WEBVTT - Is infinity real or just mathematical? 0:00:08.760 --> 0:00:11.319 Hey, Katie, how many pets do you have living with 0:00:11.360 --> 0:00:14.280 you at home? That depends on your definition. Does it 0:00:14.400 --> 0:00:17.280 include all living creatures in my house? Does it include 0:00:17.280 --> 0:00:20.720 the mites in my eyelashes? Maybe not, like all the 0:00:20.800 --> 0:00:23.439 microobs and the tardy grades and all the bacteria in 0:00:23.480 --> 0:00:26.759 your gut? What about my husband? That's your call. Depends 0:00:26.760 --> 0:00:30.880 on your relationship, personal choice. Okay, then I've got a 0:00:30.960 --> 0:00:35.360 dog named Cookie, I've got to fish. I'm going to 0:00:35.479 --> 0:00:39.120 name them right now, called Bill and Head. So three 0:00:39.280 --> 0:00:41.879 or four, depending on the definition of my husband. All right, 0:00:41.920 --> 0:00:43.600 well that's more than I've got. And I got a 0:00:43.680 --> 0:00:46.400 question for you about pets. At what point do you 0:00:46.479 --> 0:00:49.680 say you have too many pets? I think that's the 0:00:49.680 --> 0:00:52.640 wrong way to look at it. Once you have, you know, 0:00:53.520 --> 0:00:56.720 two or three, like one more, there's no difference, right, 0:00:56.760 --> 0:00:59.760 It's the cat principle, like once you have two cats, 0:00:59.840 --> 0:01:03.440 there's no difference between two cats for cats tin cats. 0:01:03.680 --> 0:01:06.960 I think that sounds dangerous. You're gonna mathematically induce yourself 0:01:07.000 --> 0:01:10.160 to having an infinite number of cats. Exactly if you 0:01:10.200 --> 0:01:13.039 already have infinite pets, you can add more and more 0:01:13.160 --> 0:01:31.800 and still have infinite pets. Hi, I'm Daniel, I'm a 0:01:31.840 --> 0:01:35.800 particle physicist, and I have an infinite number of questions 0:01:35.880 --> 0:01:39.600 about the nature of our universe. And I am Katie Golden. 0:01:40.000 --> 0:01:43.480 I like pets, but I am not a particle physicist, 0:01:43.640 --> 0:01:48.560 and I still have infinite questions about the universe and 0:01:48.640 --> 0:01:50.960 about how many cats I can have and get away with. 0:01:51.320 --> 0:01:54.040 How many cats can there be in the universe? Could 0:01:54.120 --> 0:01:57.720 we have an infinite universe field with infinite cats. I 0:01:57.760 --> 0:02:01.840 think that if you replace the sort of building blocks 0:02:01.840 --> 0:02:04.280 of the universe with cats, it would be a lot cuter. 0:02:05.520 --> 0:02:07.640 But yeah, I come to the show with a little 0:02:07.640 --> 0:02:11.320 bit of background in a psychology and evolutionary biology, and 0:02:11.360 --> 0:02:13.800 that's why I like to talk about cats all the time. 0:02:14.280 --> 0:02:18.480 And welcome then to the podcast Daniel and Jorge Explain 0:02:18.560 --> 0:02:22.280 the Universe with its Infinite variety of Cats, a podcast 0:02:22.320 --> 0:02:25.440 in which we talk about all the crazy, amazing things 0:02:25.480 --> 0:02:27.560 that are out there in the universe. Not just the 0:02:27.680 --> 0:02:30.120 things in the universe. We talk about the very nature 0:02:30.360 --> 0:02:33.080 of the universe. We talk about the universe that's in 0:02:33.160 --> 0:02:36.480 our minds, that mental model of what's going on outside, 0:02:36.639 --> 0:02:39.440 and we talk about the real, actual universe, if it 0:02:39.560 --> 0:02:43.160 actually exists, and what it means what's in it, how 0:02:43.200 --> 0:02:45.440 it all works, and we try to explain all of 0:02:45.440 --> 0:02:48.640 it to you. We tackled the biggest questions, the even 0:02:48.639 --> 0:02:52.520 bigger questions, and the absolute biggest of big questions on 0:02:52.560 --> 0:02:54.520 the podcast. I think we can do all of that 0:02:54.600 --> 0:02:56.880 in an hour. I think we can. You know, I 0:02:56.919 --> 0:02:59.120 was thinking as we were talking about this question of 0:02:59.160 --> 0:03:02.040 an infinite universe is filled with infinite cats. And you know, 0:03:02.280 --> 0:03:05.720 probably there was some like ancient Greek philosopher who first 0:03:05.800 --> 0:03:08.240 raised this question. You know how the Greek philosophers had 0:03:08.280 --> 0:03:11.480 like the craziest ideas about how the universe might work, 0:03:11.720 --> 0:03:14.360 and basically every idea that is out there, some old 0:03:14.440 --> 0:03:16.560 dude in a robe thought of his first thousands of 0:03:16.639 --> 0:03:20.480 years ago. Yeah, just Plato surrounded by his cats thinking 0:03:20.520 --> 0:03:23.560 about infinity as they're all me owing. The thing that 0:03:23.600 --> 0:03:25.600 bugs me about the Greeks is that they had so 0:03:25.639 --> 0:03:28.840 many ideas that even though only like one or two 0:03:28.880 --> 0:03:32.000 of them turned out to be correct. Like Democratus, he's 0:03:32.000 --> 0:03:33.800 the first one to name the adam and think about 0:03:33.800 --> 0:03:35.960 the universe is made of tiny bits. People give him 0:03:36.000 --> 0:03:38.320 credit for that because he had a right idea, but 0:03:38.360 --> 0:03:42.320 he had like a million wrong totally idea. If you 0:03:42.440 --> 0:03:45.000 just like spew ideas, eventually one of them is going 0:03:45.040 --> 0:03:49.200 to be right. Statistically speaking, exactly, It's like Nostradamus, like 0:03:49.280 --> 0:03:52.440 he predicted everything because he made a bunch of false 0:03:52.480 --> 0:03:55.680 predictions as well, exactly. And so if I wrote down 0:03:55.720 --> 0:03:58.120 literally every possible thing that could happen, if I made 0:03:58.160 --> 0:04:01.120 an infinite number of predictions, then one of them would 0:04:01.160 --> 0:04:03.400 be correct. I could predict the whole future of the 0:04:03.480 --> 0:04:06.480 university if I had an infinite number predictions, and one 0:04:06.520 --> 0:04:09.320 of those would be the actual theories of events of 0:04:09.360 --> 0:04:12.280 the future, right exactly. If you think about, like if 0:04:12.320 --> 0:04:15.560 you make a random noise in your head, like, the 0:04:15.640 --> 0:04:18.040 chances are someone else is going to think that same 0:04:18.160 --> 0:04:20.760 random noise in your head at some point in an 0:04:20.760 --> 0:04:24.520 infinite universe. And that's what we're gonna talk about today 0:04:24.600 --> 0:04:28.040 on the podcast. These big concepts, the concepts that are 0:04:28.040 --> 0:04:30.800 bigger than big, the ones that are so hard that 0:04:30.839 --> 0:04:34.080 we can only grapple with them in our minds. Today's 0:04:34.120 --> 0:04:38.240 episode will be focused on this concept of infinity. Katie, 0:04:38.240 --> 0:04:40.560 what do you think of when I say the word infinity? 0:04:41.200 --> 0:04:45.840 Just real big, you know, really really big, actually, you know, 0:04:46.160 --> 0:04:48.479 in a serious answer to that. When I was a 0:04:48.600 --> 0:04:53.039 kid first learning about infinity, it was almost like the 0:04:53.160 --> 0:04:56.800 concepts of infinity and nothing were very similar to me 0:04:56.880 --> 0:05:00.560 because both kind of made my brain feel fuzzy. So 0:05:00.600 --> 0:05:04.360 thinking about an infinite amount of stuff and thinking about 0:05:04.400 --> 0:05:08.160 the absence of stuff, it's like a similar feeling to me, 0:05:08.160 --> 0:05:10.760 where it's like my brain just gets this prickly static 0:05:10.760 --> 0:05:13.040 feeling when I try to think too hard about it. 0:05:13.040 --> 0:05:15.320 It's amazing that you can hold an idea in your 0:05:15.320 --> 0:05:18.960 mind that also sort of confuses your mind. I also 0:05:19.000 --> 0:05:21.640 remember learning about infinity as a kid, and you know, 0:05:21.720 --> 0:05:23.520 my dad was really into math, and so this is 0:05:23.520 --> 0:05:25.480 a very simple proof of like, all right, give me 0:05:25.520 --> 0:05:27.800 a number. I can give you a bigger number, and 0:05:27.800 --> 0:05:29.320 then I can give you a bigger number than that 0:05:29.640 --> 0:05:33.560 shows you pretty quickly that there's like no biggest number. 0:05:34.120 --> 0:05:36.599 And to me that was a crazy concept lying in 0:05:36.600 --> 0:05:39.200 bed thinking about super big numbers and imagining, well, you know, 0:05:39.240 --> 0:05:41.599 there's always one bigger than that. So it's super fun 0:05:41.640 --> 0:05:44.159 to think about, like how your brain gets stretched by 0:05:44.200 --> 0:05:46.960 the ideas that you stick inside of it. Yeah, it's like, 0:05:47.400 --> 0:05:50.719 how do you fit a concept like infinity inside of 0:05:50.720 --> 0:05:54.000 a finite brain with a finite number of neurons, so 0:05:54.000 --> 0:05:57.440 you can't It's not something that you can really conceive 0:05:57.520 --> 0:06:01.080 of without using some kind of shortcut, because our brains 0:06:01.080 --> 0:06:03.960 aren't infinite, so you can't fit infinity in there. Well, 0:06:04.000 --> 0:06:05.400 I haven't met you in person, so I don't know 0:06:05.480 --> 0:06:07.240 if your brain is infinite or not, but I definitely 0:06:07.240 --> 0:06:10.240 know that mine is finite in its capacity. But this 0:06:10.320 --> 0:06:12.320 really goes to the heart of what we're trying to 0:06:12.440 --> 0:06:15.160 do with the whole project of physics or of science, 0:06:15.160 --> 0:06:19.440 which is to understand that external universe out there and 0:06:19.560 --> 0:06:22.760 represent it in our minds. And so we do that 0:06:22.839 --> 0:06:26.360 often using mathematics. Right, we have seen so many beautiful 0:06:26.440 --> 0:06:29.120 moments when crazy things happening out there in the universe 0:06:29.200 --> 0:06:34.120 can be represented with mathematical representations in our minds. Newton's 0:06:34.160 --> 0:06:36.839 laws of motion, for example, tell us about the crazy 0:06:36.920 --> 0:06:38.960 motions of all the planets and sort of lock them 0:06:39.000 --> 0:06:42.200 into place to some mathematical rigor. So we know that 0:06:42.279 --> 0:06:46.120 mathematics is very helpful in helping us understand what's going 0:06:46.160 --> 0:06:49.400 on out there, that we can represent the craziness, the insanity, 0:06:49.440 --> 0:06:52.279 the beauty, the complexity of the chaos of the universe 0:06:52.520 --> 0:06:56.839 using simpler mathematical representations in our minds. But if we 0:06:56.920 --> 0:07:00.400 know that everything that's physical has to be mathematical, do 0:07:00.440 --> 0:07:04.000 we know that everything that's mathematical has to be physical? Like? 0:07:04.200 --> 0:07:08.360 Is everything that you can imagine physically possible? Right? Right? 0:07:08.400 --> 0:07:14.000 Because you can have something in math that's theoretical, but 0:07:14.080 --> 0:07:17.920 it's not necessarily a tangible thing that you can like 0:07:18.360 --> 0:07:21.640 observe or hold in your hands, Like can you pick 0:07:21.760 --> 0:07:25.720 up a chunk of infinity and have that in your hand? Yeah, 0:07:25.800 --> 0:07:28.680 And it seems like math is definitely sort of larger 0:07:28.720 --> 0:07:32.320 than physics. A very simple example is I can come 0:07:32.400 --> 0:07:36.720 up with a theory of particles that's wrong. It would 0:07:36.720 --> 0:07:39.679 be mathematically correct, it could work, you know, it doesn't 0:07:39.840 --> 0:07:43.680 generate nonsense or whatever. It just doesn't describe our universe. 0:07:43.960 --> 0:07:46.720 So there are more theories. There are more mathematical theories 0:07:46.720 --> 0:07:51.160 of the universe than actual physical laws that govern the universe. Right, 0:07:51.160 --> 0:07:53.200 And so far we don't even think we found the one. 0:07:53.240 --> 0:07:55.880 We have lots of really nice theories of the universe 0:07:56.000 --> 0:07:59.360 that are beautiful but are probably not correct. So it 0:07:59.440 --> 0:08:03.600 seems like not everything that's mathematical has to be physical, right. 0:08:03.920 --> 0:08:06.280 I think for me. One of the simplest examples is, 0:08:06.360 --> 0:08:09.200 just as you were saying before, things that are perfect, 0:08:09.360 --> 0:08:12.120 like a circle, like a perfect circle. You can hold 0:08:12.160 --> 0:08:15.440 that idea in your mind somehow there's a representation in 0:08:15.480 --> 0:08:18.520 your brain to this concept of a perfect idealized circle. 0:08:18.800 --> 0:08:23.920 But I've never seen a perfect circle in reality of you. Ah, well, 0:08:24.280 --> 0:08:26.840 when I eat enough food, maybe, but no, I guess 0:08:26.960 --> 0:08:30.200 that any kind of drawing or depiction of a circle, 0:08:30.240 --> 0:08:33.600 even if it looks perfect, if you look at it 0:08:33.640 --> 0:08:37.880 with like an electron microscope, you're gonna find imperfections. Yeah, 0:08:37.960 --> 0:08:40.840 and everything is made out of particles, out of atoms 0:08:40.880 --> 0:08:43.440 at a discrete chunks. Right, So you can't have a 0:08:43.880 --> 0:08:47.720 perfectly continuous circle in reality. But of course you can 0:08:47.760 --> 0:08:49.920 hold one in your mind, and you can write down 0:08:49.920 --> 0:08:53.040 in just a few letters the equation for one right 0:08:53.480 --> 0:08:55.720 x where it puss wise square, it equals four. That's 0:08:55.720 --> 0:08:59.200 the equation that describes a perfect circle. But where is 0:08:59.280 --> 0:09:01.559 that circle? Right? Have the equation? I have the idea 0:09:01.600 --> 0:09:04.440 in my mind is this set of neurons in my 0:09:04.520 --> 0:09:07.440 brain that's now representing that concept, But there is no 0:09:07.559 --> 0:09:10.640 circle actually out there in the universe. It's sort of 0:09:10.640 --> 0:09:13.200 hard to grapple with and even in our brains, like 0:09:13.280 --> 0:09:17.480 that circle that you can kind of visualize is just 0:09:17.640 --> 0:09:21.040 made up of the firing of your neurons, and so 0:09:21.240 --> 0:09:24.520 is that circle in your brain even perfect? Who knows? 0:09:24.600 --> 0:09:26.760 Who knows? And we'll dig into that in another episode. 0:09:26.800 --> 0:09:29.600 But today I want to focus on this biggest concept 0:09:29.600 --> 0:09:33.200 in mathematics, infinity, because it's something that pops up all 0:09:33.240 --> 0:09:36.120 the time in physics. It's something that really tickles our 0:09:36.160 --> 0:09:38.679 brain and makes us ask these really big questions. So 0:09:38.800 --> 0:09:47.280 today on the podcast, we'll be asking is infinity real? 0:09:47.559 --> 0:09:51.560 Is it physical? Or is it just mathematical? And people 0:09:51.640 --> 0:09:54.240 have some answers to this question, which we're going to 0:09:54.320 --> 0:09:57.120 listen to right now. That's right. Thank you very much 0:09:57.160 --> 0:09:59.960 to those of you who answered my tough physics questions 0:10:00.040 --> 0:10:03.240 with no preparation for the podcast. If you like to 0:10:03.280 --> 0:10:06.920 participate for a future virtual Person on the Street interviews, 0:10:06.960 --> 0:10:10.079 please write to me two questions at Daniel and Jorge 0:10:10.240 --> 0:10:13.559 dot com. I promise it's fun. Here's what people have 0:10:13.720 --> 0:10:18.240 to say. I think that infinity is definitely real. I 0:10:18.280 --> 0:10:21.559 I believe that the universe is infinite in its size, 0:10:21.640 --> 0:10:24.439 so that there is the real infinite in the universe 0:10:24.520 --> 0:10:26.480 that we live in. And also if we believe in 0:10:26.520 --> 0:10:30.560 the Einstein's theory about black holes, that also inside a 0:10:30.559 --> 0:10:34.240 black hole we have a singularity that is a point 0:10:34.360 --> 0:10:37.080 of infinite density. So there we go. We have another 0:10:37.400 --> 0:10:41.120 very real infinite The only situation in which I can 0:10:41.160 --> 0:10:44.280 think of um from a physics point of view, where 0:10:44.320 --> 0:10:50.520 infinity might be real is if the universe has been internal, 0:10:50.559 --> 0:10:56.440 if it's been around forever. Maybe if there is a 0:10:56.600 --> 0:11:02.439 singularity in black holes and that might also be considered 0:11:02.520 --> 0:11:06.800 a point of infinite density. I guess, well, that was 0:11:07.160 --> 0:11:11.000 very interesting. I do like the idea of the universe 0:11:11.240 --> 0:11:13.600 being eternal. I think that's kind of one of my 0:11:13.760 --> 0:11:19.960 concepts of infinity, is that existence of something some you know, universe, 0:11:20.080 --> 0:11:22.880 even if it's not our universe, but just the existence 0:11:22.920 --> 0:11:25.480 of stuff has to be infinite, right, Like, there can't 0:11:25.559 --> 0:11:29.640 be a start and an endpoint to stuff being around. 0:11:30.000 --> 0:11:32.480 There can't be such an appeal to naturalism. Right, you're 0:11:32.520 --> 0:11:35.360 just saying, like it's obvious that it must be this way. 0:11:35.520 --> 0:11:38.280 You're basically a Greek philosopher. That's how they argued. Also, right, 0:11:38.760 --> 0:11:41.520 I don't know whether to be insulted or not. But yeah, 0:11:41.600 --> 0:11:44.280 I mean that's how I feel. This is not based 0:11:44.360 --> 0:11:47.000 on science. It's just how I feel of like, well, 0:11:47.040 --> 0:11:51.080 we can't just have stuff come out of nothing, go 0:11:51.320 --> 0:11:53.840 do its thing for a while and then end and 0:11:53.840 --> 0:11:58.240 then there'll be nothing again. But there's not any scientific 0:11:58.280 --> 0:12:02.479 reason for that. That's coming from my human feelings of existence. 0:12:02.480 --> 0:12:05.160 And maybe even that's what I want to believe, because 0:12:05.160 --> 0:12:08.480 it's very comforting to think there's an infinite universe that 0:12:08.520 --> 0:12:11.559 doesn't die, and I think it tells you something about 0:12:11.640 --> 0:12:14.920 yourself and how your mind works and your understanding of 0:12:14.920 --> 0:12:17.840 the universe to answer these questions that it feels natural 0:12:17.880 --> 0:12:19.880 to you that the universe is infinite, or that it's 0:12:19.920 --> 0:12:23.200 not that there are infinities or that there aren't. But remember, 0:12:23.200 --> 0:12:26.080 of course, that the process of science is to question 0:12:26.200 --> 0:12:28.959 all that intuition, is to go the opposite direction and 0:12:29.000 --> 0:12:31.680 to start from the data and say, well, universe, tell 0:12:31.760 --> 0:12:34.439 us what's out there, and then from that will build 0:12:34.440 --> 0:12:37.160 our model of the universe, instead of saying like, well, 0:12:37.200 --> 0:12:39.520 this just has to be this way. But it's hard 0:12:39.559 --> 0:12:42.400 to sort of understand where those limits are. You can't 0:12:42.440 --> 0:12:45.280 like question every single thing all the time. You can 0:12:45.320 --> 0:12:48.160 do experiments every day about every tiny detail. So it's 0:12:48.160 --> 0:12:50.800 hard to know sometimes when a question is this hard 0:12:50.840 --> 0:12:53.320 to grapple with exactly how to figure it out. And 0:12:53.360 --> 0:12:55.480 that's what I think we hear our listeners grappling with 0:12:55.720 --> 0:12:58.080 whether it can be, whether it must be, whether it 0:12:58.200 --> 0:13:02.240 can't be. Tricky stuff. Are you saying that I, Katie Golden, 0:13:02.400 --> 0:13:05.200 could be wrong about one of the most difficult concepts 0:13:05.200 --> 0:13:09.520 in the universe. I'm saying, basically, everybody is probably wrong 0:13:09.559 --> 0:13:12.400 about almost everything you know. But those are the best 0:13:12.440 --> 0:13:15.760 parts of science, right When your intuition is confronted with data, 0:13:15.960 --> 0:13:18.560 that's when you learn something. That's when the universe is 0:13:18.600 --> 0:13:21.240 telling you are n't I get why you think this 0:13:21.360 --> 0:13:24.360 is true, but it turns out it's not. Works a 0:13:24.400 --> 0:13:26.640 totally different way, And that's why we do science, right, 0:13:26.640 --> 0:13:29.840 That's why we have this methodological approach to revealing the 0:13:29.880 --> 0:13:32.680 truth about the universe, rather than just you know, eating 0:13:32.800 --> 0:13:35.760 olives and lying around in robes and saying maybe it's 0:13:35.760 --> 0:13:39.719 made of tiny cats, as fun as that is. So 0:13:39.800 --> 0:13:44.440 say I'm Plato and I'm trying to conceive of infinity. 0:13:44.520 --> 0:13:48.240 Am I just thinking of like a really really long 0:13:48.559 --> 0:13:52.839 line of olives? What's going on with that. No, you're right, 0:13:52.920 --> 0:13:56.439 the Greeks actually do have records of thinking about infinity. 0:13:56.520 --> 0:13:59.360 And you know, probably people before that thought about infinity. 0:13:59.360 --> 0:14:02.400 Who knows, But the earliest records we have about infinity 0:14:02.640 --> 0:14:06.160 go back to about six hundred BC. And you know, 0:14:06.200 --> 0:14:08.640 there are some examples there are people thinking about infinity, 0:14:08.720 --> 0:14:11.320 not necessarily in terms of an infinite number of olives. 0:14:11.559 --> 0:14:14.440 But for example Zeno's paradox. You know, when you say 0:14:14.559 --> 0:14:18.360 I'm gonna walk first halfway home, and then how much 0:14:18.400 --> 0:14:19.880 distance do I have to get to the rest of 0:14:19.920 --> 0:14:22.760 my house, Well half the distance, and before I can 0:14:22.880 --> 0:14:25.160 get home, I have to then walk half of that distance. 0:14:25.360 --> 0:14:27.280 Then I have to walk half of that distance and 0:14:27.320 --> 0:14:29.520 half of that distance, And there's no limit to the 0:14:29.560 --> 0:14:32.400 number of times you can divide the rest of your 0:14:32.480 --> 0:14:36.040 distance between yourself and your home. So in principle, there's 0:14:36.080 --> 0:14:38.920 an infinite number of steps you need to take between 0:14:38.960 --> 0:14:41.560 yourself and your house if every step is half as 0:14:41.600 --> 0:14:44.960 long as the previous one. This is Zeno's famous paradox. 0:14:45.040 --> 0:14:47.240 Of course, right, if there's an infinite number of steps 0:14:47.240 --> 0:14:49.160 between you and your house, how do you ever get there? 0:14:49.200 --> 0:14:51.640 You can't, But of course you do. This is how 0:14:51.680 --> 0:14:54.880 I approach eating a cookie, because I think, well, I'm 0:14:54.880 --> 0:14:56.680 not going to eat this whole cookie, so I'll just 0:14:56.720 --> 0:14:59.040 eat half of this cookie. And I'm like, well, I 0:14:59.080 --> 0:15:01.080 want a little more of this cookie, so we'll eat 0:15:01.120 --> 0:15:03.400 half of the half of the cookie, and then half 0:15:03.400 --> 0:15:05.960 of that half and so on and so that way 0:15:05.960 --> 0:15:07.400 I get to say, like, well, I didn't eat the 0:15:07.440 --> 0:15:11.240 whole cookie, even though there's only an infintestimally small chrome 0:15:11.360 --> 0:15:14.920 left of the cookie. Yeah, that's a mathematical proof of 0:15:14.960 --> 0:15:18.160 converging series. Right. You could have an infinite number of 0:15:18.240 --> 0:15:21.760 terms and still have it converged to a finite number. Unfortunately, 0:15:21.760 --> 0:15:24.680 my cookie eating strategy is a divergent series. I started 0:15:24.800 --> 0:15:26.840 with great intentions. I'm like, I'm just gonna have one, 0:15:27.320 --> 0:15:28.760 and then I'm like, well I had one, so mighty 0:15:28.760 --> 0:15:30.280 will just have another one. And then I'm like, well 0:15:30.320 --> 0:15:32.520 I've had two, so i can just grab a couple more. 0:15:32.840 --> 0:15:35.000 And then I'm like, might it will just finished the box? 0:15:35.080 --> 0:15:37.400 Because we all know where this is going, so let's 0:15:37.440 --> 0:15:42.000 just get there right now. But the Greeks really struggled 0:15:42.000 --> 0:15:45.400 with this idea. They didn't know if infinity was real, 0:15:45.720 --> 0:15:48.960 and they had two different concepts. They talked about actual 0:15:49.040 --> 0:15:52.840 infinity and potential infinity, and they separated these two concepts. 0:15:52.840 --> 0:15:55.320 They're like, well, this potential infinity, we can think about 0:15:55.400 --> 0:15:58.120 that for like proofs and geometry and stuff like that. 0:15:58.400 --> 0:16:01.520 But you know, actual infinity, that's something they actually shied 0:16:01.560 --> 0:16:04.000 away from. They didn't even want to talk about it. 0:16:04.120 --> 0:16:06.320 They try to avoid it ever coming up in their 0:16:06.360 --> 0:16:08.680 proofs and in their logic and in their thought. That 0:16:08.920 --> 0:16:12.360 like saying sort of a naughty word, like what about 0:16:12.480 --> 0:16:15.320 actual infinity? And then you get kicked out of class, 0:16:15.320 --> 0:16:18.080 out of Greek class. Yeah, and they called it the 0:16:18.240 --> 0:16:21.360 horror of the infinite. They did all these proofs where 0:16:21.560 --> 0:16:24.680 you could have, you know, used infinity as a mathematical step, 0:16:24.920 --> 0:16:27.360 but they took all these extra steps to make these 0:16:27.360 --> 0:16:31.680 proofs happen without needing to resort infinity, without infinity appearing 0:16:31.760 --> 0:16:34.560 anywhere in their math, because they thought, well, that must 0:16:34.560 --> 0:16:38.800 be nonsense. That's so interesting. So they had a concept 0:16:38.840 --> 0:16:44.840 of the potential for infinity, but actually discussing infinity itself 0:16:45.000 --> 0:16:48.920 was taboo because it makes your brain feel bad. Because 0:16:48.960 --> 0:16:51.320 I think they thought it wasn't real, you know. I 0:16:51.360 --> 0:16:54.920 think they thought as a mathematical curiosity, and if you 0:16:55.000 --> 0:16:56.760 invoked it, if you had to invoke it, then it's 0:16:56.760 --> 0:17:00.320 somehow made your proofs invalid. And now, of course, you know, 0:17:00.400 --> 0:17:02.600 we have the same sort of relationship with infinity, Like 0:17:02.640 --> 0:17:04.840 it's a thing. We can think about it, we can 0:17:04.920 --> 0:17:07.320 understand it, but it is difficult for us to wrap 0:17:07.320 --> 0:17:09.240 our minds around, at least to all sorts of like 0:17:09.600 --> 0:17:14.280 counterintuitive concepts, and infinity has many, many interesting layers. You know. 0:17:14.320 --> 0:17:16.679 The basic notion of infinity is just the one we 0:17:16.680 --> 0:17:20.040 talked about before, Like the number of numbers. Right, if 0:17:20.040 --> 0:17:22.480 you start from zero and you keep going, you'll never 0:17:22.520 --> 0:17:25.439 get to a point where you can't add another number. 0:17:25.720 --> 0:17:29.040 Our number system just has that flexibility is no limit, 0:17:29.320 --> 0:17:31.600 so that you can always make a bigger number. And 0:17:31.640 --> 0:17:34.040 that's like a really powerful concept. It also makes me 0:17:34.040 --> 0:17:37.480 wonder sometimes, like what's the biggest number any human has 0:17:37.520 --> 0:17:40.119 ever thought of? Right? There isn't the biggest number in 0:17:40.119 --> 0:17:42.879 principle that there might be a biggest number anybody has 0:17:42.880 --> 0:17:45.280 ever like written down or held in their head. Yeah, 0:17:45.359 --> 0:17:47.600 I mean, but then once you find that out, you 0:17:47.600 --> 0:17:50.120 can just add one to it and then you've got 0:17:50.160 --> 0:17:52.840 the record. Right, you're the human who's thought the biggest 0:17:52.920 --> 0:17:56.280 number ever. That would be super cool. Yeah, like a googleplex, 0:17:56.400 --> 0:17:59.840 a google plex plus one, that's it. You're the chain, 0:18:00.640 --> 0:18:03.440 So that's super fun. And there's all these really interesting 0:18:03.480 --> 0:18:06.800 exercises to get your brain used to the counterintuitive nature 0:18:06.800 --> 0:18:10.480 of infinity, because the mathematics of infinity are quite slippery, right, 0:18:10.520 --> 0:18:13.040 you can think about infinity plus one, but like infinity 0:18:13.040 --> 0:18:16.280 plus one really is just infinity. One of my favorite 0:18:16.280 --> 0:18:19.600 examples comes from David Hilbert. The guy invented Hilbert Space 0:18:19.680 --> 0:18:23.000 like hundred years ago, and this is called Hilbert's Hotel. 0:18:23.480 --> 0:18:26.360 And in Hilbert's Hotel, he's got a hotel with an 0:18:26.400 --> 0:18:29.399 infinite number of rooms, right, but he's also got an 0:18:29.440 --> 0:18:32.400 infinite number of guests. So every room in the hotel 0:18:32.640 --> 0:18:35.320 has somebody in it. It's all occupied. And you might think, well, 0:18:35.359 --> 0:18:37.479 if I had a thousand rooms and a thousand guests, 0:18:37.560 --> 0:18:39.720 then I'm filled up, and if somebody shows up and 0:18:39.720 --> 0:18:41.359 asked for a room, I would have to say no. 0:18:41.680 --> 0:18:45.040 But Hilbert's Hotel is different because infinity is really different 0:18:45.080 --> 0:18:47.400 from other numbers. If he's in the office, he's got 0:18:47.440 --> 0:18:50.840 infinite rooms and infinite guests, and somebody shows up, he 0:18:50.880 --> 0:18:53.240 can make room. What he has to do is this 0:18:53.320 --> 0:18:55.720 special trick and sell everybody to pack up, leave your room, 0:18:55.800 --> 0:18:58.879 and move over to the next room. So the person 0:18:58.920 --> 0:19:01.320 in room number one goes to number two. Person room 0:19:01.400 --> 0:19:04.280 number five hundred forty seven goes through room five hundred, 0:19:05.359 --> 0:19:08.720 and in this way, room number one now becomes vacant. 0:19:08.880 --> 0:19:12.440 What happens at the very very end, like in room infinity, 0:19:12.480 --> 0:19:16.000 will room infinity goes through Room infinity plus one, right, 0:19:16.280 --> 0:19:19.040 and I think, well that room didn't exist before. Well 0:19:19.040 --> 0:19:21.640 it did because there were an infinite number of rooms. 0:19:22.200 --> 0:19:24.520 And so in this weird way you can still make 0:19:24.600 --> 0:19:27.240 room even if you had an infinite number of guests. 0:19:27.480 --> 0:19:29.040 But you know, it sounds like this is going to 0:19:29.160 --> 0:19:31.520 only get a two point five stars on Yell because 0:19:32.080 --> 0:19:34.840 Albert's making you pack up and move your rooms all 0:19:34.880 --> 0:19:38.560 the time. Terrible stay. Yeah, and he can even accommodate 0:19:38.560 --> 0:19:41.680 an infinite number of guests, Like if an infinite number 0:19:41.680 --> 0:19:43.920 of people show up and want rooms, he can just say, 0:19:43.960 --> 0:19:47.200 all right, everybody, leave your room and go to twice 0:19:47.320 --> 0:19:50.240 your room number right to then person in room four 0:19:50.359 --> 0:19:53.720 goes to room eight for example, leaving three rooms open, 0:19:54.200 --> 0:19:57.560 and so there's a lot of flexibility there left in infinity. 0:19:57.560 --> 0:20:00.280 It's really fascinating, but of course it feels unfa of all, 0:20:00.359 --> 0:20:03.720 this doesn't feel like something that could actually exist because 0:20:03.760 --> 0:20:06.119 hotels have a certain number of rooms. Even if there 0:20:06.160 --> 0:20:09.879 are very very large numbers, there is a number, and 0:20:10.000 --> 0:20:12.600 so that shows you that, like infinity really not a 0:20:12.760 --> 0:20:15.840 number in the same way that other big numbers are. Right, 0:20:15.960 --> 0:20:21.359 when we try to apply something like infinity to something 0:20:21.440 --> 0:20:24.119 that in our human minds is a finite thing, like 0:20:24.400 --> 0:20:28.280 physical space of a hotel rooms, you know, those little 0:20:28.359 --> 0:20:31.639 mints they put under your pillow, it's very difficult to 0:20:31.720 --> 0:20:35.800 actually try to translate that concept into infinity, Like I 0:20:35.920 --> 0:20:38.920 can't think of infinity mints. That's just not you know, 0:20:39.080 --> 0:20:41.760 infinite meus. Like, although that would be a good brand 0:20:41.880 --> 0:20:44.919 for those little hotel ments in the infinity, we are 0:20:44.960 --> 0:20:47.280 copyrighting that right here. And you can play other fun 0:20:47.359 --> 0:20:49.679 games to really blow your mind, because it turns out 0:20:49.760 --> 0:20:53.200 that there are different sizes of infinity. Like you know, 0:20:53.280 --> 0:20:55.600 I think a whole lot of infinity is everything is 0:20:55.640 --> 0:20:57.720 the biggest thing. There can't be anything bigger than that. 0:20:58.400 --> 0:21:01.679 But you can't actually take an infinite list and then 0:21:01.880 --> 0:21:05.600 make something bigger than that infinite list. There's another really 0:21:05.720 --> 0:21:08.920 clever example in mathematics, like say, for example, you write 0:21:08.960 --> 0:21:12.760 down all of the numbers between one and two. How 0:21:12.800 --> 0:21:15.200 many numbers are there between one and two? Well, is 0:21:15.200 --> 0:21:18.399 an infinite number because you can always add another digit 0:21:18.480 --> 0:21:21.400 to subdivice, like Zeno's paradox. If you have one point 0:21:21.480 --> 0:21:23.439 seven and one point eight, you can always go one 0:21:23.480 --> 0:21:25.760 point seven five, and then one point seven two five 0:21:25.840 --> 0:21:28.560