WEBVTT - Boltzmann Brains Revisited 0:00:03.040 --> 0:00:05.840 Welcome to Stuff to Blow Your Mind from how Stuff 0:00:05.840 --> 0:00:14.239 Works dot com. Hey you welcome to Stuff to Blow 0:00:14.280 --> 0:00:17.200 your Mind. My name is Robert Lamb and I'm Joe McCormick. 0:00:17.280 --> 0:00:20.520 And today we're going to be revisiting a topic that 0:00:20.640 --> 0:00:23.640 came up in an episode last month. So last month, 0:00:23.760 --> 0:00:29.280 on March, we released a podcast on a fascinating subject 0:00:29.400 --> 0:00:31.880 that had been much requested by listeners in the past, 0:00:32.280 --> 0:00:36.640 known as the Boltzman Brain argument, and we wanted to 0:00:36.640 --> 0:00:39.320 revisit this topic for a couple of reasons. One is 0:00:39.400 --> 0:00:42.400 that in the original version of the episode we published 0:00:42.400 --> 0:00:46.320 on March fifteen, I said something that was completely wrong, 0:00:46.440 --> 0:00:50.240 not about Boltzman brains themselves, but about gay ord Cantor 0:00:50.479 --> 0:00:54.080 and the nature of infinity as a mathematical concept. And 0:00:54.160 --> 0:00:57.400 when we became aware of this, we edited and republished 0:00:57.440 --> 0:00:59.959 the episode so not to leave a mistake and assumption 0:01:00.080 --> 0:01:02.440 floating out there on the Internet. But ever since then, 0:01:02.880 --> 0:01:04.959 I wanted to come back to the topic because I 0:01:04.959 --> 0:01:07.120 know some of you out there probably downloaded the early 0:01:07.240 --> 0:01:10.000 version of the episode with an arrow left in. I 0:01:10.040 --> 0:01:13.160 wanted to acknowledge my mistake, make a clear correction, and 0:01:13.200 --> 0:01:16.039 then take that as an opportunity to talk a little 0:01:16.040 --> 0:01:21.560 bit about the amazing nature of mathematical infinity on its own. Yeah, 0:01:21.640 --> 0:01:24.160 I mean it's this is one of those topics that 0:01:24.360 --> 0:01:28.320 combines both our attempt to understand consciousness and what it is, 0:01:28.720 --> 0:01:34.240 and also the the this mind boggling concept well actually 0:01:34.440 --> 0:01:38.080 a number of different concepts regarding the nature of infinity. 0:01:38.480 --> 0:01:41.800 Uh So, you know, it stands to reason that that 0:01:41.600 --> 0:01:44.280 that there would be a few holes left to fill, 0:01:44.440 --> 0:01:46.080 that there would be room to revisit it in a 0:01:46.120 --> 0:01:49.600 future episode, And here we are absolutely So we're going 0:01:49.640 --> 0:01:52.280 to explore the thing that I mentioned that was wrong 0:01:52.360 --> 0:01:54.800 in the first episode, and we're gonna explore a lot 0:01:54.840 --> 0:01:58.200 of the actually mind blowing reality underlying that claim, and 0:01:58.240 --> 0:02:00.320 that that's going to have to do with can Her 0:02:00.440 --> 0:02:04.480 and the idea of accountable infinities and unaccountable infinities. But 0:02:04.520 --> 0:02:06.400 we're also just going to have a chance to explore 0:02:06.440 --> 0:02:08.600 the idea of Boltzman Brains a little more, including a 0:02:08.600 --> 0:02:11.679 little bit of feedback from listeners on the idea and 0:02:11.720 --> 0:02:14.160 so forth. So if you never listen to the original 0:02:14.200 --> 0:02:17.200 Boltzman Brain episode from back in March, you might want 0:02:17.240 --> 0:02:18.960 to go back and check that one out first. But 0:02:19.000 --> 0:02:21.160 I'd say it's not strictly necessary if you want to 0:02:21.240 --> 0:02:23.600 jump right in with us here if you did listen, 0:02:23.680 --> 0:02:25.680 but you're you're a little bit fuzzy on the concept 0:02:25.720 --> 0:02:27.839 because it's been a little while. Don't worry. We will 0:02:27.840 --> 0:02:31.040 give you a brief refresher on the Boltzman brain argument. Yeah, 0:02:31.080 --> 0:02:32.760 and if you're not even sure what the deal is 0:02:32.800 --> 0:02:36.680 with infinity, I'll try and roll through some of the 0:02:36.720 --> 0:02:39.240 basics there as well. But let's get back to these brains, 0:02:39.320 --> 0:02:44.080 these marvelous floating space brains that could destroy us. All. Okay, So, 0:02:44.160 --> 0:02:47.679 the standard Boltzman brain argument, as we explored the last time, 0:02:48.040 --> 0:02:51.919 states that if you assume yourself to be a typical observer, 0:02:52.200 --> 0:02:54.960 which you should, right, why wouldn't you assume yourself to 0:02:54.960 --> 0:02:57.800 be typical? Yeah, well, there's a danger in assuming that 0:02:57.880 --> 0:03:01.640 you were a privileged observer. Right. In fact, by definition, 0:03:01.680 --> 0:03:04.520 if you assume yourself to be a typical the odds 0:03:04.520 --> 0:03:07.160 are that you're wrong. Right, that's the definition of what 0:03:07.200 --> 0:03:10.480 it means to be a typical. If you assume yourself 0:03:10.560 --> 0:03:14.160 as you should, to be a typical observer, it is 0:03:14.280 --> 0:03:20.160 more likely that you are something like a disembodied hallucinating 0:03:20.320 --> 0:03:26.320 brain that randomly fluctuated into existence in empty space, floating there, 0:03:26.360 --> 0:03:30.800 hallucinating your life, your memories, your current sense experience, and 0:03:31.000 --> 0:03:33.640 all of that kind of stuff. It's more likely that 0:03:33.680 --> 0:03:37.200 you're one of those than that you're a normal mammalian 0:03:37.320 --> 0:03:41.560 organism that evolved on a rocky planet. This random isolated 0:03:41.640 --> 0:03:45.640 floating brain in space is known as the Boltzman brain, 0:03:46.080 --> 0:03:50.240 and normal conscious organisms that exist through biochemical evolution, like 0:03:50.280 --> 0:03:53.400 we assume ourselves to be, are called in these arguments 0:03:53.440 --> 0:03:56.920 usually ordinary observers. So you've got these these two different 0:03:56.960 --> 0:04:00.520 types of beings. You could be right and techly, you 0:04:00.560 --> 0:04:04.000 wouldn't know the difference whether you were one or the other. Yeah, 0:04:04.080 --> 0:04:06.280 And if you really stop and think long and hard 0:04:06.320 --> 0:04:09.960 about either possibility, they're they're they're both. They both feel 0:04:10.040 --> 0:04:12.480 kind of far fetched, you know, like they're they're they're 0:04:12.560 --> 0:04:16.840 kind of equally believable and equally fantastic, because it's just 0:04:16.920 --> 0:04:20.080 the the O O argument here that I'm just this 0:04:20.200 --> 0:04:24.279 mammal on this rock and I'm thinking about thinking is crazy. 0:04:24.520 --> 0:04:26.440 That's the core of the show. I mean, we spend 0:04:26.720 --> 0:04:28.800 most of our time on stuff to blow your mind, 0:04:29.120 --> 0:04:32.039 exploring the weirdness of what it means to be a 0:04:32.120 --> 0:04:35.040 mammal that lives on a rock floating in space, even 0:04:35.080 --> 0:04:37.680 though that's what all our best science tells us we 0:04:37.800 --> 0:04:41.520 actually are. Yeah, but according to the Boltzman brain argument, 0:04:42.240 --> 0:04:46.080 you should believe that it's more likely that you're actually 0:04:46.279 --> 0:04:49.400 a brain floating in space hallucinating everything about your life 0:04:49.760 --> 0:04:52.480 then that you're one of these creatures living on a 0:04:52.560 --> 0:04:55.039 rocky planet. And the way it works is pretty simple. 0:04:55.080 --> 0:04:57.520 You look at different models of the universe and you 0:04:57.640 --> 0:05:00.800 roughly calculate how many of each type of being you'd 0:05:00.800 --> 0:05:03.599 expect to find given the model of the universe that 0:05:03.640 --> 0:05:06.240 you're looking at. Now, if you're looking at our universe, 0:05:06.279 --> 0:05:08.400 you might still think, well, how could this be a 0:05:08.440 --> 0:05:11.920 place where there are more random space brains than animals 0:05:11.920 --> 0:05:14.960 on rocky planets? After all, there are at the very 0:05:15.040 --> 0:05:18.800 least billions of ordinary observers alive today here on Earth, 0:05:19.080 --> 0:05:21.599 and there have been billions more in the past, and 0:05:21.640 --> 0:05:24.320 there hopefully will be billions more in the future. And 0:05:24.360 --> 0:05:26.719 that's just Earth. I mean, we could think about millions, 0:05:26.839 --> 0:05:29.440 billions of other planets out there that are full of 0:05:29.560 --> 0:05:32.760 ordinary observers. To all those aliens, and so we have 0:05:32.839 --> 0:05:37.240 evidence of at the very very least billions of ordinary observers, 0:05:37.279 --> 0:05:41.040 and we don't have evidence of a single random brain 0:05:41.240 --> 0:05:45.520 floating in space assembled out of random particles ever existing 0:05:45.560 --> 0:05:47.840 in the history of the universe. So how could these 0:05:47.960 --> 0:05:51.839 random space brains out number normal evolved organisms That, on 0:05:51.880 --> 0:05:54.400 the face of it doesn't seem to make any sense, right, 0:05:54.640 --> 0:05:57.200 But this is where our old friend time comes in, 0:05:57.839 --> 0:06:00.520 and time, as we know kind of us, is up 0:06:00.640 --> 0:06:03.400 our ideas of probability with all kinds of things. Like 0:06:03.440 --> 0:06:06.400 if somebody tells you, well, I was drawing poker hands 0:06:06.400 --> 0:06:08.839 and I just drew five royal flushes in a row, 0:06:09.320 --> 0:06:12.400 you wouldn't believe them because that's impossible, that that will 0:06:12.440 --> 0:06:15.400 never happen in the history of Earth. But if you 0:06:15.480 --> 0:06:18.560 also know that they had been attempting to draw hands 0:06:18.760 --> 0:06:20.760 for I don't know, tend to the ten to the 0:06:20.800 --> 0:06:25.280 ten to the ten to the ten years, then okay, sure, yeah, 0:06:25.320 --> 0:06:27.680 it becomes a little more believable. Now. Right now, our 0:06:27.800 --> 0:06:30.520 best looking scientific model of the universe, the one that 0:06:30.600 --> 0:06:33.200 the most evidence today seems to be pointing toward, is 0:06:33.240 --> 0:06:37.000 the lambda cold dark matter model. And this model includes 0:06:37.040 --> 0:06:39.120 everything we know about the past and present, so it's 0:06:39.160 --> 0:06:42.240 got the idea that our local universe is about thirteen 0:06:42.279 --> 0:06:44.840 point eight billion years old, that it began in an 0:06:44.839 --> 0:06:48.080 incredibly hot, dense state, and then for about the past 0:06:48.120 --> 0:06:51.640 thirteen point eight billion years, it's been expanding and cooling. 0:06:52.320 --> 0:06:54.520 And if we look at what this universe is doing 0:06:54.600 --> 0:06:56.559 and what it's made of right now, we can actually 0:06:56.640 --> 0:06:59.120 make pretty decent predictions about what it's going to do 0:06:59.200 --> 0:07:02.440 in the future. And it appears that what the universe 0:07:02.440 --> 0:07:05.120 will do in the future is that it will continue 0:07:05.240 --> 0:07:08.279 to do what it's doing. It will expand and cool, 0:07:08.880 --> 0:07:12.840 and entropy will steadily increase, meaning that order is going 0:07:12.920 --> 0:07:16.840 to tend to disorder, and specialness and uniqueness will tend 0:07:16.880 --> 0:07:21.160 to unspecialness and equilibrium. An energy that can be used 0:07:21.200 --> 0:07:24.520 to do work will turn into useless ambient heat that 0:07:24.640 --> 0:07:27.280 can't do anything, and stars are going to use up 0:07:27.320 --> 0:07:30.600 their fuel and burn out and they'll collapse into black holes, 0:07:30.640 --> 0:07:34.360 and black holes eventually themselves will dissipate due to hawking radiation, 0:07:34.720 --> 0:07:38.160 and everything will just run down and cool and even out, 0:07:38.520 --> 0:07:43.600 until the universe eventually becomes a vast, undifferentiated, cold grave 0:07:43.680 --> 0:07:47.400 of thermal energy, no stars, no planets, no animals, and 0:07:47.680 --> 0:07:51.760 no ordinary observers going on into the future for more 0:07:51.840 --> 0:07:54.280 time than you can imagine, and it sounds like a 0:07:55.240 --> 0:08:00.000 grim outcome for things. But again, we're talking about very 0:08:00.080 --> 0:08:03.960 vast measurements of time. Yeah, I mean, you don't personally 0:08:03.960 --> 0:08:05.760 really need to worry about this. This is so many 0:08:05.760 --> 0:08:07.760 billions of years away that it's not going to be 0:08:07.800 --> 0:08:10.440 a problem for you. Yeah, it's more of a blow 0:08:10.520 --> 0:08:14.280 to one's sort of a loose worldview than it is 0:08:15.240 --> 0:08:19.040 an actual existential threat to you specifically. Yes, but I 0:08:19.040 --> 0:08:22.200 would guess more worldviews are actually compatible with this model 0:08:22.200 --> 0:08:24.880 of things, then you would guess at first blush. Well, 0:08:24.880 --> 0:08:27.320 I mean, I guess it certainly lines up with Ragnarok, 0:08:27.560 --> 0:08:31.640 but sure. But also, I mean, people it can feel 0:08:31.720 --> 0:08:34.600 kind of depressing when people say, like, oh, you mean, 0:08:34.679 --> 0:08:38.719 like human civilization can't exist forever. At some point all 0:08:38.800 --> 0:08:41.520 the usable energy would run out and we'd have to 0:08:41.559 --> 0:08:45.000 go extinct somewhere out there in the dark. Well, yeah, 0:08:45.080 --> 0:08:48.360 technically that does appear to be physically true, but on 0:08:48.400 --> 0:08:51.640 the time scales we're talking about, human civilization wouldn't be 0:08:51.760 --> 0:08:54.640 human civilization anymore. It would be some kind of thing 0:08:54.840 --> 0:08:58.800 evolved so far beyond what human civilization is now that 0:08:58.920 --> 0:09:02.480 you wouldn't recognize it at all. We would be will 0:09:02.520 --> 0:09:05.840 be so pretentious by that point that we're able to 0:09:05.840 --> 0:09:08.959 to glimpse these future post humans would say, oh, screw 0:09:09.040 --> 0:09:13.880 those guys often too, the twilight uh entropy with them. Well, 0:09:13.920 --> 0:09:18.079 I mean, if existence is endless variation on change, eventually 0:09:18.120 --> 0:09:20.760 one of the changes worth exploring might be the change 0:09:20.760 --> 0:09:24.560 of not existing. Yeah yeah, or or kind of to 0:09:24.559 --> 0:09:27.760 to invoke Ian and banks of sublime ng and taking 0:09:27.760 --> 0:09:31.640 this kind of alternate mode of existence that that that 0:09:31.800 --> 0:09:34.920 comes after you've you've you've had your full shot at 0:09:34.960 --> 0:09:39.520 the imperial conquest game. I like that sublime NG. That's nice. 0:09:39.720 --> 0:09:42.560 It's like a chemical process almost. Yeah. That in in 0:09:42.720 --> 0:09:46.960 his um uh constructed universe of the of the of 0:09:47.040 --> 0:09:49.559 the culture. That's what you see these super advanced elder 0:09:49.600 --> 0:09:53.760 civilizations doing. Uh. They they don't destroy themselves. They reach 0:09:53.760 --> 0:09:56.200 a point where they just sublime and they they just 0:09:56.280 --> 0:09:59.480 kind of leave everything that they've built behind. Well, as 0:09:59.520 --> 0:10:02.000 more and more of our existence tends to be uh 0:10:02.760 --> 0:10:06.679 trending toward being less physical and being more encoded as 0:10:06.720 --> 0:10:10.720 digital information, one wonder is if the ultimate transcendence is 0:10:10.760 --> 0:10:14.440 just to sort of like become a a radiation imprint 0:10:14.520 --> 0:10:17.120 on the background of the universe. Yeah, I mean that's 0:10:17.160 --> 0:10:19.800 kind of a beautiful afterlife in its own right. But anyway, 0:10:19.800 --> 0:10:23.600 back to Boltzmon brains. Um, So what what? What? What 0:10:23.800 --> 0:10:26.760 happens when the universe just goes on and on and 0:10:26.840 --> 0:10:29.200 on like that for periods of time you can't even 0:10:29.240 --> 0:10:32.960 begin to comprehend. Well, here's where the weird statistical argument 0:10:33.000 --> 0:10:35.120 comes in, that you're more likely to be a space 0:10:35.160 --> 0:10:37.679 brain than one of the kinds of creatures we're pretty 0:10:37.720 --> 0:10:41.120 sure we actually are. During all that vast time in 0:10:41.160 --> 0:10:45.960 the future, random fluctuations in this empty universe will occasionally 0:10:46.040 --> 0:10:50.719 depart from this dead equilibrium before returning to it again. Now, 0:10:50.760 --> 0:10:53.720 how will that happen? Well, Sean Carroll, the physicist who 0:10:53.760 --> 0:10:55.760 has written a good bit about Boltzmon brains, who we 0:10:55.800 --> 0:10:58.320 quoted in the last episode, will continue to quote today. 0:10:58.559 --> 0:11:01.320 He writes about this in his paper why Boltzmann Brains 0:11:01.360 --> 0:11:04.640 are Bad. Just to give one example of what's going 0:11:04.679 --> 0:11:07.520 on here, so he says, quote, there can be collisions 0:11:07.559 --> 0:11:11.400 between rare high energy photons or gravitons, which could pair 0:11:11.520 --> 0:11:15.880 produce electrons and positrons or protons and anti protons and 0:11:15.920 --> 0:11:19.160 so forth, and the general tendency of such pairs would 0:11:19.200 --> 0:11:22.440 be to re annihilate rather quickly. They would annihilate each other, 0:11:22.920 --> 0:11:25.760 he writes. But occasionally the new particles will have enough 0:11:25.800 --> 0:11:30.120 momentum to travel far apart from each other. Sometimes rarely, 0:11:30.240 --> 0:11:34.600 as should henceforth be understood, many such collisions will happen nearby, 0:11:34.800 --> 0:11:39.160 producing enough nearby matter to assemble itself into a macroscopic 0:11:39.200 --> 0:11:43.000 object such as a brain. Now again that sounds that's 0:11:43.040 --> 0:11:46.679 so improbable. So you're just saying random particles that are 0:11:46.679 --> 0:11:50.640 getting pair produced out there in space by random events 0:11:51.040 --> 0:11:55.480 are going to collect into enough atomic matter that they 0:11:55.520 --> 0:11:59.199 would form an object like a brain, like a specified 0:11:59.200 --> 0:12:02.600 object like that. Again, that is super improbable. If you're 0:12:02.640 --> 0:12:06.200 thinking that would never happen, you're right, Except you're never 0:12:06.480 --> 0:12:09.760 is not including enough time, and including enough time, it 0:12:09.840 --> 0:12:12.319 actually would happen. As I think we mentioned in the 0:12:12.400 --> 0:12:15.000 last episode, one is instantly reminded, of course, of the 0:12:15.520 --> 0:12:18.480 monkeys pounding on the typewriters to produce the complete works 0:12:18.480 --> 0:12:21.880 of William Shakespeare. Right, and in given normal time, that 0:12:21.920 --> 0:12:25.320 will never happen. But given enough time, if you just 0:12:25.640 --> 0:12:29.520 stretch the t variable out to arbitrary lengths that actually 0:12:29.559 --> 0:12:32.720 will happen. It's almost guaranteed to happen. But in fact, 0:12:33.040 --> 0:12:35.920 Carol writes that it it doesn't really matter exactly what 0:12:36.040 --> 0:12:38.720 the process of producing brains is as long as you've 0:12:38.720 --> 0:12:42.000 got two assumptions. And these two assumptions that give us 0:12:42.000 --> 0:12:45.560 the Boltzman Brand universe are that the universe either last 0:12:45.640 --> 0:12:49.560 forever or last for an extraordinary long time, much longer 0:12:49.559 --> 0:12:53.440 than ten to the ten to the sixty six power years, 0:12:53.480 --> 0:12:56.800 and that is a long long time. And then the 0:12:56.840 --> 0:13:01.080 second criterion is that it undergoes random fluctionations that could 0:13:01.080 --> 0:13:06.000 potentially create conscious observers. Now, as long as those conditions 0:13:06.000 --> 0:13:10.640 are met, randomly assembled brains or conscious computers or whatever 0:13:10.720 --> 0:13:13.480 other type of conscious agent you want to imagine, will 0:13:13.559 --> 0:13:18.000 eventually become more numerous in the history of the universe 0:13:18.280 --> 0:13:22.120 than normal conscious animals like us. And so Carol points 0:13:22.120 --> 0:13:24.720 out in his paper that the lambda cold dark matter model, 0:13:24.800 --> 0:13:27.960 the best current scientific model of the universe, looks like 0:13:28.000 --> 0:13:31.880 it fulfills exactly those criteria. It's laws of physics are 0:13:31.920 --> 0:13:35.760 going to allow random fluctuations that can randomly assemble particles 0:13:35.800 --> 0:13:40.520 into macroscopic objects. Like dust, like rocks, stars, planets, but 0:13:40.600 --> 0:13:45.640 also lobsters, blue rays of Jean Claude Van Damn, movies, uh, 0:13:46.120 --> 0:13:50.320 Christoph Lambert, even human brains. And it appears that the 0:13:50.440 --> 0:13:53.880 universe will go on existing for so long that there 0:13:53.880 --> 0:13:57.319 will be an unbelievably huge number of opportunities for random 0:13:57.360 --> 0:14:01.080 fluctuation objects like this to come and do existence. And 0:14:01.120 --> 0:14:03.480 so this gives us the argument there's going to be 0:14:03.559 --> 0:14:06.640 more time out there for random brains to as symbol 0:14:06.720 --> 0:14:09.640 out of particles in the void than there is going 0:14:09.679 --> 0:14:14.480 to be opportunities for ordinary observers to exist in the 0:14:14.559 --> 0:14:17.400 time space of the universe where there's usable energy to 0:14:17.440 --> 0:14:21.520 power things like stars and planets and evolution. Our evolution 0:14:21.640 --> 0:14:25.840 window is actually extremely small compared to the whole life 0:14:25.880 --> 0:14:28.880 of the universe, a lifespan that is so vast that 0:14:29.280 --> 0:14:32.400 from our finite perspective it might as well be infinite. 0:14:32.480 --> 0:14:34.520 And for all we know, we can't prove that it's 0:14:34.560 --> 0:14:37.160 not infinite. But of course, the question of what it 0:14:37.160 --> 0:14:39.920 would mean for time to be infinite is I guess 0:14:40.000 --> 0:14:43.920 kind of Uh, that's a vexing question itself, right, what 0:14:43.920 --> 0:14:46.400 would that mean in reality? All? Right, Well, let's take 0:14:46.440 --> 0:14:48.640 a quick break, and when we come back we will 0:14:48.680 --> 0:14:55.520 discuss the identity of infinity. Than alright, we're back, all right, 0:14:55.600 --> 0:14:57.920 So today in the podcast, we're gonna be discussing the 0:14:57.960 --> 0:15:02.120 concept of infinity and infinite sets, since this is where 0:15:02.160 --> 0:15:04.440 I made an erroneous statement in the first time we 0:15:04.560 --> 0:15:07.400 covered Boltzmann brains. The last time, we were in the 0:15:07.440 --> 0:15:10.520 middle of trying to illustrate the idea that it can 0:15:10.560 --> 0:15:14.080 be proved that some infinities are larger than others. Because 0:15:14.080 --> 0:15:16.080 if you've got a universe going on and on in 0:15:16.080 --> 0:15:19.560 an equilibrium, how can you compare how numerous two different 0:15:19.600 --> 0:15:22.920 sets of objects within them, like Boltzmann brains and ordinary 0:15:22.960 --> 0:15:25.680 observers are. If it just keeps going on, you'd want 0:15:25.720 --> 0:15:27.600 to have some way of saying, well, even though it 0:15:27.680 --> 0:15:30.800 keeps going on, one group within the universe is bigger 0:15:30.800 --> 0:15:33.440 than the other group within the universe. And so the 0:15:33.520 --> 0:15:37.040 idea here is that you look at relative frequencies. Right, 0:15:37.280 --> 0:15:40.080 So here's what I said last time. I said, some 0:15:40.160 --> 0:15:42.760 infinities are larger than others. And the example I gave 0:15:42.880 --> 0:15:46.600 was comparing two sets of integers. Those sets were even 0:15:46.720 --> 0:15:51.240 numbers everything divisible by two and numbers divisible by seven. 0:15:51.760 --> 0:15:55.240 So even though you could keep naming items in each 0:15:55.280 --> 0:15:58.560 of these sets forever, I claimed that there were more 0:15:58.760 --> 0:16:02.880 even numbers the numbers divisible by seven. And that seems 0:16:02.920 --> 0:16:05.120 quite intuitive, right, I mean, it made sense to me 0:16:05.200 --> 0:16:07.200 when I said it, and I thought I had actually 0:16:07.240 --> 0:16:10.040 read that in the past. But that turns out to 0:16:10.120 --> 0:16:14.280 be not true. Now on a sort of face value level, 0:16:14.280 --> 0:16:16.560 it's kind of hard to see how that's not true. Right, 0:16:16.880 --> 0:16:20.320 Even numbers happen more often if you just keep counting 0:16:20.400 --> 0:16:23.680 up the number line, you will hit even numbers more 0:16:23.760 --> 0:16:26.760 often than you hit numbers divisible by seven. Yeah. I 0:16:26.800 --> 0:16:29.440 think I kind of played off of this by saying 0:16:29.440 --> 0:16:33.200 with that Connor McCloud from Highlander, if he lived forever, 0:16:33.960 --> 0:16:36.720 he's going to cut off heads and he's going to 0:16:36.360 --> 0:16:40.200 use the bathroom, and he is going to use the 0:16:40.240 --> 0:16:43.160 bathroom more than he cuts off heads, right, right, And 0:16:43.240 --> 0:16:44.800 so that was that I had said, Yeah, when I 0:16:44.880 --> 0:16:46.760 when I when I take your example and I line 0:16:46.800 --> 0:16:50.400 it up with my Highlander heavy example, it seems to 0:16:50.480 --> 0:16:54.080 be true in both cases. We were right until you 0:16:54.120 --> 0:16:58.240 invoke the concept of infinity, and if you actually say 0:16:58.360 --> 0:17:03.800 this process extends infinitely, then everything gets ruined. Everything gets 0:17:03.880 --> 0:17:06.960 jammed up because, believe it or not, if the process 0:17:07.119 --> 0:17:10.560 mathematically were to go on for infinite time, and each 0:17:10.600 --> 0:17:15.000 set of events happens infinitely, then it's provable by something 0:17:15.040 --> 0:17:18.320 called set theory that you can show that those sets 0:17:18.320 --> 0:17:21.320 are of equal size. And that might be hard to understand, 0:17:21.359 --> 0:17:23.480 but we will explain that. It comes down to the 0:17:23.480 --> 0:17:26.919 work of a German mathematician named gay Org Canter that 0:17:27.000 --> 0:17:29.399 there are ways in which some infinite sets that seem 0:17:29.480 --> 0:17:32.159 like they should be of different sizes are actually the 0:17:32.200 --> 0:17:35.640 same size. And yet at the same time there actually 0:17:35.720 --> 0:17:40.400 are different sizes of infinite sets those so some infinities 0:17:40.480 --> 0:17:43.639 really are bigger than others, just not in the example 0:17:43.720 --> 0:17:46.880 that I gave of of integers. And I guess it's 0:17:46.880 --> 0:17:49.800 more arguable when you're talking about events like what Connor 0:17:49.880 --> 0:17:53.040 McCloud does, because then you're talking about things happening in 0:17:53.080 --> 0:17:56.400 the physical world, where the concept of infinity gets even 0:17:56.400 --> 0:17:58.960 more complicated. What would it mean for events to be 0:17:59.080 --> 0:18:02.520 recurring infinitely? Well, I think the problem is that by 0:18:02.640 --> 0:18:05.840 by dragging infinity into it, we're basically dragging God into 0:18:05.920 --> 0:18:08.680 the scenario, right, and you know how God reacts to 0:18:08.960 --> 0:18:13.639 mortals trying to to boss uh it around. Well, you know, 0:18:14.160 --> 0:18:17.600 infinity has historically been a concept of that's very, very 0:18:17.640 --> 0:18:21.639 wrapped up in ideas about theology and religion and the 0:18:21.760 --> 0:18:25.760 universe and the afterlife. Uh I just mentioned Georg Cantor. 0:18:25.800 --> 0:18:29.080 I mean, Georg Cantor had a lot of strange religious 0:18:29.080 --> 0:18:31.719 beliefs that kind of read to us as eccentric today, 0:18:32.119 --> 0:18:35.440 deeply wrapped up in his mathematical discoveries about the nature 0:18:35.440 --> 0:18:37.560 of infinity. Yeah. Yeah, the more you look at it, 0:18:37.560 --> 0:18:40.080 the more intertwined they are. Well, I guess we should 0:18:40.080 --> 0:18:42.720 come back to Cantor in a bit and talk specifically 0:18:42.760 --> 0:18:45.760 about the types of infinity that he dealt with, but 0:18:46.080 --> 0:18:49.240 more generally, maybe let's look at the concept of infinity 0:18:49.280 --> 0:18:53.040 and in cultures throughout history. All right, Yeah, so this 0:18:53.119 --> 0:18:56.399 is it's really fascinating when you think about it, because 0:18:57.000 --> 0:19:00.840 we obviously as humans live limited lives, finite lives of 0:19:00.960 --> 0:19:05.919 finite observation within a finite world. Yet we advanced to 0:19:06.000 --> 0:19:10.000 think beyond these limitations and in doing so envision to 0:19:10.200 --> 0:19:14.239 some degree infinity, or at least two toy with the 0:19:14.320 --> 0:19:18.360 concept of something being infinite. In fact, through the use 0:19:18.359 --> 0:19:21.959 of language. In particular, we used a to quote our 0:19:22.000 --> 0:19:25.920 old friend Julian James, finite set of terms that by metaphor, 0:19:26.040 --> 0:19:29.440 is able to stretch out over an infinite set of circumstances. Oh, 0:19:29.480 --> 0:19:32.040 I mean that's essentially the library of Babel, right. I 0:19:32.040 --> 0:19:36.200 mean that you have a finite set of encoding features. 0:19:36.240 --> 0:19:39.399 But those bits of code can make every statement in 0:19:39.400 --> 0:19:42.040 the world. Yeah, and we alone, of all the animals, 0:19:42.080 --> 0:19:45.440 are able to stand in wonder and horror of infinity. 0:19:45.600 --> 0:19:48.120 You look over your cat, your dog, you check out 0:19:48.160 --> 0:19:51.560 the smartest dolphin in the sea. They don't. They don't 0:19:51.560 --> 0:19:55.359 have any concept of infinity, and maybe they're happier for it. Well, 0:19:55.400 --> 0:19:57.280 I mean you might wonder if they have some kind 0:19:57.320 --> 0:20:01.680 of intuitive concept, not of in infinity, but of the 0:20:01.760 --> 0:20:06.320 idea of recurrence. There are different ways of picturing something 0:20:06.400 --> 0:20:08.280 is infinite. One of the things that I often think 0:20:08.280 --> 0:20:10.919 about is that if you go back into ancient times, 0:20:10.920 --> 0:20:14.520 you'll see these different ways of imagining, say time, even 0:20:14.560 --> 0:20:16.919 if you had the idea that there's not necessarily a 0:20:16.960 --> 0:20:20.000 beginning and end of time. Of course, many religions have that. 0:20:20.320 --> 0:20:24.920 They are totally different ways of emphasizing how that lack 0:20:24.960 --> 0:20:29.119 of beginning and end works. Is there a boundless infinity 0:20:29.160 --> 0:20:32.399 where time stretches all the way back forever and goes 0:20:32.520 --> 0:20:35.640 all the way into the future forever, or is there 0:20:35.640 --> 0:20:40.000 a bounded infinity where time endlessly repeats like along a 0:20:40.040 --> 0:20:42.920 loop that never stops going. You know, one of our 0:20:43.240 --> 0:20:45.960 probably the you know, the earliest ideas that gets tied 0:20:46.080 --> 0:20:48.240 up and eventually becomes adfinity is that, of course, of 0:20:48.320 --> 0:20:52.000 the ocean. And we still invoke this, uh, the old saying. 0:20:52.000 --> 0:20:54.880 You know, somebody breaks up with their high school breaks 0:20:54.920 --> 0:20:57.600 up with his girlfriend. He's he's he or she they're 0:20:57.600 --> 0:21:00.240 they're distraught about it. What's the nuggeta whizd them that 0:21:00.320 --> 0:21:03.080 is leveled at them? They're always more fish in the sea. 0:21:03.200 --> 0:21:05.320 But there's a finite number of fish in the sea, right, 0:21:05.359 --> 0:21:08.119 But there are so many fish that it's almost a 0:21:08.200 --> 0:21:11.000 kind of infinity, like there's a there's a boundlessness to 0:21:11.119 --> 0:21:14.399 it um And likewise, just the idea of the ocean 0:21:14.440 --> 0:21:18.359 itself is certainly a finite realm, but it is so vast, 0:21:18.800 --> 0:21:21.480 it is so hard, especially for for ancient people to 0:21:21.560 --> 0:21:25.639 comprehend that it was considered almost a kind of boundless realm. 0:21:25.680 --> 0:21:27.880 It was and it was tied up in these notions 0:21:27.880 --> 0:21:31.600 of primordial chaos. Well, it's one of those pseudo effective 0:21:31.640 --> 0:21:34.160 concepts that's useful to us in mathematics in the same 0:21:34.200 --> 0:21:37.800 way that our lives are full of useful pseudo random numbers. 0:21:37.840 --> 0:21:41.880 Like people cannot actually generate truly random sequences of numbers. 0:21:41.880 --> 0:21:44.919 We talked about this in the Eaching episode, But you 0:21:44.960 --> 0:21:47.800 can generate what feel like pseudo random numbers. You know, 0:21:47.880 --> 0:21:50.520 it's like at least seems kind of random, even if 0:21:50.520 --> 0:21:53.240 it's not truly mathematically random. And we use that kind 0:21:53.240 --> 0:21:56.439 of inference to you know, guess things and come up 0:21:56.480 --> 0:21:58.879 with a random solutions to things all the time in 0:21:58.920 --> 0:22:01.720 our life, in the same way we use pseudo infinities 0:22:01.760 --> 0:22:04.920 as a metaphor for understanding all kinds of things. Uh. 0:22:04.960 --> 0:22:07.200 You know, the stars in the sky or the fish 0:22:07.240 --> 0:22:10.639 in the sea were constantly picturing infinities that are not 0:22:10.720 --> 0:22:13.720 truly infinite, their finite numbers of things, but they're so 0:22:13.840 --> 0:22:17.320 big they stand in for infinity, which we can't conceive. 0:22:17.720 --> 0:22:21.000 But we've tried to conceive infinity. Uh, and we've been 0:22:21.000 --> 0:22:24.360 doing so for hundreds and hundreds of years from millennia. Uh. 0:22:24.359 --> 0:22:27.159 So I wanted to run through some sort of I 0:22:27.160 --> 0:22:30.960