1 00:00:00,060 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high-quality educational resources for free. 5 00:00:10,120 --> 00:00:12,660 To make a donation, or to view additional materials 6 00:00:12,660 --> 00:00:16,332 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,332 --> 00:00:18,420 at ocw.mit.edu. 8 00:00:18,420 --> 00:00:21,000 NICHOLAS DIBELLA: OK. 9 00:00:21,000 --> 00:00:24,750 So last week, I described the quest 10 00:00:24,750 --> 00:00:29,370 that theoretical physicists are on today in coming up 11 00:00:29,370 --> 00:00:31,680 with a theory of everything. 12 00:00:31,680 --> 00:00:33,930 We don't know what the theory of everything is yet. 13 00:00:33,930 --> 00:00:35,700 Nobody knows. 14 00:00:35,700 --> 00:00:38,100 It would somehow have to reconcile 15 00:00:38,100 --> 00:00:41,630 the conflict between quantum theory and general relativity. 16 00:00:41,630 --> 00:00:44,910 It would have to be some kind of theory of quantum gravity. 17 00:00:44,910 --> 00:00:46,720 But many people are certainly trying. 18 00:00:46,720 --> 00:00:49,260 And many of the optimistic people 19 00:00:49,260 --> 00:00:52,420 think that we're almost there. 20 00:00:52,420 --> 00:00:56,100 I personally have no idea because I'm not 21 00:00:56,100 --> 00:00:58,560 an expert in any candidate theories of quantum gravity. 22 00:00:58,560 --> 00:01:01,140 But there's a sense that we've definitely 23 00:01:01,140 --> 00:01:05,172 been making progress over the past hundred years. 24 00:01:05,172 --> 00:01:07,380 And so when you think of all the progress we've made, 25 00:01:07,380 --> 00:01:09,900 it seems maybe it's conceivable that sometime 26 00:01:09,900 --> 00:01:11,771 in the future, one day, we'll finally have 27 00:01:11,771 --> 00:01:13,020 the theory of quantum gravity. 28 00:01:13,020 --> 00:01:15,780 We'll finally have the theory of everything. 29 00:01:15,780 --> 00:01:21,200 Whether that's in 50 years, or 100 years, or 500 years, 30 00:01:21,200 --> 00:01:22,570 I have no idea. 31 00:01:22,570 --> 00:01:25,090 Nobody knows. 32 00:01:25,090 --> 00:01:27,851 But that day might eventually come. 33 00:01:27,851 --> 00:01:29,850 So it seems like we can understand the universe. 34 00:01:29,850 --> 00:01:32,070 The universe is an extremely complicated thing, 35 00:01:32,070 --> 00:01:34,374 but someday we might be able to understand it. 36 00:01:34,374 --> 00:01:38,610 One, in fact, famous quote of Einstein 37 00:01:38,610 --> 00:01:39,990 is that the most incomprehensible 38 00:01:39,990 --> 00:01:46,080 thing about the universe is that it is comprehensible. 39 00:01:46,080 --> 00:01:48,690 But is the universe really comprehensible? 40 00:01:48,690 --> 00:01:51,210 Can we really understand everything? 41 00:01:51,210 --> 00:01:53,529 We certainly understand some things about the universe. 42 00:01:53,529 --> 00:01:56,070 Actually, we think we know a lot about the universe, just not 43 00:01:56,070 --> 00:01:56,740 everything. 44 00:01:56,740 --> 00:02:00,482 But could we really understand everything about the universe? 45 00:02:00,482 --> 00:02:01,440 What do you guys think? 46 00:02:01,440 --> 00:02:03,640 Do you think we can actually understand everything? 47 00:02:03,640 --> 00:02:06,020 Have the theory of everything? 48 00:02:06,020 --> 00:02:06,749 No? 49 00:02:06,749 --> 00:02:07,290 AUDIENCE: No. 50 00:02:07,290 --> 00:02:08,020 NICHOLAS DIBELLA: No. 51 00:02:08,020 --> 00:02:08,520 Why not? 52 00:02:08,520 --> 00:02:11,494 Why are you all so pessimistic? 53 00:02:11,494 --> 00:02:13,035 AUDIENCE: I'll be the optimist today. 54 00:02:13,035 --> 00:02:13,619 AUDIENCE: Yes. 55 00:02:13,619 --> 00:02:14,868 NICHOLAS DIBELLA: Yes, we can. 56 00:02:14,868 --> 00:02:15,540 OK. 57 00:02:15,540 --> 00:02:17,382 Why are you so optimistic? 58 00:02:17,382 --> 00:02:19,280 AUDIENCE: Because I'm an optimistic person. 59 00:02:19,280 --> 00:02:20,155 NICHOLAS DIBELLA: OK. 60 00:02:20,155 --> 00:02:22,400 That's a valid position to take. 61 00:02:22,400 --> 00:02:24,120 Why are you so pessimistic, though? 62 00:02:24,120 --> 00:02:27,990 I mean, we've been able to describe things 63 00:02:27,990 --> 00:02:29,340 that we can't even see. 64 00:02:29,340 --> 00:02:34,320 Things that are a billion or a million billion times smaller 65 00:02:34,320 --> 00:02:36,750 than a centimeter. 66 00:02:36,750 --> 00:02:38,760 I mean, we can see things that are so small. 67 00:02:38,760 --> 00:02:41,520 We have theories about them that really seem to work. 68 00:02:41,520 --> 00:02:44,930 We can see things that are incredibly small. 69 00:02:44,930 --> 00:02:47,010 We've been making headway in these questions, 70 00:02:47,010 --> 00:02:48,750 like [INAUDIBLE] the beginning of time. 71 00:02:48,750 --> 00:02:49,749 Is time travel possible? 72 00:02:52,170 --> 00:02:54,930 So it's conceivable that someday we'll understand everything. 73 00:02:54,930 --> 00:02:56,027 It's conceivable. 74 00:02:58,650 --> 00:03:02,360 I mean, I'm not going to-- 75 00:03:02,360 --> 00:03:05,690 I wouldn't put forth an opinion. 76 00:03:05,690 --> 00:03:07,260 But yeah, it's natural. 77 00:03:07,260 --> 00:03:10,470 I mean, it's natural to think that maybe we won't be able to. 78 00:03:10,470 --> 00:03:14,820 I mean, the universe is a really big thing, right? 79 00:03:14,820 --> 00:03:17,100 We're small things. 80 00:03:17,100 --> 00:03:19,290 So maybe something really small can't find out 81 00:03:19,290 --> 00:03:21,323 everything about something that's really big. 82 00:03:21,323 --> 00:03:21,948 AUDIENCE: Wait. 83 00:03:21,948 --> 00:03:22,800 What was the question? 84 00:03:22,800 --> 00:03:25,258 NICHOLAS DIBELLA: Do you think it's impossible in principle 85 00:03:25,258 --> 00:03:27,030 to know everything about the universe? 86 00:03:27,030 --> 00:03:31,530 I mean, maybe not information about every particle, 87 00:03:31,530 --> 00:03:32,610 every bit of matter. 88 00:03:32,610 --> 00:03:34,050 But the laws at least. 89 00:03:34,050 --> 00:03:37,560 Is it possible in principle to know all the laws 90 00:03:37,560 --> 00:03:40,683 with 100% accuracy? 91 00:03:40,683 --> 00:03:42,099 AUDIENCE: I thought we [INAUDIBLE] 92 00:03:42,099 --> 00:03:44,920 only scratched at the surface of [INAUDIBLE].. 93 00:03:44,920 --> 00:03:45,930 NICHOLAS DIBELLA: OK. 94 00:03:45,930 --> 00:03:48,657 Yeah, it's true that we've kind of scratched the surface. 95 00:03:48,657 --> 00:03:50,990 But many people think that we're actually getting close. 96 00:03:50,990 --> 00:03:52,573 Many people worked on quantum gravity, 97 00:03:52,573 --> 00:03:55,012 like string theory and things like that. 98 00:03:55,012 --> 00:03:57,220 But also, many people don't think we're close at all. 99 00:03:57,220 --> 00:03:58,590 Many people don't think we've scratched the surface. 100 00:03:58,590 --> 00:04:00,256 We're going in all the wrong directions. 101 00:04:03,240 --> 00:04:05,490 OK. 102 00:04:05,490 --> 00:04:06,420 Yeah. 103 00:04:06,420 --> 00:04:08,490 Pessimism is certainly a valid position to take. 104 00:04:08,490 --> 00:04:11,160 Optimism might be. 105 00:04:11,160 --> 00:04:13,630 I don't know. 106 00:04:13,630 --> 00:04:15,640 OK. 107 00:04:15,640 --> 00:04:17,300 So today, I'd like to-- 108 00:04:17,300 --> 00:04:19,050 well, for the first part of today's class, 109 00:04:19,050 --> 00:04:21,425 I'd like to look at the question of whether it's actually 110 00:04:21,425 --> 00:04:24,666 possible in principle to have a theory of everything. 111 00:04:24,666 --> 00:04:27,100 Could we really know everything? 112 00:04:27,100 --> 00:04:29,830 So first, we need to backtrack. 113 00:04:29,830 --> 00:04:32,890 We need to backtrack. 114 00:04:32,890 --> 00:04:36,440 Take a look at these theories. 115 00:04:36,440 --> 00:04:40,410 So one of my aims for this class has 116 00:04:40,410 --> 00:04:43,080 been to describe these physical theories-- 117 00:04:43,080 --> 00:04:46,880 quantum mechanics, relativity, string theory, and so forth-- 118 00:04:46,880 --> 00:04:51,040 as conceptually as I could, just using ideas. 119 00:04:51,040 --> 00:04:52,610 And I think to a large extent, it's 120 00:04:52,610 --> 00:04:55,440 possible to convey the sense of ideas 121 00:04:55,440 --> 00:05:00,470 that you have in these theories, like extra dimensions or time 122 00:05:00,470 --> 00:05:03,320 going shorter for one person than for another. 123 00:05:03,320 --> 00:05:06,420 But deep down, these theories are mathematical in nature. 124 00:05:06,420 --> 00:05:10,700 I mean, deep down when you look at the theories, 125 00:05:10,700 --> 00:05:13,107 they're just a bunch of equations. 126 00:05:13,107 --> 00:05:15,440 Strictly speaking, physics is just a bunch of equations. 127 00:05:18,530 --> 00:05:21,620 But what physicists do is they look at these equations 128 00:05:21,620 --> 00:05:23,569 and they try to make meaning out of them. 129 00:05:23,569 --> 00:05:25,610 They try to interpret them in some kind of a way. 130 00:05:25,610 --> 00:05:26,401 And they say, a-ha. 131 00:05:26,401 --> 00:05:28,610 These mathematical ideas actually 132 00:05:28,610 --> 00:05:32,190 correspond to something in reality. 133 00:05:32,190 --> 00:05:34,940 But deep down, they're mathematical in nature. 134 00:05:34,940 --> 00:05:37,653 And the real way to understand them is mathematically. 135 00:05:40,900 --> 00:05:45,120 So if anything I've said over the past six weeks 136 00:05:45,120 --> 00:05:46,280 has sounded vague-- 137 00:05:46,280 --> 00:05:48,113 I'm sure there are a lot of things I've said 138 00:05:48,113 --> 00:05:49,850 that have sounded vague, or anything 139 00:05:49,850 --> 00:05:52,460 that sounded imprecise or unclear, 140 00:05:52,460 --> 00:05:56,245 you'll just have to take my word that if you studied 141 00:05:56,245 --> 00:05:58,370 the theories mathematically, then eventually things 142 00:05:58,370 --> 00:05:59,690 would make a lot more sense. 143 00:05:59,690 --> 00:06:00,800 I tried to explain them as clearly 144 00:06:00,800 --> 00:06:01,799 as I could without math. 145 00:06:01,799 --> 00:06:03,170 But eventually, you need math. 146 00:06:05,960 --> 00:06:08,720 Actually, more speculative to theories, 147 00:06:08,720 --> 00:06:11,090 like theories of quantum gravity, 148 00:06:11,090 --> 00:06:12,860 even the math is not clear. 149 00:06:12,860 --> 00:06:15,830 We don't really know what the right math to use is. 150 00:06:15,830 --> 00:06:20,360 So there's still many ideas and our current candidate theories 151 00:06:20,360 --> 00:06:22,849 of everything that are still very vague and imprecise. 152 00:06:22,849 --> 00:06:24,890 But definitely everything about quantum mechanics 153 00:06:24,890 --> 00:06:26,362 and relativity, those are all-- 154 00:06:26,362 --> 00:06:28,070 pretty much everything about those, those 155 00:06:28,070 --> 00:06:32,355 are all very clear and precise. 156 00:06:36,740 --> 00:06:38,990 OK. 157 00:06:38,990 --> 00:06:41,381 Everyone fine with that? 158 00:06:41,381 --> 00:06:41,880 OK. 159 00:06:41,880 --> 00:06:46,830 Now, I'd like to describe one of the most surprising results 160 00:06:46,830 --> 00:06:49,709 of modern mathematics. 161 00:06:49,709 --> 00:06:51,750 But first, I just want to ask the question, well, 162 00:06:51,750 --> 00:06:53,670 what is mathematics? 163 00:06:53,670 --> 00:06:56,730 Most of us when we're first exposed to math, 164 00:06:56,730 --> 00:06:59,620 we just see it as a bunch of tools or a bag of tricks. 165 00:06:59,620 --> 00:07:03,450 Just some interesting things about numbers, shapes. 166 00:07:03,450 --> 00:07:05,070 I mean, a lot of these things seem 167 00:07:05,070 --> 00:07:09,335 to find their place in the real world. 168 00:07:09,335 --> 00:07:11,210 I mean, we see shapes around us, for example. 169 00:07:11,210 --> 00:07:14,520 I mean, here is a rectangle. 170 00:07:14,520 --> 00:07:15,730 There is a circle. 171 00:07:15,730 --> 00:07:17,580 The sun looks like a circle. 172 00:07:17,580 --> 00:07:18,600 We can count objects-- 173 00:07:18,600 --> 00:07:19,490 1, 2, 3, 4. 174 00:07:19,490 --> 00:07:22,430 We can count things. 175 00:07:22,430 --> 00:07:24,030 So when we're first exposed to math, 176 00:07:24,030 --> 00:07:26,550 we didn't really see it as much more than something that's 177 00:07:26,550 --> 00:07:29,350 not really worth studying. 178 00:07:29,350 --> 00:07:31,090 Well, many of us at least. 179 00:07:31,090 --> 00:07:35,790 But there do exist people that study math for a living. 180 00:07:35,790 --> 00:07:37,285 They're called mathematicians. 181 00:07:37,285 --> 00:07:38,910 And to them, math is much more exciting 182 00:07:38,910 --> 00:07:43,280 than just a bag of tricks or just a set of tools. 183 00:07:43,280 --> 00:07:44,842 Although, many people do get excited 184 00:07:44,842 --> 00:07:46,050 about certain bags of tricks. 185 00:07:48,580 --> 00:07:52,890 But to mathematicians, to a pure mathematician, someone 186 00:07:52,890 --> 00:07:55,710 who studies purely math, who doesn't care about what it has 187 00:07:55,710 --> 00:07:59,460 to do with the real world, the ideas in math, 188 00:07:59,460 --> 00:08:02,400 the objects and notions that mathematicians study, 189 00:08:02,400 --> 00:08:04,110 they really take on a life of their own. 190 00:08:08,250 --> 00:08:10,061 There are a lot of things about math 191 00:08:10,061 --> 00:08:12,060 that seem to find their place in the real world. 192 00:08:12,060 --> 00:08:15,150 But mathematicians often tend to pride themselves 193 00:08:15,150 --> 00:08:17,640 on things that actually don't have anything whatsoever 194 00:08:17,640 --> 00:08:19,250 to do with reality. 195 00:08:19,250 --> 00:08:23,100 Now, I said before that there are a lot of things that-- 196 00:08:23,100 --> 00:08:25,750 a lot of mathematical concepts that find a place in reality. 197 00:08:25,750 --> 00:08:27,312 Like we can count. 198 00:08:27,312 --> 00:08:29,090 There are integers or whole numbers. 199 00:08:29,090 --> 00:08:30,350 We can count 1, 2, 3. 200 00:08:30,350 --> 00:08:31,620 There are shapes. 201 00:08:31,620 --> 00:08:34,230 But there are many things, many abstract mathematical ideas, 202 00:08:34,230 --> 00:08:37,090 that don't find themselves in reality. 203 00:08:37,090 --> 00:08:38,940 For example, imaginary numbers. 204 00:08:53,940 --> 00:08:54,700 What's that? 205 00:08:54,700 --> 00:08:55,620 AUDIENCE: [INAUDIBLE] 206 00:08:55,620 --> 00:08:55,867 NICHOLAS DIBELLA: Yeah. 207 00:08:55,867 --> 00:08:57,540 So imaginary numbers are numbers that 208 00:08:57,540 --> 00:08:59,100 are square roots of negative numbers. 209 00:09:01,515 --> 00:09:03,265 They're these things that are square roots 210 00:09:03,265 --> 00:09:04,690 of negative numbers. 211 00:09:04,690 --> 00:09:08,280 So when we first learn about multiplication, 212 00:09:08,280 --> 00:09:11,100 we first learn about numbers that aren't imaginary. 213 00:09:11,100 --> 00:09:12,700 We call them real numbers. 214 00:09:12,700 --> 00:09:20,310 So numbers like 3, negative 4, 6.9. 215 00:09:20,310 --> 00:09:22,300 And we can take a number and we can square it. 216 00:09:22,300 --> 00:09:25,420 And to square a number, it just means to multiply it by itself. 217 00:09:25,420 --> 00:09:30,510 So for example, 3 times 3 is 9. 218 00:09:30,510 --> 00:09:33,720 We say that 9 is the square root of 3. 219 00:09:33,720 --> 00:09:38,640 Negative 1 times negative 1 is positive 1. 220 00:09:38,640 --> 00:09:41,730 So 1 is the square of negative 1. 221 00:09:41,730 --> 00:09:44,924 And so whenever you multiply a positive number, a number 222 00:09:44,924 --> 00:09:47,340 greater than 0 by itself, you get another positive number. 223 00:09:47,340 --> 00:09:49,131 And whenever you multiply a negative number 224 00:09:49,131 --> 00:09:51,880 by another negative number, you always get a positive number. 225 00:09:51,880 --> 00:09:53,338 So the square of a number is always 226 00:09:53,338 --> 00:09:54,900 something that's positive. 227 00:09:54,900 --> 00:09:55,920 Or else it's 0. 228 00:09:55,920 --> 00:09:57,210 0 times 0 is 0. 229 00:09:57,210 --> 00:09:59,680 We call those real numbers. 230 00:09:59,680 --> 00:10:01,110 So these are the real numbers. 231 00:10:04,090 --> 00:10:05,315 This sort of number. 232 00:10:05,315 --> 00:10:06,190 AUDIENCE: [INAUDIBLE] 233 00:10:06,190 --> 00:10:07,440 NICHOLAS DIBELLA: What's that? 234 00:10:07,440 --> 00:10:08,430 AUDIENCE: [INAUDIBLE] 235 00:10:08,430 --> 00:10:09,970 NICHOLAS DIBELLA: Oh. 236 00:10:09,970 --> 00:10:11,080 Reals. 237 00:10:11,080 --> 00:10:11,580 OK. 238 00:10:11,580 --> 00:10:13,150 Those are the real numbers. 239 00:10:13,150 --> 00:10:15,240 But you can ask the question, does there 240 00:10:15,240 --> 00:10:18,150 exist any kind of number which when you square 241 00:10:18,150 --> 00:10:19,830 it gives a negative number? 242 00:10:19,830 --> 00:10:24,330 Like does negative 1 have-- we call it the square root. 243 00:10:24,330 --> 00:10:26,430 So we say 3 is the square root of 9. 244 00:10:26,430 --> 00:10:28,706 Negative 1 is a square root of positive 1. 245 00:10:28,706 --> 00:10:30,330 But does negative 1 have a square root? 246 00:10:30,330 --> 00:10:32,920 Is there any number, which when you multiply by itself 247 00:10:32,920 --> 00:10:34,650 gives negative 1? 248 00:10:34,650 --> 00:10:36,720 And there doesn't seem to be any real number that 249 00:10:36,720 --> 00:10:41,040 gives negative 1 as its square. 250 00:10:41,040 --> 00:10:44,520 So you can invent a new kind of number. 251 00:10:44,520 --> 00:10:47,324 You can say, well, let's invent a new kind of number. 252 00:10:47,324 --> 00:10:48,740 Let's call it an imaginary number. 253 00:10:48,740 --> 00:10:50,300 And let's give it the symbol i. 254 00:10:50,300 --> 00:10:55,070 Then me say i times i equals negative 1. 255 00:10:55,070 --> 00:10:55,737 Problem solved. 256 00:10:55,737 --> 00:10:57,570 You can get square roots of negative numbers 257 00:10:57,570 --> 00:11:00,250 by making imaginary numbers. 258 00:11:00,250 --> 00:11:07,590 Similarly, 2i times 2i equals negative 4. 259 00:11:07,590 --> 00:11:10,560 So these are the imaginary numbers. 260 00:11:10,560 --> 00:11:15,630 Now, real numbers seem to appear everywhere in the world. 261 00:11:15,630 --> 00:11:17,240 I can count people-- 262 00:11:17,240 --> 00:11:18,600 1, 2, 3. 263 00:11:18,600 --> 00:11:24,150 I can measure the length of some stick. 264 00:11:24,150 --> 00:11:29,020 Maybe this is 2.3 feet long. 265 00:11:29,020 --> 00:11:31,801 So real numbers appear everywhere. 266 00:11:31,801 --> 00:11:32,550 Imaginary numbers? 267 00:11:32,550 --> 00:11:34,326 Not so much. 268 00:11:34,326 --> 00:11:35,970 Does anyone see imaginary numbers 269 00:11:35,970 --> 00:11:38,944 when they look around the world? 270 00:11:38,944 --> 00:11:40,360 Maybe if you see imaginary people, 271 00:11:40,360 --> 00:11:43,420 then they might remind you of imaginary numbers. 272 00:11:43,420 --> 00:11:45,690 So imaginary numbers don't seem to find 273 00:11:45,690 --> 00:11:47,760 any place in the real world. 274 00:11:47,760 --> 00:11:49,560 But mathematicians studying things like-- 275 00:11:49,560 --> 00:11:50,233 yeah? 276 00:11:50,233 --> 00:11:52,649 AUDIENCE: The only place I've seen imaginary numbers would 277 00:11:52,649 --> 00:11:54,930 be my math team. 278 00:11:54,930 --> 00:11:56,734 We were all numbered imaginary numbers. 279 00:11:56,734 --> 00:12:00,710 I was number square root of negative 5. 280 00:12:00,710 --> 00:12:01,860 NICHOLAS DIBELLA: OK. 281 00:12:01,860 --> 00:12:04,440 Yeah, so imaginary numbers might show up 282 00:12:04,440 --> 00:12:05,970 in recreational activities involving 283 00:12:05,970 --> 00:12:08,840 math people in everyday life. 284 00:12:08,840 --> 00:12:10,764 Yeah, certainly. 285 00:12:10,764 --> 00:12:12,930 But you definitely don't count in imaginary numbers. 286 00:12:12,930 --> 00:12:14,805 We don't measure things in imaginary numbers. 287 00:12:14,805 --> 00:12:17,090 That's definitely true. 288 00:12:17,090 --> 00:12:20,761 But imaginary numbers are things that mathematicians study. 289 00:12:20,761 --> 00:12:22,260 And they don't seem to have anything 290 00:12:22,260 --> 00:12:25,070 to do with the real world. 291 00:12:25,070 --> 00:12:26,284 It's just an abstract idea. 292 00:12:26,284 --> 00:12:27,450 The imaginary number system. 293 00:12:27,450 --> 00:12:30,060 It's this abstract idea that mathematicians study. 294 00:12:30,060 --> 00:12:32,220 The very simple abstract system. 295 00:12:32,220 --> 00:12:35,790 I mean, this is all there is to it. 296 00:12:35,790 --> 00:12:38,100 But there are many other abstract mathematical systems 297 00:12:38,100 --> 00:12:40,950 that mathematicians study that don't seem to have anything 298 00:12:40,950 --> 00:12:43,190 to do with the real world. 299 00:12:43,190 --> 00:12:45,090 Now, imaginary numbers actually turn out 300 00:12:45,090 --> 00:12:50,220 to be very useful in quantum mechanics. 301 00:12:50,220 --> 00:12:52,720 But you definitely don't measure anything to be imaginary. 302 00:12:52,720 --> 00:12:54,480 In the theory, imaginary numbers actually 303 00:12:54,480 --> 00:12:58,410 show up in the quantum states, in the wave function. 304 00:12:58,410 --> 00:13:01,080 It's actually an imaginary thing. 305 00:13:01,080 --> 00:13:02,160 OK. 306 00:13:02,160 --> 00:13:05,639 So to a mathematician, there exists a mathematical kind 307 00:13:05,639 --> 00:13:06,180 of existence. 308 00:13:06,180 --> 00:13:09,760 There is some abstract mathematical truth 309 00:13:09,760 --> 00:13:13,180 that kind of exists out there independently of us. 310 00:13:13,180 --> 00:13:15,180 But I'll have a lot more to say about the nature 311 00:13:15,180 --> 00:13:16,513 of mathematical existence later. 312 00:13:19,551 --> 00:13:20,050 OK. 313 00:13:20,050 --> 00:13:28,370 So all mathematical systems have some basic axioms 314 00:13:28,370 --> 00:13:31,960 or some basic postulates that make the systems. 315 00:13:31,960 --> 00:13:33,520 So remember a couple of weeks ago 316 00:13:33,520 --> 00:13:35,224 when I talked about special relativity? 317 00:13:35,224 --> 00:13:37,390 Special relativity had what I called two postulates, 318 00:13:37,390 --> 00:13:40,170 There are these two fundamental principles. 319 00:13:40,170 --> 00:13:41,920 And from these two fundamental principles, 320 00:13:41,920 --> 00:13:45,320 everything about the theory you can derive. 321 00:13:45,320 --> 00:13:48,590 You can derive the fact that moving clocks run slow. 322 00:13:48,590 --> 00:13:51,710 The fact that lengths gets shorter for moving objects. 323 00:13:51,710 --> 00:13:54,110 All these facts can be derived from two 324 00:13:54,110 --> 00:13:55,130 fundamental principles. 325 00:13:55,130 --> 00:13:57,560 We call them postulates. 326 00:13:57,560 --> 00:14:00,470 The situation is similar in math. 327 00:14:00,470 --> 00:14:02,600 All mathematical systems also are 328 00:14:02,600 --> 00:14:06,350 built from fundamental postulates or axioms. 329 00:14:06,350 --> 00:14:08,140 So I'll use both for it. 330 00:14:08,140 --> 00:14:11,050 So axioms. 331 00:14:11,050 --> 00:14:13,210 That's a cool word, I think. 332 00:14:13,210 --> 00:14:22,120 So fundamental premises of math. 333 00:14:29,789 --> 00:14:31,830 How many of you ever studied any geometry at all? 334 00:14:34,490 --> 00:14:35,446 OK. 335 00:14:35,446 --> 00:14:37,480 A good number of you. 336 00:14:37,480 --> 00:14:40,740 So how many of you have heard of Euclid's axioms 337 00:14:40,740 --> 00:14:42,000 or Euclid's postulates? 338 00:14:44,790 --> 00:14:47,610 Well, it turns out that all of geometry 339 00:14:47,610 --> 00:14:51,230 that you've probably studied can be traced back to some-- 340 00:14:51,230 --> 00:14:54,240 to a certain number of fundamental postulates. 341 00:14:54,240 --> 00:15:01,140 So all the geometry that is that you do on flat surfaces, 342 00:15:01,140 --> 00:15:02,160 like on this chalkboard. 343 00:15:02,160 --> 00:15:03,630 You can draw a circle. 344 00:15:03,630 --> 00:15:04,320 Draw a line. 345 00:15:04,320 --> 00:15:05,800 A triangle. 346 00:15:05,800 --> 00:15:08,120 You can draw geometrical objects. 347 00:15:08,120 --> 00:15:11,340 All geometry that you can draw on a flat surface, 348 00:15:11,340 --> 00:15:13,190 we call that plane geometry. 349 00:15:13,190 --> 00:15:17,910 And all the facts that you can derive about these things 350 00:15:17,910 --> 00:15:21,180 can all be traced back down to some fundamental axioms. 351 00:15:21,180 --> 00:15:26,220 And the axioms of plane geometry are called Euclid's axioms. 352 00:15:26,220 --> 00:15:30,810 So this is one type of mathematical system. 353 00:15:40,610 --> 00:15:42,320 I guess that's OK. 354 00:15:42,320 --> 00:15:47,510 So the first axiom is that any two points can be 355 00:15:47,510 --> 00:15:50,670 connected by a straight line. 356 00:15:50,670 --> 00:15:53,200 So I'll just write down these axioms in pictures 357 00:15:53,200 --> 00:15:55,880 instead of words if I can. 358 00:15:55,880 --> 00:15:59,270 So you have two points. 359 00:15:59,270 --> 00:16:01,610 The first axiom says you can take these two points 360 00:16:01,610 --> 00:16:02,680 and form a line. 361 00:16:05,400 --> 00:16:07,330 It's very simple. 362 00:16:07,330 --> 00:16:10,420 A line. 363 00:16:10,420 --> 00:16:13,705 The second axiom says that if you have a line segment-- 364 00:16:13,705 --> 00:16:15,240 not an infinitely long thing. 365 00:16:15,240 --> 00:16:18,090 But if you have a line segment like this, 366 00:16:18,090 --> 00:16:19,090 there are two endpoints. 367 00:16:19,090 --> 00:16:22,110 You can always take a line segment 368 00:16:22,110 --> 00:16:23,110 and turn it into a line. 369 00:16:27,320 --> 00:16:32,154 The third axiom says that given a line segment-- 370 00:16:32,154 --> 00:16:33,320 move this down a little bit. 371 00:16:37,120 --> 00:16:44,037 Given a line segment, you can always take this line segment 372 00:16:44,037 --> 00:16:45,120 and turn it into a circle. 373 00:16:49,220 --> 00:16:51,590 It's supposed to be a circle. 374 00:16:51,590 --> 00:16:57,470 The fourth axiom says that all right angles are the same. 375 00:16:57,470 --> 00:16:59,460 Here. 376 00:16:59,460 --> 00:17:01,620 So right angle is just 90 degrees. 377 00:17:01,620 --> 00:17:06,569 So this thing is the same as-- 378 00:17:06,569 --> 00:17:07,619 I can do a little better. 379 00:17:07,619 --> 00:17:09,160 This thing is the same as this thing. 380 00:17:09,160 --> 00:17:10,540 All right angles are the same. 381 00:17:10,540 --> 00:17:13,910 And then, the fifth axiom is actually 382 00:17:13,910 --> 00:17:16,130 a more interesting axiom. 383 00:17:16,130 --> 00:17:17,960 Because if you change it, then you 384 00:17:17,960 --> 00:17:21,170 can actually look at geometry on curved surfaces. 385 00:17:21,170 --> 00:17:23,210 Like general relativity is based on geometries 386 00:17:23,210 --> 00:17:24,640 of curved surfaces. 387 00:17:24,640 --> 00:17:27,500 But on plane services, like on a chalkboard, 388 00:17:27,500 --> 00:17:32,000 the fifth postulate says that parallel lines never intersect. 389 00:17:34,399 --> 00:17:35,440 I'll write this in words. 390 00:17:46,510 --> 00:17:52,830 So this line and this line. 391 00:17:52,830 --> 00:17:55,557 Use your imagination to pretend that they go forever. 392 00:17:55,557 --> 00:17:56,640 They never hit each other. 393 00:17:56,640 --> 00:17:58,310 They never intersect. 394 00:17:58,310 --> 00:17:59,900 So those are the two postulates. 395 00:17:59,900 --> 00:18:04,850 Sorry, the five postulates of Euclidean geometry. 396 00:18:04,850 --> 00:18:09,290 This is the geometry that Euclid, a Greek mathematician 397 00:18:09,290 --> 00:18:14,040 from the BCs, wrote down. 398 00:18:14,040 --> 00:18:15,830 These were his axioms. 399 00:18:15,830 --> 00:18:16,330 OK. 400 00:18:16,330 --> 00:18:18,790 Are there any questions about Euclid's axioms? 401 00:18:21,310 --> 00:18:22,530 OK. 402 00:18:22,530 --> 00:18:25,860 From these axioms, you can then derive things. 403 00:18:25,860 --> 00:18:29,895 You can then prove things, like the Pythagorean theorem. 404 00:18:32,950 --> 00:18:35,760 Just take a right triangle. 405 00:18:35,760 --> 00:18:37,270 Triangle with a right angle. 406 00:18:37,270 --> 00:18:37,950 Call this a. 407 00:18:37,950 --> 00:18:39,440 Call the length of this b. 408 00:18:39,440 --> 00:18:39,940 Call this c. 409 00:18:39,940 --> 00:18:47,859 And then, you can prove this fact just from these axioms. 410 00:18:47,859 --> 00:18:49,650 If you're smart enough, you can prove them. 411 00:18:52,240 --> 00:18:54,046 Makes sense, right? 412 00:18:54,046 --> 00:18:55,170 Yeah, something you can do. 413 00:18:55,170 --> 00:18:57,330 If you're smart enough, you can use these axioms. 414 00:18:57,330 --> 00:19:01,170 Use logic and deduce these facts. 415 00:19:01,170 --> 00:19:05,870 You can also prove that the area of a circle-- 416 00:19:05,870 --> 00:19:08,124 the way to find the area of a circle. 417 00:19:08,124 --> 00:19:11,450 You look at the radius of a circle, r. 418 00:19:11,450 --> 00:19:15,350 And then it turns out you can prove that the area was 419 00:19:15,350 --> 00:19:19,634 just pi, 3.14159, et cetera. 420 00:19:19,634 --> 00:19:21,360 It's pi times r squared. 421 00:19:21,360 --> 00:19:22,950 That's the area of the circle. 422 00:19:22,950 --> 00:19:25,170 And if you're smart enough, you can-- 423 00:19:25,170 --> 00:19:27,400 there's no limit to what-- 424 00:19:27,400 --> 00:19:29,050 who knows what you can do? 425 00:19:29,050 --> 00:19:30,900 You'll prove all sorts of crazy things 426 00:19:30,900 --> 00:19:35,040 about circles, triangles, heptagons, lines, line 427 00:19:35,040 --> 00:19:36,420 segments, angles. 428 00:19:36,420 --> 00:19:40,210 You can prove all sorts of crazy things about it. 429 00:19:40,210 --> 00:19:41,944 OK. 430 00:19:41,944 --> 00:19:42,735 So that's geometry. 431 00:19:45,480 --> 00:19:49,634 You can also write down some axioms for arithmetic. 432 00:19:58,040 --> 00:20:00,440 And I'm going to define arithmetic 433 00:20:00,440 --> 00:20:03,000 as the math of whole numbers. 434 00:20:03,000 --> 00:20:04,630 I'll just define it very loosely. 435 00:20:19,630 --> 00:20:24,240 0, 1, 2, 3, et cetera. 436 00:20:24,240 --> 00:20:26,310 Those are all whole numbers. 437 00:20:26,310 --> 00:20:27,000 And the math. 438 00:20:27,000 --> 00:20:28,770 What kinds of things can you do with them? 439 00:20:28,770 --> 00:20:31,125 Well, you can add them. 440 00:20:31,125 --> 00:20:33,870 You can multiply them. 441 00:20:33,870 --> 00:20:39,690 And you can have a lot of fun. 442 00:20:39,690 --> 00:20:40,890 So that's arithmetic. 443 00:20:40,890 --> 00:20:42,950 That's how I define arithmetic. 444 00:20:42,950 --> 00:20:44,650 There are a certain number of axioms. 445 00:20:44,650 --> 00:20:46,025 I'm not going to write them down. 446 00:20:46,025 --> 00:20:48,780 They're a little more complicated than these axioms. 447 00:20:48,780 --> 00:20:50,380 These axioms are very intuitive. 448 00:20:50,380 --> 00:20:52,590 We can just look around us and see 449 00:20:52,590 --> 00:20:57,570 that Euclid's axioms seem to be similar to the real world. 450 00:20:57,570 --> 00:21:00,030 The axioms of arithmetic are a little more complicated, 451 00:21:00,030 --> 00:21:03,730 but the story's pretty much the same. 452 00:21:03,730 --> 00:21:05,390 There's a certain number of them. 453 00:21:05,390 --> 00:21:07,740 And then, from those axioms you can prove things. 454 00:21:07,740 --> 00:21:09,680 Prove things about whole numbers. 455 00:21:09,680 --> 00:21:18,400 Like, an odd number plus an odd number is an even number. 456 00:21:18,400 --> 00:21:26,140 An even number plus an odd number is an odd. 457 00:21:26,140 --> 00:21:28,790 And a lot more complicated things, too. 458 00:21:28,790 --> 00:21:30,890 A lot more complicated things, but I'll just 459 00:21:30,890 --> 00:21:31,950 leave them like that. 460 00:21:36,720 --> 00:21:37,710 OK. 461 00:21:37,710 --> 00:21:40,383 So that's how I define arithmetic. 462 00:21:42,845 --> 00:21:44,720 Now, are you ready to have your mind's blown? 463 00:21:44,720 --> 00:21:46,920 Blown away? 464 00:21:46,920 --> 00:21:48,210 Ready? 465 00:21:48,210 --> 00:21:48,710 OK. 466 00:21:48,710 --> 00:21:50,010 No, you're not ready? 467 00:21:50,010 --> 00:21:52,172 Well, you have to be ready soon. 468 00:21:52,172 --> 00:21:57,628 AUDIENCE: [INAUDIBLE] 469 00:21:57,628 --> 00:21:59,917 NICHOLAS DIBELLA: OK. 470 00:21:59,917 --> 00:22:00,500 I just had to. 471 00:22:00,500 --> 00:22:01,416 I couldn't control it. 472 00:22:03,570 --> 00:22:05,890 You were spread, and now you're destroyed. 473 00:22:12,890 --> 00:22:13,880 OK. 474 00:22:13,880 --> 00:22:17,901 Now, it turns out when you look at arithmetic, 475 00:22:17,901 --> 00:22:20,400 you look at these things that you can prove from arithmetic. 476 00:22:20,400 --> 00:22:21,960 We call them theorems. 477 00:22:21,960 --> 00:22:23,230 We just call them facts. 478 00:22:23,230 --> 00:22:23,730 Theorems. 479 00:22:23,730 --> 00:22:25,320 Facts about whole numbers. 480 00:22:25,320 --> 00:22:30,900 Turns out that there exists true things about these numbers 481 00:22:30,900 --> 00:22:32,767 that you can't prove. 482 00:22:32,767 --> 00:22:34,100 You can't prove from the axioms. 483 00:22:36,770 --> 00:22:44,000 So in a sense, truth is more fundamental than provability. 484 00:22:44,000 --> 00:22:48,680 And this is called Godel's incompleteness theorem. 485 00:22:48,680 --> 00:22:51,986 Very loosely put. 486 00:22:51,986 --> 00:22:57,360 AUDIENCE: Don't you need provability to establish truth? 487 00:22:57,360 --> 00:23:00,110 NICHOLAS DIBELLA: Well, yeah. 488 00:23:00,110 --> 00:23:02,100 So how did I prove this you mean? 489 00:23:02,100 --> 00:23:03,646 [INAUDIBLE] 490 00:23:03,646 --> 00:23:05,854 AUDIENCE: Or to prove things that are true, don't you 491 00:23:05,854 --> 00:23:06,990 have to prove it? 492 00:23:06,990 --> 00:23:07,865 NICHOLAS DIBELLA: OK. 493 00:23:07,865 --> 00:23:09,630 So there are two kinds of proofs going on. 494 00:23:09,630 --> 00:23:12,880 There are proofs inside of arithmetic. 495 00:23:12,880 --> 00:23:15,630 So you're working with the axioms of arithmetic 496 00:23:15,630 --> 00:23:17,280 and you prove things about the numbers. 497 00:23:17,280 --> 00:23:19,650 Well, then you can step outside or the arithmetic. 498 00:23:19,650 --> 00:23:21,330 Step outside the system. 499 00:23:21,330 --> 00:23:25,050 Then, you can prove things about arithmetic, not things about-- 500 00:23:25,050 --> 00:23:29,250 you can prove things about arithmetic, not the stuff 501 00:23:29,250 --> 00:23:32,950 that arithmetic describes, but arithmetic itself. 502 00:23:32,950 --> 00:23:35,930 So this is called doing meta-mathematics. 503 00:23:35,930 --> 00:23:37,962 Kind of just take a step outside of math. 504 00:23:37,962 --> 00:23:40,170 And so you can prove things about the way mathematics 505 00:23:40,170 --> 00:23:41,370 is done itself. 506 00:23:41,370 --> 00:23:44,970 The way proof happens itself. 507 00:23:44,970 --> 00:23:48,690 And this was a major discovery that happened 508 00:23:48,690 --> 00:23:50,710 in the early 20th century. 509 00:23:50,710 --> 00:23:53,190 And this result was proven by a German mathematician 510 00:23:53,190 --> 00:23:55,590 named Godel. 511 00:23:55,590 --> 00:23:56,980 I'm sure I'm mispronouncing it. 512 00:23:56,980 --> 00:23:58,395 That's the best approximation of the pronunciation 513 00:23:58,395 --> 00:23:59,070 that I can do. 514 00:24:02,409 --> 00:24:04,200 It's called Godel's incompleteness theorem. 515 00:24:16,610 --> 00:24:19,040 Arithmetic is in the sense incomplete. 516 00:24:19,040 --> 00:24:23,070 It doesn't describe all the nature of mathematical truth. 517 00:24:23,070 --> 00:24:24,550 There are some mathematical truths. 518 00:24:24,550 --> 00:24:27,250 Some truths about whole numbers in arithmetic 519 00:24:27,250 --> 00:24:28,350 that you just can't reach. 520 00:24:28,350 --> 00:24:31,900 They're just unreachable truths. 521 00:24:31,900 --> 00:24:34,720 So there's a much more precise meaning to this all, 522 00:24:34,720 --> 00:24:39,610 but the gist of it is that there exists-- 523 00:24:39,610 --> 00:24:46,670 there are true mathematical statements. 524 00:24:46,670 --> 00:25:00,600 Well, there are true arithmetical statements 525 00:25:00,600 --> 00:25:01,590 that can't be proven. 526 00:25:08,950 --> 00:25:10,740 And what I mean by "proven," what 527 00:25:10,740 --> 00:25:12,300 I mean by when you prove something, 528 00:25:12,300 --> 00:25:14,850 the way you prove a theorem, what I mean by prove 529 00:25:14,850 --> 00:25:16,500 is you start off with the axioms. 530 00:25:16,500 --> 00:25:18,660 You start off with Euclid's axioms, 531 00:25:18,660 --> 00:25:20,880 and then you just apply deductive reasoning. 532 00:25:20,880 --> 00:25:22,760 You just apply logic to those axioms, 533 00:25:22,760 --> 00:25:24,450 and then you see what you can get. 534 00:25:24,450 --> 00:25:26,530 That's what I mean by prove. 535 00:25:26,530 --> 00:25:29,220 There exists true arithmetical statements 536 00:25:29,220 --> 00:25:30,150 that can't be proven. 537 00:25:33,300 --> 00:25:35,660 That's the famous Godel's incompleteness theorem. 538 00:25:35,660 --> 00:25:40,980 Now, why it's true is a vast, complicated story 539 00:25:40,980 --> 00:25:46,310 that I can't describe now. 540 00:25:46,310 --> 00:25:52,570 But just to kind of tantalize you about why it's true, 541 00:25:52,570 --> 00:25:57,240 an important idea that comes up in the proof is self-reference. 542 00:25:59,880 --> 00:26:02,760 So there exists statements that can 543 00:26:02,760 --> 00:26:05,780 kind of refer to themselves. 544 00:26:05,780 --> 00:26:19,500 Consider the sentence, this sentence is false. 545 00:26:19,500 --> 00:26:22,902 The statement that's referring to itself. 546 00:26:22,902 --> 00:26:24,860 It's also a very paradoxical-sounding sentence. 547 00:26:24,860 --> 00:26:28,854 Because if it's true, then it says it's false. 548 00:26:28,854 --> 00:26:30,020 So if it's true, it's false. 549 00:26:30,020 --> 00:26:32,260 But if it's false, then it's saying it's false, 550 00:26:32,260 --> 00:26:33,320 so it's true. 551 00:26:33,320 --> 00:26:37,066 So it's the famous liar's paradox. 552 00:26:37,066 --> 00:26:41,630 I'll just tantalize you all with that for now. 553 00:26:41,630 --> 00:26:42,130 OK. 554 00:26:44,690 --> 00:26:47,600 Godel's theorem is actually more general than just 555 00:26:47,600 --> 00:26:50,350 applying to arithmetic. 556 00:26:50,350 --> 00:26:53,810 It turns out not only is arithmetic incomplete, 557 00:26:53,810 --> 00:27:01,410 but any system is incomplete if it satisfies three conditions. 558 00:27:01,410 --> 00:27:04,070 So the first condition is that there 559 00:27:04,070 --> 00:27:05,495 are a finite number of axioms. 560 00:27:22,110 --> 00:27:24,090 There are just five axioms of Euclid. 561 00:27:24,090 --> 00:27:25,750 So that's finite. 562 00:27:25,750 --> 00:27:28,590 There's also just a finite number of axioms of arithmetic. 563 00:27:28,590 --> 00:27:30,630 So that's good. 564 00:27:30,630 --> 00:27:34,590 The second condition is that the system 565 00:27:34,590 --> 00:27:37,270 has to be large enough to include arithmetic. 566 00:28:00,910 --> 00:28:03,130 I defined arithmetic with all these whole numbers, 567 00:28:03,130 --> 00:28:04,240 and then with a plus-- 568 00:28:04,240 --> 00:28:06,600 addition and multiplication. 569 00:28:06,600 --> 00:28:08,650 A system that's not as large as arithmetic 570 00:28:08,650 --> 00:28:12,300 is just a system that only has plus and no multiplication, 571 00:28:12,300 --> 00:28:15,190 or a system that just has times and no addition. 572 00:28:15,190 --> 00:28:17,840 So that would be an example of a system that's 573 00:28:17,840 --> 00:28:19,280 not as big as arithmetic. 574 00:28:19,280 --> 00:28:20,580 It's simpler than arithmetic. 575 00:28:20,580 --> 00:28:22,310 It's not as complex. 576 00:28:22,310 --> 00:28:24,540 And so for that kind of a system, 577 00:28:24,540 --> 00:28:27,010 Godel's theorem actually doesn't apply. 578 00:28:27,010 --> 00:28:29,920 So if you have arithmetic with just multiplication, 579 00:28:29,920 --> 00:28:34,090 then every mathematical truth about that arithmetic 580 00:28:34,090 --> 00:28:35,050 can be proven. 581 00:28:35,050 --> 00:28:37,082 All those mathematical truths are reachable. 582 00:28:39,615 --> 00:28:42,240 Of course, you could also make a system larger than arithmetic. 583 00:28:42,240 --> 00:28:44,250 You can add extra stuff. 584 00:28:44,250 --> 00:28:46,780 That would be a system larger than arithmetic. 585 00:28:46,780 --> 00:28:50,760 The final condition is that it has to be consistent. 586 00:28:50,760 --> 00:28:52,700 The system has to be consistent. 587 00:29:06,334 --> 00:29:08,250 And by "consistent," what I mean is that there 588 00:29:08,250 --> 00:29:10,500 are no contradictions. 589 00:29:10,500 --> 00:29:13,439 For example, one way that Euclid's axioms 590 00:29:13,439 --> 00:29:15,480 could contain a contradiction, one way they could 591 00:29:15,480 --> 00:29:19,560 be inconsistent, is, well, let's say axiom 4 I said 592 00:29:19,560 --> 00:29:21,420 was all right triangles are congruent. 593 00:29:21,420 --> 00:29:23,100 They're all the same. 594 00:29:23,100 --> 00:29:25,860 Maybe there's a sixth axiom that says all right triangles are 595 00:29:25,860 --> 00:29:27,250 different. 596 00:29:27,250 --> 00:29:30,780 So that's an obvious contradiction. 597 00:29:30,780 --> 00:29:32,450 So if you had a system like that, 598 00:29:32,450 --> 00:29:33,616 then it won't be consistent. 599 00:29:33,616 --> 00:29:37,650 And so Godel's theorem wouldn't apply. 600 00:29:37,650 --> 00:29:40,770 So any system that has the three properties 601 00:29:40,770 --> 00:29:42,390 contains truths about it. 602 00:29:42,390 --> 00:29:46,670 So arithmetical then is a generalization. 603 00:29:46,670 --> 00:29:49,612 A generalization depending on the type of system. 604 00:29:49,612 --> 00:29:51,570 If the system satisfies these three conditions, 605 00:29:51,570 --> 00:29:54,970 then there are truths about it that can't be proven. 606 00:29:54,970 --> 00:29:55,470 OK. 607 00:29:55,470 --> 00:29:58,600 Are there any questions about what Godel's theorem says? 608 00:29:58,600 --> 00:30:02,151 Not about why it's true, but what it says? 609 00:30:02,151 --> 00:30:04,440 Yes. 610 00:30:04,440 --> 00:30:09,022 AUDIENCE: So if the theorem says that there are true-- 611 00:30:11,788 --> 00:30:13,640 OK. 612 00:30:13,640 --> 00:30:17,602 For any system of mathematics with postulates, 613 00:30:17,602 --> 00:30:21,065 by the definition of a postulate is that it is true, 614 00:30:21,065 --> 00:30:22,894 but cannot be proven? 615 00:30:22,894 --> 00:30:23,810 NICHOLAS DIBELLA: Yes. 616 00:30:23,810 --> 00:30:25,170 Yes, good point. 617 00:30:25,170 --> 00:30:27,600 So these postulates, these axioms, they 618 00:30:27,600 --> 00:30:29,214 themselves are unprovable. 619 00:30:29,214 --> 00:30:31,380 So for any system, you actually have some statements 620 00:30:31,380 --> 00:30:32,832 that are just unprovable. 621 00:30:32,832 --> 00:30:34,290 I mean, you accept your postulates, 622 00:30:34,290 --> 00:30:35,880 and then you work from there. 623 00:30:35,880 --> 00:30:37,584 That's a good point. 624 00:30:37,584 --> 00:30:39,000 So what I should have said here is 625 00:30:39,000 --> 00:30:41,850 there are true arithmetical statements that 626 00:30:41,850 --> 00:30:43,020 aren't postulates. 627 00:30:43,020 --> 00:30:44,850 That aren't axioms. 628 00:30:44,850 --> 00:30:46,200 You can never prove the axiom. 629 00:30:46,200 --> 00:30:48,690 You just accept them for whatever reasons you want. 630 00:30:48,690 --> 00:30:50,189 Maybe they model the physical world. 631 00:30:50,189 --> 00:30:52,194 Maybe you think they're kind of cute. 632 00:30:52,194 --> 00:30:53,610 But yeah, more precisely there are 633 00:30:53,610 --> 00:30:57,435 true arithmetical statements that aren't axioms. 634 00:31:06,560 --> 00:31:08,810 Good point. 635 00:31:08,810 --> 00:31:11,300 Any other questions about Godel's theorem? 636 00:31:14,223 --> 00:31:14,723 OK. 637 00:31:19,130 --> 00:31:21,160 As I said before, the theories that we 638 00:31:21,160 --> 00:31:24,520 use to describe nature, all the theories we've ever used, 639 00:31:24,520 --> 00:31:26,420 they've always been mathematical. 640 00:31:26,420 --> 00:31:28,090 I mean, math seems to be the language 641 00:31:28,090 --> 00:31:29,290 that the university uses. 642 00:31:29,290 --> 00:31:30,998 There's just no getting around using math 643 00:31:30,998 --> 00:31:32,870 to describe the universe. 644 00:31:32,870 --> 00:31:36,250 And so the system that described the theory of everything that 645 00:31:36,250 --> 00:31:38,830 describes the universe would have 646 00:31:38,830 --> 00:31:41,200 to have some kind of mathematical system. 647 00:31:41,200 --> 00:31:44,360 And so you could ask the question, 648 00:31:44,360 --> 00:31:48,370 does that mathematical system satisfy these three conditions? 649 00:31:48,370 --> 00:31:51,820 If it does, then it would seem that there 650 00:31:51,820 --> 00:31:56,980 would be true statements about the universe that 651 00:31:56,980 --> 00:32:01,290 simply can't be explained. 652 00:32:01,290 --> 00:32:03,640 We're used to explaining things that 653 00:32:03,640 --> 00:32:08,290 happen by ultimately going back down 654 00:32:08,290 --> 00:32:10,250 to a fundamental deep level. 655 00:32:10,250 --> 00:32:15,010 So for example, if I take a pen and I drop it, 656 00:32:15,010 --> 00:32:18,980 you can ask my why that happened and I can give you an answer. 657 00:32:18,980 --> 00:32:25,351 I can say, well, pens and pencils, they have this thing. 658 00:32:25,351 --> 00:32:26,350 They have this property. 659 00:32:26,350 --> 00:32:29,380 We call it mass. 660 00:32:29,380 --> 00:32:32,100 And there are a lot of things that have mass. 661 00:32:32,100 --> 00:32:33,430 Pens have mass. 662 00:32:33,430 --> 00:32:35,440 The earth also has mass. 663 00:32:35,440 --> 00:32:38,740 Now, there is a law of physics. 664 00:32:38,740 --> 00:32:40,930 There's a law of physics about the universe that 665 00:32:40,930 --> 00:32:43,717 says that anything that has mass attracts each other. 666 00:32:43,717 --> 00:32:45,800 Any two objects that have mass attract each other. 667 00:32:45,800 --> 00:32:49,690 This is the law of gravitation that Newton first described. 668 00:32:49,690 --> 00:32:52,480 And when you ask me to explain why that happened, I can say, 669 00:32:52,480 --> 00:32:54,790 well, the pen has mass. 670 00:32:54,790 --> 00:32:56,320 The earth has mass. 671 00:32:56,320 --> 00:33:00,190 I'm no longer holding it, so they attract each other, 672 00:33:00,190 --> 00:33:02,080 and that's it. 673 00:33:02,080 --> 00:33:03,490 We can explain what happened. 674 00:33:03,490 --> 00:33:05,730 We can explain the observation. 675 00:33:05,730 --> 00:33:11,410 If the theory of everything, if the mathematical system 676 00:33:11,410 --> 00:33:13,780 that describes the theory of everything 677 00:33:13,780 --> 00:33:16,510 satisfies the conditions of Godel's theorem, 678 00:33:16,510 --> 00:33:18,744 that there will be true things about the universe 679 00:33:18,744 --> 00:33:19,660 that we can't explain. 680 00:33:23,670 --> 00:33:28,780 If that's the case, then that's really not 681 00:33:28,780 --> 00:33:33,280 good for people who want a theory of everything. 682 00:33:33,280 --> 00:33:36,820 If the theory is incomplete, then we 683 00:33:36,820 --> 00:33:38,170 couldn't explain everything. 684 00:33:38,170 --> 00:33:40,480 So it wouldn't be the theory of everything. 685 00:33:40,480 --> 00:33:42,040 And so if Godel's theorem applies 686 00:33:42,040 --> 00:33:43,748 to the mathematical system that described 687 00:33:43,748 --> 00:33:46,431 the theory of everything, then there 688 00:33:46,431 --> 00:33:47,930 is actually no theory of everything. 689 00:33:47,930 --> 00:33:50,720 There is stuff that we know about the universe, 690 00:33:50,720 --> 00:33:53,261 but then there's always going to be stuff that we don't know. 691 00:33:54,720 --> 00:33:57,440 So now you can ask the question, well, 692 00:33:57,440 --> 00:34:00,850 does Godel's theorem apply? 693 00:34:00,850 --> 00:34:03,130 And the answer is maybe and maybe not. 694 00:34:03,130 --> 00:34:06,910 So we don't yet know what the mathematical system 695 00:34:06,910 --> 00:34:08,820 is that describes the universe. 696 00:34:08,820 --> 00:34:11,434 People are certainly working on it. 697 00:34:11,434 --> 00:34:13,600 Maybe it's the mathematical system of string theory. 698 00:34:13,600 --> 00:34:16,388 Maybe it's the mathematical system of loop quantum gravity 699 00:34:16,388 --> 00:34:17,679 that I talked about last class. 700 00:34:20,590 --> 00:34:23,170 There are certainly ways that these conditions 701 00:34:23,170 --> 00:34:25,750 could fail to be met. 702 00:34:25,750 --> 00:34:29,920 So the first condition that there 703 00:34:29,920 --> 00:34:35,192 are a finite number of axioms, that certainly seems like it 704 00:34:35,192 --> 00:34:37,150 would be the case for the theory of everything. 705 00:34:37,150 --> 00:34:40,090 I mean, every theory that we've ever written down 706 00:34:40,090 --> 00:34:42,010 has had a finite number of axioms. 707 00:34:42,010 --> 00:34:44,366 Remember, special relativity has two. 708 00:34:44,366 --> 00:34:45,699 Quantum mechanics has a few-ish. 709 00:34:47,735 --> 00:34:50,110 All these physical theories that we've ever written down, 710 00:34:50,110 --> 00:34:52,360 they've always had a finite number of axioms. 711 00:34:52,360 --> 00:34:54,004 So there's good reason to believe 712 00:34:54,004 --> 00:34:55,420 that the theory of everything also 713 00:34:55,420 --> 00:34:56,500 has a finite number of axioms. 714 00:34:56,500 --> 00:34:58,970 But it could very well not have a finite number of axioms. 715 00:34:58,970 --> 00:35:01,261 And if it does, if it has an infinite number of axioms, 716 00:35:01,261 --> 00:35:03,190 then Godel's theorem wouldn't apply 717 00:35:03,190 --> 00:35:06,880 and it would be possible to know everything about the universe. 718 00:35:06,880 --> 00:35:09,790 It would be possible to have a theory of everything. 719 00:35:09,790 --> 00:35:12,460 But of course, how you would actually 720 00:35:12,460 --> 00:35:14,030 store that theory of everything if it 721 00:35:14,030 --> 00:35:19,160 has an infinite number of items in it, that's another issue. 722 00:35:19,160 --> 00:35:21,374 OK. 723 00:35:21,374 --> 00:35:23,290 The third condition could also fail to be met. 724 00:35:25,810 --> 00:35:29,492 Well, it could, in principle. 725 00:35:29,492 --> 00:35:31,450 But I personally have a hard time understanding 726 00:35:31,450 --> 00:35:33,850 what a system that's consistent would mean. 727 00:35:33,850 --> 00:35:35,890 If a system is inconsistent, then it 728 00:35:35,890 --> 00:35:38,290 would say that contradictions are true. 729 00:35:38,290 --> 00:35:42,970 So something is true and it's not true simultaneously. 730 00:35:42,970 --> 00:35:45,910 I just have a hard time understanding that. 731 00:35:45,910 --> 00:35:49,840 Does anybody have an intuition for understanding 732 00:35:49,840 --> 00:35:52,490 true contradictions? 733 00:35:52,490 --> 00:35:54,340 Yeah, I have a hard time. 734 00:35:54,340 --> 00:35:59,581 But actually, there exists philosophers and logicians. 735 00:35:59,581 --> 00:36:01,330 People that study logic that have actually 736 00:36:01,330 --> 00:36:05,650 come up with meaningful ways of understanding 737 00:36:05,650 --> 00:36:09,070 true contradictions. 738 00:36:09,070 --> 00:36:13,480 I don't know that much about it, but I do know it has a name. 739 00:36:13,480 --> 00:36:16,150 So one way it could fail to be met 740 00:36:16,150 --> 00:36:25,030 is if there's this aspect about the mathematical system called 741 00:36:25,030 --> 00:36:31,362 dialetheism, or something like that. 742 00:36:31,362 --> 00:36:33,070 I don't actually know how it's pronounced 743 00:36:33,070 --> 00:36:35,800 because I haven't really talked about this in person 744 00:36:35,800 --> 00:36:37,210 with experts. 745 00:36:37,210 --> 00:36:40,510 But I found this word before, and I've 746 00:36:40,510 --> 00:36:43,410 read a little bit about it. 747 00:36:43,410 --> 00:36:45,100 So I mean, there are some people that 748 00:36:45,100 --> 00:36:47,260 think you could have true contradictions 749 00:36:47,260 --> 00:36:49,960 and have contradictions manifest themselves 750 00:36:49,960 --> 00:36:52,650 in the real world in a meaningful way. 751 00:36:52,650 --> 00:36:53,590 I don't know. 752 00:36:53,590 --> 00:36:56,110 It's hard for me to understand what a true contradiction so. 753 00:36:56,110 --> 00:37:00,250 So it seemed that the mathematical system 754 00:37:00,250 --> 00:37:01,570 should be consistent. 755 00:37:01,570 --> 00:37:03,207 I mean, just intuitively. 756 00:37:03,207 --> 00:37:05,290 It might not be meaningful to have an inconsistent 757 00:37:05,290 --> 00:37:08,320 system, that's all I'm saying. 758 00:37:08,320 --> 00:37:09,520 Finally, the second axiom. 759 00:37:09,520 --> 00:37:11,740 The second axiom is the one that's 760 00:37:11,740 --> 00:37:15,490 most likely to have a chance of not being met. 761 00:37:18,650 --> 00:37:22,550 So if the system is large enough to include arithmetic, 762 00:37:22,550 --> 00:37:24,467 and if the other two axioms are met-- 763 00:37:24,467 --> 00:37:26,300 I'm sorry, the other two conditions are met, 764 00:37:26,300 --> 00:37:29,550 then Godel's theorem would apply. 765 00:37:29,550 --> 00:37:32,110 So it turns out that real numbers are actually 766 00:37:32,110 --> 00:37:35,860 simpler than whole numbers. 767 00:37:35,860 --> 00:37:37,390 Whole numbers seem pretty simple. 768 00:37:37,390 --> 00:37:39,850 Just 0, 1, 2, 3. 769 00:37:39,850 --> 00:37:42,970 But if you tried doing arithmetic with real numbers, 770 00:37:42,970 --> 00:37:48,569 with numbers like negative 5, 6.8, pi, square root of 2. 771 00:37:48,569 --> 00:37:50,110 Rational numbers, irrational numbers, 772 00:37:50,110 --> 00:37:52,900 you put them all in the mix, and tried to do arithmetic 773 00:37:52,900 --> 00:37:55,620 with those numbers, it turns out that it's actually 774 00:37:55,620 --> 00:37:57,190 simpler than arithmetic. 775 00:37:57,190 --> 00:38:01,630 And the system ends up being complete, it turns out. 776 00:38:01,630 --> 00:38:04,462 And actually for a similar reason, 777 00:38:04,462 --> 00:38:05,920 Euclidean geometry, Euclid's axiom, 778 00:38:05,920 --> 00:38:09,220 that's also a complete system. 779 00:38:09,220 --> 00:38:11,470 So it could be that the mathematical system just 780 00:38:11,470 --> 00:38:13,780 isn't very complex. 781 00:38:13,780 --> 00:38:15,370 It's not as complex as arithmetic. 782 00:38:15,370 --> 00:38:20,155 And so it could fail to meet that requirement. 783 00:38:20,155 --> 00:38:22,030 There are a lot of other mathematical systems 784 00:38:22,030 --> 00:38:24,847 that are simpler than arithmetic. 785 00:38:24,847 --> 00:38:27,430 And if that system is the system that the theory of everything 786 00:38:27,430 --> 00:38:29,724 uses, then Godel's theorem wouldn't apply 787 00:38:29,724 --> 00:38:32,140 and there would be hope in getting a theory of everything. 788 00:38:32,140 --> 00:38:34,630 But there are many other systems that 789 00:38:34,630 --> 00:38:36,760 are more complex than arithmetic, 790 00:38:36,760 --> 00:38:39,232 and that might be the system that you use 791 00:38:39,232 --> 00:38:40,440 for the theory of everything. 792 00:38:40,440 --> 00:38:42,160 And so then, Godel's theorem would apply if the other two 793 00:38:42,160 --> 00:38:43,359 conditions were also met. 794 00:38:43,359 --> 00:38:45,400 And so then the dream of the theory of everything 795 00:38:45,400 --> 00:38:50,890 would be down the hall and impossible. 796 00:38:50,890 --> 00:38:51,640 OK. 797 00:38:51,640 --> 00:38:53,350 Anybody have any questions about that? 798 00:38:59,040 --> 00:39:00,750 Some people would be sad, I think, 799 00:39:00,750 --> 00:39:03,830 if it's impossible to have a theory of everything. 800 00:39:03,830 --> 00:39:07,530 Some people want to know everything about the universe. 801 00:39:07,530 --> 00:39:10,080 But Godel's theorem says that there's a chance that that 802 00:39:10,080 --> 00:39:11,010 might not be possible. 803 00:39:11,010 --> 00:39:12,551 It might not be possible in principle 804 00:39:12,551 --> 00:39:13,940 to have a theory of everything. 805 00:39:13,940 --> 00:39:15,900 And so a lot of people, I'm sure would be sad, if we 806 00:39:15,900 --> 00:39:17,020 found that out to be the case. 807 00:39:17,020 --> 00:39:18,510 I mean, we don't know if it's the case. 808 00:39:18,510 --> 00:39:19,120 It might be the case. 809 00:39:19,120 --> 00:39:19,661 It might not. 810 00:39:19,661 --> 00:39:23,340 It depends on the mathematical structure of the universe 811 00:39:23,340 --> 00:39:24,600 how it all works out. 812 00:39:24,600 --> 00:39:25,244 Yeah. 813 00:39:25,244 --> 00:39:28,852 AUDIENCE: I think people need a quest for something. 814 00:39:28,852 --> 00:39:33,409 If we find out everything, we find out everything 815 00:39:33,409 --> 00:39:34,700 about the universe [INAUDIBLE]. 816 00:39:34,700 --> 00:39:36,030 NICHOLAS DIBELLA: Yeah, that's a good point. 817 00:39:36,030 --> 00:39:37,773 AUDIENCE: Because I think we all need-- 818 00:39:40,506 --> 00:39:41,130 whatever quest. 819 00:39:41,130 --> 00:39:43,018 We always need to attain something. 820 00:39:43,018 --> 00:39:44,885 It doesn't have to be a physical thing, 821 00:39:44,885 --> 00:39:48,370 but we need to reach a goal. 822 00:39:48,370 --> 00:39:50,922 If there are no goals, people will quickly 823 00:39:50,922 --> 00:39:52,860 become kind of bored or-- 824 00:39:52,860 --> 00:39:54,820 NICHOLAS DIBELLA: Yeah, that's a good point. 825 00:39:54,820 --> 00:39:57,270 I mean, what's more important? 826 00:39:57,270 --> 00:39:59,520 Is it the conclusion of the journey or the journey 827 00:39:59,520 --> 00:40:00,480 itself that's more important? 828 00:40:00,480 --> 00:40:00,780 AUDIENCE: [INAUDIBLE] 829 00:40:00,780 --> 00:40:02,130 NICHOLAS DIBELLA: Yeah. 830 00:40:02,130 --> 00:40:05,070 So that would be good news for physicists. 831 00:40:05,070 --> 00:40:08,429 Physicists would always have something to do. 832 00:40:08,429 --> 00:40:09,720 Similarly, with mathematicians. 833 00:40:09,720 --> 00:40:13,350 I mean, mathematicians knew this a long time ago. 834 00:40:13,350 --> 00:40:17,880 So mathematicians have this set of truths 835 00:40:17,880 --> 00:40:19,580 that they can always-- 836 00:40:19,580 --> 00:40:21,479 I mean, there's always more math to discover. 837 00:40:21,479 --> 00:40:23,020 There's always more math to discover. 838 00:40:23,020 --> 00:40:24,280 And the same might be the same for physicists. 839 00:40:24,280 --> 00:40:25,655 There might always be new physics 840 00:40:25,655 --> 00:40:29,028 to discover, new things to learn about the universe. 841 00:40:29,028 --> 00:40:31,410 AUDIENCE: You wouldn't need physicists [INAUDIBLE].. 842 00:40:31,410 --> 00:40:32,810 NICHOLAS DIBELLA: Yeah. 843 00:40:32,810 --> 00:40:34,060 Physics would be out of a job. 844 00:40:34,060 --> 00:40:35,610 AUDIENCE: Exactly. 845 00:40:35,610 --> 00:40:36,450 NICHOLAS DIBELLA: But then again, physicists 846 00:40:36,450 --> 00:40:37,866 are very clever and very skillful. 847 00:40:37,866 --> 00:40:39,830 So they'd have no problem finding a new job. 848 00:40:39,830 --> 00:40:42,152 But they might not be happy. 849 00:40:42,152 --> 00:40:42,652 OK. 850 00:40:46,596 --> 00:40:47,582 Let's see. 851 00:40:55,490 --> 00:40:56,540 OK. 852 00:40:56,540 --> 00:41:00,200 So [INAUDIBLE] mentioned a good point before, 853 00:41:00,200 --> 00:41:05,000 which is that even if the system is complete, 854 00:41:05,000 --> 00:41:07,610 even if Godel's theorem doesn't apply, 855 00:41:07,610 --> 00:41:09,680 then there would still be some things that 856 00:41:09,680 --> 00:41:13,130 were left explaining, some unprovable things 857 00:41:13,130 --> 00:41:14,910 according to the theory. 858 00:41:14,910 --> 00:41:15,520 The axioms. 859 00:41:15,520 --> 00:41:18,040 There would still be the axioms leftover. 860 00:41:18,040 --> 00:41:19,941 You still might be wondering, well, 861 00:41:19,941 --> 00:41:21,440 why is this the mathematical system? 862 00:41:21,440 --> 00:41:22,340 Why these axioms? 863 00:41:22,340 --> 00:41:25,250 Why not others? 864 00:41:25,250 --> 00:41:27,290 The theory itself has no explanation for it. 865 00:41:27,290 --> 00:41:30,290 The theory explains the physical universe, 866 00:41:30,290 --> 00:41:32,240 but it doesn't explain itself. 867 00:41:32,240 --> 00:41:34,550 So to explain it, to explain the theory, 868 00:41:34,550 --> 00:41:37,730 you need some other kind of reasoning. 869 00:41:37,730 --> 00:41:42,650 And I'll get to that. 870 00:41:42,650 --> 00:41:47,810 I'll get to that probably next class why those axioms, 871 00:41:47,810 --> 00:41:48,960 not others. 872 00:41:48,960 --> 00:41:52,820 So now we'd like to take a brief digression 873 00:41:52,820 --> 00:41:56,420 and talk about types of existence. 874 00:41:56,420 --> 00:42:00,020 So up to now, I've been pretty casual in referring 875 00:42:00,020 --> 00:42:02,600 to the theory of everything. 876 00:42:02,600 --> 00:42:04,915 And last class, I said there's matter. 877 00:42:04,915 --> 00:42:05,540 There's energy. 878 00:42:05,540 --> 00:42:06,900 There's forces. 879 00:42:06,900 --> 00:42:07,570 That's it. 880 00:42:07,570 --> 00:42:08,850 That's everything. 881 00:42:08,850 --> 00:42:10,220 It's everything. 882 00:42:10,220 --> 00:42:13,176 Let's get a theory for that stuff. 883 00:42:13,176 --> 00:42:15,300 But you can still ask the question, what is energy? 884 00:42:15,300 --> 00:42:16,120 What is matter? 885 00:42:16,120 --> 00:42:19,040 What are forces? 886 00:42:19,040 --> 00:42:21,440 And these are actually the types of questions, 887 00:42:21,440 --> 00:42:24,530 these deep questions-- "what is" questions. 888 00:42:24,530 --> 00:42:26,570 These kinds of questions, scientists 889 00:42:26,570 --> 00:42:30,080 tend to have a habit of shying away from them. 890 00:42:30,080 --> 00:42:36,310 Because they'll say, oh, it's not a scientific question. 891 00:42:36,310 --> 00:42:39,110 We only care about things that are scientific. 892 00:42:39,110 --> 00:42:42,680 We only care about experiments and observations. 893 00:42:42,680 --> 00:42:45,011 The questions like, what are forces, what is energy-- 894 00:42:45,011 --> 00:42:47,510 those Kind of questions-- those are philosophical questions, 895 00:42:47,510 --> 00:42:48,676 so we don't care about them. 896 00:42:48,676 --> 00:42:52,220 That's what many scientists say 897 00:42:52,220 --> 00:42:56,180 but I personally am kind of confused about this response. 898 00:43:02,212 --> 00:43:04,670 How can anybody do science when you don't really understand 899 00:43:04,670 --> 00:43:06,215 what science is about? 900 00:43:06,215 --> 00:43:08,090 I mean, if I have a theory about an electron, 901 00:43:08,090 --> 00:43:10,200 I'd like to know what the electron is. 902 00:43:10,200 --> 00:43:14,690 I'd be a lot more comfortable knowing what the electron is. 903 00:43:14,690 --> 00:43:18,680 So this leads to a big question. 904 00:43:18,680 --> 00:43:20,330 Namely, what is reality? 905 00:43:20,330 --> 00:43:21,560 What is all this stuff? 906 00:43:24,344 --> 00:43:26,120 Where shall I write it? 907 00:43:29,051 --> 00:43:31,550 Because it's very big, I'm going to write it in big letters. 908 00:43:38,460 --> 00:43:41,290 What is reality? 909 00:43:41,290 --> 00:43:43,850 So now, I'd like to discuss a few kinds of existence, 910 00:43:43,850 --> 00:43:46,870 types of existence. 911 00:43:46,870 --> 00:43:51,250 So one type of existence that people think about 912 00:43:51,250 --> 00:43:53,780 is mathematical existence. 913 00:43:53,780 --> 00:43:56,480 Well, let's start with mathematical existence. 914 00:43:56,480 --> 00:43:58,750 One type of existence is mathematical existence. 915 00:44:10,606 --> 00:44:12,582 This is types of existence. 916 00:44:23,490 --> 00:44:28,480 So before, when I was describing math, how mathematicians think 917 00:44:28,480 --> 00:44:32,110 of it, a lot of mathematicians like 918 00:44:32,110 --> 00:44:34,530 to think of math as something that exists out there. 919 00:44:34,530 --> 00:44:38,316 There are these mathematical truths that we try to discover. 920 00:44:38,316 --> 00:44:41,635 There are these abstract things that are true. 921 00:44:41,635 --> 00:44:44,470 There's the abstract concept of a line, the abstract concept 922 00:44:44,470 --> 00:44:48,250 of a number that exists out there in a mathematical world. 923 00:44:48,250 --> 00:44:51,130 So there is a mathematical kind of existence. 924 00:44:51,130 --> 00:44:53,267 The second kind of existence? 925 00:44:53,267 --> 00:44:55,215 A physical existence. 926 00:45:04,470 --> 00:45:06,540 Well, let's just abbreviate this by math. 927 00:45:13,527 --> 00:45:14,860 Let's abbreviate this by matter. 928 00:45:20,090 --> 00:45:22,490 When we look around us, we see things. 929 00:45:22,490 --> 00:45:23,390 We see matter. 930 00:45:23,390 --> 00:45:23,992 We see energy. 931 00:45:23,992 --> 00:45:24,575 We see forces. 932 00:45:27,230 --> 00:45:29,320 And generally, we call them physical things. 933 00:45:29,320 --> 00:45:30,740 We just give them a name. 934 00:45:30,740 --> 00:45:32,240 Not really explaining what they are, 935 00:45:32,240 --> 00:45:33,614 we just kind of give them a name. 936 00:45:33,614 --> 00:45:35,930 We call them physical things. 937 00:45:35,930 --> 00:45:38,704 So the theory of everything aims to describe 938 00:45:38,704 --> 00:45:39,620 the physical universe. 939 00:45:39,620 --> 00:45:42,600 It tries to describe physical existence. 940 00:45:42,600 --> 00:45:45,860 So I'll abbreviate that by matter. 941 00:45:45,860 --> 00:45:50,030 And finally, a third kind of existence that people think 942 00:45:50,030 --> 00:45:52,360 about is mental existence. 943 00:46:00,770 --> 00:46:01,270 Mind. 944 00:46:05,250 --> 00:46:09,390 This is the world inside of our heads. 945 00:46:09,390 --> 00:46:11,280 We have a conscious experience. 946 00:46:11,280 --> 00:46:12,690 We have perceptions. 947 00:46:12,690 --> 00:46:13,860 We have thoughts. 948 00:46:13,860 --> 00:46:15,000 We have ideas. 949 00:46:15,000 --> 00:46:16,580 We have emotions. 950 00:46:16,580 --> 00:46:19,290 And it seems like this is like a completely separate kind 951 00:46:19,290 --> 00:46:23,075 of thing from physical existence or a mathematical existence, 952 00:46:23,075 --> 00:46:24,200 at least on the face of it. 953 00:46:24,200 --> 00:46:27,520 It seems to be a wholly separate kind of thing. 954 00:46:27,520 --> 00:46:29,730 So mental existence. 955 00:46:29,730 --> 00:46:33,780 So we have math, matter, and mind. 956 00:46:33,780 --> 00:46:36,450 Now, there are a lot of ways that these different kinds 957 00:46:36,450 --> 00:46:40,890 of existence can be related to each other. 958 00:46:40,890 --> 00:46:44,980 I'll start off with mind and matter. 959 00:46:44,980 --> 00:46:50,100 So probably most scientists who study the brain, most 960 00:46:50,100 --> 00:46:54,880 neuroscientists, most cognitive scientists, they 961 00:46:54,880 --> 00:46:57,570 would say that the mind is actually 962 00:46:57,570 --> 00:47:00,950 something that emerges from something that's physical. 963 00:47:00,950 --> 00:47:02,770 So there are these physical systems. 964 00:47:02,770 --> 00:47:04,230 We call them humans. 965 00:47:04,230 --> 00:47:06,960 There's this part, this region inside of humans-- 966 00:47:06,960 --> 00:47:10,082 this area has a physical thing called the brain, which 967 00:47:10,082 --> 00:47:11,790 consists of all these atoms and molecules 968 00:47:11,790 --> 00:47:14,470 and electrical signals, and so forth. 969 00:47:14,470 --> 00:47:16,590 And when everything's put together the right way, 970 00:47:16,590 --> 00:47:18,070 you get a mind. 971 00:47:18,070 --> 00:47:20,790 That's how a mind forms. 972 00:47:20,790 --> 00:47:23,550 It's true that there's still a lot to be learned, 973 00:47:23,550 --> 00:47:25,200 the scientists say, about the nature 974 00:47:25,200 --> 00:47:26,950 of the conscious experience, how it really 975 00:47:26,950 --> 00:47:31,890 emerges from physical things. 976 00:47:31,890 --> 00:47:37,800 Nonetheless, the view among most neuroscientists and cognitive 977 00:47:37,800 --> 00:47:41,080 scientists is that the mind is a physical thing. 978 00:47:41,080 --> 00:47:44,990 And so mind actually arises from matter. 979 00:47:44,990 --> 00:47:49,020 So matter gives rise to mind. 980 00:47:49,020 --> 00:47:52,160 Now, there are some people that don't believe this. 981 00:47:52,160 --> 00:47:54,390 Probably, minority. 982 00:47:54,390 --> 00:47:58,230 They believe that mind and body are actually separate things. 983 00:47:58,230 --> 00:48:01,560 This is the mind-body problem. 984 00:48:01,560 --> 00:48:03,360 AUDIENCE: [INAUDIBLE] talking about a soul? 985 00:48:03,360 --> 00:48:05,100 NICHOLAS DIBELLA: Maybe it's some kind of a soul 986 00:48:05,100 --> 00:48:06,390 or some kind of a spirit. 987 00:48:06,390 --> 00:48:10,082 Or maybe it's just some kind of-- just a different thing. 988 00:48:10,082 --> 00:48:11,040 Just a different thing. 989 00:48:11,040 --> 00:48:13,736 Who knows what kind of thing it is? 990 00:48:13,736 --> 00:48:15,360 It just has a different kind of exist-- 991 00:48:15,360 --> 00:48:16,920 it's a different kind of substance, 992 00:48:16,920 --> 00:48:21,970 different kind of existence. 993 00:48:21,970 --> 00:48:26,210 I won't try to go into the mind-body debate. 994 00:48:26,210 --> 00:48:28,140 It's sufficiently complex to make 995 00:48:28,140 --> 00:48:32,366 me afraid of trying to describe it all in full detail. 996 00:48:32,366 --> 00:48:33,240 But there's a debate. 997 00:48:33,240 --> 00:48:39,660 And most scientists would say that it's in this direction-- 998 00:48:39,660 --> 00:48:42,542 mind emerges from matter. 999 00:48:42,542 --> 00:48:45,000 But some people would say that these are actually separate. 1000 00:48:45,000 --> 00:48:47,250 They call themselves dualist. 1001 00:48:47,250 --> 00:48:49,320 There's a dual nature to mind and body. 1002 00:48:52,210 --> 00:48:55,590 Let's look at another possible connection. 1003 00:48:55,590 --> 00:48:57,600 Let's see. 1004 00:48:57,600 --> 00:49:02,070 Mathematical existence and mental existence. 1005 00:49:02,070 --> 00:49:05,130 So there's a question that philosophers of mathematics 1006 00:49:05,130 --> 00:49:08,580 have asked for a long time, which is, is math 1007 00:49:08,580 --> 00:49:11,410 discovered or invented? 1008 00:49:11,410 --> 00:49:14,520 Is math something that exists out there 1009 00:49:14,520 --> 00:49:16,300 as a lot of mathematicians believe? 1010 00:49:16,300 --> 00:49:18,760 A lot of non-mathematicians as well. 1011 00:49:18,760 --> 00:49:21,210 Is it something out there that is uncovered, 1012 00:49:21,210 --> 00:49:24,450 the truth is uncovered, it's discovered, exists out there? 1013 00:49:24,450 --> 00:49:26,570 Or is math something that's invented? 1014 00:49:26,570 --> 00:49:28,470 Is it a product of the human mind? 1015 00:49:28,470 --> 00:49:33,000 Is it simply something that creative people come up with? 1016 00:49:33,000 --> 00:49:37,650 Something that emerges from human creativity? 1017 00:49:37,650 --> 00:49:44,460 I'm also not going to try to go into this in any detail, 1018 00:49:44,460 --> 00:49:49,960 but there's a debate whether math is discovered or invented. 1019 00:49:49,960 --> 00:49:53,790 If it's invented-- that is, if math is something that is 1020 00:49:53,790 --> 00:49:55,140 a product of the mind-- 1021 00:49:55,140 --> 00:49:58,572 then it would seem to be that mind gives rise to math. 1022 00:49:58,572 --> 00:50:00,030 So you can draw an arrow like this. 1023 00:50:02,744 --> 00:50:04,410 On the other hand, it could be that math 1024 00:50:04,410 --> 00:50:05,701 is something that's discovered. 1025 00:50:05,701 --> 00:50:07,470 It's something that's completely different 1026 00:50:07,470 --> 00:50:09,060 from mental existence. 1027 00:50:09,060 --> 00:50:13,830 And so they could actually be separate from one another. 1028 00:50:13,830 --> 00:50:19,099 So we can draw an interesting diagram of these three 1029 00:50:19,099 --> 00:50:19,890 kinds of existence. 1030 00:50:19,890 --> 00:50:23,622 We could put mind here. 1031 00:50:23,622 --> 00:50:25,620 A little more space. 1032 00:50:25,620 --> 00:50:26,340 Mind here. 1033 00:50:30,020 --> 00:50:31,422 Let's put math here. 1034 00:50:35,490 --> 00:50:39,562 And matter here. 1035 00:50:43,424 --> 00:50:45,090 So if both of these types of connections 1036 00:50:45,090 --> 00:50:49,770 are true for example, if matter gives rise to mind and mind 1037 00:50:49,770 --> 00:50:55,230 gives rise to math, then we can draw it an arrow this way. 1038 00:50:55,230 --> 00:50:58,580 Matter gives rise to mind, and then mind gives rise to math, 1039 00:50:58,580 --> 00:51:00,757 so we can draw an arrow this way. 1040 00:51:00,757 --> 00:51:03,090 And then, this would be the connection between the three 1041 00:51:03,090 --> 00:51:05,640 of them. 1042 00:51:05,640 --> 00:51:10,660 Now, you can also draw other possible combinations 1043 00:51:10,660 --> 00:51:13,350 or get rid of the arrows all together. 1044 00:51:13,350 --> 00:51:19,440 You could say, for example, that matter and mind are different, 1045 00:51:19,440 --> 00:51:21,240 but mind does give rise to matter. 1046 00:51:24,102 --> 00:51:26,070 You could say something weird. 1047 00:51:26,070 --> 00:51:30,717 Most people wouldn't say it, but mind gives rise to matter. 1048 00:51:30,717 --> 00:51:32,050 The math is something different. 1049 00:51:32,050 --> 00:51:34,290 You can do all sorts of combinations. 1050 00:51:34,290 --> 00:51:35,155 All sorts of them. 1051 00:51:38,630 --> 00:51:40,110 But there's a big question. 1052 00:51:40,110 --> 00:51:42,450 There's a big question confronting all these kinds 1053 00:51:42,450 --> 00:51:46,380 of connections, which is, why is math 1054 00:51:46,380 --> 00:51:50,190 so darn effective at describing the universe? 1055 00:51:50,190 --> 00:51:54,180 Why is it so effective at describing reality? 1056 00:51:54,180 --> 00:52:00,030 It's puzzling in pretty much all types of the combinations. 1057 00:52:00,030 --> 00:52:02,370 If math is separate from matter, that's 1058 00:52:02,370 --> 00:52:04,530 probably the most puzzling. 1059 00:52:04,530 --> 00:52:06,210 Probably the most puzzling case. 1060 00:52:06,210 --> 00:52:08,400 Because in that case, there is a physical universe, 1061 00:52:08,400 --> 00:52:12,090 which we usually think of as being real, more real 1062 00:52:12,090 --> 00:52:14,430 than all the other kinds of existences, 1063 00:52:14,430 --> 00:52:16,890 if they're real at all. 1064 00:52:16,890 --> 00:52:21,370 And there is this abstract thing called math that describes it. 1065 00:52:21,370 --> 00:52:23,620 Why should the real universe be described by something 1066 00:52:23,620 --> 00:52:25,000 that doesn't really exist? 1067 00:52:25,000 --> 00:52:27,083 Why should it be described as some abstract thing? 1068 00:52:30,200 --> 00:52:33,330 So you have this puzzling question, 1069 00:52:33,330 --> 00:52:38,880 why is math so effective at describing reality? 1070 00:52:38,880 --> 00:52:40,629 Can anybody give an explanation? 1071 00:52:44,929 --> 00:52:47,220 Well, don't worry if you can't because many people have 1072 00:52:47,220 --> 00:52:49,860 thought about this over the years 1073 00:52:49,860 --> 00:52:52,050 and they've all been pretty puzzled s it. 1074 00:52:52,050 --> 00:52:54,870 They've all been really puzzled by the unreasonable 1075 00:52:54,870 --> 00:53:03,420 effectiveness of mathematics as Eugene Wigner put it. 1076 00:53:03,420 --> 00:53:04,690 So you have a choice. 1077 00:53:04,690 --> 00:53:07,470 You can either accept this puzzling fact 1078 00:53:07,470 --> 00:53:11,080 or you can try to explain it. 1079 00:53:11,080 --> 00:53:18,955 So I'd like to give one possible explanation for it now. 1080 00:53:18,955 --> 00:53:21,310 Hopefully, I'll have enough time to explain it. 1081 00:53:21,310 --> 00:53:23,100 But first-- OK. 1082 00:53:23,100 --> 00:53:26,760 So I'll give one explanation for it, 1083 00:53:26,760 --> 00:53:32,500 which is essentially that the universe is mathematical. 1084 00:53:32,500 --> 00:53:33,990 It's a mathematical thing. 1085 00:53:33,990 --> 00:53:35,910 And therefore, you shouldn't be surprised 1086 00:53:35,910 --> 00:53:37,870 that math describes it so well. 1087 00:53:37,870 --> 00:53:40,230 And this is called the mathematical universe 1088 00:53:40,230 --> 00:53:42,570 hypothesis. 1089 00:53:42,570 --> 00:53:44,100 It's proposed by Max Tegmark, who's 1090 00:53:44,100 --> 00:53:45,420 a cosmologist here at MIT. 1091 00:54:11,870 --> 00:54:16,270 So it starts out by assuming that mental existence isn't 1092 00:54:16,270 --> 00:54:18,492 the only kind of existence. 1093 00:54:18,492 --> 00:54:20,950 So it starts off with something called the external reality 1094 00:54:20,950 --> 00:54:21,989 hypothesis. 1095 00:54:38,150 --> 00:54:41,440 I mean in principle, strictly speaking, 1096 00:54:41,440 --> 00:54:44,490 the only thing we know for sure exists is our mind. 1097 00:54:44,490 --> 00:54:47,220 I think, therefore I am. 1098 00:54:47,220 --> 00:54:49,800 Famous words of Descartes. 1099 00:54:49,800 --> 00:54:56,100 So it's quite conceivable that the mind is all that exists. 1100 00:54:56,100 --> 00:54:59,520 Ordinarily, we like to think that there's 1101 00:54:59,520 --> 00:55:05,400 a whole lot of reality in addition to just our mind. 1102 00:55:05,400 --> 00:55:09,090 I like to think there are people out there that also have minds. 1103 00:55:09,090 --> 00:55:10,910 There are things that-- 1104 00:55:10,910 --> 00:55:14,610 the things that exist in addition to me. 1105 00:55:14,610 --> 00:55:17,030 So there is this external reality hypothesis. 1106 00:55:17,030 --> 00:55:18,930 An external world exists. 1107 00:55:27,810 --> 00:55:30,810 So it's based on this hypothesis. 1108 00:55:30,810 --> 00:55:33,490 But you might not believe this hypothesis. 1109 00:55:33,490 --> 00:55:35,860 You might want to believe that your mind is 1110 00:55:35,860 --> 00:55:37,360 the only thing that exists. 1111 00:55:37,360 --> 00:55:39,443 In other words, might want to believe in something 1112 00:55:39,443 --> 00:55:42,015 called solipsism. 1113 00:55:49,780 --> 00:55:51,660 I'm all that exists, and that's it. 1114 00:55:56,310 --> 00:55:57,430 I exist and nothing else. 1115 00:56:13,770 --> 00:56:16,110 So the external reality hypothesis rejects solipsism. 1116 00:56:18,720 --> 00:56:20,812 How many of you are solipsists? 1117 00:56:20,812 --> 00:56:24,090 How many of you believe in the solipsism? 1118 00:56:24,090 --> 00:56:25,350 You in the green shirt? 1119 00:56:28,299 --> 00:56:29,340 Or is your friend joking? 1120 00:56:32,690 --> 00:56:33,190 OK. 1121 00:56:33,190 --> 00:56:36,810 So most of us don't believe it. 1122 00:56:36,810 --> 00:56:37,740 So why not? 1123 00:56:37,740 --> 00:56:39,330 Why don't you believe in solipsism? 1124 00:56:39,330 --> 00:56:41,413 Why don't you believe that you're all that exists? 1125 00:56:43,490 --> 00:56:44,121 Yeah. 1126 00:56:44,121 --> 00:56:45,079 AUDIENCE: I don't know. 1127 00:56:45,079 --> 00:56:47,876 I think that's kind of an egocentric way 1128 00:56:47,876 --> 00:56:48,950 to look at the universe. 1129 00:56:48,950 --> 00:56:50,241 I'm the only thing that exists. 1130 00:56:50,241 --> 00:56:53,090 And therefore, nothing else does. 1131 00:56:53,090 --> 00:56:54,775 I mean, I understand it-- 1132 00:56:54,775 --> 00:56:57,405 the idea. 1133 00:56:57,405 --> 00:57:00,240 But the whole idea that you're the only thing that exists 1134 00:57:00,240 --> 00:57:01,820 is nothing else. 1135 00:57:01,820 --> 00:57:04,898 And there's no [INAUDIBLE] possibility 1136 00:57:04,898 --> 00:57:08,585 of something else existing, be it another person or matter, 1137 00:57:08,585 --> 00:57:09,480 whatever. 1138 00:57:09,480 --> 00:57:11,610 I find that a little egocentric. 1139 00:57:11,610 --> 00:57:14,494 You know, I'm the only thing. 1140 00:57:14,494 --> 00:57:15,910 NICHOLAS DIBELLA: So one objection 1141 00:57:15,910 --> 00:57:18,400 is that solipsism is egocentric. 1142 00:57:18,400 --> 00:57:18,927 Yeah. 1143 00:57:18,927 --> 00:57:20,260 Just in case they can't hear it. 1144 00:57:23,240 --> 00:57:23,740 Yeah. 1145 00:57:23,740 --> 00:57:27,460 I mean, it's kind of an arrogant position to take. 1146 00:57:27,460 --> 00:57:29,620 But what's wrong with arrogance? 1147 00:57:29,620 --> 00:57:32,620 What's wrong with egocentrism? 1148 00:57:32,620 --> 00:57:35,734 AUDIENCE: [INAUDIBLE] 1149 00:57:35,734 --> 00:57:37,400 NICHOLAS DIBELLA: Another justification? 1150 00:57:37,400 --> 00:57:38,495 AUDIENCE: I don't think it's egocentric. 1151 00:57:38,495 --> 00:57:39,161 It's depressing. 1152 00:57:39,161 --> 00:57:42,550 Who wants to be the only guy out there? 1153 00:57:42,550 --> 00:57:44,790 NICHOLAS DIBELLA: So yeah, it's depressing. 1154 00:57:44,790 --> 00:57:50,822 Why should reality care about what's depressing for you? 1155 00:57:50,822 --> 00:57:52,030 Why is that a valid argument? 1156 00:57:55,230 --> 00:57:59,746 AUDIENCE: I reserve my right to respond-- 1157 00:57:59,746 --> 00:58:01,940 never mind. 1158 00:58:01,940 --> 00:58:05,410 NICHOLAS DIBELLA: I mean, you have to think a bit about this 1159 00:58:05,410 --> 00:58:09,040 to really give a good explanation for or against him. 1160 00:58:09,040 --> 00:58:11,770 The first responses we get are usually emotional ones. 1161 00:58:11,770 --> 00:58:14,980 I mean, it's egocentric or that would be depressing. 1162 00:58:14,980 --> 00:58:16,650 I'm not going to believe it. 1163 00:58:16,650 --> 00:58:18,250 But let me ask for some others. 1164 00:58:18,250 --> 00:58:18,843 Yeah. 1165 00:58:18,843 --> 00:58:20,262 AUDIENCE: It seems like really unreal. 1166 00:58:20,262 --> 00:58:22,012 Because like, if I'm the only that exists, 1167 00:58:22,012 --> 00:58:24,050 then everything [INAUDIBLE]. 1168 00:58:24,050 --> 00:58:25,800 NICHOLAS DIBELLA: Everything else is what? 1169 00:58:25,800 --> 00:58:29,920 AUDIENCE: Everything else that we're seeing must be false. 1170 00:58:29,920 --> 00:58:31,800 NICHOLAS DIBELLA: So if solipsism is true, 1171 00:58:31,800 --> 00:58:35,371 then our reality is kind of like an illusion? 1172 00:58:35,371 --> 00:58:37,120 Why should reality care about whether it's 1173 00:58:37,120 --> 00:58:38,692 an illusion for you? 1174 00:58:38,692 --> 00:58:40,106 AUDIENCE: Well, it's sad. 1175 00:58:40,106 --> 00:58:41,480 NICHOLAS DIBELLA: It's sad, yeah. 1176 00:58:41,480 --> 00:58:43,282 AUDIENCE: [INAUDIBLE] lied to and stuff. 1177 00:58:43,282 --> 00:58:44,240 NICHOLAS DIBELLA: Yeah. 1178 00:58:44,240 --> 00:58:46,650 Well, maybe reality is a liar. 1179 00:58:46,650 --> 00:58:50,920 AUDIENCE: [INAUDIBLE] then don't you exist to? 1180 00:58:50,920 --> 00:58:54,030 NICHOLAS DIBELLA: Well, I mean, I can certainly say I exist. 1181 00:58:54,030 --> 00:58:55,230 I don't know if you exist. 1182 00:58:55,230 --> 00:58:57,100 I mean, I definitely know I exist, 1183 00:58:57,100 --> 00:58:59,630 but I don't know if you exist. 1184 00:58:59,630 --> 00:59:00,786 Yeah. 1185 00:59:00,786 --> 00:59:03,671 AUDIENCE: It's not possible to have two [INAUDIBLE].. 1186 00:59:03,671 --> 00:59:05,712 NICHOLAS DIBELLA: It's not possible to have what? 1187 00:59:05,712 --> 00:59:06,917 AUDIENCE: It's not possible to have 1188 00:59:06,917 --> 00:59:09,042 two people that believe in that because one of them 1189 00:59:09,042 --> 00:59:12,904 shouldn't exist [INAUDIBLE]. 1190 00:59:12,904 --> 00:59:14,820 NICHOLAS DIBELLA: So it's not possible to have 1191 00:59:14,820 --> 00:59:17,100 two people believe in solipsism you're saying? 1192 00:59:17,100 --> 00:59:20,430 Well, the way that solipsists would respond is by saying, 1193 00:59:20,430 --> 00:59:22,680 well, I'm the only person that exists. 1194 00:59:22,680 --> 00:59:24,930 I mean, there are all these other figments 1195 00:59:24,930 --> 00:59:26,517 of my imagination essentially. 1196 00:59:26,517 --> 00:59:27,600 They don't actually exist. 1197 00:59:27,600 --> 00:59:28,725 They just exist in my mind. 1198 00:59:28,725 --> 00:59:30,720 And these things, these objects in my mind, 1199 00:59:30,720 --> 00:59:31,650 think that they exist. 1200 00:59:31,650 --> 00:59:33,530 But really, I'm all that exist. 1201 00:59:33,530 --> 00:59:34,950 Define exist. 1202 00:59:34,950 --> 00:59:36,000 Hard question. 1203 00:59:36,000 --> 00:59:36,630 I can't define. 1204 00:59:36,630 --> 00:59:37,963 I don't know how to define that. 1205 00:59:41,231 --> 00:59:42,480 I don't know how to define it. 1206 00:59:42,480 --> 00:59:45,510 But that's how solipsists would respond 1207 00:59:45,510 --> 00:59:48,971 to your defense [INAUDIBLE]. 1208 00:59:48,971 --> 00:59:49,470 Yeah. 1209 00:59:49,470 --> 00:59:51,448 AUDIENCE: There's no way to prove that anything 1210 00:59:51,448 --> 00:59:52,568 but yourself exists. 1211 00:59:52,568 --> 00:59:55,485 So isn't more solipsism more plausible 1212 00:59:55,485 --> 00:59:58,045 than the external reality? 1213 00:59:58,045 --> 00:59:58,920 NICHOLAS DIBELLA: OK. 1214 00:59:58,920 --> 01:00:01,584 So the only thing that we know for sure exists, the only thing 1215 01:00:01,584 --> 01:00:03,750 that we can definitely prove, at least to ourselves, 1216 01:00:03,750 --> 01:00:04,870 is that we exist. 1217 01:00:04,870 --> 01:00:07,150 So you're saying, well, isn't it more plausible? 1218 01:00:07,150 --> 01:00:09,790 Well, it's definitely more conservative. 1219 01:00:09,790 --> 01:00:10,936 I mean, it's actually-- 1220 01:00:10,936 --> 01:00:12,060 I mean, it's quite a leap-- 1221 01:00:12,060 --> 01:00:14,640 you say, well, I exist. 1222 01:00:14,640 --> 01:00:15,990 Does anything else exist? 1223 01:00:15,990 --> 01:00:18,880 It's quite a leap to say, well, other things also exist. 1224 01:00:18,880 --> 01:00:22,392 So solipsism is actually a very conservative viewpoint to take. 1225 01:00:22,392 --> 01:00:23,850 The external reality of all this is 1226 01:00:23,850 --> 01:00:25,980 quite liberal in this respect. 1227 01:00:25,980 --> 01:00:29,180 So you might want to stick just with conservative positions. 1228 01:00:29,180 --> 01:00:34,180 I mean, you don't want to make any radical assumptions. 1229 01:00:34,180 --> 01:00:35,640 Yeah. 1230 01:00:35,640 --> 01:00:37,390 AUDIENCE: Well, one way I look at 1231 01:00:37,390 --> 01:00:39,590 is there's a lot of things in the world 1232 01:00:39,590 --> 01:00:42,860 that I don't know how to do, or languages I don't know. 1233 01:00:42,860 --> 01:00:44,800 And I don't think my mind would invent things 1234 01:00:44,800 --> 01:00:45,720 that I don't know. 1235 01:00:45,720 --> 01:00:49,050 Like, why would my mind come up with French 1236 01:00:49,050 --> 01:00:51,424 if I don't speak French? 1237 01:00:51,424 --> 01:00:53,840 Why would my mind come up with something going to the moon 1238 01:00:53,840 --> 01:00:55,350 if I can't go to the moon. 1239 01:00:55,350 --> 01:00:56,682 NICHOLAS DIBELLA: Yeah. 1240 01:00:56,682 --> 01:00:58,140 So you're saying, why would my mind 1241 01:00:58,140 --> 01:01:00,011 come up with all this stuff that's seems 1242 01:01:00,011 --> 01:01:02,302 to be out of my control, or something like that, right? 1243 01:01:05,640 --> 01:01:08,130 That's also a good response. 1244 01:01:08,130 --> 01:01:11,160 Any other takes? 1245 01:01:11,160 --> 01:01:12,306 Yeah. 1246 01:01:12,306 --> 01:01:14,431 AUDIENCE: Things don't go the way we want it to go. 1247 01:01:14,431 --> 01:01:16,889 NICHOLAS DIBELLA: Things don't go the way we want it to go. 1248 01:01:16,889 --> 01:01:19,540 But then, it could be that there are really two types of-- 1249 01:01:19,540 --> 01:01:20,520 there are really two parts about it. 1250 01:01:20,520 --> 01:01:22,680 There's a conscious part and there's an unconscious part. 1251 01:01:22,680 --> 01:01:24,430 The conscious part is what we can control, 1252 01:01:24,430 --> 01:01:27,070 the unconscious part is what we can't control. 1253 01:01:27,070 --> 01:01:30,027 I mean, there's a thing that we call "we" or "I." 1254 01:01:30,027 --> 01:01:32,610 It can control some things, but it can't control other things. 1255 01:01:32,610 --> 01:01:34,234 So that would be a way of explaining it 1256 01:01:34,234 --> 01:01:38,590 from a solipsist's perspective. 1257 01:01:38,590 --> 01:01:41,040 Yeah. 1258 01:01:41,040 --> 01:01:45,170 AUDIENCE: It's for our mind to develop this amazingly complex 1259 01:01:45,170 --> 01:01:50,740 reality where the human race quote unquote, 1260 01:01:50,740 --> 01:01:55,770 "knows everything," or at least a whole darn lot. 1261 01:01:55,770 --> 01:02:01,066 Our minds would need to be super-powerful, or something. 1262 01:02:01,066 --> 01:02:03,262 AUDIENCE: [INAUDIBLE] 1263 01:02:03,262 --> 01:02:04,980 AUDIENCE: I don't think that's true. 1264 01:02:04,980 --> 01:02:05,521 AUDIENCE: No. 1265 01:02:05,521 --> 01:02:09,684 It actually-- that actually has been debunked. 1266 01:02:09,684 --> 01:02:12,242 I'm just saying, the thing is we use 10% of our brains. 1267 01:02:12,242 --> 01:02:14,950 But we use different parts of it at different times [INAUDIBLE].. 1268 01:02:14,950 --> 01:02:18,224 NICHOLAS DIBELLA: I think we only use 10% of our hearts. 1269 01:02:18,224 --> 01:02:19,865 I forgot what that was from. 1270 01:02:19,865 --> 01:02:20,990 AUDIENCE: Wedding Crashers. 1271 01:02:20,990 --> 01:02:21,865 NICHOLAS DIBELLA: OK. 1272 01:02:21,865 --> 01:02:23,340 AUDIENCE: Anyway. 1273 01:02:23,340 --> 01:02:24,670 NICHOLAS DIBELLA: Yeah, anyway. 1274 01:02:24,670 --> 01:02:25,580 Yeah. 1275 01:02:25,580 --> 01:02:29,490 So one response is that, well, the mind 1276 01:02:29,490 --> 01:02:32,530 would have to be this really complex thing to account 1277 01:02:32,530 --> 01:02:34,090 for what we observe, right? 1278 01:02:36,970 --> 01:02:39,910 Physics gives a quite simple explanation of everything. 1279 01:02:39,910 --> 01:02:45,160 Whereas, proposing some kind of a very complex mind that 1280 01:02:45,160 --> 01:02:47,830 has a conscious part, an unconscious part, 1281 01:02:47,830 --> 01:02:50,310 that's a very complicated explanation of everything. 1282 01:02:50,310 --> 01:02:52,402 And remember what I talked about last time. 1283 01:02:52,402 --> 01:02:53,860 I talked about Occam's razor, which 1284 01:02:53,860 --> 01:02:57,020 is that the simplest theory is more likely to be true. 1285 01:02:57,020 --> 01:02:59,140 And so you might say, well, solipsism 1286 01:02:59,140 --> 01:03:01,990 is actually a more complicated theory 1287 01:03:01,990 --> 01:03:06,784 than non-solipsism-- a theory that has an external reality. 1288 01:03:06,784 --> 01:03:08,200 And so by Occam's razor, you would 1289 01:03:08,200 --> 01:03:10,074 say that solipsism is less likely to be true. 1290 01:03:12,860 --> 01:03:14,632 AUDIENCE: How is that more complex? 1291 01:03:14,632 --> 01:03:16,340 NICHOLAS DIBELLA: How is it more complex? 1292 01:03:16,340 --> 01:03:16,840 OK. 1293 01:03:16,840 --> 01:03:20,810 So according to solipsism-- 1294 01:03:20,810 --> 01:03:23,930 well, as you pointed out before, there 1295 01:03:23,930 --> 01:03:26,030 are parts of our experience that we can control 1296 01:03:26,030 --> 01:03:27,405 and there are parts that we don't 1297 01:03:27,405 --> 01:03:28,810 seem to be able to control. 1298 01:03:28,810 --> 01:03:32,360 We have a conscious part and an unconscious part. 1299 01:03:32,360 --> 01:03:36,170 The unconscious part would seem to be what a non-solipsist 1300 01:03:36,170 --> 01:03:38,420 would call the external world. 1301 01:03:38,420 --> 01:03:44,749 It's a part of the world that we do scientific experiments on 1302 01:03:44,749 --> 01:03:46,040 and that we try to learn about. 1303 01:03:49,557 --> 01:03:51,890 To a solipsist, he would call that an unconscious thing. 1304 01:03:51,890 --> 01:03:53,473 He would call it the unconscious mind. 1305 01:03:57,900 --> 01:04:02,870 So I'm going to actually explain my argument against solipsism, 1306 01:04:02,870 --> 01:04:04,390 and then I'll slightly-- 1307 01:04:04,390 --> 01:04:07,130 that will in the process explain-- 1308 01:04:07,130 --> 01:04:08,760 answer your question. 1309 01:04:08,760 --> 01:04:09,260 Yeah. 1310 01:04:09,260 --> 01:04:10,820 So I personally feel that solipsism 1311 01:04:10,820 --> 01:04:14,160 is more complicated, more complex than non-solipsism. 1312 01:04:14,160 --> 01:04:16,580 So you start off by looking at the conscious 1313 01:04:16,580 --> 01:04:19,580 versus the unconscious. 1314 01:04:19,580 --> 01:04:21,830 The unconscious being the part that you can't control. 1315 01:04:21,830 --> 01:04:25,310 And it's similar to what non-solipsists would 1316 01:04:25,310 --> 01:04:28,520 call the external reality. 1317 01:04:28,520 --> 01:04:31,130 Well, if they're essentially the same. 1318 01:04:31,130 --> 01:04:34,192 I mean, the unconscious mind, and then the external reality 1319 01:04:34,192 --> 01:04:35,900 just then seem to be just different names 1320 01:04:35,900 --> 01:04:38,420 for the same thing then. 1321 01:04:38,420 --> 01:04:42,590 There's still the conscious mind in both cases, 1322 01:04:42,590 --> 01:04:48,470 but the external reality just gets a different name. 1323 01:04:48,470 --> 01:04:52,430 So then, solipsism reduces to a case 1324 01:04:52,430 --> 01:04:56,450 of non-solipsism with respect to the unconscious mind. 1325 01:04:56,450 --> 01:04:59,710 But solipsism has a different view for the conscious mind. 1326 01:04:59,710 --> 01:05:02,080 In non-solipsism, the conscious mind 1327 01:05:02,080 --> 01:05:05,550 is a physical thing that emerges from physical laws. 1328 01:05:05,550 --> 01:05:08,850 When physical things are put together in some sort of way-- 1329 01:05:08,850 --> 01:05:12,890 in the right way, you get a conscious mind. 1330 01:05:12,890 --> 01:05:16,550 And so you have a simple interplay between the conscious 1331 01:05:16,550 --> 01:05:19,460 and the unconscious in the non-solipsist theory. 1332 01:05:19,460 --> 01:05:22,169 In solipsist theory, you have this unconscious mind. 1333 01:05:22,169 --> 01:05:23,460 You also have a conscious mind. 1334 01:05:26,190 --> 01:05:29,690 And to have a comprehensive solipsist theory, 1335 01:05:29,690 --> 01:05:32,160 you have to have a theory for how the conscious mind works, 1336 01:05:32,160 --> 01:05:33,980 a theory for how the unconscious mind works, 1337 01:05:33,980 --> 01:05:36,021 and also a theory for how they work together, how 1338 01:05:36,021 --> 01:05:37,370 they interact with each other. 1339 01:05:37,370 --> 01:05:39,590 Whereas, in the non-solipsist theory, you just have physics. 1340 01:05:39,590 --> 01:05:41,180 Everything is physics, and that's all. 1341 01:05:41,180 --> 01:05:42,242 AUDIENCE: [INAUDIBLE] 1342 01:05:42,242 --> 01:05:43,200 NICHOLAS DIBELLA: Yeah. 1343 01:05:43,200 --> 01:05:46,190 So in that respect then, solipsism 1344 01:05:46,190 --> 01:05:48,901 seems to be more complicated than non-solipsism. 1345 01:05:48,901 --> 01:05:50,400 Non-solipsism, you just get physics. 1346 01:05:50,400 --> 01:05:51,620 Gets the job done. 1347 01:05:51,620 --> 01:05:54,240 Solipsism, you need all these extra rules. 1348 01:05:54,240 --> 01:05:56,570 Well, then you could say, well, maybe 1349 01:05:56,570 --> 01:05:58,640 there's an overarching rule that describes 1350 01:05:58,640 --> 01:06:01,070 how the conscious and the unconscious both work 1351 01:06:01,070 --> 01:06:02,440 and how they both interact. 1352 01:06:02,440 --> 01:06:05,180 Well, then you might as well call it physics. 1353 01:06:05,180 --> 01:06:10,090 And then if you have these overall arching rules, 1354 01:06:10,090 --> 01:06:11,840 then you've essentially got physics again. 1355 01:06:11,840 --> 01:06:15,170 So then, solipsism became non-solipsism. 1356 01:06:15,170 --> 01:06:17,956 So that's my argument against solipsism. 1357 01:06:21,764 --> 01:06:23,770 Does that makes sense? 1358 01:06:23,770 --> 01:06:25,125 It has more complex. 1359 01:06:25,125 --> 01:06:27,685 Were you going to say something? 1360 01:06:27,685 --> 01:06:32,616 AUDIENCE: [INAUDIBLE] dreams at night sort of a [INAUDIBLE]?? 1361 01:06:32,616 --> 01:06:34,365 NICHOLAS DIBELLA: Are dreams at night sort 1362 01:06:34,365 --> 01:06:36,044 of a solipsist thing? 1363 01:06:36,044 --> 01:06:36,952 AUDIENCE: Yeah. 1364 01:06:36,952 --> 01:06:39,680 [INAUDIBLE] 1365 01:06:39,680 --> 01:06:40,796 NICHOLAS DIBELLA: What? 1366 01:06:40,796 --> 01:06:45,540 AUDIENCE: [INAUDIBLE] 1367 01:06:45,540 --> 01:06:47,710 NICHOLAS DIBELLA: So I guess when you're dreaming, 1368 01:06:47,710 --> 01:06:52,337 you kind of become an effective solipsist, sort of. 1369 01:06:52,337 --> 01:06:53,170 Effective solipsist. 1370 01:06:53,170 --> 01:06:57,430 And actually, sometimes it's fun to be recreational solipsist. 1371 01:06:57,430 --> 01:06:58,782 Like someone talks to you. 1372 01:06:58,782 --> 01:06:59,740 They ask you questions. 1373 01:06:59,740 --> 01:07:03,500 You just-- I'm a solipsist. 1374 01:07:03,500 --> 01:07:04,520 I'm a solipsist. 1375 01:07:04,520 --> 01:07:06,640 I'm a solipsist. 1376 01:07:06,640 --> 01:07:08,120 Sometimes, it's fun. 1377 01:07:08,120 --> 01:07:10,030 You know, if you're stressed or if you 1378 01:07:10,030 --> 01:07:11,710 have nothing else better to do. 1379 01:07:11,710 --> 01:07:15,220 Just be a recreational solipsist. 1380 01:07:15,220 --> 01:07:17,810 OK. 1381 01:07:17,810 --> 01:07:21,090 Any questions about solipsism? 1382 01:07:21,090 --> 01:07:25,040 I mean, I gave my justification for why I think-- 1383 01:07:25,040 --> 01:07:26,900 I've read a number of justifications 1384 01:07:26,900 --> 01:07:28,785 and I've kind of combined them all together 1385 01:07:28,785 --> 01:07:32,480 to get this one, what I think is reasonable. 1386 01:07:32,480 --> 01:07:35,630 But maybe you're smarter than me and you 1387 01:07:35,630 --> 01:07:39,790 can think of a better argument for solipsism 1388 01:07:39,790 --> 01:07:42,500 than for against it. 1389 01:07:42,500 --> 01:07:44,930 If you can, then I'd be very interested in hearing it. 1390 01:07:44,930 --> 01:07:48,710 Because I think solipsism is a very cool idea. 1391 01:07:48,710 --> 01:07:51,380 It'd be cool if I'm the only thing that exists. 1392 01:07:51,380 --> 01:07:53,840 But yeah, maybe it would also be depressing. 1393 01:07:53,840 --> 01:07:55,670 And what was the other? 1394 01:07:55,670 --> 01:07:58,490 Egocentric. 1395 01:07:58,490 --> 01:08:00,660 It's interesting in any case, though. 1396 01:08:00,660 --> 01:08:01,160 OK. 1397 01:08:01,160 --> 01:08:04,220 So the mathematical universe hypothesis 1398 01:08:04,220 --> 01:08:06,470 starts off by rejecting solipsism and accepting 1399 01:08:06,470 --> 01:08:09,020 the external reality hypothesis. 1400 01:08:12,420 --> 01:08:15,200 So according to the way that this argument works, 1401 01:08:15,200 --> 01:08:17,689 you start off by saying there's an external reality. 1402 01:08:17,689 --> 01:08:23,160 So we humans can't perceive this external reality. 1403 01:08:23,160 --> 01:08:25,939 But if extraterrestrial intelligent aliens exist, 1404 01:08:25,939 --> 01:08:30,140 then they'd also be able to observe this external reality. 1405 01:08:30,140 --> 01:08:32,456 If it's possible to create artificial intelligence, 1406 01:08:32,456 --> 01:08:33,830 then that artificial intelligence 1407 01:08:33,830 --> 01:08:39,300 would be able to observe this external reality. 1408 01:08:39,300 --> 01:08:41,899 So there's this external reality that's 1409 01:08:41,899 --> 01:08:45,209 accessible to all intelligent beings. 1410 01:08:45,209 --> 01:08:47,000 Now, we, humans, have a very particular way 1411 01:08:47,000 --> 01:08:50,029 of describing that external reality. 1412 01:08:50,029 --> 01:08:55,880 We've defined concepts like mass and energy, protons. 1413 01:08:55,880 --> 01:08:58,430 And so we have this set of language 1414 01:08:58,430 --> 01:09:00,799 that we use in describing the world. 1415 01:09:00,799 --> 01:09:02,840 Aliens could very well have a different language. 1416 01:09:02,840 --> 01:09:04,926 I mean, who knows how alien psychology works? 1417 01:09:04,926 --> 01:09:07,550 They could have a very different set of concepts that they use. 1418 01:09:10,080 --> 01:09:13,350 If humans and aliens and all-- everyone else 1419 01:09:13,350 --> 01:09:16,370 finally comes across the theory of everything, 1420 01:09:16,370 --> 01:09:17,757 they should all be similar. 1421 01:09:17,757 --> 01:09:20,090 The theories that they form, they should all be similar. 1422 01:09:20,090 --> 01:09:26,399 And in particular, the theory should be free of the language 1423 01:09:26,399 --> 01:09:28,569 that we try to use to describe the theories. 1424 01:09:28,569 --> 01:09:30,735 It shouldn't matter whether a human is describing it 1425 01:09:30,735 --> 01:09:33,470 or whether an alien describing, and so forth. 1426 01:09:33,470 --> 01:09:37,964 So when we try to describe the external reality, we see things 1427 01:09:37,964 --> 01:09:39,380 and we see things that are related 1428 01:09:39,380 --> 01:09:41,764 to each other in sorts of ways. 1429 01:09:41,764 --> 01:09:43,430 Things that have relations between them. 1430 01:09:46,441 --> 01:09:47,899 The theory of everything would have 1431 01:09:47,899 --> 01:09:50,569 to be free from human language or alien language, 1432 01:09:50,569 --> 01:09:52,649 free from what we call baggage. 1433 01:09:52,649 --> 01:09:54,800 And so therefore, it would have to be abstract. 1434 01:09:54,800 --> 01:09:57,008 It would have to be an abstract description of things 1435 01:09:57,008 --> 01:10:00,340 that are related to each other in some sort of way. 1436 01:10:00,340 --> 01:10:06,170 Now, it turns out that the notion 1437 01:10:06,170 --> 01:10:08,840 of a set of things, a set of abstract things 1438 01:10:08,840 --> 01:10:11,870 that are related to each other in a particular way, 1439 01:10:11,870 --> 01:10:13,640 has a name for it. 1440 01:10:13,640 --> 01:10:15,456 It's called the mathematical structure. 1441 01:10:31,040 --> 01:10:35,766 Abstract objects related in some way. 1442 01:10:41,850 --> 01:10:45,807 For example, a square is a very primitive type 1443 01:10:45,807 --> 01:10:46,890 of mathematical structure. 1444 01:10:46,890 --> 01:10:49,050 There exists these things called points, 1445 01:10:49,050 --> 01:10:51,256 or things called line segments. 1446 01:10:51,256 --> 01:10:53,380 And they're related to each other in a sort of way. 1447 01:10:53,380 --> 01:10:55,505 So you have a very primitive mathematical structure 1448 01:10:55,505 --> 01:10:58,170 of a square. 1449 01:10:58,170 --> 01:11:03,335 If an external reality exists, then-- 1450 01:11:03,335 --> 01:11:05,509 oh, I'll be done. 1451 01:11:05,509 --> 01:11:06,550 Give me two more minutes. 1452 01:11:09,260 --> 01:11:11,710 If an external reality exists, then it 1453 01:11:11,710 --> 01:11:12,910 should be free from baggage. 1454 01:11:12,910 --> 01:11:15,340 And so it should be some kind of abstract description 1455 01:11:15,340 --> 01:11:20,320 of nature with objects related in some sort of way. 1456 01:11:20,320 --> 01:11:26,430 So the external realities should be a mathematical structure. 1457 01:11:26,430 --> 01:11:28,480 And so the mathematical universe hypothesis 1458 01:11:28,480 --> 01:11:31,570 is that the universe is actually a mathematical structure. 1459 01:11:31,570 --> 01:11:33,310 We live in a mathematical structure. 1460 01:11:33,310 --> 01:11:35,980 We're mathematical stuff. 1461 01:11:35,980 --> 01:11:37,840 And so the mathematical universe hypothesis 1462 01:11:37,840 --> 01:11:39,940 has a very simple explanation for why 1463 01:11:39,940 --> 01:11:42,570 math is so effective at describing the world. 1464 01:11:42,570 --> 01:11:44,090 It's a very simple answer. 1465 01:11:44,090 --> 01:11:46,180 The world is mathematical, so it's no surprise 1466 01:11:46,180 --> 01:11:49,945 that it should describe it very well. 1467 01:11:49,945 --> 01:11:52,120 We ran out of time today. 1468 01:11:52,120 --> 01:11:54,750 But next class, I'll continue. 1469 01:11:54,750 --> 01:11:59,844 I'll continue from here and discuss another level 1470 01:11:59,844 --> 01:12:00,760 of parallel universes. 1471 01:12:00,760 --> 01:12:02,350 Discuss something even crazier, which 1472 01:12:02,350 --> 01:12:06,140 is that our universe is actually a computer simulation. 1473 01:12:06,140 --> 01:12:08,620 So that's next class. 1474 01:12:08,620 --> 01:12:10,770 OK, see you next week.