1 00:00:03,040 --> 00:00:05,840 Speaker 1: Welcome to Stuff to Blow Your Mind from how Stuff 2 00:00:05,840 --> 00:00:14,239 Speaker 1: Works dot com. Hey you welcome to Stuff to Blow 3 00:00:14,280 --> 00:00:17,200 Speaker 1: your Mind. My name is Robert Lamb and I'm Joe McCormick. 4 00:00:17,280 --> 00:00:20,520 Speaker 1: And today we're going to be revisiting a topic that 5 00:00:20,640 --> 00:00:23,640 Speaker 1: came up in an episode last month. So last month, 6 00:00:23,760 --> 00:00:29,280 Speaker 1: on March, we released a podcast on a fascinating subject 7 00:00:29,400 --> 00:00:31,880 Speaker 1: that had been much requested by listeners in the past, 8 00:00:32,280 --> 00:00:36,640 Speaker 1: known as the Boltzman Brain argument, and we wanted to 9 00:00:36,640 --> 00:00:39,320 Speaker 1: revisit this topic for a couple of reasons. One is 10 00:00:39,400 --> 00:00:42,400 Speaker 1: that in the original version of the episode we published 11 00:00:42,400 --> 00:00:46,320 Speaker 1: on March fifteen, I said something that was completely wrong, 12 00:00:46,440 --> 00:00:50,240 Speaker 1: not about Boltzman brains themselves, but about gay ord Cantor 13 00:00:50,479 --> 00:00:54,080 Speaker 1: and the nature of infinity as a mathematical concept. And 14 00:00:54,160 --> 00:00:57,400 Speaker 1: when we became aware of this, we edited and republished 15 00:00:57,440 --> 00:00:59,959 Speaker 1: the episode so not to leave a mistake and assumption 16 00:01:00,080 --> 00:01:02,440 Speaker 1: floating out there on the Internet. But ever since then, 17 00:01:02,880 --> 00:01:04,959 Speaker 1: I wanted to come back to the topic because I 18 00:01:04,959 --> 00:01:07,120 Speaker 1: know some of you out there probably downloaded the early 19 00:01:07,240 --> 00:01:10,000 Speaker 1: version of the episode with an arrow left in. I 20 00:01:10,040 --> 00:01:13,160 Speaker 1: wanted to acknowledge my mistake, make a clear correction, and 21 00:01:13,200 --> 00:01:16,039 Speaker 1: then take that as an opportunity to talk a little 22 00:01:16,040 --> 00:01:21,560 Speaker 1: bit about the amazing nature of mathematical infinity on its own. Yeah, 23 00:01:21,640 --> 00:01:24,160 Speaker 1: I mean it's this is one of those topics that 24 00:01:24,360 --> 00:01:28,320 Speaker 1: combines both our attempt to understand consciousness and what it is, 25 00:01:28,720 --> 00:01:34,240 Speaker 1: and also the the this mind boggling concept well actually 26 00:01:34,440 --> 00:01:38,080 Speaker 1: a number of different concepts regarding the nature of infinity. 27 00:01:38,480 --> 00:01:41,800 Speaker 1: Uh So, you know, it stands to reason that that 28 00:01:41,600 --> 00:01:44,280 Speaker 1: that there would be a few holes left to fill, 29 00:01:44,440 --> 00:01:46,080 Speaker 1: that there would be room to revisit it in a 30 00:01:46,120 --> 00:01:49,600 Speaker 1: future episode, And here we are absolutely So we're going 31 00:01:49,640 --> 00:01:52,280 Speaker 1: to explore the thing that I mentioned that was wrong 32 00:01:52,360 --> 00:01:54,800 Speaker 1: in the first episode, and we're gonna explore a lot 33 00:01:54,840 --> 00:01:58,200 Speaker 1: of the actually mind blowing reality underlying that claim, and 34 00:01:58,240 --> 00:02:00,320 Speaker 1: that that's going to have to do with can Her 35 00:02:00,440 --> 00:02:04,480 Speaker 1: and the idea of accountable infinities and unaccountable infinities. But 36 00:02:04,520 --> 00:02:06,400 Speaker 1: we're also just going to have a chance to explore 37 00:02:06,440 --> 00:02:08,600 Speaker 1: the idea of Boltzman Brains a little more, including a 38 00:02:08,600 --> 00:02:11,679 Speaker 1: little bit of feedback from listeners on the idea and 39 00:02:11,720 --> 00:02:14,160 Speaker 1: so forth. So if you never listen to the original 40 00:02:14,200 --> 00:02:17,200 Speaker 1: Boltzman Brain episode from back in March, you might want 41 00:02:17,240 --> 00:02:18,960 Speaker 1: to go back and check that one out first. But 42 00:02:19,000 --> 00:02:21,160 Speaker 1: I'd say it's not strictly necessary if you want to 43 00:02:21,240 --> 00:02:23,600 Speaker 1: jump right in with us here if you did listen, 44 00:02:23,680 --> 00:02:25,680 Speaker 1: but you're you're a little bit fuzzy on the concept 45 00:02:25,720 --> 00:02:27,839 Speaker 1: because it's been a little while. Don't worry. We will 46 00:02:27,840 --> 00:02:31,040 Speaker 1: give you a brief refresher on the Boltzman brain argument. Yeah, 47 00:02:31,080 --> 00:02:32,760 Speaker 1: and if you're not even sure what the deal is 48 00:02:32,800 --> 00:02:36,680 Speaker 1: with infinity, I'll try and roll through some of the 49 00:02:36,720 --> 00:02:39,240 Speaker 1: basics there as well. But let's get back to these brains, 50 00:02:39,320 --> 00:02:44,080 Speaker 1: these marvelous floating space brains that could destroy us. All. Okay, So, 51 00:02:44,160 --> 00:02:47,679 Speaker 1: the standard Boltzman brain argument, as we explored the last time, 52 00:02:48,040 --> 00:02:51,919 Speaker 1: states that if you assume yourself to be a typical observer, 53 00:02:52,200 --> 00:02:54,960 Speaker 1: which you should, right, why wouldn't you assume yourself to 54 00:02:54,960 --> 00:02:57,800 Speaker 1: be typical? Yeah, well, there's a danger in assuming that 55 00:02:57,880 --> 00:03:01,640 Speaker 1: you were a privileged observer. Right. In fact, by definition, 56 00:03:01,680 --> 00:03:04,520 Speaker 1: if you assume yourself to be a typical the odds 57 00:03:04,520 --> 00:03:07,160 Speaker 1: are that you're wrong. Right, that's the definition of what 58 00:03:07,200 --> 00:03:10,480 Speaker 1: it means to be a typical. If you assume yourself 59 00:03:10,560 --> 00:03:14,160 Speaker 1: as you should, to be a typical observer, it is 60 00:03:14,280 --> 00:03:20,160 Speaker 1: more likely that you are something like a disembodied hallucinating 61 00:03:20,320 --> 00:03:26,320 Speaker 1: brain that randomly fluctuated into existence in empty space, floating there, 62 00:03:26,360 --> 00:03:30,800 Speaker 1: hallucinating your life, your memories, your current sense experience, and 63 00:03:31,000 --> 00:03:33,640 Speaker 1: all of that kind of stuff. It's more likely that 64 00:03:33,680 --> 00:03:37,200 Speaker 1: you're one of those than that you're a normal mammalian 65 00:03:37,320 --> 00:03:41,560 Speaker 1: organism that evolved on a rocky planet. This random isolated 66 00:03:41,640 --> 00:03:45,640 Speaker 1: floating brain in space is known as the Boltzman brain, 67 00:03:46,080 --> 00:03:50,240 Speaker 1: and normal conscious organisms that exist through biochemical evolution, like 68 00:03:50,280 --> 00:03:53,400 Speaker 1: we assume ourselves to be, are called in these arguments 69 00:03:53,440 --> 00:03:56,920 Speaker 1: usually ordinary observers. So you've got these these two different 70 00:03:56,960 --> 00:04:00,520 Speaker 1: types of beings. You could be right and techly, you 71 00:04:00,560 --> 00:04:04,000 Speaker 1: wouldn't know the difference whether you were one or the other. Yeah, 72 00:04:04,080 --> 00:04:06,280 Speaker 1: And if you really stop and think long and hard 73 00:04:06,320 --> 00:04:09,960 Speaker 1: about either possibility, they're they're they're both. They both feel 74 00:04:10,040 --> 00:04:12,480 Speaker 1: kind of far fetched, you know, like they're they're they're 75 00:04:12,560 --> 00:04:16,840 Speaker 1: kind of equally believable and equally fantastic, because it's just 76 00:04:16,920 --> 00:04:20,080 Speaker 1: the the O O argument here that I'm just this 77 00:04:20,200 --> 00:04:24,279 Speaker 1: mammal on this rock and I'm thinking about thinking is crazy. 78 00:04:24,520 --> 00:04:26,440 Speaker 1: That's the core of the show. I mean, we spend 79 00:04:26,720 --> 00:04:28,800 Speaker 1: most of our time on stuff to blow your mind, 80 00:04:29,120 --> 00:04:32,039 Speaker 1: exploring the weirdness of what it means to be a 81 00:04:32,120 --> 00:04:35,040 Speaker 1: mammal that lives on a rock floating in space, even 82 00:04:35,080 --> 00:04:37,680 Speaker 1: though that's what all our best science tells us we 83 00:04:37,800 --> 00:04:41,520 Speaker 1: actually are. Yeah, but according to the Boltzman brain argument, 84 00:04:42,240 --> 00:04:46,080 Speaker 1: you should believe that it's more likely that you're actually 85 00:04:46,279 --> 00:04:49,400 Speaker 1: a brain floating in space hallucinating everything about your life 86 00:04:49,760 --> 00:04:52,480 Speaker 1: then that you're one of these creatures living on a 87 00:04:52,560 --> 00:04:55,039 Speaker 1: rocky planet. And the way it works is pretty simple. 88 00:04:55,080 --> 00:04:57,520 Speaker 1: You look at different models of the universe and you 89 00:04:57,640 --> 00:05:00,800 Speaker 1: roughly calculate how many of each type of being you'd 90 00:05:00,800 --> 00:05:03,599 Speaker 1: expect to find given the model of the universe that 91 00:05:03,640 --> 00:05:06,240 Speaker 1: you're looking at. Now, if you're looking at our universe, 92 00:05:06,279 --> 00:05:08,400 Speaker 1: you might still think, well, how could this be a 93 00:05:08,440 --> 00:05:11,920 Speaker 1: place where there are more random space brains than animals 94 00:05:11,920 --> 00:05:14,960 Speaker 1: on rocky planets? After all, there are at the very 95 00:05:15,040 --> 00:05:18,800 Speaker 1: least billions of ordinary observers alive today here on Earth, 96 00:05:19,080 --> 00:05:21,599 Speaker 1: and there have been billions more in the past, and 97 00:05:21,640 --> 00:05:24,320 Speaker 1: there hopefully will be billions more in the future. And 98 00:05:24,360 --> 00:05:26,719 Speaker 1: that's just Earth. I mean, we could think about millions, 99 00:05:26,839 --> 00:05:29,440 Speaker 1: billions of other planets out there that are full of 100 00:05:29,560 --> 00:05:32,760 Speaker 1: ordinary observers. To all those aliens, and so we have 101 00:05:32,839 --> 00:05:37,240 Speaker 1: evidence of at the very very least billions of ordinary observers, 102 00:05:37,279 --> 00:05:41,040 Speaker 1: and we don't have evidence of a single random brain 103 00:05:41,240 --> 00:05:45,520 Speaker 1: floating in space assembled out of random particles ever existing 104 00:05:45,560 --> 00:05:47,840 Speaker 1: in the history of the universe. So how could these 105 00:05:47,960 --> 00:05:51,839 Speaker 1: random space brains out number normal evolved organisms That, on 106 00:05:51,880 --> 00:05:54,400 Speaker 1: the face of it doesn't seem to make any sense, right, 107 00:05:54,640 --> 00:05:57,200 Speaker 1: But this is where our old friend time comes in, 108 00:05:57,839 --> 00:06:00,520 Speaker 1: and time, as we know kind of us, is up 109 00:06:00,640 --> 00:06:03,400 Speaker 1: our ideas of probability with all kinds of things. Like 110 00:06:03,440 --> 00:06:06,400 Speaker 1: if somebody tells you, well, I was drawing poker hands 111 00:06:06,400 --> 00:06:08,839 Speaker 1: and I just drew five royal flushes in a row, 112 00:06:09,320 --> 00:06:12,400 Speaker 1: you wouldn't believe them because that's impossible, that that will 113 00:06:12,440 --> 00:06:15,400 Speaker 1: never happen in the history of Earth. But if you 114 00:06:15,480 --> 00:06:18,560 Speaker 1: also know that they had been attempting to draw hands 115 00:06:18,760 --> 00:06:20,760 Speaker 1: for I don't know, tend to the ten to the 116 00:06:20,800 --> 00:06:25,280 Speaker 1: ten to the ten to the ten years, then okay, sure, yeah, 117 00:06:25,320 --> 00:06:27,680 Speaker 1: it becomes a little more believable. Now. Right now, our 118 00:06:27,800 --> 00:06:30,520 Speaker 1: best looking scientific model of the universe, the one that 119 00:06:30,600 --> 00:06:33,200 Speaker 1: the most evidence today seems to be pointing toward, is 120 00:06:33,240 --> 00:06:37,000 Speaker 1: the lambda cold dark matter model. And this model includes 121 00:06:37,040 --> 00:06:39,120 Speaker 1: everything we know about the past and present, so it's 122 00:06:39,160 --> 00:06:42,240 Speaker 1: got the idea that our local universe is about thirteen 123 00:06:42,279 --> 00:06:44,840 Speaker 1: point eight billion years old, that it began in an 124 00:06:44,839 --> 00:06:48,080 Speaker 1: incredibly hot, dense state, and then for about the past 125 00:06:48,120 --> 00:06:51,640 Speaker 1: thirteen point eight billion years, it's been expanding and cooling. 126 00:06:52,320 --> 00:06:54,520 Speaker 1: And if we look at what this universe is doing 127 00:06:54,600 --> 00:06:56,559 Speaker 1: and what it's made of right now, we can actually 128 00:06:56,640 --> 00:06:59,120 Speaker 1: make pretty decent predictions about what it's going to do 129 00:06:59,200 --> 00:07:02,440 Speaker 1: in the future. And it appears that what the universe 130 00:07:02,440 --> 00:07:05,120 Speaker 1: will do in the future is that it will continue 131 00:07:05,240 --> 00:07:08,279 Speaker 1: to do what it's doing. It will expand and cool, 132 00:07:08,880 --> 00:07:12,840 Speaker 1: and entropy will steadily increase, meaning that order is going 133 00:07:12,920 --> 00:07:16,840 Speaker 1: to tend to disorder, and specialness and uniqueness will tend 134 00:07:16,880 --> 00:07:21,160 Speaker 1: to unspecialness and equilibrium. An energy that can be used 135 00:07:21,200 --> 00:07:24,520 Speaker 1: to do work will turn into useless ambient heat that 136 00:07:24,640 --> 00:07:27,280 Speaker 1: can't do anything, and stars are going to use up 137 00:07:27,320 --> 00:07:30,600 Speaker 1: their fuel and burn out and they'll collapse into black holes, 138 00:07:30,640 --> 00:07:34,360 Speaker 1: and black holes eventually themselves will dissipate due to hawking radiation, 139 00:07:34,720 --> 00:07:38,160 Speaker 1: and everything will just run down and cool and even out, 140 00:07:38,520 --> 00:07:43,600 Speaker 1: until the universe eventually becomes a vast, undifferentiated, cold grave 141 00:07:43,680 --> 00:07:47,400 Speaker 1: of thermal energy, no stars, no planets, no animals, and 142 00:07:47,680 --> 00:07:51,760 Speaker 1: no ordinary observers going on into the future for more 143 00:07:51,840 --> 00:07:54,280 Speaker 1: time than you can imagine, and it sounds like a 144 00:07:55,240 --> 00:08:00,000 Speaker 1: grim outcome for things. But again, we're talking about very 145 00:08:00,080 --> 00:08:03,960 Speaker 1: vast measurements of time. Yeah, I mean, you don't personally 146 00:08:03,960 --> 00:08:05,760 Speaker 1: really need to worry about this. This is so many 147 00:08:05,760 --> 00:08:07,760 Speaker 1: billions of years away that it's not going to be 148 00:08:07,800 --> 00:08:10,440 Speaker 1: a problem for you. Yeah, it's more of a blow 149 00:08:10,520 --> 00:08:14,280 Speaker 1: to one's sort of a loose worldview than it is 150 00:08:15,240 --> 00:08:19,040 Speaker 1: an actual existential threat to you specifically. Yes, but I 151 00:08:19,040 --> 00:08:22,200 Speaker 1: would guess more worldviews are actually compatible with this model 152 00:08:22,200 --> 00:08:24,880 Speaker 1: of things, then you would guess at first blush. Well, 153 00:08:24,880 --> 00:08:27,320 Speaker 1: I mean, I guess it certainly lines up with Ragnarok, 154 00:08:27,560 --> 00:08:31,640 Speaker 1: but sure. But also, I mean, people it can feel 155 00:08:31,720 --> 00:08:34,600 Speaker 1: kind of depressing when people say, like, oh, you mean, 156 00:08:34,679 --> 00:08:38,719 Speaker 1: like human civilization can't exist forever. At some point all 157 00:08:38,800 --> 00:08:41,520 Speaker 1: the usable energy would run out and we'd have to 158 00:08:41,559 --> 00:08:45,000 Speaker 1: go extinct somewhere out there in the dark. Well, yeah, 159 00:08:45,080 --> 00:08:48,360 Speaker 1: technically that does appear to be physically true, but on 160 00:08:48,400 --> 00:08:51,640 Speaker 1: the time scales we're talking about, human civilization wouldn't be 161 00:08:51,760 --> 00:08:54,640 Speaker 1: human civilization anymore. It would be some kind of thing 162 00:08:54,840 --> 00:08:58,800 Speaker 1: evolved so far beyond what human civilization is now that 163 00:08:58,920 --> 00:09:02,480 Speaker 1: you wouldn't recognize it at all. We would be will 164 00:09:02,520 --> 00:09:05,840 Speaker 1: be so pretentious by that point that we're able to 165 00:09:05,840 --> 00:09:08,959 Speaker 1: to glimpse these future post humans would say, oh, screw 166 00:09:09,040 --> 00:09:13,880 Speaker 1: those guys often too, the twilight uh entropy with them. Well, 167 00:09:13,920 --> 00:09:18,079 Speaker 1: I mean, if existence is endless variation on change, eventually 168 00:09:18,120 --> 00:09:20,760 Speaker 1: one of the changes worth exploring might be the change 169 00:09:20,760 --> 00:09:24,560 Speaker 1: of not existing. Yeah yeah, or or kind of to 170 00:09:24,559 --> 00:09:27,760 Speaker 1: to invoke Ian and banks of sublime ng and taking 171 00:09:27,760 --> 00:09:31,640 Speaker 1: this kind of alternate mode of existence that that that 172 00:09:31,800 --> 00:09:34,920 Speaker 1: comes after you've you've you've had your full shot at 173 00:09:34,960 --> 00:09:39,520 Speaker 1: the imperial conquest game. I like that sublime NG. That's nice. 174 00:09:39,720 --> 00:09:42,560 Speaker 1: It's like a chemical process almost. Yeah. That in in 175 00:09:42,720 --> 00:09:46,960 Speaker 1: his um uh constructed universe of the of the of 176 00:09:47,040 --> 00:09:49,559 Speaker 1: the culture. That's what you see these super advanced elder 177 00:09:49,600 --> 00:09:53,760 Speaker 1: civilizations doing. Uh. They they don't destroy themselves. They reach 178 00:09:53,760 --> 00:09:56,200 Speaker 1: a point where they just sublime and they they just 179 00:09:56,280 --> 00:09:59,480 Speaker 1: kind of leave everything that they've built behind. Well, as 180 00:09:59,520 --> 00:10:02,000 Speaker 1: more and more of our existence tends to be uh 181 00:10:02,760 --> 00:10:06,679 Speaker 1: trending toward being less physical and being more encoded as 182 00:10:06,720 --> 00:10:10,720 Speaker 1: digital information, one wonder is if the ultimate transcendence is 183 00:10:10,760 --> 00:10:14,440 Speaker 1: just to sort of like become a a radiation imprint 184 00:10:14,520 --> 00:10:17,120 Speaker 1: on the background of the universe. Yeah, I mean that's 185 00:10:17,160 --> 00:10:19,800 Speaker 1: kind of a beautiful afterlife in its own right. But anyway, 186 00:10:19,800 --> 00:10:23,600 Speaker 1: back to Boltzmon brains. Um, So what what? What? What 187 00:10:23,800 --> 00:10:26,760 Speaker 1: happens when the universe just goes on and on and 188 00:10:26,840 --> 00:10:29,200 Speaker 1: on like that for periods of time you can't even 189 00:10:29,240 --> 00:10:32,960 Speaker 1: begin to comprehend. Well, here's where the weird statistical argument 190 00:10:33,000 --> 00:10:35,120 Speaker 1: comes in, that you're more likely to be a space 191 00:10:35,160 --> 00:10:37,679 Speaker 1: brain than one of the kinds of creatures we're pretty 192 00:10:37,720 --> 00:10:41,120 Speaker 1: sure we actually are. During all that vast time in 193 00:10:41,160 --> 00:10:45,960 Speaker 1: the future, random fluctuations in this empty universe will occasionally 194 00:10:46,040 --> 00:10:50,719 Speaker 1: depart from this dead equilibrium before returning to it again. Now, 195 00:10:50,760 --> 00:10:53,720 Speaker 1: how will that happen? Well, Sean Carroll, the physicist who 196 00:10:53,760 --> 00:10:55,760 Speaker 1: has written a good bit about Boltzmon brains, who we 197 00:10:55,800 --> 00:10:58,320 Speaker 1: quoted in the last episode, will continue to quote today. 198 00:10:58,559 --> 00:11:01,320 Speaker 1: He writes about this in his paper why Boltzmann Brains 199 00:11:01,360 --> 00:11:04,640 Speaker 1: are Bad. Just to give one example of what's going 200 00:11:04,679 --> 00:11:07,520 Speaker 1: on here, so he says, quote, there can be collisions 201 00:11:07,559 --> 00:11:11,400 Speaker 1: between rare high energy photons or gravitons, which could pair 202 00:11:11,520 --> 00:11:15,880 Speaker 1: produce electrons and positrons or protons and anti protons and 203 00:11:15,920 --> 00:11:19,160 Speaker 1: so forth, and the general tendency of such pairs would 204 00:11:19,200 --> 00:11:22,440 Speaker 1: be to re annihilate rather quickly. They would annihilate each other, 205 00:11:22,920 --> 00:11:25,760 Speaker 1: he writes. But occasionally the new particles will have enough 206 00:11:25,800 --> 00:11:30,120 Speaker 1: momentum to travel far apart from each other. Sometimes rarely, 207 00:11:30,240 --> 00:11:34,600 Speaker 1: as should henceforth be understood, many such collisions will happen nearby, 208 00:11:34,800 --> 00:11:39,160 Speaker 1: producing enough nearby matter to assemble itself into a macroscopic 209 00:11:39,200 --> 00:11:43,000 Speaker 1: object such as a brain. Now again that sounds that's 210 00:11:43,040 --> 00:11:46,679 Speaker 1: so improbable. So you're just saying random particles that are 211 00:11:46,679 --> 00:11:50,640 Speaker 1: getting pair produced out there in space by random events 212 00:11:51,040 --> 00:11:55,480 Speaker 1: are going to collect into enough atomic matter that they 213 00:11:55,520 --> 00:11:59,199 Speaker 1: would form an object like a brain, like a specified 214 00:11:59,200 --> 00:12:02,600 Speaker 1: object like that. Again, that is super improbable. If you're 215 00:12:02,640 --> 00:12:06,200 Speaker 1: thinking that would never happen, you're right, Except you're never 216 00:12:06,480 --> 00:12:09,760 Speaker 1: is not including enough time, and including enough time, it 217 00:12:09,840 --> 00:12:12,319 Speaker 1: actually would happen. As I think we mentioned in the 218 00:12:12,400 --> 00:12:15,000 Speaker 1: last episode, one is instantly reminded, of course, of the 219 00:12:15,520 --> 00:12:18,480 Speaker 1: monkeys pounding on the typewriters to produce the complete works 220 00:12:18,480 --> 00:12:21,880 Speaker 1: of William Shakespeare. Right, and in given normal time, that 221 00:12:21,920 --> 00:12:25,320 Speaker 1: will never happen. But given enough time, if you just 222 00:12:25,640 --> 00:12:29,520 Speaker 1: stretch the t variable out to arbitrary lengths that actually 223 00:12:29,559 --> 00:12:32,720 Speaker 1: will happen. It's almost guaranteed to happen. But in fact, 224 00:12:33,040 --> 00:12:35,920 Speaker 1: Carol writes that it it doesn't really matter exactly what 225 00:12:36,040 --> 00:12:38,720 Speaker 1: the process of producing brains is as long as you've 226 00:12:38,720 --> 00:12:42,000 Speaker 1: got two assumptions. And these two assumptions that give us 227 00:12:42,000 --> 00:12:45,560 Speaker 1: the Boltzman Brand universe are that the universe either last 228 00:12:45,640 --> 00:12:49,560 Speaker 1: forever or last for an extraordinary long time, much longer 229 00:12:49,559 --> 00:12:53,440 Speaker 1: than ten to the ten to the sixty six power years, 230 00:12:53,480 --> 00:12:56,800 Speaker 1: and that is a long long time. And then the 231 00:12:56,840 --> 00:13:01,080 Speaker 1: second criterion is that it undergoes random fluctionations that could 232 00:13:01,080 --> 00:13:06,000 Speaker 1: potentially create conscious observers. Now, as long as those conditions 233 00:13:06,000 --> 00:13:10,640 Speaker 1: are met, randomly assembled brains or conscious computers or whatever 234 00:13:10,720 --> 00:13:13,480 Speaker 1: other type of conscious agent you want to imagine, will 235 00:13:13,559 --> 00:13:18,000 Speaker 1: eventually become more numerous in the history of the universe 236 00:13:18,280 --> 00:13:22,120 Speaker 1: than normal conscious animals like us. And so Carol points 237 00:13:22,120 --> 00:13:24,720 Speaker 1: out in his paper that the lambda cold dark matter model, 238 00:13:24,800 --> 00:13:27,960 Speaker 1: the best current scientific model of the universe, looks like 239 00:13:28,000 --> 00:13:31,880 Speaker 1: it fulfills exactly those criteria. It's laws of physics are 240 00:13:31,920 --> 00:13:35,760 Speaker 1: going to allow random fluctuations that can randomly assemble particles 241 00:13:35,800 --> 00:13:40,520 Speaker 1: into macroscopic objects. Like dust, like rocks, stars, planets, but 242 00:13:40,600 --> 00:13:45,640 Speaker 1: also lobsters, blue rays of Jean Claude Van Damn, movies, uh, 243 00:13:46,120 --> 00:13:50,320 Speaker 1: Christoph Lambert, even human brains. And it appears that the 244 00:13:50,440 --> 00:13:53,880 Speaker 1: universe will go on existing for so long that there 245 00:13:53,880 --> 00:13:57,319 Speaker 1: will be an unbelievably huge number of opportunities for random 246 00:13:57,360 --> 00:14:01,080 Speaker 1: fluctuation objects like this to come and do existence. And 247 00:14:01,120 --> 00:14:03,480 Speaker 1: so this gives us the argument there's going to be 248 00:14:03,559 --> 00:14:06,640 Speaker 1: more time out there for random brains to as symbol 249 00:14:06,720 --> 00:14:09,640 Speaker 1: out of particles in the void than there is going 250 00:14:09,679 --> 00:14:14,480 Speaker 1: to be opportunities for ordinary observers to exist in the 251 00:14:14,559 --> 00:14:17,400 Speaker 1: time space of the universe where there's usable energy to 252 00:14:17,440 --> 00:14:21,520 Speaker 1: power things like stars and planets and evolution. Our evolution 253 00:14:21,640 --> 00:14:25,840 Speaker 1: window is actually extremely small compared to the whole life 254 00:14:25,880 --> 00:14:28,880 Speaker 1: of the universe, a lifespan that is so vast that 255 00:14:29,280 --> 00:14:32,400 Speaker 1: from our finite perspective it might as well be infinite. 256 00:14:32,480 --> 00:14:34,520 Speaker 1: And for all we know, we can't prove that it's 257 00:14:34,560 --> 00:14:37,160 Speaker 1: not infinite. But of course, the question of what it 258 00:14:37,160 --> 00:14:39,920 Speaker 1: would mean for time to be infinite is I guess 259 00:14:40,000 --> 00:14:43,920 Speaker 1: kind of Uh, that's a vexing question itself, right, what 260 00:14:43,920 --> 00:14:46,400 Speaker 1: would that mean in reality? All? Right, Well, let's take 261 00:14:46,440 --> 00:14:48,640 Speaker 1: a quick break, and when we come back we will 262 00:14:48,680 --> 00:14:55,520 Speaker 1: discuss the identity of infinity. Than alright, we're back, all right, 263 00:14:55,600 --> 00:14:57,920 Speaker 1: So today in the podcast, we're gonna be discussing the 264 00:14:57,960 --> 00:15:02,120 Speaker 1: concept of infinity and infinite sets, since this is where 265 00:15:02,160 --> 00:15:04,440 Speaker 1: I made an erroneous statement in the first time we 266 00:15:04,560 --> 00:15:07,400 Speaker 1: covered Boltzmann brains. The last time, we were in the 267 00:15:07,440 --> 00:15:10,520 Speaker 1: middle of trying to illustrate the idea that it can 268 00:15:10,560 --> 00:15:14,080 Speaker 1: be proved that some infinities are larger than others. Because 269 00:15:14,080 --> 00:15:16,080 Speaker 1: if you've got a universe going on and on in 270 00:15:16,080 --> 00:15:19,560 Speaker 1: an equilibrium, how can you compare how numerous two different 271 00:15:19,600 --> 00:15:22,920 Speaker 1: sets of objects within them, like Boltzmann brains and ordinary 272 00:15:22,960 --> 00:15:25,680 Speaker 1: observers are. If it just keeps going on, you'd want 273 00:15:25,720 --> 00:15:27,600 Speaker 1: to have some way of saying, well, even though it 274 00:15:27,680 --> 00:15:30,800 Speaker 1: keeps going on, one group within the universe is bigger 275 00:15:30,800 --> 00:15:33,440 Speaker 1: than the other group within the universe. And so the 276 00:15:33,520 --> 00:15:37,040 Speaker 1: idea here is that you look at relative frequencies. Right, 277 00:15:37,280 --> 00:15:40,080 Speaker 1: So here's what I said last time. I said, some 278 00:15:40,160 --> 00:15:42,760 Speaker 1: infinities are larger than others. And the example I gave 279 00:15:42,880 --> 00:15:46,600 Speaker 1: was comparing two sets of integers. Those sets were even 280 00:15:46,720 --> 00:15:51,240 Speaker 1: numbers everything divisible by two and numbers divisible by seven. 281 00:15:51,760 --> 00:15:55,240 Speaker 1: So even though you could keep naming items in each 282 00:15:55,280 --> 00:15:58,560 Speaker 1: of these sets forever, I claimed that there were more 283 00:15:58,760 --> 00:16:02,880 Speaker 1: even numbers the numbers divisible by seven. And that seems 284 00:16:02,920 --> 00:16:05,120 Speaker 1: quite intuitive, right, I mean, it made sense to me 285 00:16:05,200 --> 00:16:07,200 Speaker 1: when I said it, and I thought I had actually 286 00:16:07,240 --> 00:16:10,040 Speaker 1: read that in the past. But that turns out to 287 00:16:10,120 --> 00:16:14,280 Speaker 1: be not true. Now on a sort of face value level, 288 00:16:14,280 --> 00:16:16,560 Speaker 1: it's kind of hard to see how that's not true. Right, 289 00:16:16,880 --> 00:16:20,320 Speaker 1: Even numbers happen more often if you just keep counting 290 00:16:20,400 --> 00:16:23,680 Speaker 1: up the number line, you will hit even numbers more 291 00:16:23,760 --> 00:16:26,760 Speaker 1: often than you hit numbers divisible by seven. Yeah. I 292 00:16:26,800 --> 00:16:29,440 Speaker 1: think I kind of played off of this by saying 293 00:16:29,440 --> 00:16:33,200 Speaker 1: with that Connor McCloud from Highlander, if he lived forever, 294 00:16:33,960 --> 00:16:36,720 Speaker 1: he's going to cut off heads and he's going to 295 00:16:36,360 --> 00:16:40,200 Speaker 1: use the bathroom, and he is going to use the 296 00:16:40,240 --> 00:16:43,160 Speaker 1: bathroom more than he cuts off heads, right, right, And 297 00:16:43,240 --> 00:16:44,800 Speaker 1: so that was that I had said, Yeah, when I 298 00:16:44,880 --> 00:16:46,760 Speaker 1: when I when I take your example and I line 299 00:16:46,800 --> 00:16:50,400 Speaker 1: it up with my Highlander heavy example, it seems to 300 00:16:50,480 --> 00:16:54,080 Speaker 1: be true in both cases. We were right until you 301 00:16:54,120 --> 00:16:58,240 Speaker 1: invoke the concept of infinity, and if you actually say 302 00:16:58,360 --> 00:17:03,800 Speaker 1: this process extends infinitely, then everything gets ruined. Everything gets 303 00:17:03,880 --> 00:17:06,960 Speaker 1: jammed up because, believe it or not, if the process 304 00:17:07,119 --> 00:17:10,560 Speaker 1: mathematically were to go on for infinite time, and each 305 00:17:10,600 --> 00:17:15,000 Speaker 1: set of events happens infinitely, then it's provable by something 306 00:17:15,040 --> 00:17:18,320 Speaker 1: called set theory that you can show that those sets 307 00:17:18,320 --> 00:17:21,320 Speaker 1: are of equal size. And that might be hard to understand, 308 00:17:21,359 --> 00:17:23,480 Speaker 1: but we will explain that. It comes down to the 309 00:17:23,480 --> 00:17:26,919 Speaker 1: work of a German mathematician named gay Org Canter that 310 00:17:27,000 --> 00:17:29,399 Speaker 1: there are ways in which some infinite sets that seem 311 00:17:29,480 --> 00:17:32,159 Speaker 1: like they should be of different sizes are actually the 312 00:17:32,200 --> 00:17:35,640 Speaker 1: same size. And yet at the same time there actually 313 00:17:35,720 --> 00:17:40,400 Speaker 1: are different sizes of infinite sets those so some infinities 314 00:17:40,480 --> 00:17:43,639 Speaker 1: really are bigger than others, just not in the example 315 00:17:43,720 --> 00:17:46,880 Speaker 1: that I gave of of integers. And I guess it's 316 00:17:46,880 --> 00:17:49,800 Speaker 1: more arguable when you're talking about events like what Connor 317 00:17:49,880 --> 00:17:53,040 Speaker 1: McCloud does, because then you're talking about things happening in 318 00:17:53,080 --> 00:17:56,400 Speaker 1: the physical world, where the concept of infinity gets even 319 00:17:56,400 --> 00:17:58,960 Speaker 1: more complicated. What would it mean for events to be 320 00:17:59,080 --> 00:18:02,520 Speaker 1: recurring infinitely? Well, I think the problem is that by 321 00:18:02,640 --> 00:18:05,840 Speaker 1: by dragging infinity into it, we're basically dragging God into 322 00:18:05,920 --> 00:18:08,680 Speaker 1: the scenario, right, and you know how God reacts to 323 00:18:08,960 --> 00:18:13,639 Speaker 1: mortals trying to to boss uh it around. Well, you know, 324 00:18:14,160 --> 00:18:17,600 Speaker 1: infinity has historically been a concept of that's very, very 325 00:18:17,640 --> 00:18:21,639 Speaker 1: wrapped up in ideas about theology and religion and the 326 00:18:21,760 --> 00:18:25,760 Speaker 1: universe and the afterlife. Uh I just mentioned Georg Cantor. 327 00:18:25,800 --> 00:18:29,080 Speaker 1: I mean, Georg Cantor had a lot of strange religious 328 00:18:29,080 --> 00:18:31,719 Speaker 1: beliefs that kind of read to us as eccentric today, 329 00:18:32,119 --> 00:18:35,440 Speaker 1: deeply wrapped up in his mathematical discoveries about the nature 330 00:18:35,440 --> 00:18:37,560 Speaker 1: of infinity. Yeah. Yeah, the more you look at it, 331 00:18:37,560 --> 00:18:40,080 Speaker 1: the more intertwined they are. Well, I guess we should 332 00:18:40,080 --> 00:18:42,720 Speaker 1: come back to Cantor in a bit and talk specifically 333 00:18:42,760 --> 00:18:45,760 Speaker 1: about the types of infinity that he dealt with, but 334 00:18:46,080 --> 00:18:49,240 Speaker 1: more generally, maybe let's look at the concept of infinity 335 00:18:49,280 --> 00:18:53,040 Speaker 1: and in cultures throughout history. All right, Yeah, so this 336 00:18:53,119 --> 00:18:56,399 Speaker 1: is it's really fascinating when you think about it, because 337 00:18:57,000 --> 00:19:00,840 Speaker 1: we obviously as humans live limited lives, finite lives of 338 00:19:00,960 --> 00:19:05,919 Speaker 1: finite observation within a finite world. Yet we advanced to 339 00:19:06,000 --> 00:19:10,000 Speaker 1: think beyond these limitations and in doing so envision to 340 00:19:10,200 --> 00:19:14,239 Speaker 1: some degree infinity, or at least two toy with the 341 00:19:14,320 --> 00:19:18,360 Speaker 1: concept of something being infinite. In fact, through the use 342 00:19:18,359 --> 00:19:21,959 Speaker 1: of language. In particular, we used a to quote our 343 00:19:22,000 --> 00:19:25,920 Speaker 1: old friend Julian James, finite set of terms that by metaphor, 344 00:19:26,040 --> 00:19:29,440 Speaker 1: is able to stretch out over an infinite set of circumstances. Oh, 345 00:19:29,480 --> 00:19:32,040 Speaker 1: I mean that's essentially the library of Babel, right. I 346 00:19:32,040 --> 00:19:36,200 Speaker 1: mean that you have a finite set of encoding features. 347 00:19:36,240 --> 00:19:39,399 Speaker 1: But those bits of code can make every statement in 348 00:19:39,400 --> 00:19:42,040 Speaker 1: the world. Yeah, and we alone, of all the animals, 349 00:19:42,080 --> 00:19:45,440 Speaker 1: are able to stand in wonder and horror of infinity. 350 00:19:45,600 --> 00:19:48,120 Speaker 1: You look over your cat, your dog, you check out 351 00:19:48,160 --> 00:19:51,560 Speaker 1: the smartest dolphin in the sea. They don't. They don't 352 00:19:51,560 --> 00:19:55,359 Speaker 1: have any concept of infinity, and maybe they're happier for it. Well, 353 00:19:55,400 --> 00:19:57,280 Speaker 1: I mean you might wonder if they have some kind 354 00:19:57,320 --> 00:20:01,680 Speaker 1: of intuitive concept, not of in infinity, but of the 355 00:20:01,760 --> 00:20:06,320 Speaker 1: idea of recurrence. There are different ways of picturing something 356 00:20:06,400 --> 00:20:08,280 Speaker 1: is infinite. One of the things that I often think 357 00:20:08,280 --> 00:20:10,919 Speaker 1: about is that if you go back into ancient times, 358 00:20:10,920 --> 00:20:14,520 Speaker 1: you'll see these different ways of imagining, say time, even 359 00:20:14,560 --> 00:20:16,919 Speaker 1: if you had the idea that there's not necessarily a 360 00:20:16,960 --> 00:20:20,000 Speaker 1: beginning and end of time. Of course, many religions have that. 361 00:20:20,320 --> 00:20:24,920 Speaker 1: They are totally different ways of emphasizing how that lack 362 00:20:24,960 --> 00:20:29,119 Speaker 1: of beginning and end works. Is there a boundless infinity 363 00:20:29,160 --> 00:20:32,399 Speaker 1: where time stretches all the way back forever and goes 364 00:20:32,520 --> 00:20:35,640 Speaker 1: all the way into the future forever, or is there 365 00:20:35,640 --> 00:20:40,000 Speaker 1: a bounded infinity where time endlessly repeats like along a 366 00:20:40,040 --> 00:20:42,920 Speaker 1: loop that never stops going. You know, one of our 367 00:20:43,240 --> 00:20:45,960 Speaker 1: probably the you know, the earliest ideas that gets tied 368 00:20:46,080 --> 00:20:48,240 Speaker 1: up and eventually becomes adfinity is that, of course, of 369 00:20:48,320 --> 00:20:52,000 Speaker 1: the ocean. And we still invoke this, uh, the old saying. 370 00:20:52,000 --> 00:20:54,880 Speaker 1: You know, somebody breaks up with their high school breaks 371 00:20:54,920 --> 00:20:57,600 Speaker 1: up with his girlfriend. He's he's he or she they're 372 00:20:57,600 --> 00:21:00,240 Speaker 1: they're distraught about it. What's the nuggeta whizd them that 373 00:21:00,320 --> 00:21:03,080 Speaker 1: is leveled at them? They're always more fish in the sea. 374 00:21:03,200 --> 00:21:05,320 Speaker 1: But there's a finite number of fish in the sea, right, 375 00:21:05,359 --> 00:21:08,119 Speaker 1: But there are so many fish that it's almost a 376 00:21:08,200 --> 00:21:11,000 Speaker 1: kind of infinity, like there's a there's a boundlessness to 377 00:21:11,119 --> 00:21:14,399 Speaker 1: it um And likewise, just the idea of the ocean 378 00:21:14,440 --> 00:21:18,359 Speaker 1: itself is certainly a finite realm, but it is so vast, 379 00:21:18,800 --> 00:21:21,480 Speaker 1: it is so hard, especially for for ancient people to 380 00:21:21,560 --> 00:21:25,639 Speaker 1: comprehend that it was considered almost a kind of boundless realm. 381 00:21:25,680 --> 00:21:27,880 Speaker 1: It was and it was tied up in these notions 382 00:21:27,880 --> 00:21:31,600 Speaker 1: of primordial chaos. Well, it's one of those pseudo effective 383 00:21:31,640 --> 00:21:34,160 Speaker 1: concepts that's useful to us in mathematics in the same 384 00:21:34,200 --> 00:21:37,800 Speaker 1: way that our lives are full of useful pseudo random numbers. 385 00:21:37,840 --> 00:21:41,880 Speaker 1: Like people cannot actually generate truly random sequences of numbers. 386 00:21:41,880 --> 00:21:44,919 Speaker 1: We talked about this in the Eaching episode, But you 387 00:21:44,960 --> 00:21:47,800 Speaker 1: can generate what feel like pseudo random numbers. You know, 388 00:21:47,880 --> 00:21:50,520 Speaker 1: it's like at least seems kind of random, even if 389 00:21:50,520 --> 00:21:53,240 Speaker 1: it's not truly mathematically random. And we use that kind 390 00:21:53,240 --> 00:21:56,439 Speaker 1: of inference to you know, guess things and come up 391 00:21:56,480 --> 00:21:58,879 Speaker 1: with a random solutions to things all the time in 392 00:21:58,920 --> 00:22:01,720 Speaker 1: our life, in the same way we use pseudo infinities 393 00:22:01,760 --> 00:22:04,920 Speaker 1: as a metaphor for understanding all kinds of things. Uh. 394 00:22:04,960 --> 00:22:07,200 Speaker 1: You know, the stars in the sky or the fish 395 00:22:07,240 --> 00:22:10,639 Speaker 1: in the sea were constantly picturing infinities that are not 396 00:22:10,720 --> 00:22:13,720 Speaker 1: truly infinite, their finite numbers of things, but they're so 397 00:22:13,840 --> 00:22:17,320 Speaker 1: big they stand in for infinity, which we can't conceive. 398 00:22:17,720 --> 00:22:21,000 Speaker 1: But we've tried to conceive infinity. Uh, and we've been 399 00:22:21,000 --> 00:22:24,360 Speaker 1: doing so for hundreds and hundreds of years from millennia. Uh. 400 00:22:24,359 --> 00:22:27,159 Speaker 1: So I wanted to run through some sort of I 401 00:22:27,160 --> 00:22:30,960 Speaker 1: guess you could say great moments in UH infinite naval 402 00:22:31,000 --> 00:22:34,320 Speaker 1: gazing right here, just some of the UH, just a 403 00:22:34,359 --> 00:22:39,200 Speaker 1: few of the important individuals and movements. One of the 404 00:22:39,200 --> 00:22:42,639 Speaker 1: the earliest that's worth talking about, UH, it has to 405 00:22:42,640 --> 00:22:46,760 Speaker 1: do with the religion of Jainism. Okay, So after the 406 00:22:46,760 --> 00:22:49,800 Speaker 1: decline of the Vedic religion on in in on the 407 00:22:49,840 --> 00:22:53,360 Speaker 1: Indian subcontinent around a four hundred b c. To other 408 00:22:53,440 --> 00:22:57,080 Speaker 1: religions rose to prominence in India. You had you have Buddhism, 409 00:22:57,160 --> 00:22:59,640 Speaker 1: which we've talked about on the show before, as well 410 00:22:59,680 --> 00:23:02,840 Speaker 1: as Jainism. And it's a jain is um is really 411 00:23:02,880 --> 00:23:08,520 Speaker 1: its own distinct religion. It does involve some uh properties 412 00:23:08,560 --> 00:23:12,000 Speaker 1: and concepts that pop up in Hinduism or Buddhism, but uh, 413 00:23:12,359 --> 00:23:14,720 Speaker 1: you shouldn't think of it as just a meteor offshoot 414 00:23:14,720 --> 00:23:18,480 Speaker 1: of Hinduism or Buddhism. Its name derives from the Sanskrit 415 00:23:18,640 --> 00:23:21,600 Speaker 1: verb to conquer. That's interesting because I tend to think 416 00:23:21,640 --> 00:23:25,880 Speaker 1: of Jainism as embodying non violence to an incredible extent. Well, yes, 417 00:23:25,960 --> 00:23:28,080 Speaker 1: but the thing you're conquering. Think of it in terms 418 00:23:28,080 --> 00:23:33,080 Speaker 1: of conquering one's passion. It's the internal struggle. So James 419 00:23:33,400 --> 00:23:37,000 Speaker 1: don't hold up a traditional founder. Uh. They have a 420 00:23:37,080 --> 00:23:40,000 Speaker 1: number of key teachers that are called turth and Carras 421 00:23:40,119 --> 00:23:43,600 Speaker 1: or ford Makers. Uh. So, for instance, UH, there was 422 00:23:43,640 --> 00:23:46,560 Speaker 1: an important one named part of an author who may 423 00:23:46,560 --> 00:23:49,760 Speaker 1: have lived in the seventh century BC, and he would 424 00:23:49,760 --> 00:23:53,439 Speaker 1: have been the twenty three of these ford makers. But 425 00:23:53,440 --> 00:23:57,320 Speaker 1: then the twenty four and last, Uh, turthen Cara was 426 00:23:57,359 --> 00:24:00,520 Speaker 1: a man by the name of a vard Anna and 427 00:24:00,600 --> 00:24:04,159 Speaker 1: known by the title Maja Vira or great Hero, and 428 00:24:04,160 --> 00:24:06,680 Speaker 1: he would have been a contemporary of the historic Buddha. 429 00:24:07,600 --> 00:24:11,199 Speaker 1: It was during his lifetime and the years immediately to 430 00:24:11,240 --> 00:24:15,640 Speaker 1: follow some of the greatest contributions to Jane mathematics were made. Uh. 431 00:24:15,640 --> 00:24:19,679 Speaker 1: And Jane philosophy is big into pondering the enumeration of 432 00:24:19,840 --> 00:24:26,680 Speaker 1: large numbers. They classified numbers into three categories, innumerable, innumerable, 433 00:24:26,880 --> 00:24:32,720 Speaker 1: and infinite. In fact, Jainis m recognizes five different forms 434 00:24:32,760 --> 00:24:36,960 Speaker 1: of infinity. There's infinite in one and in two directions, 435 00:24:37,480 --> 00:24:41,359 Speaker 1: there's there's infinite in area, there's infinite everywhere, and there's 436 00:24:41,440 --> 00:24:46,600 Speaker 1: infinite perpetually. Infinity is also tied up in their metaphysics 437 00:24:46,800 --> 00:24:50,240 Speaker 1: of the metaphysics metaphysics of the Jains soul. There are 438 00:24:50,280 --> 00:24:53,800 Speaker 1: an infinite number of souls in the universe and liberated 439 00:24:53,880 --> 00:24:57,760 Speaker 1: souls or siddhas, which are free from the cycle of 440 00:24:57,800 --> 00:25:02,760 Speaker 1: samsara uh. They have in finite knowledge, infinite vision, infinite power, 441 00:25:02,800 --> 00:25:06,160 Speaker 1: and infinite bliss. Now I wonder what it would mean 442 00:25:06,320 --> 00:25:09,320 Speaker 1: to have infinite knowledge. This is something that's always been 443 00:25:09,400 --> 00:25:12,000 Speaker 1: kind of interesting to me about the idea of omniscience. 444 00:25:12,320 --> 00:25:15,520 Speaker 1: It seems to me that the idea of omniscience inherently 445 00:25:15,640 --> 00:25:20,399 Speaker 1: invokes or involves things that appear to be contradictions in 446 00:25:20,400 --> 00:25:24,600 Speaker 1: the same way that omnipotence does. Right, So people classically say, oh, 447 00:25:24,640 --> 00:25:28,000 Speaker 1: if you've got an omnipotent soul or an omnipotent being, 448 00:25:28,119 --> 00:25:30,920 Speaker 1: could it create a rock that it itself could not lift? 449 00:25:31,000 --> 00:25:32,800 Speaker 1: You know that that's the classic one. The same thing 450 00:25:32,840 --> 00:25:37,359 Speaker 1: would come through with omniscience, meaning could an omniscient being 451 00:25:37,400 --> 00:25:40,399 Speaker 1: with all knowledge know what it is like to not 452 00:25:40,600 --> 00:25:44,879 Speaker 1: know things well? Or would you know things if to 453 00:25:44,960 --> 00:25:48,200 Speaker 1: be complete knowledge? Is that maybe just a complete absence 454 00:25:48,200 --> 00:25:50,240 Speaker 1: of knowledge in a way, like you if you try 455 00:25:50,280 --> 00:25:53,000 Speaker 1: and imagine a person that is defined, but you know 456 00:25:53,040 --> 00:25:56,400 Speaker 1: what we have finite knowledge, finite vision, finite power, it's 457 00:25:56,400 --> 00:25:59,439 Speaker 1: certainly finite bliss. If you ramp all those things up 458 00:25:59,440 --> 00:26:02,520 Speaker 1: to the infinite, then do we just bleed into the 459 00:26:02,560 --> 00:26:05,760 Speaker 1: fabric of reality? We vanished completely? We I mean maybe 460 00:26:05,800 --> 00:26:09,240 Speaker 1: that that is that's the ultimate liberation right there. Well, 461 00:26:09,280 --> 00:26:12,280 Speaker 1: all of these concepts that are being taken to the 462 00:26:12,280 --> 00:26:16,480 Speaker 1: transcendent level of infinity here, like vision, power, bliss, knowledge, 463 00:26:16,720 --> 00:26:19,199 Speaker 1: they're all in a way kind of defined by the 464 00:26:19,240 --> 00:26:25,800 Speaker 1: bounds of limitation. What what would it mean to see everything? Then? 465 00:26:26,200 --> 00:26:28,679 Speaker 1: Are you even really seeing? I mean seeing is an 466 00:26:28,720 --> 00:26:32,280 Speaker 1: act of like focusing and an act of perceiving, And 467 00:26:32,320 --> 00:26:35,119 Speaker 1: so if it's everything, I don't know what what it 468 00:26:35,119 --> 00:26:37,679 Speaker 1: would necessarily mean. What is bliss of? Is it if 469 00:26:37,680 --> 00:26:43,080 Speaker 1: it is experienced outside of the contrast with with suffering, right, 470 00:26:43,200 --> 00:26:47,560 Speaker 1: or even something that's slightly less than bliss. Now I 471 00:26:47,560 --> 00:26:49,440 Speaker 1: want to be clear, I'm not trying to like rag 472 00:26:49,520 --> 00:26:53,000 Speaker 1: On Jane metaphysics or whatever. I mean. These are kind 473 00:26:53,000 --> 00:26:55,000 Speaker 1: of beliefs that appear in all kinds of religions, but 474 00:26:55,359 --> 00:26:59,439 Speaker 1: I think they highlight some of the inherent qualities of 475 00:26:59,480 --> 00:27:02,800 Speaker 1: contem lating the infinite that makes it seem wholly and 476 00:27:02,880 --> 00:27:05,720 Speaker 1: kind of mind boggling to begin with. It's exactly these 477 00:27:05,760 --> 00:27:10,040 Speaker 1: types of contradictions that make it so attractive as an idea. Yeah, certainly. 478 00:27:10,040 --> 00:27:12,680 Speaker 1: And of course the Greek scrappled with infinity as well. 479 00:27:13,080 --> 00:27:15,880 Speaker 1: In fact, if we look back to the to the 480 00:27:15,880 --> 00:27:20,200 Speaker 1: the ideas of an Aximander of Militiasts who lived six 481 00:27:20,240 --> 00:27:24,320 Speaker 1: ten through five b c. Uh, he's considered the the 482 00:27:24,359 --> 00:27:27,480 Speaker 1: earliest Greek philosopher to commit his ideas to writing. The 483 00:27:27,560 --> 00:27:30,359 Speaker 1: only fragments of his work remain, and he introduced the 484 00:27:30,359 --> 00:27:36,479 Speaker 1: principle of the apron or boundless, and this is uh 485 00:27:37,080 --> 00:27:39,840 Speaker 1: what he would discussed as the original state of the universe. 486 00:27:40,280 --> 00:27:43,200 Speaker 1: And he's certainly describing infinity that you can also see 487 00:27:43,400 --> 00:27:47,159 Speaker 1: clear the clear influence of the Greek cosmological concept of 488 00:27:47,200 --> 00:27:50,120 Speaker 1: primordial chaos here as well. Oh yeah, and this would 489 00:27:50,119 --> 00:27:53,000 Speaker 1: be this would be an important concept that that subsequent 490 00:27:53,840 --> 00:27:57,920 Speaker 1: Greek philosophers would toy with as well. One of the 491 00:27:58,400 --> 00:28:01,960 Speaker 1: big names is of or says Zeno of Elia, who 492 00:28:01,960 --> 00:28:04,919 Speaker 1: lived four ninety through b c. And he played with 493 00:28:04,960 --> 00:28:08,280 Speaker 1: this concept of the apron uh, and he found that 494 00:28:08,400 --> 00:28:13,520 Speaker 1: infinities breed paradoxes such as the fabulous paradox of Achilles 495 00:28:13,880 --> 00:28:17,520 Speaker 1: and the tortoise. Uh and uh, this this is a 496 00:28:17,640 --> 00:28:20,680 Speaker 1: this is a fun little thought experiment that I think 497 00:28:20,720 --> 00:28:24,120 Speaker 1: will that lines up rather nicely with the Boltzmann brains 498 00:28:24,760 --> 00:28:27,800 Speaker 1: thought experiment, because the idea here is that you have 499 00:28:27,840 --> 00:28:30,840 Speaker 1: a tortoise and then you have the the the god 500 00:28:30,920 --> 00:28:35,600 Speaker 1: blessed um hero Achilles. And the tortoise says, hey, I 501 00:28:35,600 --> 00:28:38,240 Speaker 1: would like to challenge you to a foot race, and 502 00:28:38,320 --> 00:28:41,240 Speaker 1: Achilles is like, what are you doing. I'm gonna I'm 503 00:28:41,240 --> 00:28:44,160 Speaker 1: gonna of course, I'm gonna beat you. I am Achilles. Uh, 504 00:28:44,240 --> 00:28:47,680 Speaker 1: you know, nothing stands in my way, and I'm certainly 505 00:28:47,680 --> 00:28:51,760 Speaker 1: going to outrun a tortoise. And the tortoise says, well, uh, 506 00:28:52,240 --> 00:28:54,720 Speaker 1: I would need a head start, of course, because I 507 00:28:54,720 --> 00:28:56,600 Speaker 1: am a tortoise and you are Achilles. But I'll still 508 00:28:56,640 --> 00:28:58,640 Speaker 1: beat you if you just give me a reasonable head start, 509 00:28:59,080 --> 00:29:01,560 Speaker 1: and they agree on a head start something to the 510 00:29:01,560 --> 00:29:04,880 Speaker 1: effect of like ten feet or so. There's nothing crazy, 511 00:29:04,880 --> 00:29:06,480 Speaker 1: and Achilles is like, well, I'm still gonna be you, 512 00:29:06,600 --> 00:29:09,720 Speaker 1: don't I don't understand, and the tortoise explains, well, think 513 00:29:09,720 --> 00:29:12,760 Speaker 1: of it this way, You're gonna have to catch up 514 00:29:12,800 --> 00:29:16,440 Speaker 1: with me. Before you pass me, and Achilles says, sure, yeah, 515 00:29:16,480 --> 00:29:18,040 Speaker 1: I'm gonna have to I'm gonna have to run to 516 00:29:18,120 --> 00:29:21,920 Speaker 1: where you start. And then the tortoise says, but by 517 00:29:21,960 --> 00:29:24,720 Speaker 1: the time you reach my starting point, I'll be ahead 518 00:29:24,720 --> 00:29:26,960 Speaker 1: of you, and uh, and then you're gonna have to 519 00:29:27,000 --> 00:29:30,200 Speaker 1: catch up with where I've gotten to. And and he keeps, 520 00:29:30,240 --> 00:29:33,680 Speaker 1: you know, carrying this out one step further, one step further, 521 00:29:33,960 --> 00:29:36,800 Speaker 1: until he convince his Achilles that he can never catch 522 00:29:36,880 --> 00:29:39,520 Speaker 1: up with the tortoise, and a Kelly's just concedes the 523 00:29:39,520 --> 00:29:42,560 Speaker 1: whole race, and the tortoise wins. I remember reading about 524 00:29:42,560 --> 00:29:46,520 Speaker 1: this paradox when I was younger, and I love stuff 525 00:29:46,560 --> 00:29:49,040 Speaker 1: like this. It always like, I mean, it seemed right. 526 00:29:49,600 --> 00:29:52,120 Speaker 1: It's one of those things that seems very true in 527 00:29:52,120 --> 00:29:54,120 Speaker 1: the same way that it certainly seems like there must 528 00:29:54,160 --> 00:29:57,520 Speaker 1: be more even numbers than there are than there are 529 00:29:57,800 --> 00:30:00,920 Speaker 1: numbers divisible by seven, right, Yeah, yeah, I mean, like 530 00:30:01,080 --> 00:30:03,840 Speaker 1: another version of this would be imagine how long it 531 00:30:03,880 --> 00:30:07,600 Speaker 1: would take you to walk across a basketball court. Yeah, well, 532 00:30:07,600 --> 00:30:10,280 Speaker 1: first you have to walk halfway across it. Before you 533 00:30:10,280 --> 00:30:12,720 Speaker 1: reach that you gotta walk halfway across that distance, and 534 00:30:12,760 --> 00:30:15,600 Speaker 1: then halfway across that distance. So you can take something 535 00:30:15,680 --> 00:30:20,840 Speaker 1: finite and if you divide it up into infinite portions, 536 00:30:21,440 --> 00:30:25,440 Speaker 1: then something that is very possible seems impossible. Uh. One 537 00:30:25,480 --> 00:30:27,440 Speaker 1: of the take homes is that, according to the argument 538 00:30:27,520 --> 00:30:31,600 Speaker 1: made by the tortoise, movement is impossible. Yeah, exactly. Now 539 00:30:31,600 --> 00:30:34,080 Speaker 1: to move the analogy over to the Boltzmon brain argument, 540 00:30:34,120 --> 00:30:37,280 Speaker 1: I would say, actually, if you wanna say, like the 541 00:30:37,320 --> 00:30:42,480 Speaker 1: tortoise represents Boltzman brains and Achilles represents ordinary observers, all 542 00:30:42,520 --> 00:30:45,000 Speaker 1: you don't really have to do is say, okay, tortoise 543 00:30:45,000 --> 00:30:47,720 Speaker 1: does not get ahead, start, they start at the same time. 544 00:30:48,200 --> 00:30:52,400 Speaker 1: Achilles runs, say a thousand times faster than the tortoise, 545 00:30:52,760 --> 00:30:55,000 Speaker 1: and they both start the race and they just go. 546 00:30:55,480 --> 00:30:59,840 Speaker 1: Except the difference is the tortoise is immortal, all right, 547 00:31:00,040 --> 00:31:02,240 Speaker 1: So Achilles is gonna win the early part of the 548 00:31:02,320 --> 00:31:05,640 Speaker 1: race by a long shot. Like tortoise isn't gonna come 549 00:31:05,680 --> 00:31:09,560 Speaker 1: close until Achilles gets really exhausted and dies. I know, 550 00:31:09,840 --> 00:31:12,320 Speaker 1: if if only, if only he was full God, he 551 00:31:12,400 --> 00:31:15,000 Speaker 1: might have a shot. But then you've got eternity for 552 00:31:15,040 --> 00:31:19,520 Speaker 1: the tortoise to just keep walking out ahead. Yeah, slow 553 00:31:19,520 --> 00:31:23,400 Speaker 1: and steady right right, wins the race. Uh. The Great thinkers, 554 00:31:23,400 --> 00:31:28,400 Speaker 1: of course, tackled infinity as well. Aristotle three twenty two 555 00:31:28,560 --> 00:31:32,880 Speaker 1: BC he considered the the apperon as well. Um. Yeah, 556 00:31:32,880 --> 00:31:36,960 Speaker 1: here he is in physics talking about the boundless quote. 557 00:31:36,960 --> 00:31:40,120 Speaker 1: Everything has an origin or is an origin. The boundless 558 00:31:40,240 --> 00:31:43,400 Speaker 1: has no origin, for then it would have a limit. Moreover, 559 00:31:43,480 --> 00:31:46,760 Speaker 1: it is both unborn and immortal, being a kind of origin. 560 00:31:47,160 --> 00:31:50,680 Speaker 1: For that which has become has also necessarily an end, 561 00:31:51,000 --> 00:31:54,400 Speaker 1: and there is a termination to every process of destruction. 562 00:31:54,640 --> 00:31:59,080 Speaker 1: Another thinker of note, h Thomas Aquinas, came along twelve 563 00:31:59,120 --> 00:32:02,760 Speaker 1: seventy four. He focused on the quality of existence rather 564 00:32:02,800 --> 00:32:06,440 Speaker 1: than the quantity. Of course he didn't uh. So he considered, 565 00:32:06,440 --> 00:32:10,200 Speaker 1: you know, God is infinite in quality more so than 566 00:32:10,200 --> 00:32:12,840 Speaker 1: in quantity. So he saw infinity as a mode of 567 00:32:12,880 --> 00:32:18,040 Speaker 1: existence and identified a separation between mathematical infinity and religious infinity. 568 00:32:18,520 --> 00:32:21,840 Speaker 1: So we see a curious principle emerge. Even an infinite 569 00:32:21,840 --> 00:32:25,720 Speaker 1: God cannot create an infinite object. He talks about this 570 00:32:25,760 --> 00:32:30,160 Speaker 1: at length in his work Assuma Theologica, uh, sort of 571 00:32:30,200 --> 00:32:32,200 Speaker 1: taking both sides, like saying, all right, well if it 572 00:32:32,320 --> 00:32:35,160 Speaker 1: if God is infinite, then X. If God is fine eye, 573 00:32:35,200 --> 00:32:38,600 Speaker 1: then why But I think he says that nothing except 574 00:32:38,760 --> 00:32:41,320 Speaker 1: God can be infinite, right right, Well, that kind of 575 00:32:41,320 --> 00:32:43,000 Speaker 1: point of view is going to hold some sway for 576 00:32:43,040 --> 00:32:48,800 Speaker 1: a while. Um. Another individual worth noting Nicholas of Cusa 577 00:32:49,160 --> 00:32:52,239 Speaker 1: fourteen o one through fourteen sixty four. He argued that 578 00:32:52,320 --> 00:32:56,360 Speaker 1: everything is within the infinite. The world itself must be 579 00:32:56,480 --> 00:32:59,960 Speaker 1: within God. So he used mathematical examples to describe the 580 00:33:00,040 --> 00:33:04,560 Speaker 1: relationship ship between God and the world. There's no circumference 581 00:33:04,600 --> 00:33:07,520 Speaker 1: to God, and the center is everywhere. This is the 582 00:33:07,560 --> 00:33:12,880 Speaker 1: concept of a maximum God, unlimited, transcendent, and also unreachable 583 00:33:12,880 --> 00:33:16,680 Speaker 1: and unfathomable to a species that is defined by its 584 00:33:16,720 --> 00:33:19,960 Speaker 1: finite limitations. So this say, is starting to sound kind 585 00:33:20,000 --> 00:33:23,000 Speaker 1: of like the like God is the universe type belief. 586 00:33:24,360 --> 00:33:29,200 Speaker 1: Then you have Spinosa comes around two through seven and 587 00:33:29,240 --> 00:33:31,520 Speaker 1: he says, if God is infinite, then God is the 588 00:33:31,520 --> 00:33:35,280 Speaker 1: one substance. The substance must have infinite attributes, and we 589 00:33:35,360 --> 00:33:39,520 Speaker 1: must be modes of this one entity. So God is 590 00:33:39,560 --> 00:33:42,840 Speaker 1: not the not a personal God. God and Nature are 591 00:33:42,920 --> 00:33:45,720 Speaker 1: one and the highest ethic, according to Spinoza, is to 592 00:33:45,800 --> 00:33:47,760 Speaker 1: live in accordance with the laws of nature, to be 593 00:33:47,840 --> 00:33:50,680 Speaker 1: part of the infinite. So the ideas that were beings 594 00:33:50,720 --> 00:33:53,440 Speaker 1: of mind and body, both of which are composed of 595 00:33:53,520 --> 00:33:58,320 Speaker 1: this universal substance. That's an interesting idea that if if 596 00:33:58,440 --> 00:34:01,800 Speaker 1: one thinks of God as infinite, then how could there 597 00:34:01,920 --> 00:34:06,520 Speaker 1: be things that weren't God? Right? Because wouldn't God necessarily 598 00:34:06,640 --> 00:34:09,879 Speaker 1: encompass those things? If God had no limit and went 599 00:34:09,960 --> 00:34:13,279 Speaker 1: on forever, what what could be outside of him that 600 00:34:13,320 --> 00:34:17,359 Speaker 1: would seem to suggest that there was a limit to him? Yeah? Yeah, 601 00:34:17,400 --> 00:34:21,319 Speaker 1: what could what could be outside of the absolute, the 602 00:34:21,440 --> 00:34:25,319 Speaker 1: maximum God? Yet again, I feel like infinity is one 603 00:34:25,320 --> 00:34:27,960 Speaker 1: of those things where you start playing with language in 604 00:34:27,960 --> 00:34:29,880 Speaker 1: a way, it's kind of a game that you you 605 00:34:29,920 --> 00:34:32,839 Speaker 1: start using certain words thinking you know what they mean. 606 00:34:33,480 --> 00:34:37,360 Speaker 1: But when you use words like infinity or everything or 607 00:34:37,440 --> 00:34:40,399 Speaker 1: forever thinking you know what they mean, they end up 608 00:34:40,520 --> 00:34:44,160 Speaker 1: kicking up conclusions that you couldn't expect because you you 609 00:34:44,280 --> 00:34:47,799 Speaker 1: contemplate them more and more deeply all the time. Uh, 610 00:34:47,840 --> 00:34:51,600 Speaker 1: and they tend to transcend the ways you originally invoked them. 611 00:34:51,640 --> 00:34:53,319 Speaker 1: Does that make any sense? Yeah? Well, I mean I 612 00:34:53,320 --> 00:34:55,080 Speaker 1: think that the big one is. Of course, it's one 613 00:34:55,080 --> 00:35:00,480 Speaker 1: thing to talk about infinity in terms of just pure philosophy, 614 00:35:00,960 --> 00:35:03,560 Speaker 1: but then when you start lining it up with mathematics, right, 615 00:35:03,640 --> 00:35:06,480 Speaker 1: and when you start bringing the raw numbers in and 616 00:35:06,480 --> 00:35:09,400 Speaker 1: crunching those numbers. Uh, that's where you get into some 617 00:35:09,480 --> 00:35:12,680 Speaker 1: of these these real conundrums. That's where you you you 618 00:35:12,880 --> 00:35:16,920 Speaker 1: hear these arguments of someone saying, well, you're talking about infinity, 619 00:35:16,960 --> 00:35:20,239 Speaker 1: but you're not talking about mathematical and infinity. Because here's 620 00:35:20,280 --> 00:35:22,480 Speaker 1: what happens when you throw the numbers in, right, And 621 00:35:22,520 --> 00:35:25,680 Speaker 1: this brings us back to Gayard Cantor. So Gayard Canter 622 00:35:26,000 --> 00:35:28,000 Speaker 1: was a I think I mentioned this earlier, but he 623 00:35:28,040 --> 00:35:30,719 Speaker 1: was a German mathematician. He was born in Russia in 624 00:35:30,760 --> 00:35:34,239 Speaker 1: eighteen forty five and he lived until nineteen eighteen, and 625 00:35:34,280 --> 00:35:38,840 Speaker 1: he is the main person responsible for modern set theory 626 00:35:38,960 --> 00:35:42,320 Speaker 1: and the theory of what are now called transfinite numbers, 627 00:35:42,680 --> 00:35:46,400 Speaker 1: which was revolutionary and very controversial in its time, but 628 00:35:46,560 --> 00:35:49,280 Speaker 1: in many ways is widely accepted now and has proven 629 00:35:49,320 --> 00:35:53,879 Speaker 1: extremely useful. So in writing about infinity, Cantor made use 630 00:35:54,000 --> 00:35:58,040 Speaker 1: of the idea of sets. Sets are both simple in 631 00:35:58,080 --> 00:36:02,120 Speaker 1: the core principle and extra ordinarily complicated and powerful as 632 00:36:02,120 --> 00:36:05,480 Speaker 1: a tool for mathematical reasoning. And so set theory is 633 00:36:05,520 --> 00:36:10,200 Speaker 1: basically just it's a framework for grouping items into sets. 634 00:36:10,239 --> 00:36:12,920 Speaker 1: So you've got some items, you could group them together, 635 00:36:13,080 --> 00:36:14,840 Speaker 1: and then you've got a set of items, and you 636 00:36:14,840 --> 00:36:17,680 Speaker 1: can treat that set as a thing. And one of 637 00:36:17,719 --> 00:36:20,319 Speaker 1: the things set theory does is that it gives you 638 00:36:20,400 --> 00:36:24,200 Speaker 1: a way of comparing the size of sets of things 639 00:36:24,600 --> 00:36:27,759 Speaker 1: by matching the items in those sets in a one 640 00:36:27,840 --> 00:36:31,320 Speaker 1: to one pairing off process. And the size of a set, 641 00:36:31,680 --> 00:36:35,719 Speaker 1: meaning how many items it contains, is known as its cardinality. No, 642 00:36:35,880 --> 00:36:39,200 Speaker 1: so I know that's a lot of terminology. We want 643 00:36:39,239 --> 00:36:42,000 Speaker 1: to try to avoid getting too abstract here. But basically, 644 00:36:42,080 --> 00:36:45,440 Speaker 1: if you think, okay, I've got a set with uh 645 00:36:45,680 --> 00:36:48,640 Speaker 1: five objects in it, my five fingers on my right hand, 646 00:36:48,960 --> 00:36:52,560 Speaker 1: and then I've got another set that has all my 647 00:36:52,640 --> 00:36:56,920 Speaker 1: fingers on my left hand, are those sets equal in size? Well? 648 00:36:57,000 --> 00:36:59,319 Speaker 1: I can check. I mean, obviously I can tell because 649 00:36:59,360 --> 00:37:01,840 Speaker 1: I can count to five. But even if I couldn't 650 00:37:01,880 --> 00:37:05,120 Speaker 1: count to five, I can check by pairing off the 651 00:37:05,160 --> 00:37:08,640 Speaker 1: items in each set and seeing if they pair up 652 00:37:08,680 --> 00:37:11,480 Speaker 1: in the same extent, or do I have leftover items 653 00:37:11,520 --> 00:37:13,279 Speaker 1: in one set that can't go with the other. Now 654 00:37:13,360 --> 00:37:15,520 Speaker 1: I have five fingers on each hand, so yes, I 655 00:37:15,560 --> 00:37:18,959 Speaker 1: can pair up the sets. They match up all right, 656 00:37:19,480 --> 00:37:22,000 Speaker 1: So you would say that the cardinality or the size 657 00:37:22,000 --> 00:37:24,080 Speaker 1: of both of these two sets of fingers on each 658 00:37:24,120 --> 00:37:26,720 Speaker 1: hand is five, and the sets are equal in size. 659 00:37:27,080 --> 00:37:29,480 Speaker 1: But now let's go to the example that I that 660 00:37:29,600 --> 00:37:33,319 Speaker 1: I got wrong in the last boltzmon Brain podcast. So 661 00:37:33,480 --> 00:37:35,280 Speaker 1: that would be looking at the sets of all even 662 00:37:35,400 --> 00:37:39,360 Speaker 1: numbers in one set and all numbers divisible by seven 663 00:37:39,400 --> 00:37:42,720 Speaker 1: and the other set. Even though if we just count 664 00:37:42,800 --> 00:37:46,040 Speaker 1: up the natural number line, we hit way more even 665 00:37:46,160 --> 00:37:49,840 Speaker 1: numbers the number is divisible by seven. Cantor could show 666 00:37:50,400 --> 00:37:53,239 Speaker 1: that the cardinality of these two sets is exactly the 667 00:37:53,280 --> 00:37:57,440 Speaker 1: same because each item in each set can be paired 668 00:37:57,560 --> 00:38:00,799 Speaker 1: one to one with an item from another set. So 669 00:38:00,880 --> 00:38:04,400 Speaker 1: imagine you've got set A that's even numbers, and then 670 00:38:04,400 --> 00:38:07,560 Speaker 1: you've got set B that's numbers divisible by seven. Well, 671 00:38:07,640 --> 00:38:10,279 Speaker 1: let's pay off the first items in each set. You've 672 00:38:10,280 --> 00:38:13,759 Speaker 1: got Set A is to set B is seven, and 673 00:38:13,800 --> 00:38:16,319 Speaker 1: then the second item in each set that's Set A 674 00:38:16,320 --> 00:38:20,560 Speaker 1: is four, set B is fourteen. Here's the question, what 675 00:38:20,680 --> 00:38:25,000 Speaker 1: would stop you from counting forever this way? Would you 676 00:38:25,040 --> 00:38:27,600 Speaker 1: ever hit a number in set A that did not 677 00:38:27,840 --> 00:38:31,400 Speaker 1: have a corresponding number in set B. Clearly you wouldn't 678 00:38:31,520 --> 00:38:34,120 Speaker 1: you can match them off until the end of time. 679 00:38:34,640 --> 00:38:36,680 Speaker 1: Uh though, actually, if the end of time comes and 680 00:38:36,719 --> 00:38:39,560 Speaker 1: stops you from counting more, then suddenly the even numbers 681 00:38:39,560 --> 00:38:43,640 Speaker 1: set is much bigger. But because you know there's more frequency, 682 00:38:43,680 --> 00:38:46,720 Speaker 1: you hit even numbers more often. But if you grant 683 00:38:46,719 --> 00:38:48,920 Speaker 1: that they're infinite and you don't have to stop and 684 00:38:48,920 --> 00:38:52,080 Speaker 1: tallly them up, but you treat them as ongoing sets, 685 00:38:52,320 --> 00:38:56,720 Speaker 1: then they are in fact demonstrably the same size. Okay, 686 00:38:56,719 --> 00:38:59,919 Speaker 1: I'm with you. I'm surprised you are, because that's it's 687 00:39:00,040 --> 00:39:02,319 Speaker 1: messing with my brain. I mean, that seems wrong, right. 688 00:39:02,320 --> 00:39:04,560 Speaker 1: There can't be the set of even numbers and the 689 00:39:04,600 --> 00:39:07,440 Speaker 1: set of numbers visible by seven cannot be the same size. 690 00:39:07,760 --> 00:39:11,000 Speaker 1: There's clearly more of one than the other. But cantor 691 00:39:11,120 --> 00:39:13,759 Speaker 1: can prove that they're the same size. It's because we've 692 00:39:13,760 --> 00:39:16,920 Speaker 1: invoked we've invoked infinity, and that changes things. It messes 693 00:39:16,960 --> 00:39:19,760 Speaker 1: everything up. Suddenly all your intuitions go out the window 694 00:39:19,800 --> 00:39:23,320 Speaker 1: and nothing makes sense anymore. Our listener Jim in New Jersey, 695 00:39:23,320 --> 00:39:27,239 Speaker 1: who is a great, great email writer. He's always been 696 00:39:27,239 --> 00:39:30,239 Speaker 1: hearing from from Jim for years. Uh, and Jim especially 697 00:39:30,280 --> 00:39:34,560 Speaker 1: on like mathematical logical computer science type type topics. Jim 698 00:39:34,600 --> 00:39:37,520 Speaker 1: has really great emails. He sent us an excellent email, 699 00:39:38,160 --> 00:39:42,080 Speaker 1: uh gently correcting are my mistake in the first episode, 700 00:39:42,440 --> 00:39:45,120 Speaker 1: and he had a really good analogy. He said, quote, 701 00:39:45,360 --> 00:39:48,360 Speaker 1: think of this as a marathon race without a finish line. 702 00:39:48,960 --> 00:39:51,480 Speaker 1: The hair will always be ahead of the tortoise, but 703 00:39:51,520 --> 00:39:54,799 Speaker 1: they will always pass the same mile markers, just at 704 00:39:54,840 --> 00:39:58,560 Speaker 1: different times. They both cover the same distance. It's just 705 00:39:58,600 --> 00:40:02,520 Speaker 1: that one racer reaches each milestone quicker. They all reach 706 00:40:02,640 --> 00:40:05,960 Speaker 1: the same milestones since there are an infinite number of 707 00:40:06,000 --> 00:40:09,040 Speaker 1: them an infinite time. I think that's a nice way 708 00:40:09,080 --> 00:40:11,600 Speaker 1: of picturing it. Slus. We got to work the tortoise 709 00:40:11,600 --> 00:40:13,480 Speaker 1: in for like a third time here. I like it. 710 00:40:13,560 --> 00:40:15,880 Speaker 1: How many tortoises have we done so far? Three? I 711 00:40:15,880 --> 00:40:19,120 Speaker 1: guess because we talked about the tortoise and Achilles, and 712 00:40:19,120 --> 00:40:22,279 Speaker 1: then you talked about the tortoise as boltzman brain, and 713 00:40:22,360 --> 00:40:26,200 Speaker 1: now we have proper tortoise and hair example. So now 714 00:40:26,200 --> 00:40:29,799 Speaker 1: it's tortoise as numbers divisible by seven, whereas the hair 715 00:40:29,960 --> 00:40:33,120 Speaker 1: is even numbers. Yeah. So even though the hair goes faster, 716 00:40:33,280 --> 00:40:36,920 Speaker 1: they eventually go the same distance if they go forever. 717 00:40:37,239 --> 00:40:39,800 Speaker 1: But this is so weird. Right, because with this type 718 00:40:39,800 --> 00:40:43,000 Speaker 1: of reasoning, I don't know, you might begin to sense 719 00:40:43,040 --> 00:40:47,879 Speaker 1: some deeply troubling implications. One thing is, any infinite set 720 00:40:47,880 --> 00:40:51,200 Speaker 1: of things that can be counted in order is equal 721 00:40:51,239 --> 00:40:54,399 Speaker 1: in size to any other. Because if you can count 722 00:40:54,440 --> 00:40:56,759 Speaker 1: them in order and it's clear what the order is, 723 00:40:56,880 --> 00:40:59,040 Speaker 1: you can pare them off like this. And if you 724 00:40:59,080 --> 00:41:01,279 Speaker 1: can pare them off like this, you can pare them 725 00:41:01,280 --> 00:41:04,480 Speaker 1: off forever equally. I mean, I just keep coming back 726 00:41:04,560 --> 00:41:09,280 Speaker 1: to the idea. If you're willing to accept the albeit 727 00:41:09,360 --> 00:41:12,920 Speaker 1: loose idea of something going on forever that is just 728 00:41:12,960 --> 00:41:17,319 Speaker 1: going to go on into infinity, then you can you 729 00:41:17,360 --> 00:41:19,600 Speaker 1: can buy the fact that these two sets are equal. 730 00:41:20,280 --> 00:41:22,120 Speaker 1: I guess so, I mean it's cutting me, man, It's 731 00:41:22,960 --> 00:41:25,240 Speaker 1: well be because it comes back again to the idea 732 00:41:25,280 --> 00:41:28,600 Speaker 1: that we are finite beings in a finite world. That 733 00:41:28,760 --> 00:41:30,920 Speaker 1: is all we've ever evolved to be, and yet we 734 00:41:31,000 --> 00:41:34,320 Speaker 1: imagine things that are beyond that, and so of course 735 00:41:34,360 --> 00:41:38,319 Speaker 1: it breaks our our our our normal ability to to 736 00:41:38,440 --> 00:41:42,960 Speaker 1: comprehend things. Yeah, let's keep imagining. So the implications of 737 00:41:43,000 --> 00:41:47,240 Speaker 1: set theory and Cantor's work are obviously extremely profound. For example, 738 00:41:47,520 --> 00:41:50,799 Speaker 1: here's one thing that feels pretty obvious which set is 739 00:41:50,880 --> 00:41:53,799 Speaker 1: bigger All the natural numbers, and that means all the 740 00:41:53,840 --> 00:41:56,800 Speaker 1: positive integers starting with zero, so all the counting numbers, 741 00:41:56,840 --> 00:42:01,839 Speaker 1: you know, zero, what's bigger that set? Or all the 742 00:42:02,080 --> 00:42:05,520 Speaker 1: rational numbers, which is another way of saying anything that 743 00:42:05,600 --> 00:42:09,040 Speaker 1: can be expressed as a fraction with a quotation of 744 00:42:09,280 --> 00:42:12,759 Speaker 1: U with positive eneger, So like one half or two 745 00:42:12,760 --> 00:42:15,959 Speaker 1: over one or one third. Which of those sets would 746 00:42:16,000 --> 00:42:18,320 Speaker 1: you think would be bigger? Well, I mean obviously it 747 00:42:18,360 --> 00:42:21,080 Speaker 1: would seem to invoke the Akelles and Tortoise situation, right, 748 00:42:21,120 --> 00:42:24,000 Speaker 1: you would think, well, if you just start dividing everything 749 00:42:24,360 --> 00:42:27,520 Speaker 1: we've been talking about up, you can just you can 750 00:42:27,920 --> 00:42:31,600 Speaker 1: you can just have infinite divisions that just just you know, 751 00:42:31,680 --> 00:42:35,000 Speaker 1: blows that number out of proportion. Yeah, you think, obviously 752 00:42:35,120 --> 00:42:37,720 Speaker 1: the set of rational numbers is bigger, right, that includes 753 00:42:37,760 --> 00:42:41,680 Speaker 1: all the fractions, because the set of fractions includes every 754 00:42:41,760 --> 00:42:45,520 Speaker 1: natural number. Every everything in the first set is also 755 00:42:45,719 --> 00:42:48,600 Speaker 1: in the second set. One over one, two over one, 756 00:42:48,680 --> 00:42:51,319 Speaker 1: three over one, So all the natural numbers are in 757 00:42:51,440 --> 00:42:54,720 Speaker 1: the set of rational numbers, but it also includes every 758 00:42:54,800 --> 00:42:58,560 Speaker 1: fraction in between every natural number. So you've got one half, 759 00:42:59,000 --> 00:43:01,799 Speaker 1: one third, three, seventh, and nine tenths. Those are all 760 00:43:02,040 --> 00:43:05,759 Speaker 1: rational numbers. So there must be more rational numbers or 761 00:43:05,760 --> 00:43:10,840 Speaker 1: fractions than there are natural numbers. Right. Wrong, Yet again, 762 00:43:11,840 --> 00:43:15,160 Speaker 1: Cantor used set theory to prove that these sets are 763 00:43:15,200 --> 00:43:18,319 Speaker 1: of equal size. Now you might wonder again, how could 764 00:43:18,360 --> 00:43:21,319 Speaker 1: you prove that? Like, remember from a minute ago that 765 00:43:21,440 --> 00:43:24,280 Speaker 1: the key to being able to show that the items 766 00:43:24,280 --> 00:43:28,000 Speaker 1: in accountable infinite set are equal is that you can 767 00:43:28,080 --> 00:43:30,799 Speaker 1: arrange them in accountable order. You've got to be able 768 00:43:30,840 --> 00:43:32,799 Speaker 1: to order it so that you can pare them off 769 00:43:32,880 --> 00:43:36,479 Speaker 1: in an orderly way, making sure you're not missing anything. Right, 770 00:43:36,760 --> 00:43:39,320 Speaker 1: So how could you count fractions? Right? You can't start 771 00:43:39,360 --> 00:43:42,520 Speaker 1: with the lowest fraction and count up from there. That 772 00:43:42,560 --> 00:43:44,919 Speaker 1: doesn't make any sense. You know. You can't say I'll 773 00:43:44,920 --> 00:43:47,080 Speaker 1: start with one tenth and then go to two tenths 774 00:43:47,520 --> 00:43:50,200 Speaker 1: because you have lower numbers than that. So what can 775 00:43:50,239 --> 00:43:52,960 Speaker 1: you do? Well, here was a real stroke of genius. 776 00:43:53,480 --> 00:43:59,760 Speaker 1: Cantor rearranged every possible fraction into an ordered table. Robert 777 00:43:59,800 --> 00:44:02,160 Speaker 1: f a picture of this table in in our notes 778 00:44:02,200 --> 00:44:06,000 Speaker 1: for us here, So what are we looking at? Well? 779 00:44:06,000 --> 00:44:08,440 Speaker 1: I'll have to include a link to this image on 780 00:44:08,520 --> 00:44:11,239 Speaker 1: the landing page for this episode stuffable you mind dot com, 781 00:44:11,440 --> 00:44:13,920 Speaker 1: But for me, the first thing I think of is 782 00:44:13,960 --> 00:44:15,719 Speaker 1: that if you ever did those charts in like P 783 00:44:15,960 --> 00:44:19,680 Speaker 1: class that show the volleyball rotation plan when you're playing 784 00:44:20,080 --> 00:44:23,279 Speaker 1: like team volleyball, that's kind of It's like, imagine, uh, 785 00:44:23,840 --> 00:44:25,920 Speaker 1: a bunch of fractions got together to play some sort 786 00:44:25,960 --> 00:44:28,400 Speaker 1: of strange team sport and they needed a guide to 787 00:44:28,440 --> 00:44:31,120 Speaker 1: show how they're moving around. Well, it is like that 788 00:44:31,200 --> 00:44:34,840 Speaker 1: in terms of how you navigate, but it's incredibly orderly. 789 00:44:35,200 --> 00:44:37,960 Speaker 1: So here's how the table works. Is very simple. You've 790 00:44:38,000 --> 00:44:40,360 Speaker 1: got a top line in a sideline. The top line 791 00:44:40,600 --> 00:44:44,560 Speaker 1: is numbers at the bottom of the fraction, right, so, 792 00:44:44,680 --> 00:44:49,239 Speaker 1: and that goes up with every digit one to and 793 00:44:49,280 --> 00:44:52,239 Speaker 1: there's a column that goes down with fractions that have 794 00:44:52,400 --> 00:44:54,880 Speaker 1: that number on the bottom. And then you've got a 795 00:44:54,920 --> 00:44:58,560 Speaker 1: sideline that goes down with rows that are all the 796 00:44:58,600 --> 00:45:01,920 Speaker 1: integers going on for a and those integers have rose 797 00:45:02,040 --> 00:45:05,520 Speaker 1: beside them, where the top number in the fraction is 798 00:45:05,600 --> 00:45:10,760 Speaker 1: that number. So you can actually move through this list 799 00:45:10,880 --> 00:45:14,560 Speaker 1: in a diagonal back and forth pattern that allows you 800 00:45:14,600 --> 00:45:18,759 Speaker 1: to make sure you're counting every possible fraction there could be. 801 00:45:19,320 --> 00:45:21,879 Speaker 1: So it starts with one over one, then two over one, 802 00:45:21,960 --> 00:45:24,560 Speaker 1: then one over two, then one over three, then two 803 00:45:24,600 --> 00:45:28,680 Speaker 1: two three, one four one three two, and so forth. 804 00:45:28,960 --> 00:45:32,240 Speaker 1: And you actually, through this method, could make a table 805 00:45:32,239 --> 00:45:35,160 Speaker 1: where it was possible to count every fraction that you 806 00:45:35,320 --> 00:45:38,000 Speaker 1: never actually get to the end, but you can organize 807 00:45:38,040 --> 00:45:39,920 Speaker 1: them in a way that you know you're not leaving 808 00:45:39,960 --> 00:45:44,280 Speaker 1: anything out. So it's basically volleyball, because in team volleyball 809 00:45:44,520 --> 00:45:47,880 Speaker 1: everyone moves so that everyone has to serve during the 810 00:45:47,880 --> 00:45:50,640 Speaker 1: course of the game. Um, you know, unless the gamings 811 00:45:50,640 --> 00:45:54,360 Speaker 1: are like that, let's ignore that. Uh, everything must serve, 812 00:45:54,480 --> 00:45:57,480 Speaker 1: everything must be counted. And that is exactly what kills 813 00:45:57,560 --> 00:46:00,279 Speaker 1: its supremacy in terms of competing with the eyes of 814 00:46:00,320 --> 00:46:03,960 Speaker 1: the set of natural numbers, because remember from before, if 815 00:46:04,000 --> 00:46:06,440 Speaker 1: you can count them all in an order, then you 816 00:46:06,480 --> 00:46:09,680 Speaker 1: can pare them off with natural numbers like one five. So, 817 00:46:09,760 --> 00:46:12,799 Speaker 1: believe it or not, it can be mathematically shown that 818 00:46:12,880 --> 00:46:15,960 Speaker 1: the size of the set of natural numbers and the 819 00:46:16,000 --> 00:46:18,720 Speaker 1: size of the set of rational numbers is the same, 820 00:46:19,120 --> 00:46:22,480 Speaker 1: even though that makes absolutely no sense to us. Cantor 821 00:46:22,560 --> 00:46:25,040 Speaker 1: was actually writing about this to a mathematician friend of 822 00:46:25,080 --> 00:46:28,280 Speaker 1: his named Richard did kind uh, And he said about 823 00:46:28,360 --> 00:46:30,719 Speaker 1: some of his own discoveries that you know this this 824 00:46:30,800 --> 00:46:33,680 Speaker 1: type of stuff. He was coming across j evoir me 825 00:46:33,920 --> 00:46:37,000 Speaker 1: jana quapa, which means I see it, but I do 826 00:46:37,080 --> 00:46:40,200 Speaker 1: not believe. Al Right, hold that thought, Joe. We're gonna 827 00:46:40,239 --> 00:46:42,960 Speaker 1: take a quick break and when we come back back 828 00:46:43,000 --> 00:46:49,600 Speaker 1: to infinity. And we're back to infinity. Now you can 829 00:46:49,680 --> 00:46:52,520 Speaker 1: use the same logic to violate your intuitions lots of 830 00:46:52,520 --> 00:46:54,799 Speaker 1: other ways. There are plenty ways to stab your brain 831 00:46:54,880 --> 00:46:57,840 Speaker 1: with this. So here's an obvious one, which is bigger 832 00:46:58,040 --> 00:47:01,239 Speaker 1: infinity or infinity plus one. Well, on the surface, it 833 00:47:01,280 --> 00:47:03,640 Speaker 1: would sound like infinity plus one because it's all that 834 00:47:03,800 --> 00:47:06,719 Speaker 1: plus one extra dude, exactly. I mean you remember this 835 00:47:06,800 --> 00:47:10,240 Speaker 1: from playing games as a kid, right, yeah, Well, well 836 00:47:10,239 --> 00:47:12,759 Speaker 1: I am you know, I am infinity strong. Well, I'm 837 00:47:12,760 --> 00:47:15,319 Speaker 1: infinity plus one. Well, I mean it's it gets into 838 00:47:15,400 --> 00:47:18,319 Speaker 1: not to summon the specter of the infinity hotel. But 839 00:47:18,360 --> 00:47:20,480 Speaker 1: that's kind of the concept there. You have a hotel 840 00:47:20,520 --> 00:47:23,440 Speaker 1: with infinite rooms and then infinite guests are staying in it. 841 00:47:23,719 --> 00:47:25,719 Speaker 1: What do you do when one extra guest shows up? 842 00:47:25,800 --> 00:47:27,920 Speaker 1: Is there room in the hotel? Of course there is. 843 00:47:27,960 --> 00:47:30,680 Speaker 1: You just have everybody move one room over and it 844 00:47:30,719 --> 00:47:33,560 Speaker 1: opens up one room for the new guests. And apparently 845 00:47:33,640 --> 00:47:36,400 Speaker 1: that's how infinite sets work, because as crazy as it is, 846 00:47:37,239 --> 00:47:40,680 Speaker 1: an infinite set plus one is still the same size 847 00:47:40,880 --> 00:47:43,759 Speaker 1: as a regular infinite set. So you've got set A 848 00:47:44,000 --> 00:47:47,239 Speaker 1: that's an infinite number of objects. Set B contains an 849 00:47:47,239 --> 00:47:50,200 Speaker 1: infinite number of objects plus one extra object at the beginning. 850 00:47:50,520 --> 00:47:53,839 Speaker 1: Can you still pair them off forever? Yep? Will you 851 00:47:53,880 --> 00:47:57,200 Speaker 1: ever run out of items to pair off from each group? Nope? 852 00:47:57,560 --> 00:48:00,920 Speaker 1: They are somehow still equal. And this is because, I mean, 853 00:48:01,160 --> 00:48:03,000 Speaker 1: one of the things that's hard for us to remember 854 00:48:03,040 --> 00:48:06,439 Speaker 1: is we often try to treat infinity as a number, 855 00:48:06,800 --> 00:48:10,440 Speaker 1: and infinity is not a number. And infinity is a concept. 856 00:48:10,600 --> 00:48:14,280 Speaker 1: You know, it's a mathematical tool, but it's not a number. 857 00:48:14,560 --> 00:48:17,360 Speaker 1: Like if you add plus one to any number, it 858 00:48:17,520 --> 00:48:20,360 Speaker 1: is more than that number. But plus one to infinity 859 00:48:20,640 --> 00:48:23,920 Speaker 1: is not more than infinity because infinity wasn't a single 860 00:48:24,120 --> 00:48:27,120 Speaker 1: finite number to begin with, right, and and and explaining 861 00:48:27,160 --> 00:48:29,440 Speaker 1: this to my son, I don't tell him infinity is 862 00:48:29,440 --> 00:48:32,160 Speaker 1: the largest number. I tell him numbers go on forever. 863 00:48:32,560 --> 00:48:36,440 Speaker 1: That is infinity. That's good parenting. Well, I want him 864 00:48:36,440 --> 00:48:38,759 Speaker 1: to have a proper you know, understanding of the boundless. 865 00:48:39,080 --> 00:48:43,200 Speaker 1: But he's not, is he He's not? Kinna? Uh well, 866 00:48:43,440 --> 00:48:45,560 Speaker 1: a proper understanding of the boundless. That's the kind of 867 00:48:45,600 --> 00:48:47,919 Speaker 1: thing you spend your whole life trying to to wrap 868 00:48:47,960 --> 00:48:50,320 Speaker 1: your head around. Yeah, well you ought to be careful. 869 00:48:50,360 --> 00:48:51,680 Speaker 1: You might have a child that grows up to be 870 00:48:51,760 --> 00:48:56,319 Speaker 1: a philosopher, mathematician. We'll see, we'll say, playing with fire there. 871 00:48:57,360 --> 00:48:59,279 Speaker 1: All right, So where are we with with cantor though? 872 00:48:59,280 --> 00:49:01,279 Speaker 1: What what is cant are saying at this point? Well, 873 00:49:01,320 --> 00:49:03,759 Speaker 1: I mean, so we're at a weird place already, just 874 00:49:03,800 --> 00:49:06,200 Speaker 1: with what we've talked about before, because especially when you 875 00:49:06,239 --> 00:49:08,799 Speaker 1: take this and try to extrapolate it back to the 876 00:49:08,840 --> 00:49:14,920 Speaker 1: more weird less mathematical, spiritual, theological, philosophical types of ideas 877 00:49:14,920 --> 00:49:17,120 Speaker 1: of infinity, Like what does it mean to all these 878 00:49:17,120 --> 00:49:20,520 Speaker 1: people who want to invoke infinity in their religion for 879 00:49:20,600 --> 00:49:23,719 Speaker 1: us to discover that for some types of infinite quantities 880 00:49:23,760 --> 00:49:26,840 Speaker 1: in set theory, a part of the whole is actually 881 00:49:26,960 --> 00:49:31,000 Speaker 1: exactly the same size as the whole itself. Yeah, it 882 00:49:31,000 --> 00:49:32,879 Speaker 1: doesn't leave much room for God. I mean, I guess 883 00:49:32,920 --> 00:49:35,640 Speaker 1: it leads infinite room for God. That's the That's the 884 00:49:35,719 --> 00:49:39,279 Speaker 1: confounding thing about all there you go. Uh So the 885 00:49:39,320 --> 00:49:42,560 Speaker 1: other thing, though, is that Canter also showed it's absolutely 886 00:49:42,560 --> 00:49:46,960 Speaker 1: true that some infinite sets are bigger than other infinite sets. 887 00:49:47,000 --> 00:49:50,000 Speaker 1: Now we've just been messing everything up by violating our 888 00:49:50,000 --> 00:49:52,720 Speaker 1: intuitions to show that sets that seem like they should 889 00:49:52,719 --> 00:49:55,440 Speaker 1: be of different sizes if they're infinite, are actually the 890 00:49:55,480 --> 00:49:58,640 Speaker 1: same size. How could it be possible that some infinite 891 00:49:58,640 --> 00:50:01,960 Speaker 1: sets are bigger than other Cantor guides us again. Let's 892 00:50:02,000 --> 00:50:05,920 Speaker 1: think about real numbers versus rationals. So last time we 893 00:50:05,960 --> 00:50:09,239 Speaker 1: looked at rationals. That's fractions, right, anything that can be 894 00:50:09,280 --> 00:50:13,520 Speaker 1: expressed as a normal fraction. Uh, then you've got real numbers. 895 00:50:13,560 --> 00:50:18,320 Speaker 1: So real numbers include all of those numbers, but also 896 00:50:18,440 --> 00:50:23,080 Speaker 1: include irrational and transcendental numbers, which are numbers that cannot 897 00:50:23,120 --> 00:50:26,600 Speaker 1: be expressed as fractions, things like pie in the square 898 00:50:26,680 --> 00:50:29,200 Speaker 1: root of two. You know you've seen people trying to 899 00:50:29,320 --> 00:50:32,600 Speaker 1: calculate these out, like you can write out many many 900 00:50:32,680 --> 00:50:36,080 Speaker 1: digits of pie three point one, four or so on forever, 901 00:50:36,520 --> 00:50:39,160 Speaker 1: but you'll never get to the last decimal place of 902 00:50:39,200 --> 00:50:42,240 Speaker 1: pie because it does not have a last decimal place. 903 00:50:42,560 --> 00:50:44,799 Speaker 1: And if it did have a last decimal place, you 904 00:50:44,920 --> 00:50:48,040 Speaker 1: would actually somehow be able to express it as a fraction. 905 00:50:48,520 --> 00:50:51,200 Speaker 1: But despite the fact that it never terminates, pie is 906 00:50:51,239 --> 00:50:54,160 Speaker 1: a real number, you can only write it with decimals. 907 00:50:54,280 --> 00:50:57,239 Speaker 1: And there are presumably lots of numbers like this, But 908 00:50:57,520 --> 00:51:00,040 Speaker 1: how many are there you might be guessing, give and 909 00:51:00,160 --> 00:51:02,719 Speaker 1: what We've just been learning that there would be accountable 910 00:51:02,760 --> 00:51:05,600 Speaker 1: infinity of these types of numbers. So maybe there's the 911 00:51:05,719 --> 00:51:09,920 Speaker 1: same number of numbers that are rational versus numbers that 912 00:51:09,960 --> 00:51:13,920 Speaker 1: are real, but no. Cantor showed that real numbers, including 913 00:51:13,960 --> 00:51:17,439 Speaker 1: irrationals like maybe the square woradi of two, cannot be 914 00:51:17,600 --> 00:51:22,560 Speaker 1: arranged into accountable list including all of the infinite possibilities. 915 00:51:23,160 --> 00:51:26,160 Speaker 1: If you tried to make such a list. Cantor showed 916 00:51:26,200 --> 00:51:29,160 Speaker 1: that you can always point out real numbers that don't 917 00:51:29,200 --> 00:51:32,359 Speaker 1: appear on the list. So how would you show that? Well, 918 00:51:32,400 --> 00:51:34,920 Speaker 1: for a simplified version, and this is a kind of 919 00:51:35,000 --> 00:51:37,359 Speaker 1: dumb down version, Cantor was trying to do this with 920 00:51:37,400 --> 00:51:41,080 Speaker 1: like binary numbers, but uh, for for a simplified version, 921 00:51:41,239 --> 00:51:44,320 Speaker 1: imagine trying to make a list of all real numbers 922 00:51:44,640 --> 00:51:47,919 Speaker 1: lining up their decimal values. So you've got like one 923 00:51:47,960 --> 00:51:50,120 Speaker 1: point three, four, five, six, seven, eight, nine, ten, and 924 00:51:50,160 --> 00:51:53,359 Speaker 1: then one point seven to five. What you know like 925 00:51:53,440 --> 00:51:56,680 Speaker 1: that where you line up their digits in columns, and 926 00:51:56,760 --> 00:51:59,399 Speaker 1: Cantor said, whatever is in that list, I can find 927 00:51:59,440 --> 00:52:02,120 Speaker 1: a number that it's not already in it. And you 928 00:52:02,160 --> 00:52:03,919 Speaker 1: can try to make a list like that that goes 929 00:52:03,960 --> 00:52:06,439 Speaker 1: on forever. But no matter how you do it, I'll 930 00:52:06,520 --> 00:52:10,120 Speaker 1: find numbers that are real numbers that aren't on the list. 931 00:52:10,800 --> 00:52:13,280 Speaker 1: And the way he did this is is this genius 932 00:52:13,320 --> 00:52:16,200 Speaker 1: thing where he went down the list diagonally. Now we 933 00:52:16,239 --> 00:52:19,560 Speaker 1: already had a kind of diagonal squirmy line thing through 934 00:52:19,600 --> 00:52:22,520 Speaker 1: the rational numbers, but this is a straight diagonal line 935 00:52:22,800 --> 00:52:25,360 Speaker 1: through all these numbers that are listed. So what you 936 00:52:25,480 --> 00:52:27,640 Speaker 1: do is you take the first decimal digit of the 937 00:52:27,719 --> 00:52:30,319 Speaker 1: first number, and then the second decimal digit of the 938 00:52:30,320 --> 00:52:32,920 Speaker 1: second number, and then the third decimal digit of the 939 00:52:32,960 --> 00:52:35,839 Speaker 1: third number, and then take each of those digits and 940 00:52:35,960 --> 00:52:38,960 Speaker 1: change them into something else, and then put them in 941 00:52:39,040 --> 00:52:43,080 Speaker 1: sequence to create another decimal number. Now, by definition, this 942 00:52:43,200 --> 00:52:46,680 Speaker 1: number is not on your list. It can't be, because 943 00:52:46,760 --> 00:52:51,000 Speaker 1: you guaranteed at least one decimal place is off from 944 00:52:51,040 --> 00:52:53,759 Speaker 1: every possible number on the list. So no matter what, 945 00:52:54,120 --> 00:52:57,560 Speaker 1: an attempt to count real numbers fails, you can't possibly 946 00:52:57,680 --> 00:53:00,560 Speaker 1: create an ordered way of listing them all. All this 947 00:53:00,640 --> 00:53:05,560 Speaker 1: means they're not countable infinities. There's an uncountable infinity of them. 948 00:53:05,600 --> 00:53:08,359 Speaker 1: And thus you can prove that some infinities actually are 949 00:53:08,400 --> 00:53:11,840 Speaker 1: bigger than up than others. There are more real numbers 950 00:53:11,920 --> 00:53:15,520 Speaker 1: than there are rational numbers or natural numbers. Now, let's 951 00:53:15,520 --> 00:53:18,240 Speaker 1: try to bring this back to a concrete example, because 952 00:53:18,480 --> 00:53:20,240 Speaker 1: I know we've we've been in the math for a while. 953 00:53:20,360 --> 00:53:22,759 Speaker 1: I apologize for that, but I did want to try 954 00:53:22,760 --> 00:53:24,880 Speaker 1: to set that straight here. So do you want to 955 00:53:24,880 --> 00:53:28,240 Speaker 1: talk about Connor McCloud. Sure, yeah, How does Connor McLoud 956 00:53:28,239 --> 00:53:32,120 Speaker 1: fit into office? Then let me get because at this 957 00:53:32,160 --> 00:53:36,319 Speaker 1: point I'm no longer sure anymore. Certainly, I would think 958 00:53:36,640 --> 00:53:40,799 Speaker 1: during the course of a normal immortals life, ah, he 959 00:53:40,960 --> 00:53:44,759 Speaker 1: is pooping more than he is beheading. But how do 960 00:53:44,800 --> 00:53:48,080 Speaker 1: we compare those two infinities now if he's living forever, well, 961 00:53:48,120 --> 00:53:51,120 Speaker 1: that is a really difficult question. I am also in 962 00:53:51,200 --> 00:53:53,840 Speaker 1: a in a strange place with you here now, because 963 00:53:54,480 --> 00:53:58,120 Speaker 1: it's clear that the hair in this race, the one 964 00:53:58,160 --> 00:54:01,560 Speaker 1: that accumulates faster, is that the highlander goes to the 965 00:54:01,560 --> 00:54:05,440 Speaker 1: bathroom more often than he beheads. People. And if you 966 00:54:05,520 --> 00:54:09,040 Speaker 1: just keep watching this process forever, any time you stop 967 00:54:09,120 --> 00:54:12,120 Speaker 1: to check how many times each of these things has happened, 968 00:54:12,400 --> 00:54:15,239 Speaker 1: the bathroom visits will be larger, and it will just 969 00:54:15,320 --> 00:54:20,040 Speaker 1: keep getting relatively larger. It'll go on like that unless 970 00:54:20,080 --> 00:54:23,480 Speaker 1: you say it goes on forever. Now, it's hard to 971 00:54:23,480 --> 00:54:25,960 Speaker 1: say exactly what it would mean to say a sequence 972 00:54:26,000 --> 00:54:29,680 Speaker 1: like that goes on forever, because you're talking about organisms 973 00:54:29,680 --> 00:54:32,399 Speaker 1: that live in a universe that you know, even if 974 00:54:32,400 --> 00:54:34,560 Speaker 1: you call him immortal, you'd think at some point he'd 975 00:54:34,640 --> 00:54:37,560 Speaker 1: run out of usable energy in the universe. It's Highlander. 976 00:54:38,640 --> 00:54:40,640 Speaker 1: With all things Highlander, it's best just not to ask 977 00:54:40,680 --> 00:54:44,440 Speaker 1: too many questions about it. But we're in this weird conundrum, 978 00:54:44,560 --> 00:54:47,800 Speaker 1: right because we've discovered that if it were actually possible 979 00:54:47,840 --> 00:54:50,920 Speaker 1: for this to go on forever, whatever that means for 980 00:54:51,000 --> 00:54:54,200 Speaker 1: events in real time, if it were to go on forever, 981 00:54:54,560 --> 00:54:58,680 Speaker 1: then he would do those these different activities equally. But 982 00:54:59,600 --> 00:55:01,640 Speaker 1: I don't know that it makes any sense to say 983 00:55:01,719 --> 00:55:06,359 Speaker 1: events in the physical space go on forever in the 984 00:55:06,400 --> 00:55:09,279 Speaker 1: same way that say a list of integers in an 985 00:55:09,320 --> 00:55:12,800 Speaker 1: infinite set can go on forever. Yeah, I mean it, 986 00:55:12,680 --> 00:55:15,560 Speaker 1: it comes back once again. To the idea that our lives, 987 00:55:15,560 --> 00:55:19,880 Speaker 1: our experience, our world is finite numbers. This thing that 988 00:55:20,000 --> 00:55:24,200 Speaker 1: we have either created to match up with the with 989 00:55:24,400 --> 00:55:27,480 Speaker 1: the the universe, or that we have discovered in the 990 00:55:27,520 --> 00:55:30,840 Speaker 1: fabric of the universe. These go on forever, These have 991 00:55:30,920 --> 00:55:34,120 Speaker 1: true infinity. Yeah, I think that's a really great point. 992 00:55:34,320 --> 00:55:38,080 Speaker 1: But then again, what if the laws of physics don't 993 00:55:38,120 --> 00:55:42,239 Speaker 1: tell you that you can ever stop counting? You know 994 00:55:42,320 --> 00:55:44,279 Speaker 1: what if you look at the laws of physics and 995 00:55:44,280 --> 00:55:46,400 Speaker 1: you look at the universe and you say, I don't 996 00:55:46,440 --> 00:55:51,279 Speaker 1: find anything that says time stops. So what do you do? 997 00:55:51,360 --> 00:55:55,719 Speaker 1: Then you keep counting? I guess you keep pooping, you 998 00:55:55,800 --> 00:55:59,520 Speaker 1: keep beheading, and you just and things balance out. Right, 999 00:56:00,080 --> 00:56:02,799 Speaker 1: Then that seems like you've got to wonder, like, does 1000 00:56:02,920 --> 00:56:07,719 Speaker 1: that actually undercut the Boltzman brain argument then, because if 1001 00:56:07,800 --> 00:56:11,640 Speaker 1: you actually have a universe going on forever, obviously you 1002 00:56:11,760 --> 00:56:15,040 Speaker 1: have early universes where there is low entropy, and you've 1003 00:56:15,040 --> 00:56:17,399 Speaker 1: got all these stars and planets and we're we're all 1004 00:56:17,480 --> 00:56:19,920 Speaker 1: we're making aliens, you know, making all kinds of crazy 1005 00:56:20,000 --> 00:56:24,359 Speaker 1: organisms evolving through biological evolution, biochemistry, and you've got all 1006 00:56:24,400 --> 00:56:27,359 Speaker 1: these ordinary observers, and then you've got this long dead 1007 00:56:27,440 --> 00:56:30,600 Speaker 1: period where those things die out and you're generating boltzman 1008 00:56:30,719 --> 00:56:33,840 Speaker 1: brains out in space. But if it really all goes 1009 00:56:33,920 --> 00:56:36,760 Speaker 1: on forever, then it kind of doesn't matter how many 1010 00:56:36,880 --> 00:56:40,480 Speaker 1: of those things happen at either point, right, because it 1011 00:56:40,480 --> 00:56:44,440 Speaker 1: would just keep recurring. You'd eventually get an entropy fluctuation 1012 00:56:44,520 --> 00:56:46,920 Speaker 1: that would take you back to something like a big bang, 1013 00:56:47,360 --> 00:56:50,279 Speaker 1: and you just like started all over again, and then 1014 00:56:50,320 --> 00:56:53,120 Speaker 1: this would go on forever and you cannot compare them. 1015 00:56:53,560 --> 00:56:56,719 Speaker 1: That's true. So I actually wrote to Sean Carroll, did 1016 00:56:56,760 --> 00:56:59,680 Speaker 1: you ask him about Highlander infinity to poop? And it 1017 00:57:00,000 --> 00:57:02,319 Speaker 1: didn't mention a highlander. I don't know if he would 1018 00:57:02,320 --> 00:57:04,880 Speaker 1: have replied to me if I mentioned highlander, but he 1019 00:57:04,880 --> 00:57:08,400 Speaker 1: he was incredibly generous with his time to to respond 1020 00:57:08,440 --> 00:57:11,640 Speaker 1: to me. We really appreciate that. But yeah, he he 1021 00:57:11,719 --> 00:57:16,920 Speaker 1: basically acknowledged this. So Dr Carroll responded to exactly this 1022 00:57:17,000 --> 00:57:19,480 Speaker 1: kind of question and said that it's true that two 1023 00:57:19,520 --> 00:57:23,160 Speaker 1: accountable infinities are equal to each other. He said, however, 1024 00:57:23,400 --> 00:57:25,960 Speaker 1: it's also true that one is five times bigger than 1025 00:57:26,000 --> 00:57:28,520 Speaker 1: the other, or tend to a hundred times bigger than 1026 00:57:28,560 --> 00:57:31,080 Speaker 1: the other. That's how infinity works. And I think he's 1027 00:57:31,080 --> 00:57:33,760 Speaker 1: recognizing the problem there that like, you'll have these things 1028 00:57:33,760 --> 00:57:37,520 Speaker 1: where frequencies are obviously very different, but if you truly 1029 00:57:37,600 --> 00:57:41,560 Speaker 1: extend them to infinite sets of things, then they're accountably 1030 00:57:41,600 --> 00:57:45,160 Speaker 1: the same. So he says, quote, so comparing infinities is 1031 00:57:45,200 --> 00:57:47,440 Speaker 1: clearly not what you want to do. If what you 1032 00:57:47,480 --> 00:57:50,680 Speaker 1: care about is the relative frequencies of two kinds of events, 1033 00:57:51,040 --> 00:57:54,000 Speaker 1: you have two options. One throw up your hands and 1034 00:57:54,040 --> 00:57:57,880 Speaker 1: say there is no way to compare or to regularize, 1035 00:57:58,320 --> 00:58:01,400 Speaker 1: i e. Consider only a five night region of space time, 1036 00:58:01,480 --> 00:58:04,640 Speaker 1: so that all numbers are finite. Calculate the ratio of 1037 00:58:04,680 --> 00:58:08,160 Speaker 1: Boltzman brains to ordinary observers, and then take the limit 1038 00:58:08,280 --> 00:58:11,480 Speaker 1: as that region gets infinitely big. If you do the ladder, 1039 00:58:11,640 --> 00:58:15,720 Speaker 1: you will generally find Boltzman brains vastly outnumber ordinary observers. 1040 00:58:15,760 --> 00:58:18,560 Speaker 1: And and this is what they do in their papers. Now, 1041 00:58:18,600 --> 00:58:21,440 Speaker 1: to be clear, Carol is not saying he thinks we 1042 00:58:21,480 --> 00:58:26,240 Speaker 1: are Boltzman brains. He has physical arguments and to some degree, 1043 00:58:26,280 --> 00:58:29,800 Speaker 1: you might say, philosophical arguments for why we are not 1044 00:58:29,880 --> 00:58:32,760 Speaker 1: Boltzman brains, and we discuss those in the previous episode, 1045 00:58:32,800 --> 00:58:34,800 Speaker 1: and one of them was that he says, you know, 1046 00:58:35,080 --> 00:58:38,960 Speaker 1: assuming you are a Boltzman brain is cognitively unstable. It 1047 00:58:39,040 --> 00:58:42,160 Speaker 1: doesn't make any sense because what's your reason for thinking 1048 00:58:42,200 --> 00:58:45,720 Speaker 1: you're a Boltzman brain. It's like all this science and 1049 00:58:45,840 --> 00:58:48,480 Speaker 1: math and stuff that we only know because we think 1050 00:58:48,480 --> 00:58:51,600 Speaker 1: we have an accurate picture of the universe. If you 1051 00:58:51,720 --> 00:58:54,000 Speaker 1: were a Boltzman brain, you'd have no reason to think 1052 00:58:54,040 --> 00:58:56,640 Speaker 1: you had an accurate picture of the universe, and thus 1053 00:58:56,720 --> 00:58:59,680 Speaker 1: the method that you use for arriving at the conclusion 1054 00:58:59,680 --> 00:59:03,520 Speaker 1: that you're Boltzman brain would be totally worthless. So you're saying, 1055 00:59:03,520 --> 00:59:09,560 Speaker 1: there's a chance, do you everything about how subversive the 1056 00:59:09,720 --> 00:59:14,720 Speaker 1: slogan to infinity and beyond is? Yes? Unfortunately, um, it 1057 00:59:14,880 --> 00:59:17,840 Speaker 1: kept popping up during preparation for this. I'm like reading 1058 00:59:17,840 --> 00:59:21,600 Speaker 1: about these like that the Jane concepts of infinity, and 1059 00:59:21,640 --> 00:59:24,520 Speaker 1: it's just that that that stupid catchphrase from toy story 1060 00:59:24,640 --> 00:59:26,680 Speaker 1: sounding off in the back of my head, and and 1061 00:59:26,720 --> 00:59:30,160 Speaker 1: it gives me nothing. It provides no insight into what 1062 00:59:30,320 --> 00:59:32,400 Speaker 1: I'm and what into what I'm trying to wrap my 1063 00:59:32,440 --> 00:59:34,640 Speaker 1: head around. But it keeps going off like a like 1064 00:59:34,680 --> 00:59:39,680 Speaker 1: a malfunctioning and sprinkler or something that Yeah, I I 1065 00:59:39,960 --> 00:59:42,320 Speaker 1: can understand that. So I'm not one of these people 1066 00:59:42,360 --> 00:59:44,960 Speaker 1: who had like a deep emotional experience as an adult 1067 00:59:45,000 --> 00:59:47,120 Speaker 1: watching a Toy Story movie, because I don't think I 1068 00:59:47,120 --> 00:59:49,320 Speaker 1: ever saw anything after the first movie when I was 1069 00:59:49,320 --> 00:59:51,880 Speaker 1: a kid. But good, well, yeah I've heard that. But 1070 00:59:51,920 --> 00:59:54,960 Speaker 1: I remember that first movie and I didn't think about 1071 00:59:55,000 --> 00:59:57,760 Speaker 1: it back then. But now I'm thinking, wait, to infinity 1072 00:59:57,800 --> 01:00:02,960 Speaker 1: and beyond that's infinity plus one. Yeah, maybe he's he was. 1073 01:00:03,120 --> 01:00:06,000 Speaker 1: He was really contemplating infinity on a level that we 1074 01:00:06,120 --> 01:00:12,040 Speaker 1: just weren't prepared for. To infinity and incoherence to infinity 1075 01:00:12,040 --> 01:00:14,960 Speaker 1: and stay there. Well, anyway, I guess that's gonna have 1076 01:00:15,040 --> 01:00:16,960 Speaker 1: to wrap us up for today. I think we are 1077 01:00:17,040 --> 01:00:19,280 Speaker 1: we are out of time. Unfortunately, I wanted to get 1078 01:00:19,280 --> 01:00:22,120 Speaker 1: to some of the emails we've gotten about boltzmon Brains, 1079 01:00:22,120 --> 01:00:24,360 Speaker 1: but we do not have time to address the finite 1080 01:00:24,400 --> 01:00:27,960 Speaker 1: time in this podcast. We sadly do not have infinite time. 1081 01:00:28,240 --> 01:00:32,240 Speaker 1: And I'm sure also you have finite patients for for 1082 01:00:32,240 --> 01:00:35,400 Speaker 1: for infinity. You know, there's a fine line between things 1083 01:00:35,520 --> 01:00:38,520 Speaker 1: that are the most fascinating on Earth and things that 1084 01:00:38,760 --> 01:00:42,360 Speaker 1: really start to grade on your brain. It's it's like 1085 01:00:42,400 --> 01:00:44,120 Speaker 1: pulling on the tail of a of an of an 1086 01:00:44,120 --> 01:00:46,800 Speaker 1: infinite serpent, right You're never going to reach that point 1087 01:00:46,840 --> 01:00:49,800 Speaker 1: where you reach the you you find the creature's head 1088 01:00:49,800 --> 01:00:52,280 Speaker 1: and have a complete understanding of its anatomy. You're just 1089 01:00:52,280 --> 01:00:54,919 Speaker 1: gonna keep tugging and you will get pooped on. Yeah, 1090 01:00:54,920 --> 01:00:56,960 Speaker 1: you probably get pooped on because you really chose the 1091 01:00:57,000 --> 01:01:01,240 Speaker 1: wrong end to to tug. But I still think we 1092 01:01:01,240 --> 01:01:02,880 Speaker 1: we managed to have a good time here. We got 1093 01:01:02,920 --> 01:01:05,960 Speaker 1: to discuss uh infinity and a little more depth. We 1094 01:01:06,080 --> 01:01:09,600 Speaker 1: got to uh iron out some of the lingering questions 1095 01:01:09,600 --> 01:01:12,360 Speaker 1: remaining Boltzman brains, not all of them, because the mere 1096 01:01:12,480 --> 01:01:14,920 Speaker 1: concept of the Boltzman brain is kind of meant to 1097 01:01:15,480 --> 01:01:19,680 Speaker 1: uh to stir prolonged contemplation. Yeah, uh, And we I 1098 01:01:19,800 --> 01:01:21,760 Speaker 1: do not want you to walk away with the impression 1099 01:01:21,760 --> 01:01:24,320 Speaker 1: today that we're advocating the belief that you are a 1100 01:01:24,320 --> 01:01:27,120 Speaker 1: Boltzman brain. If you haven't heard that first episode, you 1101 01:01:27,120 --> 01:01:29,080 Speaker 1: should go back and listen to that where we actually 1102 01:01:29,080 --> 01:01:32,120 Speaker 1: talk about reasons you're not a Boltzman brain. Come on, yeah, 1103 01:01:32,440 --> 01:01:35,200 Speaker 1: don't uh don't lose any sleep over it. All right, Well, 1104 01:01:35,680 --> 01:01:37,760 Speaker 1: on that note, it's time for us to close out here. 1105 01:01:38,520 --> 01:01:41,240 Speaker 1: As always, I would like to direct you to stuff 1106 01:01:41,240 --> 01:01:43,040 Speaker 1: to blow your mind dot com. That's the mother ship 1107 01:01:43,080 --> 01:01:45,959 Speaker 1: that's we will find all the podcast episodes and links 1108 01:01:45,960 --> 01:01:48,200 Speaker 1: out to our various social media accounts. I'll ask you 1109 01:01:48,240 --> 01:01:50,480 Speaker 1: too that if you want to support the show, a 1110 01:01:50,520 --> 01:01:52,560 Speaker 1: great way to do it is to rate and review 1111 01:01:52,960 --> 01:01:55,960 Speaker 1: wherever you get this podcast. Thanks so much, as always 1112 01:01:56,040 --> 01:01:59,560 Speaker 1: to our excellent audio producers, Alex Williams and Tarry Harrison. 1113 01:01:59,720 --> 01:02:01,560 Speaker 1: If you'd like to get in touch with us with 1114 01:02:01,720 --> 01:02:04,880 Speaker 1: feedback on this episode or any other, just to say hi, 1115 01:02:05,080 --> 01:02:06,720 Speaker 1: let us know how you're doing, how you found out 1116 01:02:06,720 --> 01:02:08,840 Speaker 1: about the show, to suggest a topic for the future, 1117 01:02:08,920 --> 01:02:11,200 Speaker 1: any of that good stuff, you can always email us 1118 01:02:11,240 --> 01:02:24,240 Speaker 1: at blow the Mind at how stuff works dot com. 1119 01:02:24,240 --> 01:02:26,680 Speaker 1: Well more on this and thousands of other topics. Was 1120 01:02:26,720 --> 01:02:50,800 Speaker 1: it how stuff works dot com