:17.560 of virus in your bloodstream if you have HIV. It 0:10:17.600 --> 0:10:20.679 could be a population you know, of the Earth going up. 0:10:20.720 --> 0:10:23.400 Any all of these things are grist for calculus. So 0:10:23.440 --> 0:10:26.440 why is it that infinity is such an important and 0:10:26.600 --> 0:10:30.040 powerful concept that lets us now tackle new problems that 0:10:30.120 --> 0:10:32.120 we couldn't tackle before. I want to think about it 0:10:32.120 --> 0:10:34.240 in terms of like the ball flying through the air, 0:10:34.679 --> 0:10:37.320 and you we're talking about something changing about its motion, 0:10:37.600 --> 0:10:40.560 but you're also referring to like calculating the volume of spheres. 0:10:40.840 --> 0:10:43.920 What's changing in that aspect? Why do we need infinity 0:10:44.120 --> 0:10:46.480 to help us tackle these problems? I mean, the sphere 0:10:46.520 --> 0:10:49.600 is not infinitely big, the ball is not moving infinitely fast. 0:10:49.720 --> 0:10:53.079 What exactly does infinity come into play? The main point 0:10:53.200 --> 0:10:57.959 is probably infinitesimals rather than infinity. So infinitesimals. Let us 0:10:58.080 --> 0:11:02.040 pretend that us fear is made up of flat pieces. 0:11:02.559 --> 0:11:05.240 May be easier to visualize with a circle. Some of 0:11:05.480 --> 0:11:07.680 your listeners will have probably played this game. If you 0:11:07.720 --> 0:11:09.960 put a bunch of dots on a circle and connect 0:11:09.960 --> 0:11:12.720 them with straight lines, it almost looks like a you know, 0:11:12.720 --> 0:11:15.319 it'll make a polygon for instance, you could picture putting 0:11:15.360 --> 0:11:17.880 four equally space points on a circle and connect them, 0:11:17.880 --> 0:11:20.360 they'll make a square. If you put eight, then you're 0:11:20.360 --> 0:11:22.959 making an octagon. You know, the more points you put, 0:11:23.000 --> 0:11:26.120 the more it starts to look like a circle. And 0:11:26.600 --> 0:11:30.120 from ancient times people had this intuition that a circle 0:11:30.240 --> 0:11:33.160 is kind of like an infinite polygon. It's got infinitely 0:11:33.200 --> 0:11:37.960 many corners, it's connected by sides that are infinitesimally small. 0:11:38.120 --> 0:11:40.880 Now that doesn't seem right because we think of a 0:11:40.920 --> 0:11:43.920 circle as perfectly smooth it doesn't have any corners at all. 0:11:44.040 --> 0:11:46.040 But in a certain sense, it's the limit of a 0:11:46.120 --> 0:11:49.040 polygon as you take more and more points on the 0:11:49.120 --> 0:11:52.480 polygon at the corners and more and more sides. And 0:11:52.520 --> 0:11:55.280 so that was the key insight that Archimedes had, that 0:11:55.440 --> 0:11:58.720 you could calculate the area of a circle the formula 0:11:58.800 --> 0:12:00.960 we all learn in high school I R squared. He's 0:12:01.000 --> 0:12:02.880 the first one to really prove that, and he did 0:12:02.880 --> 0:12:05.320 it by thinking in this calculus way, by looking at 0:12:05.320 --> 0:12:08.880 the limit of polygons. So similarly, in the case of Galileo, 0:12:09.000 --> 0:12:12.320 in the motion of say javelin or something thrown that's 0:12:12.320 --> 0:12:15.600 going to execute parabolic flight to make the problem easier. 0:12:15.679 --> 0:12:18.079 Gal Well, actually Galileo didn't really have this idea, but 0:12:18.200 --> 0:12:21.440 later in Newton we would think of the parabola as 0:12:21.520 --> 0:12:26.720 made up of infinitely many, infinitesimally small excursions along the 0:12:26.800 --> 0:12:30.520 path that are basically straight lines in the particle or 0:12:30.520 --> 0:12:32.760 the javelin is moving at a constant speed for that 0:12:32.840 --> 0:12:36.040 infinitesimal amount of time, So it breaks the problem down 0:12:36.080 --> 0:12:38.480 into something that we already know how to solve. Everything 0:12:38.520 --> 0:12:42.320 becomes distance equals right times time again, except only over 0:12:42.360 --> 0:12:45.560 an infinitesimal segment. So you solve a problem you can't 0:12:45.600 --> 0:12:48.840 solve by turning it into an infinite number of problems 0:12:48.920 --> 0:12:52.480 that you can solve. Bingo, you've really encapsulated the heart 0:12:52.480 --> 0:12:55.559 of calculus in that sense in infinite powers. I called 0:12:55.600 --> 0:12:59.680 that the infinity principle that to solve any difficult problem 0:12:59.800 --> 0:13:03.440 in evolving curve shapes or these complicated motions, if you 0:13:03.520 --> 0:13:08.679 reconceptualize it as an infinite number of smaller, simpler problems 0:13:08.679 --> 0:13:11.199 in which you have straight lines or motion at a 0:13:11.240 --> 0:13:15.000 constant speed, you can solve incredibly hard and important problems 0:13:15.040 --> 0:13:16.760 with this trick. The only problem is you have to 0:13:16.800 --> 0:13:20.240 somehow put all those infinite testimals back together again to 0:13:20.400 --> 0:13:23.839 reconstitute the original motion or the original shape. And that's 0:13:23.880 --> 0:13:26.720 the hard part of calculus. The subdivision part is easy. 0:13:26.760 --> 0:13:29.280 It's the reassembly part that's hard, right, And so it's 0:13:29.320 --> 0:13:31.840 fascinating to me that infinity sort of appears in two 0:13:31.840 --> 0:13:34.679 places there, one as you chop it into little pieces, 0:13:35.040 --> 0:13:37.640 and then again as you put it back together. And 0:13:37.679 --> 0:13:40.839 to me, it's fascinating because the infinity appears in only 0:13:40.880 --> 0:13:44.520 the intermediate stages. Like the ball doesn't have infinite velocity 0:13:44.600 --> 0:13:48.760 or infinite acceleration or infinite anything. But we've used infinity 0:13:48.800 --> 0:13:52.880 in calculating. It's very non infinite motion. And so it's 0:13:52.880 --> 0:13:56.320 fascinating to me that infinity is such a powerful mathematical tool, 0:13:56.400 --> 0:14:00.000 yet we don't actually observe it in nature really very often. 0:14:00.120 --> 0:14:02.480 Or some people might say, ever, that's really a great 0:14:02.520 --> 0:14:04.240 point you could. I mean, if we were doing this 0:14:04.320 --> 0:14:07.560 on video, your listeners would see me smiling. I really 0:14:07.600 --> 0:14:10.280 like that. It's almost like in those old cartoons with 0:14:10.400 --> 0:14:13.480 the you know, enter stage left and exit stage right, 0:14:13.640 --> 0:14:17.000 that infinity comes onto the stage and the infinitesimals, but 0:14:17.120 --> 0:14:20.000 only as you sort of say, like an apparatus. It 0:14:20.000 --> 0:14:22.000 it lets us solve the problem, but it's sort of 0:14:22.040 --> 0:14:24.920 not really there, it's not real. In physics, we're often 0:14:24.920 --> 0:14:27.720 seeing infinities in the final answer as a sign of 0:14:27.720 --> 0:14:32.120 failure writing particle physics, prediction of infinity is unphysical. You 0:14:32.200 --> 0:14:35.920 can't have infinite probabilities for some outcome and quantum mechanics, 0:14:36.000 --> 0:14:38.000 or you can't have an infinite force on a particle. 0:14:38.120 --> 0:14:40.960 And in general relativity we think of a prediction of 0:14:41.000 --> 0:14:44.800 a singularity infinite density as the breakdown of the physical theory. 0:14:44.800 --> 0:14:47.840 We try to avoid infinities. We hide them under renormalization 0:14:47.880 --> 0:14:50.440 whatever possible. So then my question to you is in 0:14:50.440 --> 0:14:53.520 your mind are these infinities real? I mean, do they 0:14:53.600 --> 0:14:57.120 just exist in the intermediate steps of the mathematical methods 0:14:57.160 --> 0:14:59.000 we're using. Are they only in our minds? Are they 0:14:59.320 --> 0:15:03.080 half finished calculation? Or is this something real about the 0:15:03.160 --> 0:15:07.200 universe that calculus is capturing, This smooth and infinitely varying 0:15:07.440 --> 0:15:10.120 motion of a ball or changing of the velocity of 0:15:10.120 --> 0:15:13.320 a planet. Is infinity real? Is it part of our minds? 0:15:13.320 --> 0:15:15.840 Such a great deep question. I don't even know what 0:15:15.880 --> 0:15:18.440 I'm gonna say to the answer. I mean, I've been 0:15:18.480 --> 0:15:21.280 thinking about this for forty fifty years, and I still 0:15:21.320 --> 0:15:23.640 don't really know what the answer is. I I mean, 0:15:23.720 --> 0:15:29.000 the principal person in me, the philosophically tenable person, wants 0:15:29.040 --> 0:15:32.120 to say it's not real, it's a fiction, it's a 0:15:32.200 --> 0:15:35.320 useful device. Let's just try to make that argument first, 0:15:35.360 --> 0:15:38.600 before the more wild eyed person in me makes the 0:15:38.640 --> 0:15:43.200 counter argument. So the rational person would say, yeah, I mean, 0:15:43.280 --> 0:15:47.160 our from our best understanding of physics today, there are 0:15:47.200 --> 0:15:52.760 no infinitesimals. You can't subdivide matter arbitrarily finely. That's the 0:15:52.760 --> 0:15:55.880 whole concept of atoms, those things which are indivisible. You 0:15:55.920 --> 0:15:59.440 guys tell us, I say you. The physicists tell us 0:15:59.480 --> 0:16:02.320 that there is even a smallest amount of time and 0:16:02.440 --> 0:16:06.000 space that is referred to as the Planck scale, the smallest. 0:16:06.280 --> 0:16:08.760 You know, we don't really, of course, I believe understand 0:16:08.880 --> 0:16:12.920 how to unify quantum theory yet with general relativity. There 0:16:12.920 --> 0:16:15.720 are candidates. But anyway, the cool thing is that, just 0:16:15.800 --> 0:16:19.240 on dimensional grounds, if you look at the fundamental constants 0:16:19.280 --> 0:16:22.280 like the speed of light and planks constant that governs 0:16:22.360 --> 0:16:27.040 quantum phenomena, and Newton's gravitational constant for the strength of gravity, 0:16:27.080 --> 0:16:29.760 those can only be put together in one way to 0:16:29.880 --> 0:16:32.360 make a unit of length. That's the Planck length, and 0:16:32.360 --> 0:16:35.760 it's about ten to the minus thirty five meters, and 0:16:35.800 --> 0:16:39.360 that's sort of the smallest conceivable distance that has any 0:16:39.360 --> 0:16:42.600 physical meaning, wouldn't you say, whatever the theory ends up being, 0:16:42.680 --> 0:16:46.040 that's certainly an attempt to describe what might be the 0:16:46.120 --> 0:16:50.960 shortest distance. In my view, it's a not very clever attempt, 0:16:51.040 --> 0:16:53.680 but also the most clever attempt we have, and we 0:16:53.720 --> 0:16:55.880 have no better way to do it, and so this 0:16:55.960 --> 0:16:57.560 is the only thing. You know, we do this all 0:16:57.600 --> 0:16:59.480 the time in physics. We say, let's start with the 0:16:59.480 --> 0:17:01.480 most night idea and then try to build on it, 0:17:01.520 --> 0:17:04.200 and we're sort of still there with the shortest distance, 0:17:04.400 --> 0:17:05.840 you know. I mean, if you try to estimate, like 0:17:06.119 --> 0:17:09.000 how many candy bars a person eats in a year, 0:17:09.160 --> 0:17:11.800 just by combining various quantities with the right units, you 0:17:11.880 --> 0:17:14.080 might get an answer that's off by a factor a thousand, 0:17:14.240 --> 0:17:16.200 and that would feel like a pretty wrong answer in 0:17:16.240 --> 0:17:17.600 the case of the plant length, I think, which is 0:17:17.640 --> 0:17:21.040 sort of groping generally for where in the space that 0:17:21.119 --> 0:17:23.720 answer might be. But I think the point you're making 0:17:23.800 --> 0:17:27.080 is we have the sense that the universe is discreete 0:17:27.280 --> 0:17:29.960 and not continuous. Equantom mechanics tells us that you can't 0:17:30.000 --> 0:17:34.680 infinitely chop up the universe, and therefore mathematics of calculus 0:17:34.720 --> 0:17:38.679 that assumes that might not actually be describing what's happening 0:17:38.720 --> 0:17:41.120 in the universe. Fine, I mean, I take your point 0:17:41.160 --> 0:17:43.359 that we don't know that the plant length of tend 0:17:43.359 --> 0:17:45.159 to the minus thirty five is right. There could be 0:17:45.200 --> 0:17:47.880 factors of a thousand in one direction or another or more. 0:17:48.000 --> 0:17:50.479 I mean, we don't really know what the pre factor is, 0:17:50.640 --> 0:17:54.960 So okay, I accept that. Nevertheless, as you say, also 0:17:55.000 --> 0:17:57.480 from quantum theory, we have reason to think that nature 0:17:57.560 --> 0:18:01.040 is fundamentally discreet in every aspect, whether it's matter or 0:18:01.080 --> 0:18:03.199 space or time. And so if that turns out to 0:18:03.200 --> 0:18:07.760 be correct, that will mean that real numbers are not real. 0:18:08.280 --> 0:18:11.440 Real numbers are the things that we use in calculus 0:18:11.480 --> 0:18:13.920 all day long. There are numbers that have infinitely many 0:18:14.040 --> 0:18:17.080 digits after the decimal point, like pie. Right, people know, 0:18:17.200 --> 0:18:19.800 you can keep calculating and you'll never know all the 0:18:19.800 --> 0:18:22.320 digits of pie because there's infinitely many of them. Is 0:18:22.359 --> 0:18:24.639 that real? Like? In fact, you could ask that question 0:18:24.640 --> 0:18:27.119 about all of math, our circles real, you'd have to 0:18:27.119 --> 0:18:29.439 say no. Circles are not real either, because as you 0:18:29.560 --> 0:18:31.880 zoom in on them, you know what's there. It's all 0:18:31.960 --> 0:18:35.359 jiggily and there's fluctuations of the sub atomic particles. So 0:18:35.400 --> 0:18:38.720 there's no material circle in the real world. But nevertheless, 0:18:38.800 --> 0:18:41.520 going back to Plato or others, we can think about 0:18:41.520 --> 0:18:44.080 perfection in our minds. We can think about the concept 0:18:44.119 --> 0:18:46.199 of a perfect circle, and we can think about the 0:18:46.240 --> 0:18:49.440 concept of pie, and even the concept of infinity. And 0:18:49.720 --> 0:18:52.520 this is the uncanny part. These things are not real 0:18:52.760 --> 0:18:56.000 from the standpoint of physics, yet they give us our 0:18:56.040 --> 0:18:59.879 best understanding of the physical universe that we've achieved in 0:19:00.160 --> 0:19:02.320 as a species. And that's just a fact. I mean, 0:19:02.320 --> 0:19:04.800 that's just a historical fact. That calculus based on this 0:19:04.920 --> 0:19:10.520 fiction of infinitely subdivisible quantities works pretty darn well. While 0:19:10.520 --> 0:19:12.320 I was walking my dog this morning, I tried to 0:19:12.359 --> 0:19:14.680 figure out how many orders of magnitude if you tell 0:19:14.760 --> 0:19:17.280 us the universe is about ten to the twenty five 0:19:17.920 --> 0:19:21.040 meters big the visible universe, that's the estimate I looked 0:19:21.160 --> 0:19:24.000 according to When I asked Syria on my iPhone, she said, 0:19:24.000 --> 0:19:26.680 the visible universe tend to the twenty five ms, and 0:19:27.000 --> 0:19:30.560 the typical scale of a hydrogen atom is something like 0:19:30.640 --> 0:19:33.280 ten to the minus ten meters, So you've got thirty 0:19:33.280 --> 0:19:37.280 five orders of magnitude very well described by calculus, all 0:19:37.280 --> 0:19:40.320 the way from the Schroedinger equation at the lowest scale 0:19:40.320 --> 0:19:44.200 to general relativity at the highest scale, all built on calculus. 0:19:44.240 --> 0:19:47.359 So it's kind of capturing the truth. Okay, you couldn't. No, 0:19:47.560 --> 0:19:50.280 it's gonna start screwing up at the scale of quantum gravity, 0:19:50.320 --> 0:19:52.800 whatever that ends up being, we think. But I think 0:19:52.840 --> 0:19:56.320 that's a pretty good good notch in the belt of 0:19:56.400 --> 0:20:00.000 calculus that it works over such a vast range of scales. Absolutely, 0:20:00.080 --> 0:20:03.280 it's incredible because it powers not just you know, quantum 0:20:03.320 --> 0:20:06.119 field theory, which is full of integrals, but also general 0:20:06.160 --> 0:20:09.439 relativity and talking about you know, galaxies and black holes. 0:20:09.520 --> 0:20:11.960 All Right, I have a lot more questions about math 0:20:12.040 --> 0:20:14.639 and physics for our guest, but first let's take a 0:20:14.760 --> 0:20:30.080 quick break. Okay, we're back and we're talking with Professor 0:20:30.160 --> 0:20:34.480 Steve stroke Gats about why math is so important for physics. 0:20:34.840 --> 0:20:38.040 And it makes me think about the connection in physics 0:20:38.040 --> 0:20:41.160 and math of the concept of emergence. You know, some 0:20:41.359 --> 0:20:44.040 simple behavior at the scale of like me and you 0:20:44.280 --> 0:20:47.800 are a bowl of soup that neatly and compactly summarizes 0:20:48.119 --> 0:20:51.160 the almost infinite details going on underneath. I mean, even 0:20:51.200 --> 0:20:54.760 if you don't believe that a ball is infinitely divisible 0:20:54.760 --> 0:20:56.879 into bits, we know this is a huge number of 0:20:56.920 --> 0:20:59.200 bits with a huge number of details. But it's almost 0:20:59.280 --> 0:21:02.720 like the equations and the simplicity of calculus, the parabolic 0:21:02.800 --> 0:21:05.440 motion of that ball emerge somehow from all of these 0:21:05.440 --> 0:21:08.399 infinite testimals doing their bit together to tell a fairly 0:21:08.440 --> 0:21:11.199 simple story. And in physics, to me, it's something of 0:21:11.200 --> 0:21:14.680 a mystery, like why this happens? Why can we describe 0:21:14.680 --> 0:21:17.960 the universe in simple mathematical stories when we know that 0:21:18.000 --> 0:21:21.520 the details are crazy and gory in intense and calculus 0:21:21.600 --> 0:21:24.000 really wraps this up. So do you have an instinct 0:21:24.040 --> 0:21:26.440 or an intuition, or even a guest for why it's 0:21:26.480 --> 0:21:31.160 possible to wrangle these almost infinities into fairly simple stories 0:21:31.160 --> 0:21:34.880 that make sense to our human minds. That's another tough one. 0:21:37.280 --> 0:21:39.040 I don't mean to put you on this spot. Well, 0:21:39.320 --> 0:21:42.440 they're all great questions, they're so deep. My first instinct 0:21:42.440 --> 0:21:46.199 to that one after I don't know is Is it 0:21:46.280 --> 0:21:49.439 a question of the measuring apparatus meaning us that we 0:21:49.560 --> 0:21:52.880 happened to be macroscopic and so for us space has 0:21:52.920 --> 0:21:56.040 perceived as smooth, and time as smooth and so on. 0:21:56.200 --> 0:22:00.440 But if we were playing scale creatures, we wouldn't. Of course, 0:22:00.440 --> 0:22:02.359