1 00:00:00,500 --> 00:00:02,890 The following content is provided under a Creative 2 00:00:02,890 --> 00:00:04,430 Commons license. 3 00:00:04,430 --> 00:00:06,730 Your support will help MIT OpenCourseWare 4 00:00:06,730 --> 00:00:11,120 continue to offer high quality educational resources for free. 5 00:00:11,120 --> 00:00:13,720 To make a donation or view additional materials 6 00:00:13,720 --> 00:00:17,680 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,680 --> 00:00:18,850 at ocw.mit.edu. 8 00:00:20,997 --> 00:00:21,830 PROFESSOR: All guys. 9 00:00:21,830 --> 00:00:24,760 I'm going to go ahead and get started here. 10 00:00:24,760 --> 00:00:26,830 Sorry about missing last time, I had 11 00:00:26,830 --> 00:00:29,552 to go home and visit the family. 12 00:00:29,552 --> 00:00:30,760 It's been a couple of months. 13 00:00:34,120 --> 00:00:35,620 Today's going to be kind of a review 14 00:00:35,620 --> 00:00:37,060 session of a bunch of things. 15 00:00:37,060 --> 00:00:41,050 And we're also going to go through a dialogue, probably 16 00:00:41,050 --> 00:00:46,240 my favorite dialogue, maybe two, a little harmonic labyrinth. 17 00:00:46,240 --> 00:00:48,760 But I want to start out with kind 18 00:00:48,760 --> 00:00:50,290 of just entertaining any questions 19 00:00:50,290 --> 00:00:53,020 that people might have, burning to ask me right away, 20 00:00:53,020 --> 00:00:56,557 confusion over the past two lectures, or what else? 21 00:00:56,557 --> 00:00:57,140 What have you? 22 00:01:01,020 --> 00:01:02,570 Anything? 23 00:01:02,570 --> 00:01:03,070 All right. 24 00:01:03,070 --> 00:01:04,989 I'm sure questions will develop. 25 00:01:04,989 --> 00:01:05,710 All right. 26 00:01:05,710 --> 00:01:08,230 So chapter four, which I asked you 27 00:01:08,230 --> 00:01:10,990 guys to have read for the previous lecture, 28 00:01:10,990 --> 00:01:13,180 even though an entire lecture on recursion, 29 00:01:13,180 --> 00:01:15,820 was about three things-- 30 00:01:15,820 --> 00:01:30,560 consistency, completeness, and geometry. 31 00:01:34,970 --> 00:01:38,520 I'm just going to quickly define these three terms as much 32 00:01:38,520 --> 00:01:43,050 as they can be defined, and then lead into what 33 00:01:43,050 --> 00:01:45,060 the whole point of chapter four was about, 34 00:01:45,060 --> 00:01:48,180 and what the [INAUDIBLE] was trying to introduce you to. 35 00:01:48,180 --> 00:01:51,420 And that's really kind of getting these two to go 36 00:01:51,420 --> 00:01:56,020 to Godel's theorem. 37 00:02:03,840 --> 00:02:07,810 So can anyone tell me what consistency means? 38 00:02:12,210 --> 00:02:12,710 Anyone? 39 00:02:12,710 --> 00:02:13,210 Sure. 40 00:02:16,140 --> 00:02:22,520 AUDIENCE: The set is like no theorems contradict each other. 41 00:02:22,520 --> 00:02:23,590 PROFESSOR: Yeah, exactly. 42 00:02:23,590 --> 00:02:27,200 So what Felix said was no theorems contradict each other. 43 00:02:27,200 --> 00:02:28,930 So basically, if we were to put this 44 00:02:28,930 --> 00:02:32,800 in terms of a formal system, if we're deriving things, 45 00:02:32,800 --> 00:02:37,060 if we're playing with mu or whatever formal system 46 00:02:37,060 --> 00:02:39,550 we had, and we happen to derive a proposition, 47 00:02:39,550 --> 00:02:43,540 p, we couldn't somehow simultaneously derive 48 00:02:43,540 --> 00:02:46,690 from our set of axioms p and-- 49 00:02:46,690 --> 00:02:51,820 this means and-- not p, that being not p. 50 00:02:51,820 --> 00:02:55,980 So basically, if we have a set of operating assumptions 51 00:02:55,980 --> 00:02:58,540 and we are trying to somehow formally predict 52 00:02:58,540 --> 00:03:02,530 today's weather, and somehow the computer spit out, well, 53 00:03:02,530 --> 00:03:05,140 today it's going to rain and not rain 54 00:03:05,140 --> 00:03:09,700 at the same time, that would be an example 55 00:03:09,700 --> 00:03:11,194 of an inconsistent system. 56 00:03:11,194 --> 00:03:13,360 So you have things where you derived a contradiction 57 00:03:13,360 --> 00:03:15,820 directly. 58 00:03:15,820 --> 00:03:19,990 And the thing which is really interesting about this-- 59 00:03:19,990 --> 00:03:22,270 and I'm not going to go into all the details of it-- 60 00:03:22,270 --> 00:03:25,010 but if your formal system produces anywhere in it 61 00:03:25,010 --> 00:03:28,180 a contradiction, you can derive anything. 62 00:03:28,180 --> 00:03:31,280 And this is really kind of a bad thing. 63 00:03:31,280 --> 00:03:34,030 And a lot of philosophers have grappled with this question. 64 00:03:34,030 --> 00:03:38,260 Why is it that if we derive a statement, say, 65 00:03:38,260 --> 00:03:41,140 this table is red and not red, how 66 00:03:41,140 --> 00:03:45,460 can we deduce from that that the universe is infinite, right? 67 00:03:45,460 --> 00:03:47,930 But somehow when you have a contradiction, 68 00:03:47,930 --> 00:03:48,930 everything goes haywire. 69 00:03:48,930 --> 00:03:52,930 You can derive anything and problems abound. 70 00:03:52,930 --> 00:03:59,230 And this is going to be one of the things which Godel's two 71 00:03:59,230 --> 00:04:01,802 incompleteness theorems will tell us about. 72 00:04:01,802 --> 00:04:03,760 But of course, I use the word "incompleteness." 73 00:04:03,760 --> 00:04:05,170 What does "completeness" mean? 74 00:04:05,170 --> 00:04:08,740 And this is kind of a harder concept to get across. 75 00:04:08,740 --> 00:04:12,080 And it's really, really counter-intuitive at first. 76 00:04:12,080 --> 00:04:15,020 And that's something I want to talk about today. 77 00:04:15,020 --> 00:04:17,561 Does anyone have a good definition of completeness? 78 00:04:24,180 --> 00:04:25,290 Any takers? 79 00:04:25,290 --> 00:04:26,570 Sandra? 80 00:04:26,570 --> 00:04:28,590 You have an idea? 81 00:04:28,590 --> 00:04:30,030 OK. 82 00:04:30,030 --> 00:04:31,050 That's all right. 83 00:04:31,050 --> 00:04:35,360 So completeness is, I think, probably one 84 00:04:35,360 --> 00:04:36,360 of the hardest concepts. 85 00:04:36,360 --> 00:04:41,070 And it has to go back to a picture which I drew. 86 00:04:41,070 --> 00:04:42,000 Hello. 87 00:04:42,000 --> 00:04:44,270 Go ahead and come on in. 88 00:04:44,270 --> 00:04:45,910 I've got a handout for you. 89 00:04:49,910 --> 00:04:53,402 So completeness goes back to a picture 90 00:04:53,402 --> 00:04:54,860 I drew on the first day of lecture. 91 00:04:54,860 --> 00:04:56,580 And I don't know if you guys remember it. 92 00:04:56,580 --> 00:04:58,413 But it actually appears in a chapter which I 93 00:04:58,413 --> 00:04:59,960 didn't assign you all to read. 94 00:04:59,960 --> 00:05:05,350 And it's that idea that if we had a truth box 95 00:05:05,350 --> 00:05:10,330 and this truth box was somehow a graphical display of all 96 00:05:10,330 --> 00:05:13,136 of our theorems and the things we could prove, 97 00:05:13,136 --> 00:05:14,635 but things we also knew to be true-- 98 00:05:18,830 --> 00:05:22,040 so let's take this to be the true box. 99 00:05:22,040 --> 00:05:24,090 And let's take this to be the not true box. 100 00:05:26,850 --> 00:05:30,750 And as we talked in the past couple days, 101 00:05:30,750 --> 00:05:37,960 if we had some axioms, principle points 102 00:05:37,960 --> 00:05:41,380 to start building truths from, we 103 00:05:41,380 --> 00:05:50,280 can derive all sorts of ideas from these axioms. 104 00:05:50,280 --> 00:05:54,440 And this is just kind of a weird graphical tree of deductions, 105 00:05:54,440 --> 00:05:54,940 right? 106 00:05:54,940 --> 00:05:58,180 Just like when we were playing with the mu system, 107 00:05:58,180 --> 00:06:02,459 we started with mi, and then that was our axiom. 108 00:06:02,459 --> 00:06:04,500 And then we applied all of our rules of inference 109 00:06:04,500 --> 00:06:06,930 to get all the possible theorems here. 110 00:06:06,930 --> 00:06:10,820 And these are things which are provable. 111 00:06:10,820 --> 00:06:13,680 Sorry if this is incomprehensibly small. 112 00:06:13,680 --> 00:06:16,704 But it says "provable." 113 00:06:16,704 --> 00:06:18,120 But then we have all of this space 114 00:06:18,120 --> 00:06:22,080 here, which we already said were true things 115 00:06:22,080 --> 00:06:24,030 but provable things. 116 00:06:24,030 --> 00:06:27,110 And this is really kind of a counter-intuitive idea. 117 00:06:27,110 --> 00:06:30,360 And some of you might go, yeah, that's exactly 118 00:06:30,360 --> 00:06:32,394 captures what I feel. 119 00:06:32,394 --> 00:06:33,810 And that's the idea that there are 120 00:06:33,810 --> 00:06:38,910 truths, things we know to be true, which aren't provable. 121 00:06:38,910 --> 00:06:42,630 And this is really kind of hard to wrap your head around. 122 00:06:42,630 --> 00:06:45,150 Suddenly we have things which we know are true, 123 00:06:45,150 --> 00:06:46,890 but how do we know they are true? 124 00:06:46,890 --> 00:06:48,840 It's not that we have proof of them, 125 00:06:48,840 --> 00:06:51,000 we just know that they are true. 126 00:06:51,000 --> 00:06:54,850 And this really is going to go into Godel's theorem big time, 127 00:06:54,850 --> 00:06:58,360 Godel's two incompleteness theorems. 128 00:06:58,360 --> 00:07:02,460 And so completeness, if you want to give it a short definition, 129 00:07:02,460 --> 00:07:07,890 is that every true system is derive-able from the axioms, 130 00:07:07,890 --> 00:07:08,900 or from the system. 131 00:07:08,900 --> 00:07:11,400 There is no incompleteness. 132 00:07:11,400 --> 00:07:13,920 Here, based on this graphical drawing here, 133 00:07:13,920 --> 00:07:15,600 you've got obvious incompleteness. 134 00:07:15,600 --> 00:07:16,880 We have all of the space. 135 00:07:16,880 --> 00:07:19,167 We have all of these true statements which aren't 136 00:07:19,167 --> 00:07:20,250 reachable from our axioms. 137 00:07:23,630 --> 00:07:27,260 Can anyone think of something which might be true but not 138 00:07:27,260 --> 00:07:29,762 provable, or they have an idea? 139 00:07:29,762 --> 00:07:32,814 AUDIENCE: Like if there's anything outside the universe? 140 00:07:32,814 --> 00:07:34,980 PROFESSOR: If there's anything outside the universe. 141 00:07:34,980 --> 00:07:37,160 Exactly. 142 00:07:37,160 --> 00:07:38,090 That's one idea. 143 00:07:38,090 --> 00:07:41,220 I was actually just reading the other day Seth Lloyd's 144 00:07:41,220 --> 00:07:42,770 Programming of the Universe. 145 00:07:42,770 --> 00:07:48,290 That the universe, if we kind look at it, 146 00:07:48,290 --> 00:07:51,470 expanded like so from a big bang. 147 00:07:51,470 --> 00:07:54,590 But the rate at which it expands is four times the speed 148 00:07:54,590 --> 00:07:56,780 of light. 149 00:07:56,780 --> 00:07:59,720 Of course, the only things we can perceive travel as fast 150 00:07:59,720 --> 00:08:01,140 as the speed of light. 151 00:08:01,140 --> 00:08:02,560 So we've got this light cone. 152 00:08:06,420 --> 00:08:11,150 So everything inside this shaded region 153 00:08:11,150 --> 00:08:13,450 are things which we can perceive. 154 00:08:13,450 --> 00:08:17,360 But ultimately, the universe is expanding faster than that. 155 00:08:17,360 --> 00:08:20,710 So there's all sorts of these things about the universe which 156 00:08:20,710 --> 00:08:22,750 we'll never know. 157 00:08:22,750 --> 00:08:26,470 And that's a nice physical example. 158 00:08:26,470 --> 00:08:29,295 But it's not exactly what I mean in terms of formal systems 159 00:08:29,295 --> 00:08:31,420 and completeness, and things have been true but not 160 00:08:31,420 --> 00:08:31,920 provable. 161 00:08:34,370 --> 00:08:39,020 And this is the heart of Godel's theorem. 162 00:08:39,020 --> 00:08:42,309 So before we get there, I just want 163 00:08:42,309 --> 00:08:45,070 to talk briefly about geometry. 164 00:08:48,124 --> 00:08:50,290 We're going to meet it in a variety of two settings. 165 00:08:50,290 --> 00:08:52,584 In the chapter, you met Euclidean, non-Euclidean. 166 00:08:52,584 --> 00:08:54,250 And I want to elaborate on that and show 167 00:08:54,250 --> 00:08:55,720 you some of the cool things. 168 00:08:55,720 --> 00:09:02,236 So I'm going to have Euclidean and not Euclidean. 169 00:09:06,900 --> 00:09:08,070 But we'll get back to that. 170 00:09:08,070 --> 00:09:09,444 Right now, I kind of want to harp 171 00:09:09,444 --> 00:09:12,720 on what Godel's theorem is, who this guy, Kurt Godel, was, 172 00:09:12,720 --> 00:09:17,610 and why it is that we have one of our three title names named 173 00:09:17,610 --> 00:09:19,660 after him. 174 00:09:19,660 --> 00:09:24,850 So Kurt Godel was born in Vienna. 175 00:09:24,850 --> 00:09:26,890 He was a mathematician. 176 00:09:26,890 --> 00:09:32,250 And he grew up in a time where mathematics was being directly 177 00:09:32,250 --> 00:09:35,730 influenced by, really, a variety of paradoxes 178 00:09:35,730 --> 00:09:37,140 which were popping up. 179 00:09:37,140 --> 00:09:39,360 And a man named David Hilbert, who 180 00:09:39,360 --> 00:09:42,390 really wanted to clear up all these ideas of paradoxes 181 00:09:42,390 --> 00:09:45,600 which arrived in mathematics-- 182 00:09:45,600 --> 00:09:47,970 David Hilbert felt that mathematics 183 00:09:47,970 --> 00:09:50,940 was our most sure and certain source of knowledge, 184 00:09:50,940 --> 00:09:53,430 that if there was any flaws in mathematics, 185 00:09:53,430 --> 00:09:55,380 we were doomed as far as human beings 186 00:09:55,380 --> 00:09:58,980 in terms of knowing true things. 187 00:09:58,980 --> 00:10:03,540 And the paradoxes which I speak of really 188 00:10:03,540 --> 00:10:05,890 refer to two main things. 189 00:10:05,890 --> 00:10:09,060 One is this issue of Euclidean versus not Euclidean. 190 00:10:09,060 --> 00:10:11,910 This was a revolution which started happening in the 1800s, 191 00:10:11,910 --> 00:10:14,490 and people slowly started dealing with over those 100 192 00:10:14,490 --> 00:10:19,131 years entering 1900. 193 00:10:19,131 --> 00:10:20,880 But then there was, right towards the turn 194 00:10:20,880 --> 00:10:24,210 of the century, the two set theory paradoxes. 195 00:10:24,210 --> 00:10:27,160 And these were paradoxes which I talked about earlier. 196 00:10:27,160 --> 00:10:29,960 And that was the idea of the barber paradox. 197 00:10:29,960 --> 00:10:32,460 And for those of you who weren't here for the first lecture, 198 00:10:32,460 --> 00:10:34,110 the barber paradox says, suppose we 199 00:10:34,110 --> 00:10:36,630 have a town where a barber shaves all people, 200 00:10:36,630 --> 00:10:39,480 and only those people, who don't shave themselves. 201 00:10:39,480 --> 00:10:42,870 Well, then does the barber shave himself or does he not? 202 00:10:42,870 --> 00:10:46,715 And by the definition, by the way we set up the town, 203 00:10:46,715 --> 00:10:48,090 it appears to be a contradiction. 204 00:10:48,090 --> 00:10:50,280 Because if the barber does shave himself, then 205 00:10:50,280 --> 00:10:52,530 according to who the barber shaves, 206 00:10:52,530 --> 00:10:54,130 he doesn't shave himself. 207 00:10:54,130 --> 00:10:59,580 And if he doesn't shave himself, then he should. 208 00:10:59,580 --> 00:11:01,440 So it's a contradiction. 209 00:11:01,440 --> 00:11:04,290 And we thought that this was a paradox deriving 210 00:11:04,290 --> 00:11:04,970 from set theory. 211 00:11:04,970 --> 00:11:06,511 And we felt that set theory was going 212 00:11:06,511 --> 00:11:09,484 to be our sure and certain foundation of knowledge. 213 00:11:09,484 --> 00:11:11,400 We thought we could deal with these paradoxes. 214 00:11:11,400 --> 00:11:12,510 David Hilbert was huge. 215 00:11:12,510 --> 00:11:14,010 He said, guys, look. 216 00:11:14,010 --> 00:11:15,630 There is no unknown. 217 00:11:15,630 --> 00:11:19,470 Mathematics has to have a sure and certain foundation. 218 00:11:19,470 --> 00:11:20,940 And we should be able to establish 219 00:11:20,940 --> 00:11:24,210 as a model system consistency and completeness 220 00:11:24,210 --> 00:11:25,470 of number theory. 221 00:11:25,470 --> 00:11:28,740 And he felt that if anything is true, 222 00:11:28,740 --> 00:11:30,630 it's got to be number theory. 223 00:11:30,630 --> 00:11:33,040 And I always like doing this poll. 224 00:11:33,040 --> 00:11:36,130 But I want to ask you guys to vote. 225 00:11:39,370 --> 00:11:43,830 And I want you to decide the truth of the following. 226 00:11:43,830 --> 00:11:50,460 The sky is blue. 227 00:11:50,460 --> 00:11:56,470 And one plus one equals two. 228 00:11:56,470 --> 00:12:01,560 So imagine, of all the possible worlds, 229 00:12:01,560 --> 00:12:04,740 which do you feel like is more true, the fact that the sky is 230 00:12:04,740 --> 00:12:08,900 blue or that one plus one equals two? 231 00:12:08,900 --> 00:12:10,270 AUDIENCE: The second one. 232 00:12:10,270 --> 00:12:13,397 PROFESSOR: So you feel like the second one should be true. 233 00:12:13,397 --> 00:12:15,480 Do you feel like it's true in all possible worlds? 234 00:12:15,480 --> 00:12:16,730 AUDIENCE: It has to be. 235 00:12:16,730 --> 00:12:19,230 PROFESSOR: So you don't think there's any universe out there 236 00:12:19,230 --> 00:12:22,200 where one plus one could not equal two? 237 00:12:22,200 --> 00:12:23,490 OK. 238 00:12:23,490 --> 00:12:24,630 What about anybody else? 239 00:12:24,630 --> 00:12:27,670 Does anyone feel like, no, come on, this isn't even perceptual. 240 00:12:27,670 --> 00:12:29,790 What is this statement about? 241 00:12:29,790 --> 00:12:31,680 It's about these abstract entities 242 00:12:31,680 --> 00:12:35,170 which I just kind of created and wrote down on a piece of paper. 243 00:12:35,170 --> 00:12:37,854 It has absolutely no perceptual foundation. 244 00:12:37,854 --> 00:12:38,520 The sky is blue. 245 00:12:38,520 --> 00:12:40,920 I look outside and what do I see? 246 00:12:40,920 --> 00:12:42,640 I see the sky is blue. 247 00:12:42,640 --> 00:12:48,690 So surely, what I see has to be more true than this. 248 00:12:48,690 --> 00:12:51,090 Is anyone willing to defend "sky is blue" over "one 249 00:12:51,090 --> 00:12:54,305 plus one equals two?" 250 00:12:54,305 --> 00:12:57,050 AUDIENCE: [INAUDIBLE] 251 00:12:57,050 --> 00:12:58,380 PROFESSOR: True. 252 00:12:58,380 --> 00:13:00,500 So Rene Descartes was huge on this. 253 00:13:00,500 --> 00:13:02,240 He was like, what if this is all a dream? 254 00:13:02,240 --> 00:13:04,640 You know, what if this is the matrix, Neo? 255 00:13:04,640 --> 00:13:06,530 Right, like, this is exactly what he said. 256 00:13:06,530 --> 00:13:11,160 But what if we were to base our arithmetic on something else? 257 00:13:11,160 --> 00:13:14,840 Suppose we based our arithmetic on rain drops. 258 00:13:14,840 --> 00:13:21,950 So suppose we have one raindrop and another one. 259 00:13:21,950 --> 00:13:26,540 And we define addition as when they meet. 260 00:13:26,540 --> 00:13:30,210 So one raindrop plus another raindrop 261 00:13:30,210 --> 00:13:33,650 just gives me another raindrop. 262 00:13:33,650 --> 00:13:37,725 So in this system, one plus one equals one. 263 00:13:37,725 --> 00:13:39,200 And what's wrong? 264 00:13:39,200 --> 00:13:41,480 I mean, this is perceptually validated. 265 00:13:41,480 --> 00:13:44,780 When I'm driving in my car at 60 miles an hour and I 266 00:13:44,780 --> 00:13:46,850 have rain hitting my windshield, and I see 267 00:13:46,850 --> 00:13:48,710 raindrops merging together-- 268 00:13:48,710 --> 00:13:52,252 AUDIENCE: Maybe you should measure the size of yours. 269 00:13:52,252 --> 00:13:53,210 PROFESSOR: OK, exactly. 270 00:13:53,210 --> 00:13:54,810 So there's all sorts of problems with identity. 271 00:13:54,810 --> 00:13:57,143 There's problems, maybe, with how the system was formed. 272 00:13:57,143 --> 00:14:00,260 But even in mathematics, we have to be 273 00:14:00,260 --> 00:14:04,070 very clear with what we're stating, what kind of field 274 00:14:04,070 --> 00:14:06,080 we're working over here. 275 00:14:06,080 --> 00:14:09,380 Because suppose I'm actually working over the integers, 276 00:14:09,380 --> 00:14:11,497 mod two, modular arithmetic. 277 00:14:11,497 --> 00:14:13,580 And that just says if it's divisible by something, 278 00:14:13,580 --> 00:14:15,140 I say it's zero. 279 00:14:15,140 --> 00:14:22,010 So when I do modular arithmetic, when I do mod two arithmetic, 280 00:14:22,010 --> 00:14:24,680 two mod two is zero. 281 00:14:24,680 --> 00:14:29,060 So one plus one over the integers mod two is not two, 282 00:14:29,060 --> 00:14:30,671 but it's actually zero. 283 00:14:30,671 --> 00:14:33,170 So I mean, there's all sorts of things we have to deal with. 284 00:14:33,170 --> 00:14:34,461 And there's some uncertainties. 285 00:14:34,461 --> 00:14:36,530 But still, these are all rigorously defined. 286 00:14:36,530 --> 00:14:38,090 I could say, well, I'm just working over the integers. 287 00:14:38,090 --> 00:14:39,410 So I know this is fine. 288 00:14:39,410 --> 00:14:42,230 One plus one is equal to two in all possible worlds. 289 00:14:42,230 --> 00:14:45,569 So Hilbert felt like, really, this has to be it. 290 00:14:45,569 --> 00:14:46,610 Mathematics has to be it. 291 00:14:46,610 --> 00:14:49,370 Number theory has to be so sure and certain that there's 292 00:14:49,370 --> 00:14:51,570 got to be no problem. 293 00:14:51,570 --> 00:14:55,940 But what Curt Godel did is he established two things. 294 00:14:55,940 --> 00:14:58,940 And these are going to be his two incompleteness theorems. 295 00:14:58,940 --> 00:15:02,139 One we're not going to really talk about so much. 296 00:15:02,139 --> 00:15:03,680 And I'm not going to work through all 297 00:15:03,680 --> 00:15:04,680 the proofs of these. 298 00:15:04,680 --> 00:15:09,560 But we're going to try to get an idea of what each of them mean. 299 00:15:09,560 --> 00:15:21,100 The first one is that any system as powerful as number theory-- 300 00:15:27,230 --> 00:15:29,870 and I'm just going to let NT be number theory-- 301 00:15:35,250 --> 00:15:52,820 which can prove its own consistency, 302 00:15:52,820 --> 00:15:56,570 that system is necessarily inconsistent. 303 00:16:01,061 --> 00:16:02,059 AUDIENCE: [INAUDIBLE]? 304 00:16:04,560 --> 00:16:05,820 PROFESSOR: Exactly. 305 00:16:05,820 --> 00:16:08,460 Somehow, the the second a system as powerful as number 306 00:16:08,460 --> 00:16:12,390 theory is able to talk about itself, things go haywire. 307 00:16:12,390 --> 00:16:14,186 If it can actually prove that-- 308 00:16:14,186 --> 00:16:15,810 here I am, number theory, saying, look, 309 00:16:15,810 --> 00:16:20,280 I promise you guys, you can actually prove from me 310 00:16:20,280 --> 00:16:22,940 that I'm internally consistent. 311 00:16:22,940 --> 00:16:26,590 Well, any system which can talk about itself in that way 312 00:16:26,590 --> 00:16:29,580 is necessarily inconsistent. 313 00:16:29,580 --> 00:16:32,020 And you know, this is really kind of a nitty-gritty proof, 314 00:16:32,020 --> 00:16:34,230 and I can't even give you all the details. 315 00:16:34,230 --> 00:16:35,470 So the second one-- 316 00:16:35,470 --> 00:16:39,120 I'm just going to go ahead and start it over here-- 317 00:16:39,120 --> 00:16:46,600 is any system as powerful as number theory-- 318 00:16:46,600 --> 00:16:48,520 so I'm just kind of going to go "ditto"-- 319 00:16:53,400 --> 00:16:59,113 is necessarily incomplete. 320 00:17:06,020 --> 00:17:07,910 So this means that any system which 321 00:17:07,910 --> 00:17:12,230 is as powerful as number theory automatically looks like this. 322 00:17:12,230 --> 00:17:15,140 There are true statements which we can formulate 323 00:17:15,140 --> 00:17:20,099 which are not provable. 324 00:17:20,099 --> 00:17:23,450 And I'm going to go ahead and give you guys 325 00:17:23,450 --> 00:17:28,919 an English language example of a statement which is-- 326 00:17:28,919 --> 00:17:29,835 yeah, go ahead, Rishi. 327 00:17:29,835 --> 00:17:31,210 AUDIENCE: What does it mean to be 328 00:17:31,210 --> 00:17:32,770 as powerful as number theory? 329 00:17:32,770 --> 00:17:34,960 PROFESSOR: Good question. 330 00:17:34,960 --> 00:17:39,200 And I'm going to explain this a little bit. 331 00:17:39,200 --> 00:17:42,940 But the idea is that in order to prove Godel's incompleteness 332 00:17:42,940 --> 00:17:45,310 theorems, he had to use a very interesting trick. 333 00:17:45,310 --> 00:17:47,147 And that's called Godel numbering. 334 00:17:47,147 --> 00:17:48,980 But first, I want to give you this sentence. 335 00:17:48,980 --> 00:17:52,070 And then I'll tell you where that comes into play. 336 00:17:52,070 --> 00:17:53,050 So here's a statement. 337 00:17:53,050 --> 00:17:58,310 So I'm going you as star. 338 00:17:58,310 --> 00:18:01,400 And if we forget what the star means, it's Rishi's question. 339 00:18:01,400 --> 00:18:05,330 So let's just try and remember that. 340 00:18:05,330 --> 00:18:09,476 So let's consider the following statement. 341 00:18:09,476 --> 00:18:13,730 I talked about the liar paradox, right? 342 00:18:13,730 --> 00:18:17,120 I said, what happens with this sentence? 343 00:18:17,120 --> 00:18:27,085 So this statement is false. 344 00:18:27,085 --> 00:18:29,980 AUDIENCE: [INAUDIBLE] in your language, 345 00:18:29,980 --> 00:18:32,149 you can't say things like that? 346 00:18:32,149 --> 00:18:33,190 PROFESSOR: OK, very good. 347 00:18:33,190 --> 00:18:35,660 And why wouldn't we be able to say things like this? 348 00:18:35,660 --> 00:18:38,160 AUDIENCE: Because if you have a logical and perfect language 349 00:18:38,160 --> 00:18:40,886 you can say, you shouldn't be able to say things like that. 350 00:18:40,886 --> 00:18:42,344 By the end of the problem, it would 351 00:18:42,344 --> 00:18:46,110 be like, how can you tell that this is consistent? 352 00:18:46,110 --> 00:18:48,960 PROFESSOR: So what is the problem, exactly, with it? 353 00:18:48,960 --> 00:18:51,380 If you were to design your perfectly logical language, 354 00:18:51,380 --> 00:18:54,090 what would you rule out? 355 00:18:54,090 --> 00:18:56,910 What would you prevent sentences from doing that would 356 00:18:56,910 --> 00:18:58,203 prevent something like that? 357 00:18:58,203 --> 00:18:58,990 AUDIENCE: Self-reference. 358 00:18:58,990 --> 00:18:59,760 PROFESSOR: There you go. 359 00:18:59,760 --> 00:19:00,300 Exactly. 360 00:19:00,300 --> 00:19:03,030 Self-reference is key. 361 00:19:03,030 --> 00:19:06,900 And there's actually two very, very, very smart guys, 362 00:19:06,900 --> 00:19:09,460 Bertrand Russell and Alfred North Whitehead, 363 00:19:09,460 --> 00:19:10,650 who derived a book. 364 00:19:10,650 --> 00:19:11,970 They wrote a book. 365 00:19:11,970 --> 00:19:13,560 And it's not a fun book to read. 366 00:19:13,560 --> 00:19:17,910 I've heard it's about as interesting as reading 367 00:19:17,910 --> 00:19:19,859 the treads of a tire. 368 00:19:19,859 --> 00:19:21,150 So it's not interesting at all. 369 00:19:21,150 --> 00:19:26,090 But that book was called Principia Mathematica. 370 00:19:26,090 --> 00:19:28,470 And Douglas Hofstadter will talk about this book a lot. 371 00:19:31,080 --> 00:19:33,750 And in this book, they develop a system, 372 00:19:33,750 --> 00:19:35,925 they develop exactly that perfect language 373 00:19:35,925 --> 00:19:37,020 that you speak of. 374 00:19:41,440 --> 00:19:45,030 And the basic idea, if we want to formulate what you're 375 00:19:45,030 --> 00:19:51,560 thinking, is that we create a level language, L1, 376 00:19:51,560 --> 00:19:53,790 and we only allow certain terms. 377 00:19:53,790 --> 00:19:55,665 So we create a kind of bag of terms. 378 00:19:58,968 --> 00:20:00,900 In certain sentences in L1, it would 379 00:20:00,900 --> 00:20:08,250 be like "the sky is blue," et cetera, and "snow is white." 380 00:20:08,250 --> 00:20:13,560 These are perfectly upstanding citizen sentences, right? 381 00:20:13,560 --> 00:20:16,290 They never break the law. 382 00:20:16,290 --> 00:20:19,290 But then what we do is we prevent certain terms, 383 00:20:19,290 --> 00:20:22,260 we prevent these sentences from talking about themselves. 384 00:20:22,260 --> 00:20:24,330 But whenever we're in this class and we're 385 00:20:24,330 --> 00:20:26,190 talking about those sentences, we're 386 00:20:26,190 --> 00:20:28,010 actually speaking in another language-- 387 00:20:28,010 --> 00:20:28,980 L2. 388 00:20:28,980 --> 00:20:32,160 And L2 contains L1 as a subset. 389 00:20:32,160 --> 00:20:33,630 And then we can start saying things 390 00:20:33,630 --> 00:20:45,870 like "the sentence 'snow is white' is white." 391 00:20:45,870 --> 00:20:47,410 Let me do that, yeah. 392 00:20:50,740 --> 00:20:54,010 So suddenly I've got the sentence "snow is white," 393 00:20:54,010 --> 00:20:55,270 which belongs in L1. 394 00:20:55,270 --> 00:20:56,560 And I'm talking about it. 395 00:20:56,560 --> 00:20:59,720 But that's only something I can talk about in L2. 396 00:20:59,720 --> 00:21:02,380 Because in order to talk about something, 397 00:21:02,380 --> 00:21:03,820 you can't talk about yourself. 398 00:21:03,820 --> 00:21:06,580 You have to leap outside of it and then, in order refer to it, 399 00:21:06,580 --> 00:21:08,370 you have to stand from somewhere else. 400 00:21:08,370 --> 00:21:10,390 It's just like you can't really see yourself 401 00:21:10,390 --> 00:21:14,920 until you use something else to look at yourself. 402 00:21:14,920 --> 00:21:16,850 So they developed this system. 403 00:21:16,850 --> 00:21:18,040 But even this had flaws. 404 00:21:18,040 --> 00:21:19,990 And what Godel did was he actually 405 00:21:19,990 --> 00:21:23,650 used Principia Mathematica. 406 00:21:23,650 --> 00:21:28,300 And he took a statement similar to this, 407 00:21:28,300 --> 00:21:30,010 but in fact one much more clever, 408 00:21:30,010 --> 00:21:32,680 in order to prove his incompleteness theorem. 409 00:21:32,680 --> 00:21:33,824 And it's this. 410 00:21:43,020 --> 00:21:46,745 This statement is not provable. 411 00:21:50,870 --> 00:21:53,357 And we can specify in what system. 412 00:21:53,357 --> 00:21:54,940 And one of the things we'll talk about 413 00:21:54,940 --> 00:21:57,324 is in PM, which means Principia Mathematica. 414 00:21:57,324 --> 00:21:58,990 Well, we could say this statement is not 415 00:21:58,990 --> 00:22:00,760 provable in number theory. 416 00:22:00,760 --> 00:22:05,020 But the bottom line is, what does this statement say? 417 00:22:05,020 --> 00:22:11,690 It's not like this statement because what happens 418 00:22:11,690 --> 00:22:15,890 if this statement's false? 419 00:22:15,890 --> 00:22:22,400 Well, if it's false, then whatever it says about itself 420 00:22:22,400 --> 00:22:23,720 is not true. 421 00:22:23,720 --> 00:22:25,580 So that meas it's provable. 422 00:22:25,580 --> 00:22:30,410 So if we say it's false, then that means it is provable. 423 00:22:34,580 --> 00:22:37,100 But if there's one thing which we are certain of-- 424 00:22:37,100 --> 00:22:38,430 yes, go ahead. 425 00:22:38,430 --> 00:22:41,292 AUDIENCE: So the truth is, in this case, like, [INAUDIBLE].. 426 00:22:47,780 --> 00:22:51,000 PROFESSOR: Careful-- that only goes in one direction. 427 00:22:51,000 --> 00:22:53,990 So it is certainly true that a sentence-- 428 00:22:53,990 --> 00:22:55,130 OK, sorry. 429 00:22:55,130 --> 00:22:56,644 What's your name, again? 430 00:22:56,644 --> 00:22:57,620 AUDIENCE: Lativ. 431 00:22:57,620 --> 00:22:58,255 PROFESSOR: How do you say it? 432 00:22:58,255 --> 00:22:58,770 AUDIENCE: Lativ. 433 00:22:58,770 --> 00:22:59,180 PROFESSOR: Lativ? 434 00:22:59,180 --> 00:22:59,590 AUDIENCE: Yeah. 435 00:22:59,590 --> 00:23:00,173 PROFESSOR: OK. 436 00:23:00,173 --> 00:23:06,820 So one of the things Lativ said was that in this case, 437 00:23:06,820 --> 00:23:11,320 are we saying that when we're doing a derivation, when 438 00:23:11,320 --> 00:23:13,420 we have something which is provable 439 00:23:13,420 --> 00:23:16,294 we know it's true, and also vice versa. 440 00:23:16,294 --> 00:23:17,710 But what I'm cautioning against is 441 00:23:17,710 --> 00:23:19,880 that's only true in one direction. 442 00:23:19,880 --> 00:23:23,920 So we certainly know that things which are provable aren't true. 443 00:23:27,700 --> 00:23:29,740 If we can prove it, if I can say, 444 00:23:29,740 --> 00:23:32,470 I can prove to you that this is so, 445 00:23:32,470 --> 00:23:35,570 then it automatically is so. 446 00:23:35,570 --> 00:23:38,001 And this is why people put so much trust in mathematics, 447 00:23:38,001 --> 00:23:40,000 is that the second we have a proof of something, 448 00:23:40,000 --> 00:23:42,170 we know it's true. 449 00:23:42,170 --> 00:23:44,950 But what we just were asking about 450 00:23:44,950 --> 00:23:47,920 was, what about the other way? 451 00:23:47,920 --> 00:23:49,660 Does true always imply provable? 452 00:23:53,260 --> 00:23:54,990 Well, let's ask. 453 00:23:54,990 --> 00:23:57,340 Let's ask this statement. 454 00:23:57,340 --> 00:24:00,500 What if this statement is true? 455 00:24:00,500 --> 00:24:04,240 Well, if it's true, what it says about itself must be true, 456 00:24:04,240 --> 00:24:06,430 and that's that it's not provable. 457 00:24:06,430 --> 00:24:08,860 So the only way that this statement is true 458 00:24:08,860 --> 00:24:11,830 is if it's not provable. 459 00:24:11,830 --> 00:24:17,020 So suddenly, we know that we can't go the other way. 460 00:24:17,020 --> 00:24:22,030 And the trick that Godel used-- and this is why we get 461 00:24:22,030 --> 00:24:23,525 to the star question-- 462 00:24:26,200 --> 00:24:28,910 is that this is not a statement in mathematics. 463 00:24:28,910 --> 00:24:34,570 But what Godel did is he essentially 464 00:24:34,570 --> 00:24:36,530 took a statement like this and he said, 465 00:24:36,530 --> 00:24:44,540 well, we're going to let every letter and logical symbol 466 00:24:44,540 --> 00:24:45,800 stand for a number. 467 00:24:45,800 --> 00:24:53,540 So we're going to let p be 101010. 468 00:24:53,540 --> 00:24:56,030 And we're going to give a unique number to every symbol 469 00:24:56,030 --> 00:24:57,740 including spaces. 470 00:24:57,740 --> 00:25:01,160 And then once we have this, we have a Godel number 471 00:25:01,160 --> 00:25:02,960 for this statement. 472 00:25:02,960 --> 00:25:04,460 And then what we can start doing is 473 00:25:04,460 --> 00:25:07,410 we can start giving certain operations. 474 00:25:07,410 --> 00:25:08,870 Remember what we are doing miu? 475 00:25:08,870 --> 00:25:12,290 And we were saying, well, what we can always do is if we have 476 00:25:12,290 --> 00:25:14,470 three "I"s we can cancel it. 477 00:25:14,470 --> 00:25:18,450 Or if we have a string of hyphens or whatever, 478 00:25:18,450 --> 00:25:21,530 or a string of letters after m, we can double it. 479 00:25:21,530 --> 00:25:23,060 So what Godel did is he turned each 480 00:25:23,060 --> 00:25:29,270 of these rules of inference into rules of arithmetic. 481 00:25:29,270 --> 00:25:32,730 So I'm going to go ahead and hop over here. 482 00:25:41,130 --> 00:25:44,100 So what he did is he made rules of inference. 483 00:25:49,750 --> 00:25:54,490 And he made them equivalent to our special isomorphism symbol, 484 00:25:54,490 --> 00:25:58,160 to rules of arithmetic. 485 00:25:58,160 --> 00:26:00,480 So this is kind of counter-intuitive. 486 00:26:00,480 --> 00:26:03,950 And I can't go into all the details. 487 00:26:03,950 --> 00:26:07,090 But we will meet them in chapter nine. 488 00:26:07,090 --> 00:26:09,040 And that's the idea that-- 489 00:26:09,040 --> 00:26:16,710 suppose we have a logical thing like the statement P. 490 00:26:16,710 --> 00:26:23,530 And then we also have the statement that P implies Q. 491 00:26:23,530 --> 00:26:26,450 So we know that the statement if P is true. 492 00:26:26,450 --> 00:26:30,820 So if it is cloudy, then it's going to rain. 493 00:26:30,820 --> 00:26:33,470 And then if we have, well, I'm looking outside 494 00:26:33,470 --> 00:26:42,250 and it's cloudy, then this is equal to Q, right? 495 00:26:42,250 --> 00:26:45,340 So if we know that "if it's cloudy, then it will rain" 496 00:26:45,340 --> 00:26:49,150 is true, and we have that it's cloudy, 497 00:26:49,150 --> 00:26:52,960 then we can immediately deduce that it's going to rain. 498 00:26:52,960 --> 00:26:55,480 And what Godel did is he said, well, 499 00:26:55,480 --> 00:26:58,900 these statements I can actually make into numbers. 500 00:26:58,900 --> 00:27:02,590 And I can make the logical symbol "and" into an operation, 501 00:27:02,590 --> 00:27:03,700 like addition. 502 00:27:03,700 --> 00:27:05,610 And I can make implies-- 503 00:27:05,610 --> 00:27:07,870 well, this would also be a symbol. 504 00:27:07,870 --> 00:27:10,600 And I can have this total operation of detachment, 505 00:27:10,600 --> 00:27:14,800 of pulling out Q, into a statement almost like one 506 00:27:14,800 --> 00:27:16,030 plus one equals two. 507 00:27:19,012 --> 00:27:20,470 And then what he did is he captured 508 00:27:20,470 --> 00:27:23,710 the idea of provability into a property of numbers, 509 00:27:23,710 --> 00:27:26,240 like something being prime. 510 00:27:26,240 --> 00:27:29,680 So then, really, what this statement comes down to 511 00:27:29,680 --> 00:27:32,830 is such and such number, that number 512 00:27:32,830 --> 00:27:35,716 which codes for this statement, does not have a property. 513 00:27:35,716 --> 00:27:37,090 And that's why you need something 514 00:27:37,090 --> 00:27:38,548 as strong as number theory in order 515 00:27:38,548 --> 00:27:40,690 to do this numbering trick. 516 00:27:40,690 --> 00:27:43,192 But that's just kind of a first glance at Godel's theorem. 517 00:27:43,192 --> 00:27:45,400 And I don't want to go into too much detail about it. 518 00:27:48,567 --> 00:27:50,900 However, I do want to go back and talk a little bit more 519 00:27:50,900 --> 00:27:54,140 about the things which we mentioned, 520 00:27:54,140 --> 00:27:57,230 and we kind of glanced upon chapter four. 521 00:27:57,230 --> 00:28:00,950 And that was the ideas of geometry. 522 00:28:00,950 --> 00:28:02,610 And this is really cool. 523 00:28:02,610 --> 00:28:11,720 And it has something to do with interpretation. 524 00:28:11,720 --> 00:28:16,940 Now, I want you to remember what we mean by interpretation. 525 00:28:16,940 --> 00:28:24,370 And I think it's a term I briefly defined 526 00:28:24,370 --> 00:28:26,040 on the first day of lecture. 527 00:28:26,040 --> 00:28:27,790 For those of you who were here, can anyone 528 00:28:27,790 --> 00:28:29,470 tell me what interpretation's about? 529 00:28:32,690 --> 00:28:34,070 Go ahead. 530 00:28:34,070 --> 00:28:42,270 AUDIENCE: Sort of like choosing the real world [INAUDIBLE].. 531 00:28:42,270 --> 00:28:43,810 PROFESSOR: Exactly, exactly. 532 00:28:43,810 --> 00:28:48,060 So-- I'm sorry, I forgot your name. 533 00:28:48,060 --> 00:28:48,696 Is it Lativ? 534 00:28:48,696 --> 00:28:49,320 AUDIENCE: Yeah. 535 00:28:49,320 --> 00:28:50,028 PROFESSOR: Lativ. 536 00:28:50,028 --> 00:28:52,320 What Lativ said was that it's basically 537 00:28:52,320 --> 00:28:54,780 like giving an example of an interpretation, 538 00:28:54,780 --> 00:28:58,420 to be giving a real world model for what you're doing. 539 00:28:58,420 --> 00:29:00,930 And we saw an example of an interpretation 540 00:29:00,930 --> 00:29:03,000 which was not true, right? 541 00:29:03,000 --> 00:29:05,940 When we assigned to our symbols "one plus one 542 00:29:05,940 --> 00:29:10,920 equals two" to "raindrop meld with raindrop," 543 00:29:10,920 --> 00:29:12,480 it didn't give us two raindrops. 544 00:29:12,480 --> 00:29:15,330 Instead, it gave us just one. 545 00:29:15,330 --> 00:29:17,730 So that's an example of an interpretation which 546 00:29:17,730 --> 00:29:20,660 doesn't hold and doesn't work. 547 00:29:20,660 --> 00:29:23,110 But we gave some other interpretations. 548 00:29:23,110 --> 00:29:25,680 And when we were playing with the P-Q system, 549 00:29:25,680 --> 00:29:27,810 we didn't give it a real world interpretation, 550 00:29:27,810 --> 00:29:31,590 but instead we gave it a mathematical interpretation 551 00:29:31,590 --> 00:29:32,545 of addition. 552 00:29:32,545 --> 00:29:36,840 I always say that hyphen P, hyphen Q, hyphen, hyphen 553 00:29:36,840 --> 00:29:38,410 is "one plus one equals two." 554 00:29:41,520 --> 00:29:42,960 And so that's not so interesting. 555 00:29:42,960 --> 00:29:46,710 But what is interesting is that several 556 00:29:46,710 --> 00:29:48,580 thousand or so years ago-- 557 00:29:48,580 --> 00:29:53,580 not several, but at least two and 1/2-ish-- 558 00:29:53,580 --> 00:29:57,270 a guy named Euclid said OK, you know what? 559 00:29:57,270 --> 00:30:01,680 Geometry is for us as true and certain 560 00:30:01,680 --> 00:30:03,212 as anything is going to get. 561 00:30:03,212 --> 00:30:05,670 But what I want to do is go ahead and write down the rules, 562 00:30:05,670 --> 00:30:08,120 write down everything we know, so that we can proceed 563 00:30:08,120 --> 00:30:10,680 and deduce directly from these statements 564 00:30:10,680 --> 00:30:13,050 and know that everything we say is true. 565 00:30:13,050 --> 00:30:15,230 But in order to get his feet off the ground-- 566 00:30:15,230 --> 00:30:18,420 I mean, he couldn't lift himself up by his own bootstraps-- 567 00:30:18,420 --> 00:30:20,370 he made a series of requests. 568 00:30:20,370 --> 00:30:26,234 And these are known as the postulates of Euclid. 569 00:30:26,234 --> 00:30:27,750 So we've got Euclid's postulates. 570 00:30:27,750 --> 00:30:29,790 And I'm not going to go through them all. 571 00:30:29,790 --> 00:30:35,520 They're actually listed, I believe, in chapter four. 572 00:30:35,520 --> 00:30:38,850 But there was one postulate which 573 00:30:38,850 --> 00:30:41,430 really got on Euclid's nerves. 574 00:30:41,430 --> 00:30:44,490 And that was known as the fifth postulate. 575 00:30:44,490 --> 00:30:48,360 And he tried to derive it from the previous four, 576 00:30:48,360 --> 00:30:50,070 but he couldn't. 577 00:30:50,070 --> 00:30:55,830 And that's the idea that if you have a line and a point not 578 00:30:55,830 --> 00:31:01,950 on the line, I can give you a line, 579 00:31:01,950 --> 00:31:07,800 there's a unique line that goes to that point 580 00:31:07,800 --> 00:31:10,130 but never intersects this line. 581 00:31:10,130 --> 00:31:12,680 And of course, in geometry we say 582 00:31:12,680 --> 00:31:16,770 that lines extend on forever, and line segments kind of 583 00:31:16,770 --> 00:31:17,270 terminate. 584 00:31:17,270 --> 00:31:19,070 But these are good old infinite lines. 585 00:31:23,730 --> 00:31:25,160 But he could never prove it. 586 00:31:25,160 --> 00:31:31,250 And there were efforts for well over 1,500 years 587 00:31:31,250 --> 00:31:34,130 to try to prove the fifth postulate. 588 00:31:34,130 --> 00:31:39,150 But it's such an intuitive and obvious statement, right? 589 00:31:39,150 --> 00:31:42,100 I mean, does anyone feel like this isn't right, 590 00:31:42,100 --> 00:31:44,640 that there's any reason why we can't assume this? 591 00:31:47,230 --> 00:31:47,990 Good. 592 00:31:47,990 --> 00:31:51,870 AUDIENCE: What if you like [INAUDIBLE]?? 593 00:31:51,870 --> 00:31:53,400 PROFESSOR: Exactly, exactly. 594 00:31:53,400 --> 00:31:56,550 So this gives us the idea of non-Euclidean geometry. 595 00:31:56,550 --> 00:32:01,170 And just to give you a quick example, 596 00:32:01,170 --> 00:32:02,925 suppose we're on the surface of a sphere. 597 00:32:05,700 --> 00:32:11,380 And we define our lines to the great circles. 598 00:32:11,380 --> 00:32:12,870 So the way you make a great circle 599 00:32:12,870 --> 00:32:19,890 is you take your spear with your center, O, 600 00:32:19,890 --> 00:32:22,490 and you cut through it. 601 00:32:22,490 --> 00:32:25,740 And you make a plane slice. 602 00:32:25,740 --> 00:32:36,010 So that goes through the origin. 603 00:32:36,010 --> 00:32:40,120 And you define a line to be this great circle which is formed. 604 00:32:40,120 --> 00:32:44,900 So any line on here that you can form, 605 00:32:44,900 --> 00:32:47,540 any line that you necessarily have-- and these are kind of 606 00:32:47,540 --> 00:32:51,910 like your lines for longitude or latitude, 607 00:32:51,910 --> 00:32:54,980 except not necessarily because we couldn't do something 608 00:32:54,980 --> 00:32:56,650 like this. 609 00:32:56,650 --> 00:33:00,290 But if we had any line that goes through that, which 610 00:33:00,290 --> 00:33:02,990 has the same radius as our sphere, 611 00:33:02,990 --> 00:33:08,760 they necessarily intersect in at least two spots. 612 00:33:08,760 --> 00:33:11,360 So in spherical geometry, things obviously 613 00:33:11,360 --> 00:33:13,120 don't behave the same way. 614 00:33:13,120 --> 00:33:17,640 And one of the things you could derive using Euclid's axioms 615 00:33:17,640 --> 00:33:21,920 was that the sum of the internal angles of a triangle 616 00:33:21,920 --> 00:33:25,910 is always 180 degrees. 617 00:33:25,910 --> 00:33:29,990 But on a sphere, if you draw a triangle-- 618 00:33:29,990 --> 00:33:33,030 let's say we go from somewhere on the equator 619 00:33:33,030 --> 00:33:35,390 and we travel up to the North Pole. 620 00:33:35,390 --> 00:33:40,920 And we pivot 90 degrees and head back down to the equator. 621 00:33:40,920 --> 00:33:43,340 So we've got a right angle here, and a right angle here. 622 00:33:43,340 --> 00:33:45,410 And we also have a right angle here. 623 00:33:45,410 --> 00:33:48,980 So for a spherical triangle, we can actually 624 00:33:48,980 --> 00:33:52,040 get up to 90 plus 90 plus 90-- 625 00:33:52,040 --> 00:33:53,820 270 degrees. 626 00:33:53,820 --> 00:33:56,730 So obviously, this doesn't hold in the spherical geometry. 627 00:33:56,730 --> 00:34:00,380 And similarly, we have hyperbolic geometry. 628 00:34:00,380 --> 00:34:05,120 And this is something which is a very beautiful subject. 629 00:34:05,120 --> 00:34:09,230 And you have several models of how hyperbolic geology works. 630 00:34:09,230 --> 00:34:10,790 You can think of them as projections. 631 00:34:13,429 --> 00:34:15,630 And one is kind of the upper-half plane model. 632 00:34:23,760 --> 00:34:27,659 And then one's just kind of your unit disk here. 633 00:34:27,659 --> 00:34:30,270 And what we define our lines to be 634 00:34:30,270 --> 00:34:35,730 is these segments which end with right angles 635 00:34:35,730 --> 00:34:37,480 on the outside of your circle. 636 00:34:37,480 --> 00:34:41,107 So what we can do, then, is actually construct two lines. 637 00:34:41,107 --> 00:34:42,690 We can actually construct, infinitely, 638 00:34:42,690 --> 00:34:48,670 many lines that don't intersect each other. 639 00:34:48,670 --> 00:34:52,780 So in here, you had two intersection points. 640 00:34:52,780 --> 00:34:58,500 And here, if you had a line like that, 641 00:34:58,500 --> 00:35:01,200 you would only have one intersection point. 642 00:35:01,200 --> 00:35:03,690 But here, you could have a whole family of lines 643 00:35:03,690 --> 00:35:05,784 with no intersection points. 644 00:35:05,784 --> 00:35:07,200 But the weird thing is that we can 645 00:35:07,200 --> 00:35:10,500 give our same terms, our same statements, 646 00:35:10,500 --> 00:35:15,195 like point and line. 647 00:35:18,220 --> 00:35:20,640 And we can do a lot of the same geometry which Euclid 648 00:35:20,640 --> 00:35:23,940 did, except if we give them different interpretations, 649 00:35:23,940 --> 00:35:27,220 like, we'll define the line to be like this, 650 00:35:27,220 --> 00:35:29,910 or we'll define a line to be like this, 651 00:35:29,910 --> 00:35:31,141 then different things happen. 652 00:35:31,141 --> 00:35:31,640 Yes? 653 00:35:31,640 --> 00:35:33,496 AUDIENCE: So the [INAUDIBLE] necessary, 654 00:35:33,496 --> 00:35:36,750 but the way you interpret it is [INAUDIBLE].. 655 00:35:36,750 --> 00:35:40,340 PROFESSOR: So here is a fact. 656 00:35:40,340 --> 00:35:45,860 It's true that with the four original postulates of Euclid-- 657 00:35:45,860 --> 00:35:51,190 sorry, what Lativ said was that, necessarily, 658 00:35:51,190 --> 00:35:52,640 what's true in your formal system 659 00:35:52,640 --> 00:35:54,200 is the interpretation you give them. 660 00:35:58,350 --> 00:36:06,280 That is true in this example, right? 661 00:36:06,280 --> 00:36:08,770 Here, the truth of your statement 662 00:36:08,770 --> 00:36:12,430 directly depended on how you interpret your terms 663 00:36:12,430 --> 00:36:15,980 like point, and line, and things like that. 664 00:36:15,980 --> 00:36:20,890 And the problem was is that in the assumption, 665 00:36:20,890 --> 00:36:27,090 the axiom, which was Euclid's fifth postulate, 666 00:36:27,090 --> 00:36:32,150 was that what Euclid's fifth postulate said only it could 667 00:36:32,150 --> 00:36:35,420 be interpreted consistently with the other four postulates, 668 00:36:35,420 --> 00:36:38,720 and when he did it in simple, plane geometry 669 00:36:38,720 --> 00:36:41,120 like we're working on the top of this table. 670 00:36:41,120 --> 00:36:44,520 But the second you interpreted all five of his postulates 671 00:36:44,520 --> 00:36:46,940 in this setting, the fifth one was 672 00:36:46,940 --> 00:36:49,460 inconsistent with the previous four. 673 00:36:49,460 --> 00:36:53,550 And what it said was inherently wrong. 674 00:36:53,550 --> 00:36:55,160 And similarly, things with this-- 675 00:37:00,690 --> 00:37:03,950 you had to be very specific about your interpretation 676 00:37:03,950 --> 00:37:05,480 and what you assumed. 677 00:37:05,480 --> 00:37:10,340 Otherwise, you could get an internally inconsistent 678 00:37:10,340 --> 00:37:11,600 interpretation. 679 00:37:11,600 --> 00:37:14,000 So this is all part of a family of things 680 00:37:14,000 --> 00:37:15,125 called hyperbolic geometry. 681 00:37:22,260 --> 00:37:23,880 And inherently, what this has to deal 682 00:37:23,880 --> 00:37:26,880 with is the beauty of complex numbers. 683 00:37:26,880 --> 00:37:30,810 And you can do things in hyperbolic geometry which just 684 00:37:30,810 --> 00:37:33,570 completely boggle the mind. 685 00:37:33,570 --> 00:37:37,470 Like suppose you had a circle, a line-- 686 00:37:37,470 --> 00:37:39,300 this is a line, remember, because it 687 00:37:39,300 --> 00:37:43,890 ends with perpendicular points on the real axis. 688 00:37:43,890 --> 00:37:49,740 And you can find a mapping which takes it up to here 689 00:37:49,740 --> 00:37:53,112 and preserves the distance between these two. 690 00:37:53,112 --> 00:37:55,320 And there's all sorts of different things you can do. 691 00:37:55,320 --> 00:37:58,000 And it's just completely a gorgeous subject. 692 00:37:58,000 --> 00:38:01,956 I encourage you all to learn more about it. 693 00:38:01,956 --> 00:38:03,330 But this was one of the examples. 694 00:38:03,330 --> 00:38:08,520 Because what we thought for well over 1,500 years 695 00:38:08,520 --> 00:38:12,780 was that what Euclid said was as sure and certain 696 00:38:12,780 --> 00:38:15,210 as any knowledge that we could have. 697 00:38:15,210 --> 00:38:19,200 And people would often try to base their arguments 698 00:38:19,200 --> 00:38:21,000 and try to derive them to geometry. 699 00:38:23,610 --> 00:38:29,600 It's kind of a funny anecdote but Karl Marx and Frederick 700 00:38:29,600 --> 00:38:34,170 Engels, when they were writing their texts, 701 00:38:34,170 --> 00:38:36,570 they actually tried to reduce what they 702 00:38:36,570 --> 00:38:37,860 were saying to mathematics. 703 00:38:37,860 --> 00:38:41,370 Because they felt like if they could prove the system 704 00:38:41,370 --> 00:38:43,530 they were advocating in terms of mathematics, 705 00:38:43,530 --> 00:38:46,540 then people would have to accept it. 706 00:38:46,540 --> 00:38:49,090 But what happens in mathematics itself is inconsistent. 707 00:38:49,090 --> 00:38:51,000 If you get paradoxes like the set theory 708 00:38:51,000 --> 00:38:54,420 paradoxes, or you get these possible interpretations 709 00:38:54,420 --> 00:38:57,000 where things are sometimes true or not true-- 710 00:38:57,000 --> 00:39:00,630 what happens then if mathematics is not a sure footing? 711 00:39:00,630 --> 00:39:02,520 And this is a problem which I want 712 00:39:02,520 --> 00:39:07,190 you guys to think about as we move along through this book. 713 00:39:07,190 --> 00:39:11,730 And what does it mean to provide an interpretation and things 714 00:39:11,730 --> 00:39:13,150 like that? 715 00:39:13,150 --> 00:39:15,480 So what I want us to do is take a quick break, 716 00:39:15,480 --> 00:39:18,420 because we're going to go into one of my favorite dialogues. 717 00:39:18,420 --> 00:39:21,720 And you'll see the purpose of this [INAUDIBLE] later. 718 00:39:21,720 --> 00:39:23,880 But because it's a long dialogue I 719 00:39:23,880 --> 00:39:26,430 want everyone to kind of take a break 720 00:39:26,430 --> 00:39:30,160 and get some food and drink. 721 00:39:30,160 --> 00:39:32,140 And we'll then read the dialogue. 722 00:39:32,140 --> 00:39:34,080 But I'll need some volunteers for reading. 723 00:39:34,080 --> 00:39:37,980 So let's go ahead and take a five minute break 724 00:39:37,980 --> 00:39:39,380 before you start reading, OK? 725 00:39:50,649 --> 00:39:51,940 So a little harmonic labyrinth. 726 00:39:51,940 --> 00:39:53,207 Which did you guys think? 727 00:39:53,207 --> 00:39:54,430 AUDIENCE: Confusing. 728 00:39:54,430 --> 00:39:55,420 PROFESSOR: Confusing? 729 00:39:55,420 --> 00:39:56,992 Why do you say it's confusing? 730 00:39:56,992 --> 00:40:00,390 AUDIENCE: It's like it switched roles in between [INAUDIBLE].. 731 00:40:00,390 --> 00:40:02,798 PROFESSOR: So in what way do you mean "roles?" 732 00:40:02,798 --> 00:40:04,710 AUDIENCE: It's like the [INAUDIBLE].. 733 00:40:07,580 --> 00:40:09,190 PROFESSOR: So there's role flipping 734 00:40:09,190 --> 00:40:11,750 in terms of the way they treat each other, or-- 735 00:40:11,750 --> 00:40:15,436 AUDIENCE: It's like the characters [INAUDIBLE].. 736 00:40:15,436 --> 00:40:16,840 PROFESSOR: Ah, OK. 737 00:40:16,840 --> 00:40:20,320 Yeah, and more interestingly, though, 738 00:40:20,320 --> 00:40:23,800 they had an opportunity to do that by constantly going down 739 00:40:23,800 --> 00:40:25,420 to nested roles. 740 00:40:25,420 --> 00:40:28,810 And they could essentially be new people in some ways. 741 00:40:28,810 --> 00:40:29,860 So that's good. 742 00:40:29,860 --> 00:40:31,030 Does anyone else have-- 743 00:40:31,030 --> 00:40:31,926 oh, yes, Sandra? 744 00:40:31,926 --> 00:40:33,426 AUDIENCE: They talk about themselves 745 00:40:33,426 --> 00:40:35,272 in some of the dialogue. 746 00:40:35,272 --> 00:40:37,480 PROFESSOR: Right, so we had this weird playing around 747 00:40:37,480 --> 00:40:39,190 with levels. 748 00:40:39,190 --> 00:40:41,860 And in some ways, that was kind of hard to capture perfectly 749 00:40:41,860 --> 00:40:42,670 in terms of audio. 750 00:40:42,670 --> 00:40:46,240 But I'm sure most of you saw as we were reading along 751 00:40:46,240 --> 00:40:49,330 that there was indentations in the text. 752 00:40:49,330 --> 00:40:51,220 And that was a visual reminder as you're 753 00:40:51,220 --> 00:40:54,760 reading what level of the story you were at. 754 00:40:54,760 --> 00:40:57,340 And Douglas Hofstadter used all sorts of really nice tricks 755 00:40:57,340 --> 00:40:59,990 where you would have characters like, oh, I 756 00:40:59,990 --> 00:41:03,040 think he means tonic. 757 00:41:03,040 --> 00:41:07,780 And they would be talking up here on level one. 758 00:41:07,780 --> 00:41:15,370 And down on level two, they would say, oh, 759 00:41:15,370 --> 00:41:18,100 thank you, yes, tonic is exactly what I needed. 760 00:41:18,100 --> 00:41:23,710 Even though in this situation, these guys 761 00:41:23,710 --> 00:41:26,326 don't really know about their higher levels of reality. 762 00:41:26,326 --> 00:41:28,950 It's just like the same question of what happened to the weasel 763 00:41:28,950 --> 00:41:31,158 when here he was sitting in our everyday normal life, 764 00:41:31,158 --> 00:41:32,530 and he took some popping tonic? 765 00:41:32,530 --> 00:41:35,740 And he popped up to a higher level of reality. 766 00:41:35,740 --> 00:41:37,387 And it kind of makes you wonder. 767 00:41:37,387 --> 00:41:39,220 I know for me, I always get the visual image 768 00:41:39,220 --> 00:41:44,410 of playing the universe as a fractal. 769 00:41:44,410 --> 00:41:47,650 And we spend all our time living down 770 00:41:47,650 --> 00:41:52,957 in this corner of the Sierpinski gasket. 771 00:41:52,957 --> 00:41:55,540 And then one day, somebody takes a popping tonic and someone's 772 00:41:55,540 --> 00:41:58,444 like, holy mackerel, there's actually all of these levels. 773 00:41:58,444 --> 00:41:59,860 And then just the fact that you've 774 00:41:59,860 --> 00:42:02,810 had that one experience of playing around 775 00:42:02,810 --> 00:42:07,980 with two levels of reality, it makes you speculate. 776 00:42:07,980 --> 00:42:09,790 Well, why can't there be more? 777 00:42:09,790 --> 00:42:12,970 Why can't there be an infinite set of realities? 778 00:42:12,970 --> 00:42:15,550 And how do I know what mine is? 779 00:42:15,550 --> 00:42:21,220 And there's something I want us to just bring out into the open 780 00:42:21,220 --> 00:42:21,895 here. 781 00:42:21,895 --> 00:42:26,124 And it's the idea that when I first started this class, 782 00:42:26,124 --> 00:42:27,040 on the first lecture-- 783 00:42:27,040 --> 00:42:29,380 I know not all of you were here-- 784 00:42:29,380 --> 00:42:30,880 I said the fundamental thing we want 785 00:42:30,880 --> 00:42:34,070 to answer at the end of this course is, what is an "I?" 786 00:42:34,070 --> 00:42:37,480 What makes something conscious from unconscious things? 787 00:42:37,480 --> 00:42:39,460 How do we get particles and atoms 788 00:42:39,460 --> 00:42:42,910 to start talking about themselves like the way we do? 789 00:42:42,910 --> 00:42:45,010 So fundamentally, in this class we're 790 00:42:45,010 --> 00:42:49,330 going to be talking about a lot of really 791 00:42:49,330 --> 00:42:51,640 deep and profound questions. 792 00:42:51,640 --> 00:42:56,410 And I want everybody to not feel afraid that their opinion might 793 00:42:56,410 --> 00:42:58,310 be persecuted. 794 00:42:58,310 --> 00:43:00,250 Because even in this story, we managed 795 00:43:00,250 --> 00:43:03,310 to meet God during the middle of this dialogue. 796 00:43:03,310 --> 00:43:05,710 And if we can't talk about God, it's 797 00:43:05,710 --> 00:43:10,050 going to really narrow what we're allowed to talk about. 798 00:43:10,050 --> 00:43:12,130 And similarly, when I'm saying, what 799 00:43:12,130 --> 00:43:15,810 is the mind, how do we get a physical brain 800 00:43:15,810 --> 00:43:21,700 to then start operating with mental and conscious thoughts, 801 00:43:21,700 --> 00:43:24,520 that's very fundamentally asking questions about the soul. 802 00:43:24,520 --> 00:43:28,150 And what your guys opinions are on that become important. 803 00:43:28,150 --> 00:43:31,560 And I don't want to feel like anybody is learning 804 00:43:31,560 --> 00:43:32,860 in a hostile environment. 805 00:43:32,860 --> 00:43:38,050 So I encourage you guys to speak actively. 806 00:43:38,050 --> 00:43:40,490 It's interesting because Douglas Hofstadter 807 00:43:40,490 --> 00:43:44,440 presents a very unique picture of God, right? 808 00:43:44,440 --> 00:43:50,650 He picks this recursive idea of a stack of infinities. 809 00:43:50,650 --> 00:43:55,630 And just as an anecdote-- and this is actually a little 810 00:43:55,630 --> 00:43:59,240 historical fact for you-- 811 00:43:59,240 --> 00:44:01,780 the Jesuits, right around the time 812 00:44:01,780 --> 00:44:04,720 of the development of calculus, we 813 00:44:04,720 --> 00:44:07,570 had Isaac Newton, and Leibniz, and these guys 814 00:44:07,570 --> 00:44:14,860 doing their stuff in the 1670s, maybe a little later. 815 00:44:14,860 --> 00:44:23,693 But it really took everyone else into like the 1730s. 816 00:44:23,693 --> 00:44:27,110 Newton, Leibniz-- these guys that 817 00:44:27,110 --> 00:44:30,350 developed the calculus, studying the infinitesimally small. 818 00:44:30,350 --> 00:44:35,010 Remember, when we're playing around with calculus, 819 00:44:35,010 --> 00:44:41,060 we're asking what happens when we approximate a function 820 00:44:41,060 --> 00:44:42,260 originally in a finite way? 821 00:44:42,260 --> 00:44:45,197 And then what happens if we take the limit to something 822 00:44:45,197 --> 00:44:46,280 which is infinitely small? 823 00:44:46,280 --> 00:44:47,580 And what does that mean? 824 00:44:47,580 --> 00:44:50,270 And when you start playing around with calculus, 825 00:44:50,270 --> 00:44:55,410 you can find things like the area under the curve. 826 00:44:55,410 --> 00:44:59,570 And the way you approximate this is these blocks 827 00:44:59,570 --> 00:45:02,690 and taking infinitely small limits. 828 00:45:02,690 --> 00:45:06,590 The Greeks, they were really close to developing 829 00:45:06,590 --> 00:45:07,190 the calculus. 830 00:45:07,190 --> 00:45:10,310 Archimedes was, in many ways, conceptually just a few stone 831 00:45:10,310 --> 00:45:12,350 throws away from it. 832 00:45:12,350 --> 00:45:14,690 But the Greeks were also much smarter 833 00:45:14,690 --> 00:45:16,730 than Newton and Leibniz. 834 00:45:16,730 --> 00:45:20,690 Because they said, well, dumbos, when 835 00:45:20,690 --> 00:45:23,690 you take a bunch of infinitely small things, 836 00:45:23,690 --> 00:45:26,180 you can't get something finite, right? 837 00:45:26,180 --> 00:45:28,670 Like, how is it that I can take a bunch of two 838 00:45:28,670 --> 00:45:31,826 dimensional circles which are infinitely thin and then 839 00:45:31,826 --> 00:45:33,200 stacked them on top of each other 840 00:45:33,200 --> 00:45:35,900 in order to get you know a cylinder? 841 00:45:35,900 --> 00:45:37,260 It doesn't make sense. 842 00:45:37,260 --> 00:45:40,110 It does not make sense. 843 00:45:40,110 --> 00:45:42,230 So really what happened here, and then 844 00:45:42,230 --> 00:45:43,695 getting into Euler and these guys, 845 00:45:43,695 --> 00:45:45,320 is they essentially were trying to take 846 00:45:45,320 --> 00:45:49,394 our concepts of the infinite and making them rigorous. 847 00:45:49,394 --> 00:45:50,810 And the reason why I go on this is 848 00:45:50,810 --> 00:45:54,530 that the Jesuits, around this time and getting later, 849 00:45:54,530 --> 00:45:57,590 were one of the very first groups in schools 850 00:45:57,590 --> 00:46:00,050 to start teaching their students calculus. 851 00:46:00,050 --> 00:46:03,542 Because they felt that if they understood mathematically 852 00:46:03,542 --> 00:46:05,000 and rigorously, and they could deal 853 00:46:05,000 --> 00:46:07,370 with concepts of the infinite, they 854 00:46:07,370 --> 00:46:09,410 had a better understanding of God. 855 00:46:09,410 --> 00:46:12,230 So the Jesuits deeply felt that you 856 00:46:12,230 --> 00:46:14,750 understanding calculus was essential to you understanding 857 00:46:14,750 --> 00:46:16,730 God. 858 00:46:16,730 --> 00:46:20,600 And I think this very much goes in spirit 859 00:46:20,600 --> 00:46:22,730 with the little harmonic labyrinth 860 00:46:22,730 --> 00:46:28,550 where you saw this image of God over Jinn. 861 00:46:28,550 --> 00:46:33,950 And "Jinn" is actually an Arabic word for genie. 862 00:46:33,950 --> 00:46:39,800 And then this going off to God itself and then coming back. 863 00:46:42,720 --> 00:46:46,439 So this really requires wrestling 864 00:46:46,439 --> 00:46:48,230 some of the conceptual tools behind dealing 865 00:46:48,230 --> 00:46:49,820 with the infinite. 866 00:46:49,820 --> 00:46:53,139 And it's not something I can teach you fully in this class. 867 00:46:53,139 --> 00:46:54,680 But I encourage you all to pursue it. 868 00:46:57,440 --> 00:47:01,040 Quick question-- you'll notice that each of the genies 869 00:47:01,040 --> 00:47:02,420 did it in half the amount of time 870 00:47:02,420 --> 00:47:04,400 that it took the previous genie. 871 00:47:04,400 --> 00:47:07,160 Can anyone tell me why, or at least 872 00:47:07,160 --> 00:47:11,200 why Hofstadter went ahead and paid attention to that detail? 873 00:47:11,200 --> 00:47:13,630 AUDIENCE: Because you don't want to do it for everyone? 874 00:47:13,630 --> 00:47:15,671 PROFESSOR: OK, so you need something-- and Felix, 875 00:47:15,671 --> 00:47:16,885 go ahead. 876 00:47:16,885 --> 00:47:19,303 AUDIENCE: [INAUDIBLE] 877 00:47:19,303 --> 00:47:20,511 PROFESSOR: Great, one moment. 878 00:47:20,511 --> 00:47:23,082 AUDIENCE: [INAUDIBLE] 879 00:47:23,082 --> 00:47:23,790 PROFESSOR: Right. 880 00:47:23,790 --> 00:47:26,610 OK, so the amount of time it took 881 00:47:26,610 --> 00:47:30,720 was one plus 1/2 of one genie's time 882 00:47:30,720 --> 00:47:35,280 plus half of the previous genie's time plus-- 883 00:47:35,280 --> 00:47:37,570 yes, and so on. 884 00:47:42,470 --> 00:47:44,530 Yes, sorry. 885 00:47:44,530 --> 00:47:47,080 I knew that wasn't right. 886 00:47:47,080 --> 00:47:48,788 1/16, I missed a term. 887 00:47:48,788 --> 00:47:50,420 Dot, dot, dot. 888 00:47:50,420 --> 00:47:51,840 And do you know what this equals? 889 00:47:54,618 --> 00:47:56,010 AUDIENCE: [INAUDIBLE] 890 00:47:56,010 --> 00:47:57,470 PROFESSOR: OK, there you go. 891 00:47:57,470 --> 00:47:59,940 We've got some winners. 892 00:47:59,940 --> 00:48:03,904 So this is actually an example of a geometric progression. 893 00:48:11,020 --> 00:48:13,410 And this gives you an idea of-- 894 00:48:13,410 --> 00:48:15,900 we actually know that this infinite process could converge 895 00:48:15,900 --> 00:48:17,220 in a finite amount of time. 896 00:48:17,220 --> 00:48:20,370 Really, it took until this time. 897 00:48:20,370 --> 00:48:22,210 And if we were to go way back. 898 00:48:22,210 --> 00:48:24,030 Let's go ahead and have a logarithmic scale 899 00:48:24,030 --> 00:48:31,170 backwards to around, I think it was, 200 or 300 BC. 900 00:48:31,170 --> 00:48:34,770 That Zeno of Elea used, actually, 901 00:48:34,770 --> 00:48:38,580 the same argument for saying that motion was inherently 902 00:48:38,580 --> 00:48:44,070 impossible, and that we can never go anywhere, 903 00:48:44,070 --> 00:48:46,710 and all motion was an illusion because it would require 904 00:48:46,710 --> 00:48:48,469 doing an infinite amount of stuff, 905 00:48:48,469 --> 00:48:50,760 and that you could never do an infinite amount of stuff 906 00:48:50,760 --> 00:48:52,830 because it would necessarily be infinite. 907 00:48:52,830 --> 00:48:56,790 But it took us, as a human collective conscious, 908 00:48:56,790 --> 00:49:00,840 well over 1,700, close to 2000 years, 909 00:49:00,840 --> 00:49:04,110 to understand and develop the tools necessary to deal 910 00:49:04,110 --> 00:49:07,650 with infinities. 911 00:49:07,650 --> 00:49:13,110 So I think that's a really important thing to deal with. 912 00:49:13,110 --> 00:49:15,270 And I could go on and give entire courses 913 00:49:15,270 --> 00:49:18,300 about infinities, and talk about a lot 914 00:49:18,300 --> 00:49:21,070 of the important characters. 915 00:49:21,070 --> 00:49:24,410 And there's just one term I want to introduce, 916 00:49:24,410 --> 00:49:27,100 and a couple of concepts really quickly. 917 00:49:27,100 --> 00:49:29,880 That's the idea that there's never a top infinity. 918 00:49:29,880 --> 00:49:33,510 And you can always construct more infinities 919 00:49:33,510 --> 00:49:36,450 from smaller ones. 920 00:49:36,450 --> 00:49:40,500 And you can actually carry out a formal system 921 00:49:40,500 --> 00:49:44,010 for playing around with these infinities. 922 00:49:44,010 --> 00:49:47,742 You have your natural numbers, three, and then dot, dot, dot. 923 00:49:47,742 --> 00:49:49,200 And then we can just say, OK, well, 924 00:49:49,200 --> 00:49:51,060 let's take all of those guys. 925 00:49:51,060 --> 00:49:53,340 And we'll call them omega. 926 00:49:53,340 --> 00:49:55,510 Well then, we can take omega. 927 00:49:55,510 --> 00:49:58,600 And then we can take omega plus one, and dot, dot. 928 00:49:58,600 --> 00:49:59,889 Now I've got two omegas. 929 00:49:59,889 --> 00:50:01,680 And then we can actually start carrying out 930 00:50:01,680 --> 00:50:03,450 cardinal arithmetic. 931 00:50:08,710 --> 00:50:14,670 And there's kind of a mix of notations and concepts 932 00:50:14,670 --> 00:50:16,850 between different fields. 933 00:50:16,850 --> 00:50:19,560 But you might also see this first level of infinity 934 00:50:19,560 --> 00:50:24,660 as aleph, aleph not, or aleph sub zero. 935 00:50:24,660 --> 00:50:27,240 And this refers to the level of infinity which you 936 00:50:27,240 --> 00:50:29,550 get from the natural numbers. 937 00:50:29,550 --> 00:50:32,530 Of course, some of you may or may not know this, 938 00:50:32,530 --> 00:50:35,220 but we've got all sorts of infinities. 939 00:50:35,220 --> 00:50:37,290 We can start constructing tons. 940 00:50:37,290 --> 00:50:40,526 But we can even define exponentiation. 941 00:50:40,526 --> 00:50:42,150 So one of the big things is that, well, 942 00:50:42,150 --> 00:50:48,830 what happens when you have two raised to the aleph not? 943 00:50:48,830 --> 00:50:53,310 Well, the claim is that this is aleph one, which is 944 00:50:53,310 --> 00:50:54,780 roughly the size of the reals. 945 00:50:57,406 --> 00:51:00,120 And we're talking all of your numbers on the real line. 946 00:51:00,120 --> 00:51:02,870 And this is kind of a paradoxical thing 947 00:51:02,870 --> 00:51:08,610 because in here we have this many, all right? 948 00:51:08,610 --> 00:51:11,507 And what's very strange is-- yeah, go ahead. 949 00:51:11,507 --> 00:51:13,495 AUDIENCE: Isn't it like, [INAUDIBLE].. 950 00:51:16,980 --> 00:51:18,222 PROFESSOR: Yes, exactly. 951 00:51:18,222 --> 00:51:20,430 So you just actually stated that a very rigorous form 952 00:51:20,430 --> 00:51:22,500 of defining something to be infinite 953 00:51:22,500 --> 00:51:24,720 is that you can put it into one-to-one correspondence 954 00:51:24,720 --> 00:51:25,440 with itself. 955 00:51:25,440 --> 00:51:26,670 What Lativ said is-- 956 00:51:26,670 --> 00:51:29,700 he asked, is one of the properties of infinity 957 00:51:29,700 --> 00:51:31,980 the fact that you can put it in correspondence 958 00:51:31,980 --> 00:51:33,840 with a subset of itself? 959 00:51:33,840 --> 00:51:43,990 And so for example, we can take the normal integers. 960 00:51:43,990 --> 00:51:47,300 And fortunately, we have an infinite amount of those guys. 961 00:51:47,300 --> 00:51:49,640 And we can put them into one-to-one correspondence 962 00:51:49,640 --> 00:51:51,860 with the even numbers. 963 00:51:58,420 --> 00:52:01,930 And we can create a completely bijective map 964 00:52:01,930 --> 00:52:06,200 just by division or multiplication by two. 965 00:52:06,200 --> 00:52:09,910 But the weird thing is that we intuitively 966 00:52:09,910 --> 00:52:13,930 feel like all of these guys are inside of here so there should 967 00:52:13,930 --> 00:52:17,810 be less of them than these. 968 00:52:17,810 --> 00:52:20,840 But with infinity, you can do all sorts of things. 969 00:52:20,840 --> 00:52:21,970 And that's what's magical. 970 00:52:21,970 --> 00:52:27,160 And what's nice is you can also take the interval zero to one. 971 00:52:27,160 --> 00:52:29,290 And you can create a map which sends this 972 00:52:29,290 --> 00:52:31,910 to the entire infinite real line. 973 00:52:31,910 --> 00:52:34,660 So it's amazing because we can capture an infinite thing 974 00:52:34,660 --> 00:52:37,640 in a very finite way. 975 00:52:37,640 --> 00:52:46,150 And we also know, because of a good guy named Cantor, 976 00:52:46,150 --> 00:52:50,360 that these are really different quantities, 977 00:52:50,360 --> 00:52:52,310 that there's a level of infinity different 978 00:52:52,310 --> 00:52:56,250 between the natural numbers and the reals. 979 00:52:56,250 --> 00:52:58,330 And that deals with the diagonal argument, which 980 00:52:58,330 --> 00:52:59,538 I can maybe show you one day. 981 00:53:02,330 --> 00:53:09,700 So you guys read chapter five, which was recursive structures 982 00:53:09,700 --> 00:53:12,160 and processes, for today's lecture, 983 00:53:12,160 --> 00:53:14,710 although we've been talking a lot about other things. 984 00:53:14,710 --> 00:53:17,380 And this is really motivated because you 985 00:53:17,380 --> 00:53:20,290 got an excellent lecture last time from Curran Kelleher who 986 00:53:20,290 --> 00:53:23,020 showed you all of the different varieties in which infinity 987 00:53:23,020 --> 00:53:27,070 and recursion come together. 988 00:53:27,070 --> 00:53:31,600 But fundamentally, we still have a question 989 00:53:31,600 --> 00:53:34,750 which we're pursuing. 990 00:53:34,750 --> 00:53:37,810 And this derives back to, I think, our two most 991 00:53:37,810 --> 00:53:42,520 important tools for thinking which we'll meet 992 00:53:42,520 --> 00:53:44,160 in this first part of the book. 993 00:53:44,160 --> 00:53:52,070 And that's recursion and isomorphism. 994 00:53:55,280 --> 00:53:59,930 And remember, isomorphisms come about when 995 00:53:59,930 --> 00:54:02,810 you're trying to put equivalence relationships between one 996 00:54:02,810 --> 00:54:03,650 thing and the other. 997 00:54:03,650 --> 00:54:06,320 And you can do it in a well-defined way. 998 00:54:09,080 --> 00:54:11,030 And I want you to think about, what's 999 00:54:11,030 --> 00:54:13,580 the relationship between these two? 1000 00:54:13,580 --> 00:54:21,080 And what really connects these concepts and the way 1001 00:54:21,080 --> 00:54:23,490 in which we deal with things? 1002 00:54:23,490 --> 00:54:28,280 But I want to highlight just at least one 1003 00:54:28,280 --> 00:54:35,300 section or two from chapter five which 1004 00:54:35,300 --> 00:54:37,720 I don't know if you guys found interesting as well. 1005 00:54:37,720 --> 00:54:41,140 It's that idea of, well, we have recursion here. 1006 00:54:41,140 --> 00:54:43,340 And we can do all sorts of different things, right? 1007 00:54:43,340 --> 00:54:45,230 Curran showed you how we can construct 1008 00:54:45,230 --> 00:54:47,360 fractals, and trees, and mountains, and all sorts 1009 00:54:47,360 --> 00:54:48,580 of beautiful things. 1010 00:54:48,580 --> 00:54:54,080 And with the recursive transition networks, 1011 00:54:54,080 --> 00:54:59,700 we can get recursive programs to create sentences and language. 1012 00:54:59,700 --> 00:55:05,820 So then it appears that, well, if recursion 1013 00:55:05,820 --> 00:55:11,220 is at the heart of intelligence, why are humans so bad at it? 1014 00:55:11,220 --> 00:55:17,790 And two, is it really going to be what leads us to creativity 1015 00:55:17,790 --> 00:55:21,300 and creating a computer which we can't distinguish from a man 1016 00:55:21,300 --> 00:55:24,090 or from a person? 1017 00:55:24,090 --> 00:55:29,370 And it's funny, just to show the date on this book, 1018 00:55:29,370 --> 00:55:35,280 Douglas Hofstadter actually says people 1019 00:55:35,280 --> 00:55:37,950 said it would only take 10 years before we could create 1020 00:55:37,950 --> 00:55:41,940 a computer program that would be the world champion in chess. 1021 00:55:41,940 --> 00:55:43,410 And of course, from then they said 1022 00:55:43,410 --> 00:55:45,700 it would take another 10 years, and then another 10 years. 1023 00:55:45,700 --> 00:55:47,199 So he's kind of alluding to the idea 1024 00:55:47,199 --> 00:55:48,720 that it would never happen. 1025 00:55:48,720 --> 00:55:51,840 But sure enough, in the early 90's, IBM 1026 00:55:51,840 --> 00:55:55,950 had developed Deep Blue who beat Kasparov, 1027 00:55:55,950 --> 00:55:58,680 then-world champion at chess. 1028 00:55:58,680 --> 00:56:00,450 And it was kind of a triumph showing 1029 00:56:00,450 --> 00:56:03,310 that the way in which a human thinks 1030 00:56:03,310 --> 00:56:06,840 and the way that a human plays chess is very much more 1031 00:56:06,840 --> 00:56:09,585 intuitive than an analytical problem solving. 1032 00:56:13,870 --> 00:56:15,450 And he was talking about replying 1033 00:56:15,450 --> 00:56:17,520 to a recursive algorithm. 1034 00:56:17,520 --> 00:56:19,555 You basically analyze all possible moves. 1035 00:56:19,555 --> 00:56:21,180 And then you choose whichever one would 1036 00:56:21,180 --> 00:56:23,340 be worst for your opponent. 1037 00:56:23,340 --> 00:56:25,020 And then in deciding what move they're 1038 00:56:25,020 --> 00:56:28,080 going to make in the next step, you apply the same method, 1039 00:56:28,080 --> 00:56:30,750 except you take on your opponent's role and you say, 1040 00:56:30,750 --> 00:56:33,930 well, analyze all of the possible moves given that one. 1041 00:56:33,930 --> 00:56:35,920 And what would be the worst move for them? 1042 00:56:35,920 --> 00:56:37,230 And then you continue this. 1043 00:56:37,230 --> 00:56:40,920 And depending on how far out you can search this tree-- 1044 00:56:40,920 --> 00:56:43,440 we have this conceptual tree of chess moves-- 1045 00:56:43,440 --> 00:56:45,660 it really helps give you an advantage. 1046 00:56:45,660 --> 00:56:48,015 But Kasparov doesn't think like that. 1047 00:56:48,015 --> 00:56:50,846 The way Kasparov thinks is he intuits something. 1048 00:56:50,846 --> 00:56:53,220 And that's really a magical element of human intelligence 1049 00:56:53,220 --> 00:56:55,678 which we haven't yet been able to capture with our computer 1050 00:56:55,678 --> 00:56:57,900 programs. 1051 00:56:57,900 --> 00:57:01,140 Yet somehow, Deep Blue was able to beat him. 1052 00:57:01,140 --> 00:57:04,155 And I think this is an interesting example 1053 00:57:04,155 --> 00:57:06,990 of recursion and what its role in intelligence is, 1054 00:57:06,990 --> 00:57:09,570 and whether it's going to be the final answer. 1055 00:57:12,690 --> 00:57:15,570 Just to give you a quick show of things to do-- 1056 00:57:15,570 --> 00:57:17,370 I'm not talking much about chapter five 1057 00:57:17,370 --> 00:57:20,500 just because we've read that, we've done that. 1058 00:57:20,500 --> 00:57:23,170 We've had an entire lecture on recursion 1059 00:57:23,170 --> 00:57:24,480 and its possible roles. 1060 00:57:24,480 --> 00:57:28,170 I want you guys to pay careful attention to chapter six, which 1061 00:57:28,170 --> 00:57:30,140 is your reading assignment for next time, 1062 00:57:30,140 --> 00:57:31,890 because they're going to fundamentally ask 1063 00:57:31,890 --> 00:57:34,080 the question of, how do we get meaning? 1064 00:57:34,080 --> 00:57:37,150 How do we know that our words mean anything? 1065 00:57:37,150 --> 00:57:41,160 So the idea of developing a theory of meaning of language 1066 00:57:41,160 --> 00:57:43,530 goes back to the idea that, suppose 1067 00:57:43,530 --> 00:57:45,240 I were to plop you down on an island 1068 00:57:45,240 --> 00:57:46,620 in the middle of nowhere. 1069 00:57:46,620 --> 00:57:48,880 And all of these people are going, blah blah, blah. 1070 00:57:48,880 --> 00:57:51,150 And they're speaking their own kind of language. 1071 00:57:51,150 --> 00:57:53,130 And you have no idea what it means, right? 1072 00:57:55,840 --> 00:57:57,090 Yet, you're stuck there. 1073 00:57:57,090 --> 00:57:58,440 You depend on them for survival. 1074 00:57:58,440 --> 00:57:59,580 And you figure it might be a good idea 1075 00:57:59,580 --> 00:58:01,560 to go ahead and start learning their language. 1076 00:58:01,560 --> 00:58:03,994 How would you go about doing that? 1077 00:58:03,994 --> 00:58:05,910 So suppose you go out on your hunting missions 1078 00:58:05,910 --> 00:58:08,160 with this tribe. 1079 00:58:08,160 --> 00:58:11,970 And every time there's a rabbit which 1080 00:58:11,970 --> 00:58:15,450 gallops through, one of the guys says, "gavagai." 1081 00:58:15,450 --> 00:58:16,420 "Gavagai." 1082 00:58:16,420 --> 00:58:18,390 And you're like, OK, I'm going to create 1083 00:58:18,390 --> 00:58:19,627 an internal dictionary here. 1084 00:58:19,627 --> 00:58:21,710 I'm going to write a dictionary for this language. 1085 00:58:21,710 --> 00:58:26,040 And I know that "gavagai"-- 1086 00:58:26,040 --> 00:58:28,650 I don't know if that's how you spell it. 1087 00:58:28,650 --> 00:58:30,720 That's equivalent to rabbit. 1088 00:58:34,450 --> 00:58:36,880 But now let's switch roles. 1089 00:58:36,880 --> 00:58:41,440 Let's suppose you're one of those native people. 1090 00:58:41,440 --> 00:58:46,600 And you part of your culture is that you never 1091 00:58:46,600 --> 00:58:49,790 view something as greater than the sum of its parts. 1092 00:58:49,790 --> 00:58:52,810 So when you say "gavagai," you don't actually 1093 00:58:52,810 --> 00:58:55,030 mean the whole rabbit. 1094 00:58:55,030 --> 00:58:59,014 But you just mean un-detached rabbit part. 1095 00:58:59,014 --> 00:59:01,180 Because the second you start splitting up the rabbit 1096 00:59:01,180 --> 00:59:02,110 and you're cooking it, and you have 1097 00:59:02,110 --> 00:59:04,750 its leg over the open fire, it's no longer "gavagai." 1098 00:59:04,750 --> 00:59:08,440 Just like when we refer to a cow, 1099 00:59:08,440 --> 00:59:10,060 we usually talk about beef. 1100 00:59:10,060 --> 00:59:11,530 And the same with a pig, we usually 1101 00:59:11,530 --> 00:59:13,350 talk about pork once we start eating it 1102 00:59:13,350 --> 00:59:16,960 because we have this idea of connecting "gavagai" 1103 00:59:16,960 --> 00:59:18,660 to the full rabbit. 1104 00:59:18,660 --> 00:59:21,910 But they actually want that to be un-detached rabbit 1105 00:59:21,910 --> 00:59:26,110 part, which has a completely different conceptual network 1106 00:59:26,110 --> 00:59:28,760 for them than it does for us. 1107 00:59:28,760 --> 00:59:32,290 So this equivalence isn't true. 1108 00:59:32,290 --> 00:59:35,320 And this can actually be a very rigorous problem 1109 00:59:35,320 --> 00:59:38,110 which presents itself in the theory of meaning and language. 1110 00:59:38,110 --> 00:59:39,820 And we're going begin focusing on that 1111 00:59:39,820 --> 00:59:44,655 and the role of isomorphisms in the next lecture. 1112 00:59:44,655 --> 00:59:46,196 Go ahead, Lativ, you have a question? 1113 00:59:46,196 --> 00:59:48,160 AUDIENCE: So then how are you going to learn the language? 1114 00:59:48,160 --> 00:59:49,390 PROFESSOR: How are you going to learn the language? 1115 00:59:49,390 --> 00:59:51,056 Well, we can obviously develop some sort 1116 00:59:51,056 --> 00:59:54,980 of functional apparatus, like I can get close to a rabbit. 1117 00:59:54,980 --> 00:59:58,210 But how do we actually decipher meaning, right? 1118 00:59:58,210 --> 01:00:00,250 We can obviously learn a language in some ways. 1119 01:00:00,250 --> 01:00:01,780 And in some ways, it's the same. 1120 01:00:01,780 --> 01:00:03,850 How do you know what I say is exactly what you 1121 01:00:03,850 --> 01:00:04,495 want me to say? 1122 01:00:04,495 --> 01:00:04,995 And-- 1123 01:00:04,995 --> 01:00:07,610 AUDIENCE: [INAUDIBLE] 1124 01:00:07,610 --> 01:00:09,600 PROFESSOR: And that's exactly it. 1125 01:00:09,600 --> 01:00:11,856 So then how do we develop a theory of meaning? 1126 01:00:11,856 --> 01:00:13,140 But plenty of good questions. 1127 01:00:13,140 --> 01:00:14,880 Good questions for next lecture. 1128 01:00:14,880 --> 01:00:16,640 And turn in the surveys. 1129 01:00:16,640 --> 01:00:19,211 And I look forward to seeing you guys next time.