1 00:00:00,570 --> 00:00:02,990 The following content is provided under a Creative 2 00:00:02,990 --> 00:00:04,510 Commons license. 3 00:00:04,510 --> 00:00:06,850 Your support will help MIT OpenCourseWare 4 00:00:06,850 --> 00:00:11,220 continue to offer high quality educational resources for free. 5 00:00:11,220 --> 00:00:13,830 To make a donation or view additional materials 6 00:00:13,830 --> 00:00:17,707 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,707 --> 00:00:18,332 at ocw.mit.edu. 8 00:00:24,070 --> 00:00:26,540 PROFESSOR: Who can tell me what a proof is? 9 00:00:26,540 --> 00:00:29,459 Any ideas of, what is a proof, anyway? 10 00:00:29,459 --> 00:00:30,000 Any thoughts? 11 00:00:30,000 --> 00:00:30,874 Yeah? 12 00:00:30,874 --> 00:00:34,262 AUDIENCE: It's a chain of statements, each logically 13 00:00:34,262 --> 00:00:36,215 supported by the previous ones, that 14 00:00:36,215 --> 00:00:39,161 get you from a set of assumptions 15 00:00:39,161 --> 00:00:40,730 to a set of conclusions. 16 00:00:40,730 --> 00:00:41,810 PROFESSOR: Very good. 17 00:00:41,810 --> 00:00:42,640 I like that. 18 00:00:42,640 --> 00:00:47,090 That's very close to what we're going to do here, yeah. 19 00:00:47,090 --> 00:00:50,110 Now, that's a special kind of proof, though. 20 00:00:50,110 --> 00:00:52,290 That's a mathematical proof. 21 00:00:52,290 --> 00:00:54,170 And I'm going to write a definition very 22 00:00:54,170 --> 00:00:56,610 close to that in a few minutes. 23 00:00:56,610 --> 00:01:00,830 But proofs exist beyond mathematics. 24 00:01:00,830 --> 00:01:03,960 Can anybody think of a higher level 25 00:01:03,960 --> 00:01:05,061 notion of what a proof is? 26 00:01:05,061 --> 00:01:06,810 That's correct, what you said, but there's 27 00:01:06,810 --> 00:01:12,500 a higher meta level notion of what a proof is beyond that. 28 00:01:12,500 --> 00:01:16,220 It may have no logical deductions potentially. 29 00:01:16,220 --> 00:01:18,490 It may have no assumptions. 30 00:01:18,490 --> 00:01:22,710 Any thoughts about a proof? 31 00:01:22,710 --> 00:01:26,970 OK, well, I think generally, a proof 32 00:01:26,970 --> 00:01:29,400 is considered, across multiple fields, 33 00:01:29,400 --> 00:01:31,150 as a method for ascertaining the truth. 34 00:01:36,720 --> 00:01:38,190 And you described one method. 35 00:01:51,370 --> 00:01:54,720 Now, by ascertaining, I mean establishing truth, verifying 36 00:01:54,720 --> 00:01:56,340 truth. 37 00:01:56,340 --> 00:01:58,930 And there's lots of ways to ascertain truth 38 00:01:58,930 --> 00:02:02,540 in society, and even within science. 39 00:02:02,540 --> 00:02:05,050 What are some examples of ways that we 40 00:02:05,050 --> 00:02:07,395 ascertain truth in society? 41 00:02:10,210 --> 00:02:10,897 Yeah? 42 00:02:10,897 --> 00:02:13,768 AUDIENCE: Observations, like seeing that piece of chalk 43 00:02:13,768 --> 00:02:15,560 will fall to the ground. 44 00:02:15,560 --> 00:02:17,260 PROFESSOR: Observation, experiment 45 00:02:17,260 --> 00:02:18,640 and observation-- excellent. 46 00:02:18,640 --> 00:02:23,540 And that's the bedrock of physics. 47 00:02:23,540 --> 00:02:27,580 I mean, who really knows if there's gravity out there? 48 00:02:27,580 --> 00:02:29,720 Well, we observe it. 49 00:02:29,720 --> 00:02:31,730 And so we then conclude that's the truth. 50 00:02:31,730 --> 00:02:33,607 There's gravity, and we have laws about it. 51 00:02:33,607 --> 00:02:34,440 That's one good way. 52 00:02:34,440 --> 00:02:39,890 What's another way of ascertaining truth 53 00:02:39,890 --> 00:02:43,830 across scientific disciplines, or beyond science, just 54 00:02:43,830 --> 00:02:45,150 in society? 55 00:02:45,150 --> 00:02:47,700 How is truth established? 56 00:02:47,700 --> 00:02:48,450 What are the ways? 57 00:02:48,450 --> 00:02:48,950 Yeah? 58 00:02:48,950 --> 00:02:50,704 AUDIENCE: Well, establishing what's false, 59 00:02:50,704 --> 00:02:53,620 you can know what things aren't true. 60 00:02:53,620 --> 00:02:56,109 Then that helps you narrow down what is true. 61 00:02:56,109 --> 00:02:57,900 PROFESSOR: Yes, truth-- yeah, that's great. 62 00:02:57,900 --> 00:02:59,980 Truth is the opposite of falsehood. 63 00:02:59,980 --> 00:03:02,590 How do we establish falsehood? 64 00:03:02,590 --> 00:03:04,460 What are the ways in doing that? 65 00:03:04,460 --> 00:03:08,455 How you decide something is not true in every day-- yeah? 66 00:03:08,455 --> 00:03:10,100 AUDIENCE: Find counterexamples. 67 00:03:10,100 --> 00:03:13,460 PROFESSOR: Find counterexamples, yeah, that's good. 68 00:03:13,460 --> 00:03:16,890 So in fact, even a step more general, sampling. 69 00:03:20,030 --> 00:03:21,730 Counterexamples are ways. 70 00:03:26,320 --> 00:03:28,400 If you do something, an experiment, ten times, 71 00:03:28,400 --> 00:03:33,310 and every time, it comes out one way, that's truth, maybe. 72 00:03:33,310 --> 00:03:36,329 But there's fields where that becomes truth. 73 00:03:36,329 --> 00:03:37,245 What about other ways? 74 00:03:41,490 --> 00:03:44,760 How are we going to decide if Roger Clemens is 75 00:03:44,760 --> 00:03:48,410 guilty of perjury for lying about steroids to Congress? 76 00:03:51,210 --> 00:03:52,230 He did it or he didn't. 77 00:03:52,230 --> 00:03:54,300 How are we going to decide that? 78 00:03:54,300 --> 00:03:56,290 How is that truth going to be ascertained? 79 00:03:56,290 --> 00:03:56,790 Yeah? 80 00:03:56,790 --> 00:03:59,481 AUDIENCE: Would it be by examining the evidence 81 00:03:59,481 --> 00:04:00,100 that we have? 82 00:04:00,100 --> 00:04:01,350 PROFESSOR: Examining evidence. 83 00:04:01,350 --> 00:04:03,512 And who makes the conclusion there? 84 00:04:03,512 --> 00:04:04,220 AUDIENCE: Juries. 85 00:04:04,220 --> 00:04:05,540 PROFESSOR: The jury. 86 00:04:05,540 --> 00:04:07,645 Truth is established by juries or judges. 87 00:04:13,960 --> 00:04:15,859 You know, Blago-- I can never pronounce 88 00:04:15,859 --> 00:04:20,950 his name, the Illinois governor, Blagojevick-- he's guilty. 89 00:04:20,950 --> 00:04:25,880 That's the truth of-- not of conspiracy, 90 00:04:25,880 --> 00:04:27,910 trying to sell Obama a senate seat, 91 00:04:27,910 --> 00:04:30,970 but of lying to the authorities about campaign financing. 92 00:04:30,970 --> 00:04:35,640 OJ is guilty-- not of killing his wife, 93 00:04:35,640 --> 00:04:38,650 but of breaking into an apartment to steal back 94 00:04:38,650 --> 00:04:40,850 some of his merchandise. 95 00:04:40,850 --> 00:04:43,840 So judges and juries make decisions on truth. 96 00:04:43,840 --> 00:04:46,250 What are other truths-- bigger truths, even, than judges 97 00:04:46,250 --> 00:04:48,085 and juries, in society? 98 00:04:51,530 --> 00:04:54,680 There's one really big one that causes a lot of issues. 99 00:04:54,680 --> 00:04:55,180 Yeah? 100 00:04:55,180 --> 00:04:55,870 AUDIENCE: Religion. 101 00:04:55,870 --> 00:04:56,420 PROFESSOR: What is it? 102 00:04:56,420 --> 00:04:57,211 AUDIENCE: Religion. 103 00:04:57,211 --> 00:05:01,824 PROFESSOR: Religion, the word of God-- broadly construed here, 104 00:05:01,824 --> 00:05:02,365 for religion. 105 00:05:05,440 --> 00:05:08,140 Now, that one is really hard to argue about 106 00:05:08,140 --> 00:05:10,750 because you believe it. 107 00:05:10,750 --> 00:05:13,440 And especially if you're not talking to God regularly 108 00:05:13,440 --> 00:05:17,904 and somebody is, well, it's hard to argue about the truth. 109 00:05:17,904 --> 00:05:19,820 So you rely on others to interpret it for you, 110 00:05:19,820 --> 00:05:23,560 often-- a priest, or a minister, a rabbi. 111 00:05:23,560 --> 00:05:25,640 And it gets complicated, because you can end up 112 00:05:25,640 --> 00:05:29,890 with conflicting truths based on who you think you're talking to 113 00:05:29,890 --> 00:05:33,220 or who the translator is for you. 114 00:05:33,220 --> 00:05:37,080 Another one is the word of your boss. 115 00:05:37,080 --> 00:05:39,850 Whatever the boss says is right. 116 00:05:39,850 --> 00:05:42,650 Often in business, the customer is always right. 117 00:05:42,650 --> 00:05:46,680 That's the truth, whatever the customer says. 118 00:05:46,680 --> 00:05:48,920 With Donald Trump as your boss, you'd better agree 119 00:05:48,920 --> 00:05:49,586 or you're fired. 120 00:05:53,560 --> 00:05:57,110 Often in classes, the professor says it, it is true. 121 00:05:57,110 --> 00:05:58,910 Because the authority said it. 122 00:05:58,910 --> 00:06:00,290 That's not true here. 123 00:06:00,290 --> 00:06:01,600 That will not hold. 124 00:06:01,600 --> 00:06:05,340 And one of the nicest things about math that I like a lot 125 00:06:05,340 --> 00:06:09,230 is that the youngest student can stand up 126 00:06:09,230 --> 00:06:12,020 against the most oldest, most experienced professor, 127 00:06:12,020 --> 00:06:15,015 and win an argument on mathematics. 128 00:06:17,267 --> 00:06:19,850 I do get pleasure when a student comes up and proves me wrong. 129 00:06:19,850 --> 00:06:21,391 I loved it when that student came in, 130 00:06:21,391 --> 00:06:23,240 and she showed me what I said really 131 00:06:23,240 --> 00:06:25,090 wasn't right when you looked at it carefully 132 00:06:25,090 --> 00:06:27,400 or in a different light. 133 00:06:27,400 --> 00:06:30,230 Now, sometimes if I do it on the board here and it's in class, 134 00:06:30,230 --> 00:06:31,990 well, that's fun. 135 00:06:31,990 --> 00:06:34,740 I feel a little embarrassed afterwards. 136 00:06:34,740 --> 00:06:36,640 But it's a good thing about mathematics, 137 00:06:36,640 --> 00:06:41,550 is you can have that kind of dialogue. 138 00:06:41,550 --> 00:06:45,200 OK, another one, which is related to the word of God 139 00:06:45,200 --> 00:06:50,370 sometimes, is inner conviction-- very popular 140 00:06:50,370 --> 00:06:54,020 in computer science, believe it or not, with the mantra, 141 00:06:54,020 --> 00:06:57,820 there are no bugs in my program. 142 00:06:57,820 --> 00:07:00,850 I can't tell you how many times you hear that. 143 00:07:00,850 --> 00:07:06,790 Closely related is, I don't see why not something is true. 144 00:07:06,790 --> 00:07:08,940 And that's a good one, because that 145 00:07:08,940 --> 00:07:13,906 transfers the burden of proof to anybody who disagrees with you. 146 00:07:13,906 --> 00:07:14,910 You don't have to prove. 147 00:07:14,910 --> 00:07:16,550 You just say, I don't see why it's not true. 148 00:07:16,550 --> 00:07:18,883 All of a sudden, the other person who's questioning you, 149 00:07:18,883 --> 00:07:24,000 it becomes their job to disprove you, which is not so good. 150 00:07:24,000 --> 00:07:26,710 OK, now in mathematics, there's this higher level. 151 00:07:26,710 --> 00:07:28,980 And someone stated it very clearly up there. 152 00:07:28,980 --> 00:07:31,460 Let me write it up here. 153 00:07:31,460 --> 00:07:39,280 In mathematics, we have a mathematical proof 154 00:07:39,280 --> 00:07:57,700 is a verification of a proposition 155 00:07:57,700 --> 00:08:16,410 by a chain of logical deductions from a set of axioms. 156 00:08:22,030 --> 00:08:24,070 Now, that's a bit of a mouthful. 157 00:08:24,070 --> 00:08:28,500 There's three important components here-- 158 00:08:28,500 --> 00:08:34,780 propositions, logical deductions, and axioms. 159 00:08:34,780 --> 00:08:37,130 And we're going to spend the rest of the class 160 00:08:37,130 --> 00:08:38,770 today talking about each of these, 161 00:08:38,770 --> 00:08:41,144 and then give an example of a proof. 162 00:08:41,144 --> 00:08:42,394 We'll start with propositions. 163 00:08:46,910 --> 00:09:04,000 A proposition is a statement that is either true or false. 164 00:09:07,130 --> 00:09:10,800 You may not know which one, but it's one or the other. 165 00:09:10,800 --> 00:09:15,310 A simple example-- 2 plus 3 equals 5. 166 00:09:15,310 --> 00:09:18,292 Now, that's a true proposition. 167 00:09:18,292 --> 00:09:20,125 Here's one that's a little more interesting. 168 00:09:27,450 --> 00:09:32,900 For all n in the set of natural numbers, 169 00:09:32,900 --> 00:09:38,660 n squared plus n plus 41 is a prime number. 170 00:09:43,930 --> 00:09:46,520 I know I've used some notation here. 171 00:09:46,520 --> 00:09:50,465 This is-- the upside down A is the for all symbol. 172 00:09:53,200 --> 00:09:56,620 How many people have not seen that symbol before? 173 00:09:56,620 --> 00:09:57,310 A bunch of you. 174 00:09:57,310 --> 00:10:00,110 You're going to see a bunch of symbols here first week. 175 00:10:00,110 --> 00:10:03,520 And that means for every possible choice of n-- 176 00:10:03,520 --> 00:10:08,010 and this is in the natural numbers, which is the set 0, 1, 177 00:10:08,010 --> 00:10:11,970 2, 3, and so forth. 178 00:10:11,970 --> 00:10:13,644 It's the natural numbers. 179 00:10:13,644 --> 00:10:15,560 It's basically the integers, but not negative. 180 00:10:19,570 --> 00:10:22,790 So we're saying for every natural number, i.e. for 0, 181 00:10:22,790 --> 00:10:26,700 for 1, for 2, and for 3, and so forth, this expression 182 00:10:26,700 --> 00:10:27,540 is a prime. 183 00:10:27,540 --> 00:10:30,410 Now, a prime number is a number that 184 00:10:30,410 --> 00:10:35,830 is not divisible by any other number besides itself and 1. 185 00:10:35,830 --> 00:10:38,850 So 1, 3, 5, 7 are prime. 186 00:10:38,850 --> 00:10:41,890 9 is not because it's 3 times 3. 187 00:10:41,890 --> 00:10:45,700 Now, this part here is called the predicate. 188 00:10:49,390 --> 00:11:00,420 And a predicate is a proposition whose truth 189 00:11:00,420 --> 00:11:11,090 depends on the value of a variable-- in this case, n. 190 00:11:16,780 --> 00:11:21,480 All right, this is referred to as the universe of discourse. 191 00:11:21,480 --> 00:11:24,240 It's the space of all the things we're talking about. 192 00:11:24,240 --> 00:11:28,230 We're only talking about natural numbers here. 193 00:11:28,230 --> 00:11:30,140 This is called a quantifier. 194 00:11:30,140 --> 00:11:31,900 We'll see more quantifiers later. 195 00:11:34,620 --> 00:11:39,820 All right, now to see if this proposition is true, 196 00:11:39,820 --> 00:11:43,120 we need to make sure that this predicate is 197 00:11:43,120 --> 00:11:46,770 true for every natural number n. 198 00:11:46,770 --> 00:11:51,160 So let's see if we can check that. 199 00:11:51,160 --> 00:11:53,140 Let's try some values. 200 00:11:53,140 --> 00:11:58,230 So we'll try n is 1, 2, 3, and so forth. 201 00:11:58,230 --> 00:12:02,390 And we'll compute n squared plus n plus 41. 202 00:12:02,390 --> 00:12:05,890 And then we'll check, is it prime? 203 00:12:05,890 --> 00:12:12,202 So for n equals 0, n squared plus n plus 41 is 41. 204 00:12:12,202 --> 00:12:13,405 Is 41 prime? 205 00:12:16,390 --> 00:12:19,450 Yeah, nothing divides 41 but itself and 1. 206 00:12:21,980 --> 00:12:23,120 All right, let's try 1. 207 00:12:23,120 --> 00:12:25,845 1 squared plus 1 plus 41 is 43. 208 00:12:25,845 --> 00:12:28,120 Is 43 prime? 209 00:12:28,120 --> 00:12:30,400 Yes. 210 00:12:30,400 --> 00:12:32,050 Let's try 2. 211 00:12:32,050 --> 00:12:36,440 We get 4 plus 2 is 6 plus 41 is 47. 212 00:12:36,440 --> 00:12:38,230 Is 47 prime? 213 00:12:38,230 --> 00:12:39,030 AUDIENCE: Yes. 214 00:12:39,030 --> 00:12:40,650 PROFESSOR: Yes. 215 00:12:40,650 --> 00:12:41,200 Looking good. 216 00:12:41,200 --> 00:12:46,640 3-- I got 9, 12, 53. 217 00:12:46,640 --> 00:12:48,600 Is 53 prime? 218 00:12:48,600 --> 00:12:50,000 Yeah. 219 00:12:50,000 --> 00:12:51,810 And I could keep on going here. 220 00:12:51,810 --> 00:12:53,550 I could go down to 20. 221 00:12:53,550 --> 00:12:57,140 I get 420-- 461. 222 00:12:57,140 --> 00:13:00,150 In fact, that is a prime. 223 00:13:00,150 --> 00:13:02,410 And I could just keep on going here. 224 00:13:02,410 --> 00:13:03,460 Go down to 39. 225 00:13:03,460 --> 00:13:06,967 I get 1,601. 226 00:13:06,967 --> 00:13:07,550 You can check. 227 00:13:07,550 --> 00:13:09,880 That is a prime. 228 00:13:09,880 --> 00:13:13,120 The first 40 values of n, the proposition is true. 229 00:13:13,120 --> 00:13:14,700 The predicate is true. 230 00:13:14,700 --> 00:13:16,640 It is prime. 231 00:13:16,640 --> 00:13:20,520 Now, this is a great example because in a lot of fields-- 232 00:13:20,520 --> 00:13:25,560 physics, for example; statistics, often-- you 233 00:13:25,560 --> 00:13:26,720 checked 40 examples. 234 00:13:26,720 --> 00:13:29,040 That's above and beyond the call of duty. 235 00:13:29,040 --> 00:13:30,820 It's always true. 236 00:13:30,820 --> 00:13:35,160 So yeah, this must be true, right? 237 00:13:35,160 --> 00:13:37,730 No, wrong. 238 00:13:37,730 --> 00:13:40,940 Often, you'll see this in a lot of scientific fields. 239 00:13:40,940 --> 00:13:41,690 It is not true. 240 00:13:41,690 --> 00:13:43,220 Can anybody give me an example of n 241 00:13:43,220 --> 00:13:46,750 for which n squared plus n plus 41 is not prime? 242 00:13:46,750 --> 00:13:47,390 Yeah? 243 00:13:47,390 --> 00:13:48,270 AUDIENCE: 40. 244 00:13:48,270 --> 00:13:49,720 PROFESSOR: 40, good. 245 00:13:49,720 --> 00:13:51,720 Let's see about 40. 246 00:13:51,720 --> 00:13:59,990 40 squared plus 40 plus 41 is 1,681. 247 00:13:59,990 --> 00:14:02,960 What's that equal? 248 00:14:02,960 --> 00:14:06,070 41 squared. 249 00:14:06,070 --> 00:14:07,740 So it is not prime. 250 00:14:07,740 --> 00:14:10,830 Somebody give me an obvious example where it's not prime. 251 00:14:10,830 --> 00:14:11,530 AUDIENCE: 41. 252 00:14:11,530 --> 00:14:13,330 PROFESSOR: 41-- yeah, 41 squared, 253 00:14:13,330 --> 00:14:15,480 we get everything is divided by 41. 254 00:14:15,480 --> 00:14:17,990 But 40 is the first break-point. 255 00:14:17,990 --> 00:14:22,110 So the first 40 examples work, and then it failed. 256 00:14:22,110 --> 00:14:26,570 So this proposition is false, even though it 257 00:14:26,570 --> 00:14:28,657 was looking pretty good. 258 00:14:28,657 --> 00:14:29,990 There's a reason I'm doing this. 259 00:14:29,990 --> 00:14:31,990 In fact, I'm going to do it some more here. 260 00:14:31,990 --> 00:14:35,000 I'm going to beat you over the head with it. 261 00:14:35,000 --> 00:14:39,353 Here's a famous in mathematics statement. 262 00:14:41,880 --> 00:14:47,380 a to the fourth plus b to the fourth plus c to the fourth 263 00:14:47,380 --> 00:15:01,200 equals d to the fourth has no positive integer solutions. 264 00:15:01,200 --> 00:15:02,980 That is a proposition. 265 00:15:02,980 --> 00:15:07,160 Now, this proposition was conjectured to be true by Euler 266 00:15:07,160 --> 00:15:09,740 in 1769. 267 00:15:09,740 --> 00:15:11,570 Euler's a big honcho in math. 268 00:15:11,570 --> 00:15:13,970 We still talk about him a lot even though he's 269 00:15:13,970 --> 00:15:16,700 been dead for centuries. 270 00:15:16,700 --> 00:15:20,100 It was unsolved for over 2 centuries. 271 00:15:20,100 --> 00:15:21,890 Mathematicians worked on it. 272 00:15:21,890 --> 00:15:25,280 It was finally disapproved by a very clever fellow 273 00:15:25,280 --> 00:15:30,090 named Noam Elkies 218 years later after it was conjectured. 274 00:15:30,090 --> 00:15:34,100 He worked at that other school down the street. 275 00:15:34,100 --> 00:15:37,200 And he came up with this. 276 00:15:37,200 --> 00:15:41,990 a equals 95,800. 277 00:15:41,990 --> 00:15:46,510 b equals 217,519. 278 00:15:46,510 --> 00:15:49,912 c equals 414,560. 279 00:15:49,912 --> 00:15:51,620 You don't have to remember these numbers. 280 00:15:51,620 --> 00:15:57,217 We're not going to quiz you on that-- 422,481. 281 00:15:57,217 --> 00:15:59,300 Now, he claims-- I've never personally checked it, 282 00:15:59,300 --> 00:16:02,360 but presumably, people have-- you plug those in here, 283 00:16:02,360 --> 00:16:04,580 and you have an equality. 284 00:16:04,580 --> 00:16:05,850 So he says. 285 00:16:05,850 --> 00:16:08,340 So in fact, the correct proposition 286 00:16:08,340 --> 00:16:15,980 is there does exist a, b, c, d in the positive natural 287 00:16:15,980 --> 00:16:20,000 numbers such that a the fourth plus b to the fourth plus 288 00:16:20,000 --> 00:16:23,510 c to the fourth equals d to the fourth. 289 00:16:23,510 --> 00:16:28,460 I used a new quantifier here called there exists. 290 00:16:28,460 --> 00:16:29,870 Instead of an upside down A, it's 291 00:16:29,870 --> 00:16:33,030 a backwards E. Don't ask me why. 292 00:16:33,030 --> 00:16:35,960 That's what it is. 293 00:16:35,960 --> 00:16:40,040 The plus means you can't have 0 or negative numbers. 294 00:16:40,040 --> 00:16:42,280 So these are the positive natural numbers. 295 00:16:45,750 --> 00:16:51,080 And here's your predicate, which of course, the truth of this 296 00:16:51,080 --> 00:16:54,951 depends on the values of a, b, c and d. 297 00:16:54,951 --> 00:16:57,200 It took a long time to figure out that actually, there 298 00:16:57,200 --> 00:16:58,817 was a solution here. 299 00:16:58,817 --> 00:17:01,150 Obviously, everything they tried until that time failed. 300 00:17:01,150 --> 00:17:02,316 Let me give you another one. 301 00:17:08,300 --> 00:17:24,085 313 x cubed plus y cubed equals z cubed has no positive integer 302 00:17:24,085 --> 00:17:24,585 solutions. 303 00:17:29,530 --> 00:17:31,850 This turns out to be false. 304 00:17:31,850 --> 00:17:34,580 But the shortest, smallest counter-example 305 00:17:34,580 --> 00:17:37,950 has over 1,000 digits. 306 00:17:37,950 --> 00:17:38,770 This one was easy. 307 00:17:38,770 --> 00:17:40,650 It only has six digits. 308 00:17:40,650 --> 00:17:43,560 So there's no way ever you'd use a computer 309 00:17:43,560 --> 00:17:46,600 to exhaustively search 1,000 digit numbers here 310 00:17:46,600 --> 00:17:49,440 to show it's false. 311 00:17:49,440 --> 00:17:53,810 Now, of course, some of you are probably thinking, why on earth 312 00:17:53,810 --> 00:17:58,860 would I care if 313 times x cubed plus y cubed equals 313 00:17:58,860 --> 00:18:02,484 z cubed has a solution? 314 00:18:02,484 --> 00:18:04,150 And that probably won't be the last time 315 00:18:04,150 --> 00:18:07,310 that thought occurs to you during the term. 316 00:18:07,310 --> 00:18:09,635 And why on earth would anybody ever try 317 00:18:09,635 --> 00:18:13,320 to even find a solution to that? 318 00:18:13,320 --> 00:18:15,830 I mean, mathematicians are sort of a rare breed. 319 00:18:15,830 --> 00:18:18,417 Now, actually in this case, that's 320 00:18:18,417 --> 00:18:19,625 really important in practice. 321 00:18:22,580 --> 00:18:24,280 This equation is an example of what's 322 00:18:24,280 --> 00:18:26,300 called an elliptic curve-- elliptic curve. 323 00:18:26,300 --> 00:18:29,820 You study these if you're really a specialist in mathematics 324 00:18:29,820 --> 00:18:32,030 in graduate school, or if you work 325 00:18:32,030 --> 00:18:34,770 for certain three-letter agencies 326 00:18:34,770 --> 00:18:37,140 because it's central to the understanding of how 327 00:18:37,140 --> 00:18:39,670 to factor large integers. 328 00:18:39,670 --> 00:18:43,760 That means factoring, showing that-- what was it-- 329 00:18:43,760 --> 00:18:46,916 1,681 is 41 times 41. 330 00:18:46,916 --> 00:18:50,380 And I said, OK, who cares about factoring? 331 00:18:50,380 --> 00:18:54,880 Well, factoring is the way to break cryptosystems 332 00:18:54,880 --> 00:18:58,625 like RSA, which are used for everything 333 00:18:58,625 --> 00:19:01,840 that we do electronically today. 334 00:19:01,840 --> 00:19:03,130 You have a Paypal account. 335 00:19:03,130 --> 00:19:04,230 You buy something online. 336 00:19:04,230 --> 00:19:05,710 You're using SSL. 337 00:19:05,710 --> 00:19:08,750 They're all using cryptosystems, almost all of which 338 00:19:08,750 --> 00:19:10,400 are based on number theory. 339 00:19:10,400 --> 00:19:13,130 And in particular, they're based on factoring. 340 00:19:13,130 --> 00:19:16,570 And if you can find good solutions to things like this, 341 00:19:16,570 --> 00:19:19,260 or solutions to things like this, all of a sudden, 342 00:19:19,260 --> 00:19:23,325 you can get an angle and a wedge on factoring. 343 00:19:23,325 --> 00:19:24,700 And it's because of that that now 344 00:19:24,700 --> 00:19:30,940 RSA uses 1,000 digit moduluses instead of hundred 345 00:19:30,940 --> 00:19:32,844 digit moduluses like they used to use, 346 00:19:32,844 --> 00:19:34,510 because people figured out how to factor 347 00:19:34,510 --> 00:19:36,430 and how to break the cryptosystem. 348 00:19:36,430 --> 00:19:39,490 If you could break those cryptosystems, 349 00:19:39,490 --> 00:19:42,350 well, you can't rule the world, but it's close. 350 00:19:42,350 --> 00:19:44,950 All right, so we'll talk more about this 351 00:19:44,950 --> 00:19:46,830 the week after next when we do number theory, 352 00:19:46,830 --> 00:19:50,020 and we work up to RSA and how that cryptosystem works, 353 00:19:50,020 --> 00:19:52,880 and why factoring is so important. 354 00:19:52,880 --> 00:19:55,537 So yeah, you don't have to really have 355 00:19:55,537 --> 00:19:56,370 to worry about this. 356 00:19:56,370 --> 00:19:57,661 But these things are important. 357 00:19:57,661 --> 00:20:01,750 And the bigger message is that you don't just try a few cases, 358 00:20:01,750 --> 00:20:04,610 and if it works, you think it's done. 359 00:20:04,610 --> 00:20:06,906 That's not how the game works in mathematics. 360 00:20:06,906 --> 00:20:08,280 You can get an idea of maybe it's 361 00:20:08,280 --> 00:20:13,390 true, but doesn't tell you the answer. 362 00:20:13,390 --> 00:20:15,380 All right, let me give you another one. 363 00:20:15,380 --> 00:20:17,785 This is another very famous one that probably most of you 364 00:20:17,785 --> 00:20:19,830 have heard of. 365 00:20:19,830 --> 00:20:34,880 The regions in any map can be colored in four colors 366 00:20:34,880 --> 00:20:47,710 so that adjacent regions have different colors. 367 00:20:47,710 --> 00:20:49,270 Like a map of the United States-- 368 00:20:49,270 --> 00:20:50,860 every state gets a color. 369 00:20:50,860 --> 00:20:53,020 If two states share a border, they 370 00:20:53,020 --> 00:20:57,430 have different colors so you can distinguish them. 371 00:20:57,430 --> 00:20:59,530 This is known as the four color theorem. 372 00:20:59,530 --> 00:21:02,570 And it's very famous in the popular literature. 373 00:21:02,570 --> 00:21:05,060 How many people have heard of this theorem before? 374 00:21:05,060 --> 00:21:05,680 Yeah, OK. 375 00:21:05,680 --> 00:21:07,334 So you've all heard of it. 376 00:21:07,334 --> 00:21:08,250 It has a long history. 377 00:21:08,250 --> 00:21:12,240 It was conjectured by somebody named Guthrie in 1853. 378 00:21:12,240 --> 00:21:14,950 He's the first person to say this ought to be possible. 379 00:21:14,950 --> 00:21:20,630 And there were many false proofs over the ensuing century. 380 00:21:20,630 --> 00:21:23,510 One of the most convincing was a proof using pictures 381 00:21:23,510 --> 00:21:28,700 by Kempe in 1879, 26 years later. 382 00:21:28,700 --> 00:21:31,420 And this proof was believed for over a decade. 383 00:21:31,420 --> 00:21:33,890 Mathematicians thought the proof was right 384 00:21:33,890 --> 00:21:36,570 until another mathematician named Heawood found 385 00:21:36,570 --> 00:21:39,270 a fatal flaw in the argument. 386 00:21:39,270 --> 00:21:41,510 Now, this proof by Kempe consisted 387 00:21:41,510 --> 00:21:45,204 of drawing pictures of what maps have to look like. 388 00:21:45,204 --> 00:21:46,620 So he started by saying, a map has 389 00:21:46,620 --> 00:21:47,953 to look like one of these types. 390 00:21:47,953 --> 00:21:49,620 And he would draw pictures of them. 391 00:21:49,620 --> 00:21:52,690 And then he argued that those types that he drew pictures of, 392 00:21:52,690 --> 00:21:55,580 it worked for. 393 00:21:55,580 --> 00:22:01,110 Proofs by picture are often very convincing and very wrong. 394 00:22:01,110 --> 00:22:03,550 And I'm going to give you one to start lecture next time. 395 00:22:03,550 --> 00:22:06,133 It'll be a proof by PowerPoint, which is even worse than proof 396 00:22:06,133 --> 00:22:08,130 by picture. 397 00:22:08,130 --> 00:22:10,971 And it is compelling. 398 00:22:10,971 --> 00:22:13,220 And the point will to be to show you proofs by picture 399 00:22:13,220 --> 00:22:15,387 are generally not a good thing. 400 00:22:15,387 --> 00:22:16,970 Because your brain just locks in-- oh, 401 00:22:16,970 --> 00:22:19,010 that's what it has to look like. 402 00:22:19,010 --> 00:22:22,510 And you don't think about other ways that it might look like. 403 00:22:22,510 --> 00:22:24,240 Now, the four color theorem was finally 404 00:22:24,240 --> 00:22:29,120 proved by Appel and Haken in 1977, 405 00:22:29,120 --> 00:22:34,970 but they had to use a computer to check thousands of cases. 406 00:22:34,970 --> 00:22:37,510 Now, this was a little disturbing to mathematicians, 407 00:22:37,510 --> 00:22:42,989 because how do they know the computer did the right thing? 408 00:22:42,989 --> 00:22:44,780 Your colleague writes a proof on the board. 409 00:22:44,780 --> 00:22:45,630 You can check it. 410 00:22:45,630 --> 00:22:47,630 But how do you know the computer didn't mess up, 411 00:22:47,630 --> 00:22:50,130 or not do some cases? 412 00:22:50,130 --> 00:22:52,620 Now, everybody believes it's true now. 413 00:22:52,620 --> 00:22:54,780 But it's unsatisfying. 414 00:22:54,780 --> 00:23:00,420 A few years ago, a 12-page human proof was discovered, 415 00:23:00,420 --> 00:23:01,810 but it's not been verified. 416 00:23:01,810 --> 00:23:04,180 And people are very suspicious of it 417 00:23:04,180 --> 00:23:06,800 because the proof of the main lemma says, 418 00:23:06,800 --> 00:23:10,740 quote, "details of this lemma is left to the reader. 419 00:23:10,740 --> 00:23:12,509 See figure seven." 420 00:23:12,509 --> 00:23:14,300 That's what the main lemma of the proof is. 421 00:23:14,300 --> 00:23:15,674 But people think that maybe there 422 00:23:15,674 --> 00:23:19,240 were some good ideas there, but very suspicious proof. 423 00:23:19,240 --> 00:23:24,430 All right, let's do another one, another proposition-- 424 00:23:24,430 --> 00:23:27,800 also very famous. 425 00:23:27,800 --> 00:23:35,490 Every even integer but 2-- actually, 426 00:23:35,490 --> 00:23:41,875 positive integer but 2-- is the sum of two primes. 427 00:23:45,560 --> 00:23:53,025 For example, 24 is the sum of 11 and 13, which are prime. 428 00:23:57,090 --> 00:23:57,640 Anybody know? 429 00:23:57,640 --> 00:24:00,635 Is this true or false, this proposition? 430 00:24:03,870 --> 00:24:04,370 Yeah? 431 00:24:04,370 --> 00:24:05,156 AUDIENCE: I wish I knew. 432 00:24:05,156 --> 00:24:06,780 PROFESSOR: [LAUGHS] Yeah, that's right. 433 00:24:06,780 --> 00:24:08,120 Me too. 434 00:24:08,120 --> 00:24:11,140 Nobody knows if this is true or false. 435 00:24:11,140 --> 00:24:14,326 This is called Goldbach's conjecture. 436 00:24:14,326 --> 00:24:19,510 It was conjectured by Christian Goldbach in 1742. 437 00:24:19,510 --> 00:24:22,570 This is a really simple proposition. 438 00:24:22,570 --> 00:24:24,850 And it's amazing it's not known. 439 00:24:24,850 --> 00:24:28,301 In fact, I spent a couple years working on-- I thought, 440 00:24:28,301 --> 00:24:29,800 oh, well, this has to be easy enough 441 00:24:29,800 --> 00:24:34,550 to prove when I was younger, and didn't get very far. 442 00:24:34,550 --> 00:24:37,010 So people still don't know if it's true. 443 00:24:37,010 --> 00:24:40,270 And in fact, it was listed by the Globe 444 00:24:40,270 --> 00:24:42,360 as one of the great unsolved mysteries. 445 00:24:42,360 --> 00:24:45,310 So if you get out this Globe article here, one of the hand-- 446 00:24:45,310 --> 00:24:48,300 does everybody have this handout? 447 00:24:48,300 --> 00:24:48,800 You don't? 448 00:24:48,800 --> 00:24:49,800 We'll get it passed out. 449 00:24:49,800 --> 00:24:52,907 Somebody missing that handout up over there and over here? 450 00:24:52,907 --> 00:24:54,490 All right, if we get those passed out. 451 00:24:57,080 --> 00:24:59,260 Now, it lists the three conjectures. 452 00:24:59,260 --> 00:25:03,610 Do you see Goldbach's conjecture there? 453 00:25:03,610 --> 00:25:06,240 Now, can anybody point out something 454 00:25:06,240 --> 00:25:09,509 that's a little disturbing about what the Globe says 455 00:25:09,509 --> 00:25:10,675 about Goldbach's conjecture? 456 00:25:13,500 --> 00:25:14,980 AUDIENCE: 9 as a prime number. 457 00:25:14,980 --> 00:25:17,100 PROFESSOR: Yeah, it gives the example. 458 00:25:17,100 --> 00:25:19,120 Like, instead of 24 is 11 plus 13, 459 00:25:19,120 --> 00:25:24,410 it says 20 is the sum of 9 and 11. 460 00:25:24,410 --> 00:25:27,860 Now, if we're allowed to use things like 9 as primes, 461 00:25:27,860 --> 00:25:30,110 Goldbach's conjecture's pretty easy to prove is true. 462 00:25:33,540 --> 00:25:35,517 This won't be the last time we get examples 463 00:25:35,517 --> 00:25:36,350 from the literature. 464 00:25:36,350 --> 00:25:37,970 In fact, we're going to do this a lot, 465 00:25:37,970 --> 00:25:41,381 along this theme of, you cannot believe everything you read. 466 00:25:41,381 --> 00:25:43,130 Now, the Globe is easy pickings, but we'll 467 00:25:43,130 --> 00:25:45,370 do some more interesting ones later. 468 00:25:45,370 --> 00:25:47,850 Now, this article lists two other famous conjectures 469 00:25:47,850 --> 00:25:55,440 which most people believe to be true-- the Riemann hypothesis 470 00:25:55,440 --> 00:26:00,280 after an 1859 paper written by Bernard Riemann 471 00:26:00,280 --> 00:26:02,740 suggested that zeros in an infinite series of numbers 472 00:26:02,740 --> 00:26:05,170 known as a zeta function form along a straight line 473 00:26:05,170 --> 00:26:06,690 on that complex plane. 474 00:26:06,690 --> 00:26:11,350 The hypothesis has been proved to 1.5 billion zeros, not far 475 00:26:11,350 --> 00:26:13,860 enough to prove it completely. 476 00:26:13,860 --> 00:26:15,660 If they did 1.5 trillion zeros, it 477 00:26:15,660 --> 00:26:19,230 wouldn't be far enough to prove it completely, of course. 478 00:26:19,230 --> 00:26:24,510 And then the-- no, actually, the Riemann hypothesis, 479 00:26:24,510 --> 00:26:26,920 a couple years ago, somebody credible 480 00:26:26,920 --> 00:26:28,150 claimed to have proved it. 481 00:26:28,150 --> 00:26:30,320 Proof turned out not to be right. 482 00:26:30,320 --> 00:26:32,910 Then there's the Poincare conjecture. 483 00:26:32,910 --> 00:26:35,570 Now, this one was finished off. 484 00:26:35,570 --> 00:26:39,090 It was proved to be true in 2003 by a Russian named 485 00:26:39,090 --> 00:26:41,250 Grigori Perelman. 486 00:26:41,250 --> 00:26:43,560 The conjecture says, roughly speaking, 487 00:26:43,560 --> 00:26:48,180 that 3D objects without holes, like no a doughnut, 488 00:26:48,180 --> 00:26:50,200 are equivalent to the sphere. 489 00:26:50,200 --> 00:26:53,350 They can sort of be deformed into a sphere. 490 00:26:53,350 --> 00:26:55,700 This is known to be true in four dimensions and higher, 491 00:26:55,700 --> 00:26:57,616 but nobody could prove it for three dimensions 492 00:26:57,616 --> 00:26:59,490 until Perelman came along. 493 00:26:59,490 --> 00:27:02,880 Now, there's a bit of a controversy around this guy. 494 00:27:02,880 --> 00:27:06,290 He had an 80-page proof, but didn't have all the details. 495 00:27:06,290 --> 00:27:07,880 So then other teams of mathematicians 496 00:27:07,880 --> 00:27:12,770 got together and wrote 350 pages of details. 497 00:27:12,770 --> 00:27:15,080 And then most people believe now that it's right, 498 00:27:15,080 --> 00:27:16,940 and that his original proof might not 499 00:27:16,940 --> 00:27:19,420 have had all the details, but he had the right structure 500 00:27:19,420 --> 00:27:20,540 of the proof. 501 00:27:20,540 --> 00:27:23,040 So he won prizes for this. 502 00:27:23,040 --> 00:27:26,090 He won the highest prize in mathematics, the Fields Medal. 503 00:27:26,090 --> 00:27:29,660 And just earlier this year, he was awarded the $1 million 504 00:27:29,660 --> 00:27:31,450 Millennium Prize. 505 00:27:31,450 --> 00:27:33,250 And there's about six problems or so 506 00:27:33,250 --> 00:27:35,750 that if you solve one of them, the Clay Institute gives you 507 00:27:35,750 --> 00:27:36,690 a million dollars. 508 00:27:36,690 --> 00:27:40,100 And he's the first one to win the million dollars. 509 00:27:40,100 --> 00:27:41,440 Now, the guy's a little strange. 510 00:27:41,440 --> 00:27:44,820 He rejected the Fields Medal and refused to go to the ceremony 511 00:27:44,820 --> 00:27:46,350 where he was being honored. 512 00:27:46,350 --> 00:27:48,440 And he's recently rejected the Millennium prize. 513 00:27:50,960 --> 00:27:54,970 And anyway, this area's murky, and we have an expert 514 00:27:54,970 --> 00:27:56,929 to explain it all for us on video, 515 00:27:56,929 --> 00:27:57,970 which I thought I'd show. 516 00:28:06,360 --> 00:28:09,550 All right, let's do a simpler one here. 517 00:28:13,510 --> 00:28:19,620 For all n in Z, n greater than or equal to 2 518 00:28:19,620 --> 00:28:24,240 implies n squared is greater than or equal to 4. 519 00:28:24,240 --> 00:28:26,060 Now, Z, we use for the integers. 520 00:28:30,230 --> 00:28:37,440 And so that would be 0, 1, minus 1, 2, minus 2, and so forth. 521 00:28:37,440 --> 00:28:40,645 And this symbol here is implies. 522 00:28:45,310 --> 00:28:47,100 I In fact, one thing you can notice 523 00:28:47,100 --> 00:28:51,670 when you read the text is we use different notation there 524 00:28:51,670 --> 00:28:53,960 as the standard than I will use in lecture. 525 00:28:53,960 --> 00:28:56,060 And there's lots of ways of doing it. 526 00:28:56,060 --> 00:28:58,300 You could have a double arrow, a single arrow. 527 00:28:58,300 --> 00:29:00,000 You could write out implies every time 528 00:29:00,000 --> 00:29:01,200 as it's done in the text. 529 00:29:01,200 --> 00:29:02,783 And it doesn't really matter which one 530 00:29:02,783 --> 00:29:04,700 you want to use as long as you use one 531 00:29:04,700 --> 00:29:07,630 of the conventions for implies. 532 00:29:07,630 --> 00:29:10,515 And let me define what implies means. 533 00:29:26,910 --> 00:29:43,830 An implication p implies q is said to be true if p is false 534 00:29:43,830 --> 00:29:48,440 or q is true, either one. 535 00:29:48,440 --> 00:29:55,480 So we can write this down in terms of a truth table 536 00:29:55,480 --> 00:29:56,800 as follows. 537 00:29:56,800 --> 00:29:59,680 You have the values of p and q. 538 00:29:59,680 --> 00:30:03,048 And I'll give the value of p implies q. 539 00:30:03,048 --> 00:30:11,575 If p is true and q is true, what about p implies q? 540 00:30:11,575 --> 00:30:16,330 It's true, because q is true in the definition. 541 00:30:16,330 --> 00:30:19,594 If p is true and q is false? 542 00:30:19,594 --> 00:30:20,260 AUDIENCE: False. 543 00:30:20,260 --> 00:30:22,910 PROFESSOR: False. 544 00:30:22,910 --> 00:30:23,880 P is false. 545 00:30:23,880 --> 00:30:24,610 Q is true. 546 00:30:27,292 --> 00:30:28,190 True. 547 00:30:28,190 --> 00:30:32,440 What about false and false? 548 00:30:32,440 --> 00:30:34,230 It's true. 549 00:30:34,230 --> 00:30:36,760 Even though this is false, as long as p is false, 550 00:30:36,760 --> 00:30:40,790 p implies q is true. 551 00:30:40,790 --> 00:30:43,520 So this is important to remember. 552 00:30:43,520 --> 00:30:49,800 False implies anything is true, which is a little strange. 553 00:30:49,800 --> 00:30:51,880 There's a famous expression. 554 00:30:51,880 --> 00:30:55,819 If pigs could fly, I would be king. 555 00:30:55,819 --> 00:30:56,360 Is that true? 556 00:30:59,920 --> 00:31:01,020 Sort of. 557 00:31:01,020 --> 00:31:07,740 In fact, this statement, pigs fly 558 00:31:07,740 --> 00:31:18,400 implies I'm king-- that's true, because pigs don't fly. 559 00:31:18,400 --> 00:31:21,420 Doesn't matter whether or not I'm king, which I'm not. 560 00:31:21,420 --> 00:31:24,010 Since pigs don't fly, even though that's false, 561 00:31:24,010 --> 00:31:27,414 the implication is true. 562 00:31:27,414 --> 00:31:29,580 Now, some of you have worked on these things before. 563 00:31:29,580 --> 00:31:30,470 It's second nature. 564 00:31:30,470 --> 00:31:32,190 If you haven't, you want to start 565 00:31:32,190 --> 00:31:34,100 getting familiar with that. 566 00:31:36,504 --> 00:31:37,545 Let's do another example. 567 00:31:49,610 --> 00:31:50,776 What about this proposition? 568 00:31:53,850 --> 00:32:00,610 For all integers, n in Z, n greater than or equal to 2-- 569 00:32:00,610 --> 00:32:05,350 this is if and only if-- n squared greater than 570 00:32:05,350 --> 00:32:07,890 or equal to 4. 571 00:32:07,890 --> 00:32:08,590 Is that true? 572 00:32:12,830 --> 00:32:15,600 Is n only bigger than 2 if and only 573 00:32:15,600 --> 00:32:18,030 if n squared is bigger than 4? 574 00:32:18,030 --> 00:32:19,100 It's false. 575 00:32:19,100 --> 00:32:22,490 What's an example of n for which that's false? 576 00:32:22,490 --> 00:32:23,620 Negative, all right? 577 00:32:23,620 --> 00:32:25,980 So it's false. 578 00:32:25,980 --> 00:32:30,560 n equals negative 3, all right? 579 00:32:30,560 --> 00:32:32,620 Negative 3 squared is bigger than or equal to 4, 580 00:32:32,620 --> 00:32:34,620 but negative 3 is not bigger than or equal to 2. 581 00:32:34,620 --> 00:32:39,670 And in fact this if and only if means you 582 00:32:39,670 --> 00:32:43,660 have to have an implication both ways. 583 00:32:43,660 --> 00:32:45,625 So you have to check both ways for it. 584 00:32:45,625 --> 00:32:48,000 So let's do the truth table-- extend this truth table out 585 00:32:48,000 --> 00:32:55,340 here to do the truth table for p if and only if q. 586 00:33:04,740 --> 00:33:05,790 So here are p and q. 587 00:33:08,590 --> 00:33:12,800 Is q implies p true for this row? 588 00:33:12,800 --> 00:33:16,760 Does true imply true? 589 00:33:16,760 --> 00:33:18,000 Yeah. 590 00:33:18,000 --> 00:33:20,891 False implies true? 591 00:33:20,891 --> 00:33:22,120 That's true. 592 00:33:22,120 --> 00:33:23,570 True does not apply false. 593 00:33:23,570 --> 00:33:25,160 That's false. 594 00:33:25,160 --> 00:33:27,660 And false implies false. 595 00:33:27,660 --> 00:33:33,500 And so now, we can see where p is if and only if q. 596 00:33:33,500 --> 00:33:37,860 If they're both true, then it's true here. 597 00:33:37,860 --> 00:33:38,660 What about here? 598 00:33:38,660 --> 00:33:41,670 Is p true if and only if q is true in this case? 599 00:33:44,360 --> 00:33:49,190 No, because p implies q is false, but q implies p is true. 600 00:33:49,190 --> 00:33:50,980 So it's false. 601 00:33:50,980 --> 00:33:51,730 False here. 602 00:33:54,570 --> 00:33:56,490 I made a mistake there, right? 603 00:33:56,490 --> 00:34:00,070 That was true-- oops. 604 00:34:00,070 --> 00:34:01,900 And true if and only if true, OK. 605 00:34:01,900 --> 00:34:05,240 They're both true, so we're OK. 606 00:34:05,240 --> 00:34:08,260 So p if and only if q is true when they're both true 607 00:34:08,260 --> 00:34:09,499 or both false. 608 00:34:09,499 --> 00:34:10,744 And that's it. 609 00:34:10,744 --> 00:34:13,889 If they're different, then it's not true. 610 00:34:13,889 --> 00:34:17,860 The key here is to always check both ways. 611 00:34:17,860 --> 00:34:20,139 So if you're asked to prove an if and only if, 612 00:34:20,139 --> 00:34:23,037 you have to prove that way, and that way. 613 00:34:26,679 --> 00:34:30,239 We've just done about 15 propositions. 614 00:34:30,239 --> 00:34:31,620 Is every sentence a proposition? 615 00:34:36,070 --> 00:34:36,570 Yes? 616 00:34:36,570 --> 00:34:37,388 No? 617 00:34:37,388 --> 00:34:37,929 AUDIENCE: No. 618 00:34:37,929 --> 00:34:38,512 PROFESSOR: No. 619 00:34:38,512 --> 00:34:41,240 What's an example of something that's no a proposition? 620 00:34:41,240 --> 00:34:42,659 AUDIENCE: This statement is false. 621 00:34:42,659 --> 00:34:43,409 PROFESSOR: A what? 622 00:34:43,409 --> 00:34:45,420 AUDIENCE: This statement is false. 623 00:34:45,420 --> 00:34:47,719 PROFESSOR: This statement is false. 624 00:34:47,719 --> 00:34:48,590 That's true. 625 00:34:48,590 --> 00:34:52,020 Well, it's true it's not a proposition. 626 00:34:52,020 --> 00:34:53,971 Because if it were true, it wouldn't be false. 627 00:34:53,971 --> 00:34:55,429 And if was false, then it'd be true 628 00:34:55,429 --> 00:34:56,909 and you'd have a contradiction. 629 00:34:56,909 --> 00:34:58,850 So it's neither true nor false. 630 00:34:58,850 --> 00:35:00,579 What's a more simple example of something 631 00:35:00,579 --> 00:35:01,620 that's not a proposition? 632 00:35:01,620 --> 00:35:03,069 AUDIENCE: This is a tissue. 633 00:35:03,069 --> 00:35:05,484 Isn't that a [INAUDIBLE]? 634 00:35:05,484 --> 00:35:06,940 PROFESSOR: Ooh. 635 00:35:06,940 --> 00:35:11,470 Boy, I would have said that's true in some world. 636 00:35:11,470 --> 00:35:13,214 Because yeah, that's a tissue. 637 00:35:13,214 --> 00:35:14,255 So it's a true statement. 638 00:35:16,790 --> 00:35:17,670 AUDIENCE: Hello. 639 00:35:17,670 --> 00:35:18,641 PROFESSOR: Hello. 640 00:35:18,641 --> 00:35:19,140 That's good. 641 00:35:19,140 --> 00:35:21,590 That's neither true or false, yeah. 642 00:35:21,590 --> 00:35:22,720 A question. 643 00:35:22,720 --> 00:35:25,080 Who are you-- neither true nor false. 644 00:35:25,080 --> 00:35:27,000 So not everything is a proposition. 645 00:35:27,000 --> 00:35:28,750 But in this course, pretty much everything 646 00:35:28,750 --> 00:35:31,620 we deal with will be a proposition. 647 00:35:31,620 --> 00:35:34,166 All right, so that's it for propositions. 648 00:35:34,166 --> 00:35:35,415 Any questions on propositions? 649 00:35:38,460 --> 00:35:41,315 Next, we're going to talk about axioms. 650 00:35:47,540 --> 00:35:50,660 Now, the good news is that axioms are the same thing, 651 00:35:50,660 --> 00:35:52,920 really, as propositions. 652 00:35:52,920 --> 00:35:56,130 The only difference is that axioms are propositions 653 00:35:56,130 --> 00:35:57,470 that we just assume are true. 654 00:36:01,930 --> 00:36:16,470 An axiom is a proposition that is assumed to be true. 655 00:36:20,130 --> 00:36:23,570 There's no proof that an axiom is true. 656 00:36:23,570 --> 00:36:26,740 You just assume it because you think it's reasonable. 657 00:36:26,740 --> 00:36:29,430 In fact, the word "axiom" comes from Greek. 658 00:36:29,430 --> 00:36:32,250 It doesn't mean to be true. 659 00:36:32,250 --> 00:36:35,180 It means to think worthy-- something 660 00:36:35,180 --> 00:36:37,950 you think is worthy enough to be assumed to be true. 661 00:36:37,950 --> 00:36:40,780 Now, a lot of times, you'll hear people say-- sometimes, we'll 662 00:36:40,780 --> 00:36:44,000 even say it to you-- don't make assumptions 663 00:36:44,000 --> 00:36:46,150 when you're doing math. 664 00:36:46,150 --> 00:36:47,774 No, that's not true. 665 00:36:47,774 --> 00:36:49,690 You have to make assumptions when you do math. 666 00:36:49,690 --> 00:36:51,570 Otherwise, you can't do anything because you 667 00:36:51,570 --> 00:36:53,900 have to start with some axioms. 668 00:36:53,900 --> 00:36:58,240 The key in math is to identify what your assumptions are 669 00:36:58,240 --> 00:37:00,450 so people can see them. 670 00:37:00,450 --> 00:37:03,350 And the idea is that when you do a proof, 671 00:37:03,350 --> 00:37:06,634 anybody who agrees with your assumptions or your axioms 672 00:37:06,634 --> 00:37:07,550 can follow your proof. 673 00:37:07,550 --> 00:37:09,660 And they have to agree with your conclusion. 674 00:37:09,660 --> 00:37:12,210 Now, they might disagree with your axioms, 675 00:37:12,210 --> 00:37:15,810 in which case, they're not going to buy your proof. 676 00:37:15,810 --> 00:37:19,020 Now, there are lots of axioms used in math. 677 00:37:19,020 --> 00:37:31,670 For example, if a equals b and b equals c, then a equals c. 678 00:37:31,670 --> 00:37:34,250 There is no proof of that. 679 00:37:34,250 --> 00:37:36,090 But it seems pretty good. 680 00:37:36,090 --> 00:37:41,580 And so we just throw it in the bucket of axioms and use it. 681 00:37:41,580 --> 00:37:47,210 Now, axioms can be contradictory in different contexts. 682 00:37:47,210 --> 00:37:49,340 Here's a good example. 683 00:37:49,340 --> 00:37:56,220 In Euclidean geometry, there's a central axiom that 684 00:37:56,220 --> 00:38:11,420 says given a line L and a point p not on L, 685 00:38:11,420 --> 00:38:22,090 there is exactly one line through p parallel 686 00:38:22,090 --> 00:38:31,280 to L. You all saw this in geometry in middle school, 687 00:38:31,280 --> 00:38:32,020 right? 688 00:38:32,020 --> 00:38:33,228 You've got a point in a line. 689 00:38:33,228 --> 00:38:35,160 There's exactly another line through the point 690 00:38:35,160 --> 00:38:38,220 that's parallel to the line. 691 00:38:38,220 --> 00:38:40,600 Now, there's also a field called spherical geometry. 692 00:38:44,870 --> 00:38:47,820 And there, you have an axiom that contradicts this. 693 00:38:47,820 --> 00:38:52,220 It says, given a line L and a point p not on L, 694 00:38:52,220 --> 00:39:01,840 there is no line through p parallel to L on the sphere. 695 00:39:01,840 --> 00:39:09,270 There's a field called hyperbolic geometry. 696 00:39:09,270 --> 00:39:11,190 And there, there's an axiom that says, 697 00:39:11,190 --> 00:39:13,800 given a line L and a point p not on L, 698 00:39:13,800 --> 00:39:26,632 there are infinitely many lines through p parallel to L. 699 00:39:26,632 --> 00:39:27,690 So how can this be? 700 00:39:27,690 --> 00:39:29,870 Does that mean one of these fields is totally bogus, 701 00:39:29,870 --> 00:39:31,620 or two of them are? 702 00:39:31,620 --> 00:39:34,470 Because they've got contradictory axes. 703 00:39:34,470 --> 00:39:35,860 That's OK. 704 00:39:35,860 --> 00:39:39,230 Just whatever field you're in, state you're axioms. 705 00:39:39,230 --> 00:39:42,610 And they do make sense in their various fields. 706 00:39:42,610 --> 00:39:43,794 This is planar geometry. 707 00:39:43,794 --> 00:39:44,710 This is on the sphere. 708 00:39:44,710 --> 00:39:47,880 And this is on hyperbolic geometry. 709 00:39:47,880 --> 00:39:49,990 They make sense in those contexts. 710 00:39:49,990 --> 00:39:54,750 So you can have more or less whatever axioms you want. 711 00:39:54,750 --> 00:39:57,070 There are sort of two guiding principles to axioms. 712 00:40:15,440 --> 00:40:22,305 Axioms should be-- it's called consistent-- and complete. 713 00:40:25,890 --> 00:40:39,880 Now, a set of axioms is consistent 714 00:40:39,880 --> 00:40:48,300 if no proposition can be proved to be both true and false. 715 00:40:55,190 --> 00:40:57,640 And you can see why that's important. 716 00:40:57,640 --> 00:41:02,120 If you spend three weeks proving something's true, 717 00:41:02,120 --> 00:41:04,880 and the next day, somebody proves it's also false, 718 00:41:04,880 --> 00:41:07,020 I mean, the whole thing was pointless. 719 00:41:07,020 --> 00:41:09,630 So it only makes sense if your axioms, as a group, 720 00:41:09,630 --> 00:41:12,140 are consistent. 721 00:41:12,140 --> 00:41:26,610 A set of axioms is said to be complete 722 00:41:26,610 --> 00:41:35,800 if it can be used to prove every proposition is 723 00:41:35,800 --> 00:41:37,060 either true or false. 724 00:41:48,560 --> 00:41:51,930 Now, this is desirable because it means-- well, 725 00:41:51,930 --> 00:41:54,450 you can solve every problem. 726 00:41:54,450 --> 00:41:56,314 Everything is-- you can prove it's true, 727 00:41:56,314 --> 00:41:57,480 or you can prove it's false. 728 00:41:57,480 --> 00:42:00,010 You can get to the end. 729 00:42:00,010 --> 00:42:05,280 Now, you'd think it shouldn't be too 730 00:42:05,280 --> 00:42:08,230 hard to get a set of axioms that satisfies these two 731 00:42:08,230 --> 00:42:09,730 basic properties. 732 00:42:09,730 --> 00:42:12,490 You're allowed to choose whatever you want, really. 733 00:42:12,490 --> 00:42:14,990 Just, you don't want to be creating contradictions. 734 00:42:14,990 --> 00:42:16,762 And you want a set that's powerful enough 735 00:42:16,762 --> 00:42:18,970 that allows you to prove everything is true or false, 736 00:42:18,970 --> 00:42:20,740 one of the two. 737 00:42:20,740 --> 00:42:23,350 Turned out not to be so easy to do this. 738 00:42:23,350 --> 00:42:27,800 And in fact, many logicians spent their careers-- 739 00:42:27,800 --> 00:42:29,700 famous logicians-- trying to find 740 00:42:29,700 --> 00:42:32,130 a set of axioms, just one set, that 741 00:42:32,130 --> 00:42:34,529 was consistent and complete. 742 00:42:34,529 --> 00:42:36,320 In fact, Russell and Whitehead are probably 743 00:42:36,320 --> 00:42:37,820 the two most famous. 744 00:42:37,820 --> 00:42:40,080 They spent their entire careers doing this, 745 00:42:40,080 --> 00:42:42,270 and they never got there. 746 00:42:42,270 --> 00:42:45,570 Then one day, this guy named Kurt Godel showed up. 747 00:42:45,570 --> 00:42:49,940 And in the 1930s, he proved it's not possible 748 00:42:49,940 --> 00:42:52,060 that there exists any set of axioms that are 749 00:42:52,060 --> 00:42:56,450 both consistent and complete. 750 00:42:56,450 --> 00:42:59,560 Now, this discovery devastated the field. 751 00:42:59,560 --> 00:43:02,270 It was a huge discovery. 752 00:43:02,270 --> 00:43:04,670 Imagine poor Russell and Whitehead. 753 00:43:04,670 --> 00:43:08,035 They spent their entire careers going after this holy grail. 754 00:43:08,035 --> 00:43:09,660 Then Kurt shows up and said, hey, guys. 755 00:43:09,660 --> 00:43:10,580 There's no grail. 756 00:43:10,580 --> 00:43:11,910 It doesn't exist. 757 00:43:11,910 --> 00:43:14,580 And that's a little depressing-- pretty bad day 758 00:43:14,580 --> 00:43:15,840 when that happened. 759 00:43:15,840 --> 00:43:18,000 Now, it's an amazing result, because it 760 00:43:18,000 --> 00:43:21,600 says if you want consistency-- and that's a must-- 761 00:43:21,600 --> 00:43:24,630 there will be true facts that you will never 762 00:43:24,630 --> 00:43:26,052 be able to prove. 763 00:43:26,052 --> 00:43:27,510 We're not going to prove that here. 764 00:43:27,510 --> 00:43:29,210 It's proved in a logic course. 765 00:43:29,210 --> 00:43:32,740 For example, maybe Goldbach's conjecture is true 766 00:43:32,740 --> 00:43:35,960 and it is impossible to prove. 767 00:43:35,960 --> 00:43:37,810 Now, we're going to try not to assign any 768 00:43:37,810 --> 00:43:40,210 of those problems for homework. 769 00:43:40,210 --> 00:43:42,182 And in fact, they do exist. 770 00:43:42,182 --> 00:43:42,890 It's complicated. 771 00:43:42,890 --> 00:43:47,432 You can state a problem that you can't prove is true or false. 772 00:43:47,432 --> 00:43:49,390 And you may be thinking that from time to time. 773 00:43:49,390 --> 00:43:50,348 Hey, it's one of those. 774 00:43:53,500 --> 00:43:57,140 Remember when your parents told you if you work hard enough, 775 00:43:57,140 --> 00:43:59,270 you can do anything? 776 00:43:59,270 --> 00:44:01,890 They were wrong. 777 00:44:01,890 --> 00:44:03,270 All right, that's enough for now. 778 00:44:03,270 --> 00:44:06,020 And we'll do more of this next time.