1 00:00:13,010 --> 00:00:15,690 Before we begin I'd like to draw your attention 2 00:00:15,690 --> 00:00:17,910 to the picture that's being projected. 3 00:00:17,910 --> 00:00:20,310 This picture was a tribute to Ada. 4 00:00:20,310 --> 00:00:24,930 It was created by the children of Botley Primary School's code club. 5 00:00:24,930 --> 00:00:26,810 It was arranged by Peter Lister. 6 00:00:26,810 --> 00:00:28,670 Thank you very much, Peter. 7 00:00:28,670 --> 00:00:33,110 Parents of a certain age will recognize that this was implemented in a system 8 00:00:33,110 --> 00:00:35,040 called Minecraft. 9 00:00:35,040 --> 00:00:40,870 Minecraft is a plot to steal the souls of all the 10-year-olds in the country and 10 00:00:40,870 --> 00:00:42,660 I speak from experience, but 11 00:00:42,660 --> 00:00:46,580 a lovely tribute by the children of Botley Primary School. 12 00:00:46,580 --> 00:00:48,185 So we have a couple of talks. 13 00:00:48,185 --> 00:00:51,990 >> [APPLAUSE] >> We have a couple of talks this 14 00:00:51,990 --> 00:00:53,130 afternoon. 15 00:00:53,130 --> 00:00:56,228 Firstly, it's my pleasure to introduce all the way from California, 16 00:00:56,228 --> 00:00:58,180 Judith Grabiner from Pitzer College. 17 00:00:58,180 --> 00:01:00,320 She's gonna talk about mathematics in culture. 18 00:01:00,320 --> 00:01:01,612 Judith. >> Thank you. 19 00:01:01,612 --> 00:01:07,850 I wanna thank Ursula for inviting me and asking me to speak about mathematics in 20 00:01:07,850 --> 00:01:13,790 culture, and secondly I want to start by saying there will be an introduction. 21 00:01:13,790 --> 00:01:19,961 So where's the device that advances the slides? 22 00:01:19,961 --> 00:01:21,141 This is it. 23 00:01:21,141 --> 00:01:23,321 Okay, this is it. 24 00:01:23,321 --> 00:01:24,441 Okay, good. 25 00:01:24,441 --> 00:01:28,875 >> [LAUGH] All right I want to argue today that geometry 26 00:01:28,875 --> 00:01:36,330 interacts with all aspects of human thought and life. 27 00:01:36,330 --> 00:01:40,550 Now, every mathematician has some sort of relationship with Euclid, and 28 00:01:40,550 --> 00:01:46,270 in my token reference to Ada Lovelace, she began her studies of mathematics 29 00:01:46,270 --> 00:01:51,620 learning geometry and told her tutor that she could only understand the proposition, 30 00:01:51,620 --> 00:01:54,150 when she could see the figure in the air. 31 00:01:54,150 --> 00:01:59,380 Now that's a very good line that could also serves for me as a transition. 32 00:01:59,380 --> 00:02:03,340 Because geometry is long been thought of as the science of space. 33 00:02:03,340 --> 00:02:06,855 So what is space? 34 00:02:06,855 --> 00:02:10,440 Does space exist? 35 00:02:10,440 --> 00:02:12,040 What's its nature? 36 00:02:12,040 --> 00:02:16,540 The history of these questions involves a lot of important thinkers, and 37 00:02:16,540 --> 00:02:20,370 artists, and really the entire universe and that's what I want to do today. 38 00:02:20,370 --> 00:02:25,320 Now Euclid is special. 39 00:02:25,320 --> 00:02:30,953 When we use terms like truth, and proof, we're channeling Euclid. 40 00:02:30,953 --> 00:02:35,417 For 2,000 years, Euclid's geometry seemed to prove truths about 41 00:02:35,417 --> 00:02:41,870 geometrical objects and thereby to achieve certainty. 42 00:02:41,870 --> 00:02:45,117 Now, Plato thought that geometry was true and 43 00:02:45,117 --> 00:02:50,281 certain because the subject matter was eternal and helped to draw the soul 44 00:02:50,281 --> 00:02:55,130 from the imperfect world of change to the changeless eternal truth. 45 00:02:55,130 --> 00:02:57,520 But Aristotle said, no, the truth and 46 00:02:57,520 --> 00:03:03,150 certainty of geometry comes from the way it's put together, using logical 47 00:03:03,150 --> 00:03:08,010 deductions from explicit self-evident assumptions and clear definitions. 48 00:03:08,010 --> 00:03:13,000 Euclid probably looked over his shoulder at Aristotle's criteria 49 00:03:13,000 --> 00:03:15,320 while he was writing The Elements. 50 00:03:15,320 --> 00:03:19,890 And many important later thinkers believed that other subjects might come to 51 00:03:19,890 --> 00:03:26,120 share the certainty of geometry if only they followed the same method. 52 00:03:26,120 --> 00:03:31,270 For instance, Descartes said, you know, if we start with self-evident truths and 53 00:03:31,270 --> 00:03:35,800 then proceed by logically deducing more and more complex truths from these. 54 00:03:35,800 --> 00:03:39,996 Why, he says, in effect there's nothing we couldn't come to know. 55 00:03:39,996 --> 00:03:44,089 Wow. 56 00:03:44,089 --> 00:03:47,646 Following Descartes, Benedict Espinoza wrote a book, 57 00:03:47,646 --> 00:03:51,270 On Ethics Demonstrated in Geometrical Order. 58 00:03:51,270 --> 00:03:54,290 With explicitly labeled axioms and definitions, and 59 00:03:54,290 --> 00:03:59,310 including theorems like this one quote God, or a substance consisting 60 00:03:59,310 --> 00:04:06,020 of infinite attributes, necessarily exists, end quote. 61 00:04:06,020 --> 00:04:08,815 He has several proofs, he closes them with a QED. 62 00:04:08,815 --> 00:04:12,660 >> [LAUGH] >> Now the influence of the Euclidean 63 00:04:12,660 --> 00:04:16,940 ideal on science is clear 64 00:04:16,940 --> 00:04:21,320 from Isaac Newton's Mathematical Principles of Natural Philosophy. 65 00:04:21,320 --> 00:04:25,920 As you see here, Newton called his famous three laws of motion axioms, and 66 00:04:25,920 --> 00:04:32,220 he deduced even his law of gravity in the form of explicitly stated theorems. 67 00:04:32,220 --> 00:04:37,400 And Newton famously said, it is the glory of geometry that from so 68 00:04:37,400 --> 00:04:41,410 few principles it can accomplish so much. 69 00:04:41,410 --> 00:04:43,430 One more example. 70 00:04:43,430 --> 00:04:46,900 The American declaration of Independence is an argument 71 00:04:46,900 --> 00:04:52,230 whose author tried to inspire faith in its certainty by using the Euclidean form. 72 00:04:52,230 --> 00:04:56,000 Thomas Jefferson, who knew more of the mathematics of his time 73 00:04:56,000 --> 00:04:59,905 than any other American president, began his argument thus. 74 00:04:59,905 --> 00:05:03,503 >> [LAUGH] >> See, in the United States, 75 00:05:03,503 --> 00:05:04,727 people wouldn't laugh at that. 76 00:05:04,727 --> 00:05:11,124 >> [LAUGH] >> Jefferson began, 77 00:05:11,124 --> 00:05:14,975 we hold these truths to be self evident, that all right angles are equal. 78 00:05:14,975 --> 00:05:16,420 >> [LAUGH] >> Oh, oh no! 79 00:05:16,420 --> 00:05:21,570 That's not what he said, but that's what it sounds like. 80 00:05:21,570 --> 00:05:24,880 Another self evident truth in the Declaration is that if any 81 00:05:24,880 --> 00:05:27,710 form of government fails to secure human rights, 82 00:05:27,710 --> 00:05:30,318 it is the right of the people to get rid of it and set up a new government. 83 00:05:30,318 --> 00:05:35,660 The Declaration asserts that King George's government does not secure these rights. 84 00:05:35,660 --> 00:05:41,620 Then it says to prove this, let facts be submitted to a candid world. 85 00:05:41,620 --> 00:05:45,050 And once the facts have proved this, then we've got if p then q. 86 00:05:45,050 --> 00:05:51,340 p, and therefore, q. 87 00:05:51,340 --> 00:05:51,845 Thank you. 88 00:05:51,845 --> 00:05:54,030 >> [LAUGH] >> Computer scientists, I mean, 89 00:05:54,030 --> 00:05:55,275 you're supposed to know this. 90 00:05:55,275 --> 00:05:56,760 >> [LAUGH] >> All right, 91 00:05:56,760 --> 00:05:59,670 once the facts have proved this, the actual Declaration that founded 92 00:05:59,670 --> 00:06:06,120 the United States is stated explicitly as a conclusion of a logical argument. 93 00:06:06,120 --> 00:06:10,550 Beginning with a therefore, which, in one of the original printings, 94 00:06:10,550 --> 00:06:13,990 is italicized, as I have it here. 95 00:06:13,990 --> 00:06:19,080 So in philosophy, theology, science, politics, 96 00:06:19,080 --> 00:06:23,558 the idealized Euclidean model of reasoning has shaped conceptions of proof, 97 00:06:23,558 --> 00:06:26,810 truth and certainty. 98 00:06:26,810 --> 00:06:28,085 Okay, that was the introduction. 99 00:06:28,085 --> 00:06:29,310 >> [LAUGH] >> Now, for 100 00:06:29,310 --> 00:06:33,010 the first half of the actual talk, we'll look at Euclid's Elements, 101 00:06:33,010 --> 00:06:36,870 at the idea of Euclidean space, and 102 00:06:36,870 --> 00:06:40,840 some important related ideas and their wide ranging influence. 103 00:06:40,840 --> 00:06:44,360 And we'll see that it lives up to its press notices. 104 00:06:44,360 --> 00:06:46,310 But we'll also see that there was a problem. 105 00:06:46,310 --> 00:06:50,390 And in the second half of the talk, we'll see that the problem and 106 00:06:50,390 --> 00:06:54,700 its resolution have produced a non-Euclidean world, and 107 00:06:54,700 --> 00:06:59,070 that this break with the past has had a comparably great impact. 108 00:06:59,070 --> 00:07:02,850 Okay, so here we go with part one. 109 00:07:02,850 --> 00:07:05,320 These are Euclid's postulates. 110 00:07:05,320 --> 00:07:09,120 Hopefully the minimal set of self-evident truths from which 111 00:07:09,120 --> 00:07:13,830 all the true results of geometry can be logically deduced. 112 00:07:13,830 --> 00:07:17,020 Well postulates one through four are pretty straightforward. 113 00:07:17,020 --> 00:07:20,682 I hope no one wants to argue with them, but look at number five. 114 00:07:20,682 --> 00:07:23,240 Euclid's so-called parallel postulate 115 00:07:23,240 --> 00:07:27,270 which you will observe does not mention parallels at all. 116 00:07:27,270 --> 00:07:30,670 That one's complicated, so I think we'd better draw a picture, 117 00:07:30,670 --> 00:07:34,740 sort of retain in your mind how that struck you when you saw it. 118 00:07:34,740 --> 00:07:36,341 Here's a picture. 119 00:07:36,341 --> 00:07:40,159 What postulate five means is if the two green lines are cut by a third line, 120 00:07:40,159 --> 00:07:43,729 AB, such that angle A and angle B add up to less than two right angles, 121 00:07:43,729 --> 00:07:47,780 then the two green lines eventually meet on that side. 122 00:07:47,780 --> 00:07:52,750 Now I ask my students to vote on whether this postulate is self-evident. 123 00:07:52,750 --> 00:07:55,110 And they overwhelmingly vote that it is not. 124 00:07:55,110 --> 00:07:58,275 Some of them don't understand it after two runs through. 125 00:07:58,275 --> 00:08:01,830 >> [LAUGH] >> And you gotta draw a picture. 126 00:08:01,830 --> 00:08:05,500 The ancient Greeks agree with my students. 127 00:08:05,500 --> 00:08:07,904 And that is the problem that I mentioned. 128 00:08:07,904 --> 00:08:10,872 Okay well if this postulate was not self-evident, 129 00:08:10,872 --> 00:08:13,980 then maybe it can be proved from the other postulates. 130 00:08:13,980 --> 00:08:16,588 So the Greeks tried to do this For 131 00:08:16,588 --> 00:08:21,670 reasons that the mathematically literate will recognize. 132 00:08:21,670 --> 00:08:23,960 They did not succeed. 133 00:08:23,960 --> 00:08:29,070 And so, also trying did the best mathematicians in the Medieval Islamic and 134 00:08:29,070 --> 00:08:32,800 Jewish worlds. 135 00:08:32,800 --> 00:08:37,840 Them mathematicians in Christian Europe on into the 19th century. 136 00:08:37,840 --> 00:08:39,960 Including John Wallace, 137 00:08:39,960 --> 00:08:46,390 whose views on this subject are actually in a poster out in the lobby. 138 00:08:46,390 --> 00:08:48,710 On into the 19th century. 139 00:08:48,710 --> 00:08:51,550 Pretty important famous people. 140 00:08:51,550 --> 00:08:57,520 Now, while trying to prove postulate five, one thing that the Greeks proved 141 00:08:57,520 --> 00:09:01,480 right at the beginning is the Euclid's postulate is logically equivalent. 142 00:09:01,480 --> 00:09:04,490 The uniqueness of parallel lines. 143 00:09:04,490 --> 00:09:09,270 That is, given a line, then a given point in the same plane, 144 00:09:09,270 --> 00:09:13,030 there is only one parallel through that point to that line. 145 00:09:13,030 --> 00:09:17,360 It's a common alternative postulate found in many textbooks, including the one by 146 00:09:17,360 --> 00:09:23,250 John Playfair in the 18th century, hence it's called Playfair's axiom if you care. 147 00:09:23,250 --> 00:09:27,850 Euclid defined parallel lines as lines in the same plain that never meet. 148 00:09:27,850 --> 00:09:29,510 From his first four postulates, 149 00:09:29,510 --> 00:09:34,330 he can prove a lot of cool stuff, like that parallel lines can be constructed. 150 00:09:34,330 --> 00:09:40,580 But he needs that problematic 5th postulate, which he uses very rarely. 151 00:09:40,580 --> 00:09:43,880 For his proof of one key theorem. 152 00:09:43,880 --> 00:09:47,210 If two parallel lines are cut by a third line, 153 00:09:47,210 --> 00:09:50,560 the alternate interior angles, like angle 3 and angle 6, 154 00:09:50,560 --> 00:09:55,750 and the corresponding angles, like angle 2 and angle 6, are equal. 155 00:09:55,750 --> 00:10:00,070 From that theorem you could peruse not only the uniqueness of parallels but 156 00:10:00,070 --> 00:10:03,020 many other theorems that involve parallel lines 157 00:10:03,020 --> 00:10:04,680 like that they're every where equi distant. 158 00:10:04,680 --> 00:10:08,430 We expect this of decent parallel lines, or 159 00:10:08,430 --> 00:10:12,370 that the sum of the angles to a triangle is to right angles. 160 00:10:12,370 --> 00:10:16,465 But Euclid doesn't talk about space. 161 00:10:16,465 --> 00:10:20,138 Nevertheless starting in the Renaissance people talk a lot about space. 162 00:10:20,138 --> 00:10:24,420 An infinite homogeneous space that's the same in all directions 163 00:10:24,420 --> 00:10:27,910 in which every point is just like every other point. 164 00:10:27,910 --> 00:10:31,830 This is the space that Euclid's geometry happens in, and 165 00:10:31,830 --> 00:10:37,080 the role of philosophy in describing that space turns out to be really huge. 166 00:10:37,080 --> 00:10:42,160 The idea that space must be the same in all directions comes from what is called 167 00:10:42,160 --> 00:10:45,480 the principle of sufficient reason. 168 00:10:45,480 --> 00:10:48,850 For everything that is, there's a reason why it is as it is and not otherwise. 169 00:10:48,850 --> 00:10:51,790 That sounds pretty trivial. 170 00:10:51,790 --> 00:10:53,720 Turns out, not to be quite so trivial. 171 00:10:53,720 --> 00:11:00,350 Principles at least as old as Archimedes, here's what Archimedes noticed with it. 172 00:11:00,350 --> 00:11:02,815 Have a lever with equal weights and 173 00:11:02,815 --> 00:11:06,558 equal distances from the fulcrum it has to balance. 174 00:11:06,558 --> 00:11:09,332 Why? 175 00:11:09,332 --> 00:11:10,040 Why not? 176 00:11:10,040 --> 00:11:14,278 >> [LAUGH] >> Because there's no reason for it to go 177 00:11:14,278 --> 00:11:18,851 down to one side or the other because the sides are completely equivalent and 178 00:11:18,851 --> 00:11:22,520 that's why the symmetric lever necessarily misbalanced. 179 00:11:22,520 --> 00:11:26,368 Furthermore, this principle of sufficient reason, explains why space is infinite. 180 00:11:26,368 --> 00:11:30,730 [INAUDIBLE] Bruno said, space must be infinite, 181 00:11:30,730 --> 00:11:34,460 because there's no reason for it to stop, at any particular point. 182 00:11:34,460 --> 00:11:37,030 All the points are the same. 183 00:11:37,030 --> 00:11:40,850 The greatest advocate of the principle of sufficient reason, was Liebniz. 184 00:11:40,850 --> 00:11:44,750 Leibniz said it's the principle of sufficient reason and 185 00:11:44,750 --> 00:11:48,810 the laws of logic that God used in making the universe. 186 00:11:48,810 --> 00:11:53,250 God made the universe optimal in the best possible way. 187 00:11:53,250 --> 00:11:57,510 Many scientific laws still use the language of optimization, 188 00:11:57,510 --> 00:12:01,788 like that light is refracted in the least possible time. 189 00:12:01,788 --> 00:12:06,320 Now for Leibniz sufficient reason gives us the simplest 190 00:12:06,320 --> 00:12:11,680 scientific laws and the best possible laws and universe. 191 00:12:11,680 --> 00:12:14,420 And if the principle of sufficient reason is true 192 00:12:14,420 --> 00:12:16,880 the universe is transparent to reason. 193 00:12:16,880 --> 00:12:20,350 God made it rationally so we humans can figure it out and 194 00:12:20,350 --> 00:12:24,350 here is one striking example. 195 00:12:24,350 --> 00:12:28,900 Consider Newton's first law, which was first formulated 50 years before Newton, 196 00:12:28,900 --> 00:12:31,560 independently by Descartes and Pierre Gassendi. 197 00:12:31,560 --> 00:12:33,270 And here's how they found it. 198 00:12:33,270 --> 00:12:37,510 The first law of motion says, a body with no forces acting on it, 199 00:12:37,510 --> 00:12:39,810 continues in a straight line at a constant speed. 200 00:12:39,810 --> 00:12:40,830 Why? 201 00:12:40,830 --> 00:12:43,870 Well, it goes in a straight line because every direction is the same, 202 00:12:43,870 --> 00:12:47,250 so there's no reason for it to turn to the left or to the right. 203 00:12:47,250 --> 00:12:51,200 It goes at a constant speed because every point's the same, so there's no reason for 204 00:12:51,200 --> 00:12:55,480 it to prefer this point to this point and speed up to get there. 205 00:12:55,480 --> 00:12:59,530 And you can see that the same argument can be made 206 00:12:59,530 --> 00:13:05,610 why a body that is at rest with no forces acting on it stays where it is. 207 00:13:05,610 --> 00:13:11,620 Okay, now back to parallel lines. 208 00:13:11,620 --> 00:13:14,610 The great mathematician Joseph-Louis Lagrange, 209 00:13:14,610 --> 00:13:18,250 decided that he could prove the uniqueness of parallels. 210 00:13:18,250 --> 00:13:21,180 Logically equivalent to Euclid's 5th postulate 211 00:13:21,180 --> 00:13:24,300 by using the principle of sufficient reason. 212 00:13:24,300 --> 00:13:26,890 Okay? 213 00:13:26,890 --> 00:13:32,330 Well, this is what he's trying to prove, the uniqueness of parallels. 214 00:13:32,330 --> 00:13:36,710 Well wait a minute, who says you can use the principles of sufficient reason, 215 00:13:36,710 --> 00:13:38,870 in a mathematical proof? 216 00:13:38,870 --> 00:13:41,020 Lagrange says so. 217 00:13:41,020 --> 00:13:45,180 He says it is just as obvious and true as are the laws of logic, and 218 00:13:45,180 --> 00:13:50,570 I hope I've convinced in that little brief thing I said that it was all over since 219 00:13:50,570 --> 00:13:52,820 in the 18th century principle sufficient reason. 220 00:13:52,820 --> 00:13:55,510 I found this manuscript in which Lagrange does this. 221 00:13:55,510 --> 00:13:57,470 I think it's really, really interesting. 222 00:13:57,470 --> 00:13:58,920 He's going to prove that theorem. 223 00:13:58,920 --> 00:14:01,990 And it's a proof by contradiction. 224 00:14:01,990 --> 00:14:04,090 Given that we have one parallel through point P, 225 00:14:04,090 --> 00:14:08,200 which we have here, suppose there were another parallel. 226 00:14:08,200 --> 00:14:12,730 The original line through point P might look like this. 227 00:14:12,730 --> 00:14:17,260 But by the principle of sufficient reason, there's no reason that the new parallel 228 00:14:17,260 --> 00:14:20,820 line should be below the point P on the left and above it on the right. 229 00:14:20,820 --> 00:14:23,630 It could equally well be drawn the other way. 230 00:14:23,630 --> 00:14:28,040 So the principle of sufficient reason says, there must be another symmetric 231 00:14:28,040 --> 00:14:32,870 parallel line that goes the other way as in this new diagram. 232 00:14:32,870 --> 00:14:35,880 Okay, Lagrange repeats this exact same argument for 233 00:14:35,880 --> 00:14:38,370 lines symmetric to his new parallels. 234 00:14:38,370 --> 00:14:41,780 So for instance you get a third line on the other side of his new parallel, and 235 00:14:41,780 --> 00:14:44,050 it looks like this. 236 00:14:44,050 --> 00:14:48,380 And so on, lots and lots more, and I will show you Lagrange's own picture, 237 00:14:48,380 --> 00:14:52,520 of the final situation, from his manuscript. 238 00:14:52,520 --> 00:14:54,625 Which Lagrange says is absurd. 239 00:14:54,625 --> 00:14:57,100 >> [LAUGH] >> So he concludes, 240 00:14:57,100 --> 00:14:58,818 there can only be one parallel, QED. 241 00:14:58,818 --> 00:15:00,572 >> [LAUGH] >> Now, 242 00:15:00,572 --> 00:15:03,620 Lagrange never published this manuscript. 243 00:15:03,620 --> 00:15:06,670 Maybe he came to see that when he used the principle of sufficient reason at 244 00:15:06,670 --> 00:15:11,230 the beginning to produce the new first set of new symmetric parallels, 245 00:15:11,230 --> 00:15:15,520 that he was assuming that, this one here, 246 00:15:15,520 --> 00:15:19,110 he was assuming that the two parallel lines were everywhere equidistant and 247 00:15:19,110 --> 00:15:23,010 thus assuming the very Euclidian nature space that he was trying to prove. 248 00:15:23,010 --> 00:15:25,770 But the fact that a great mathematician like Lagrange got up 249 00:15:25,770 --> 00:15:29,610 in the Public Institute of France and linked space being Euclidian 250 00:15:29,610 --> 00:15:33,800 with Leibniz's principal of sufficient reason demonstrates how closely linked 251 00:15:33,800 --> 00:15:39,470 sufficient reason was with the necessity of space being Euclidian. 252 00:15:39,470 --> 00:15:40,210 Oh yes. 253 00:15:40,210 --> 00:15:42,810 And space has to be real. 254 00:15:42,810 --> 00:15:44,370 Newton insisted on that. 255 00:15:44,370 --> 00:15:45,520 Why? 256 00:15:45,520 --> 00:15:48,420 So he could distinguish real accelerations, 257 00:15:48,420 --> 00:15:53,890 that is accelerations with respect to space, from the parent accelerations. 258 00:15:53,890 --> 00:15:57,470 Real accelerations involve real forces. 259 00:15:57,470 --> 00:16:02,204 So from real acceleration, like a falling apple or 260 00:16:02,204 --> 00:16:08,260 a planet orbiting the sun, real acceleration require forces and 261 00:16:08,260 --> 00:16:12,460 thus he can show that gravity is a real force. 262 00:16:12,460 --> 00:16:17,175 Furthermore, that real space for Newton's physics turns out to be Euclidean. 263 00:16:17,175 --> 00:16:18,560 Why? 264 00:16:18,560 --> 00:16:23,301 Well, one reason is that you have to use parallelograms of force all over 265 00:16:23,301 --> 00:16:24,960 Newton's physics. 266 00:16:24,960 --> 00:16:27,320 And proving the properties of parallelograms, 267 00:16:27,320 --> 00:16:32,110 requires Euclid's theory of parallels, and therefore, postulate five. 268 00:16:32,110 --> 00:16:33,870 Well okay, that's Newton. 269 00:16:33,870 --> 00:16:34,625 There's another guy. 270 00:16:34,625 --> 00:16:37,364 Oops, wrong guy. 271 00:16:37,364 --> 00:16:39,520 All right, I'm sorry. 272 00:16:39,520 --> 00:16:41,870 I thought I had a picture of Leibniz again, but I didn't. 273 00:16:41,870 --> 00:16:44,370 Leibniz disagreed with Newton about space. 274 00:16:44,370 --> 00:16:47,450 Leibniz essentially said there is no such thing. 275 00:16:47,450 --> 00:16:51,900 Space is just the relations between bodies. 276 00:16:51,900 --> 00:16:54,950 But, Newton's view prevailed? 277 00:16:54,950 --> 00:16:57,370 Why? 278 00:16:57,370 --> 00:16:59,460 Listen to Leonhard Euler. 279 00:16:59,460 --> 00:17:03,050 Euler says the straight line constant speed motion of a single body with no 280 00:17:03,050 --> 00:17:07,720 forces acting on it, according to Newton's first law, that can't possibly 281 00:17:07,720 --> 00:17:12,530 depend on where other physical bodies just happen to be at some particular time. 282 00:17:12,530 --> 00:17:16,900 It's gotta be a straight line always, therefore a straight line not with where 283 00:17:16,900 --> 00:17:20,470 all the bodies happen to be right this second but 284 00:17:20,470 --> 00:17:23,860 with respect to something that doesn't change namely space. 285 00:17:23,860 --> 00:17:28,310 So space has to be real and Euclidian. 286 00:17:28,310 --> 00:17:32,270 Why didn't mathematicians of the 18th century want so 287 00:17:32,270 --> 00:17:34,650 much to prove the parallel postulate? 288 00:17:34,650 --> 00:17:37,050 Well precisely, because it was so 289 00:17:37,050 --> 00:17:42,830 important in establishing the correct nature of space. 290 00:17:42,830 --> 00:17:46,490 Not for just for geometry, but for all of science. 291 00:17:46,490 --> 00:17:49,520 Rested upon it. 292 00:17:49,520 --> 00:17:54,900 And now, to philosophy. 293 00:17:54,900 --> 00:17:58,470 A super influential philosopher, Immanuel Kant, agreed with Newton 294 00:17:58,470 --> 00:18:02,550 that space exists, agreed that we have to order our perceptions in space. 295 00:18:02,550 --> 00:18:05,880 But Kant said space exists in our minds. 296 00:18:05,880 --> 00:18:09,920 And we each have the same unique space in our minds. 297 00:18:09,920 --> 00:18:14,650 And it turns out for Kant that this space too has to be Euclidean and 298 00:18:14,650 --> 00:18:18,790 let me explain because he obviously never says this because it never occurs to him 299 00:18:18,790 --> 00:18:21,314 that there is a non-Euclidean alternative. 300 00:18:21,314 --> 00:18:25,968 All right, to argue Kant argued that we can come to no non-trivial truths 301 00:18:25,968 --> 00:18:30,944 about non-material things, technically synthetic [INAUDIBLE] judgments. 302 00:18:30,944 --> 00:18:34,990 Kant uses Euclid's proof that the sum of the angles of the triangle 303 00:18:34,990 --> 00:18:36,770 is two right angles. 304 00:18:36,770 --> 00:18:39,710 If it's been a long time since you did this, 305 00:18:39,710 --> 00:18:41,450 I will remind you how Euclid's proof goes. 306 00:18:41,450 --> 00:18:45,260 You start with a triangle. 307 00:18:45,260 --> 00:18:47,490 Well you can't add the angles. 308 00:18:47,490 --> 00:18:50,500 They're in different places. 309 00:18:50,500 --> 00:18:54,220 So to do the proof, first thing you do is you extend the base to D, 310 00:18:54,220 --> 00:18:58,550 and then you construct a line to the corner parallel to the opposite side. 311 00:18:58,550 --> 00:19:01,140 So CE is parallel to AB. 312 00:19:01,140 --> 00:19:05,530 And then you prove the equality of various angles in the new diagram. 313 00:19:05,530 --> 00:19:12,110 To the angles around C, because the angles around C obviously add up to right angles. 314 00:19:12,110 --> 00:19:14,130 This is the proof. 315 00:19:14,130 --> 00:19:16,540 The equal angles are specified in this slide. 316 00:19:16,540 --> 00:19:19,130 But it isn't the details of the proof that matter. 317 00:19:19,130 --> 00:19:21,010 Kant's keep point is this. 318 00:19:21,010 --> 00:19:27,200 The proof cannot work unless and until you make those constructions. 319 00:19:27,200 --> 00:19:28,210 Okay. 320 00:19:28,210 --> 00:19:31,300 Well, where'd you make those constructions. 321 00:19:31,300 --> 00:19:32,560 Not on paper. 322 00:19:32,560 --> 00:19:35,730 Geometry isn't about physical angles. 323 00:19:35,730 --> 00:19:37,720 You made them in space he said. 324 00:19:37,720 --> 00:19:41,090 Space in your mind. 325 00:19:41,090 --> 00:19:42,890 So, space. 326 00:19:42,890 --> 00:19:46,610 Now the proof needs the results that if two lines are parallel, 327 00:19:46,610 --> 00:19:49,030 the alternate interior. 328 00:19:49,030 --> 00:19:52,410 And corresponding angles are equal. 329 00:19:52,410 --> 00:19:53,452 As I said before, 330 00:19:53,452 --> 00:19:58,230 Euclid's proof of that theorem explicitly requires the fifth postulate. 331 00:19:58,230 --> 00:20:02,882 So this theorem about the sum of the angles requires space to be Euclidian, 332 00:20:02,882 --> 00:20:04,970 now Kant doesn't say this, but 333 00:20:04,970 --> 00:20:08,350 he does say there's only one space just as there's only one time. 334 00:20:08,350 --> 00:20:13,110 So for Kant, no alternative to Euclid seems conceivable. 335 00:20:13,110 --> 00:20:14,120 Well, Kant's pretty heavy. 336 00:20:14,120 --> 00:20:17,300 Let's pick somebody lighter. 337 00:20:17,300 --> 00:20:18,790 Oops. 338 00:20:18,790 --> 00:20:20,640 There you go. 339 00:20:20,640 --> 00:20:24,530 One more philosopher witness to Euclidian space as truth. 340 00:20:24,530 --> 00:20:28,120 Voltaire shared the widespread 18th century 341 00:20:28,120 --> 00:20:31,790 idea that universal agreement was a marker for truth. 342 00:20:31,790 --> 00:20:34,450 And he said of Voltaire. 343 00:20:34,450 --> 00:20:37,420 There are no sects in geometry. 344 00:20:37,420 --> 00:20:40,830 One doesn't say, I'm a Euclidean. 345 00:20:40,830 --> 00:20:44,740 Demonstrate the truth, he says, and the whole world will be of your opinion. 346 00:20:44,740 --> 00:20:49,600 Mathematics exemplifies this for Voltaire, and he wants ethics to do it too. 347 00:20:49,600 --> 00:20:53,478 Voltaire wrote in my favorite Voltaire quotation, 348 00:20:53,478 --> 00:20:58,185 quote, there is but one morality as there is but one geometry. 349 00:20:58,185 --> 00:21:01,120 >> [LAUGH] >> And I said that in class once and 350 00:21:01,120 --> 00:21:05,065 one of my students shot back wrong on both points Voltaire. 351 00:21:05,065 --> 00:21:09,455 >> [LAUGH] >> The art and architecture 352 00:21:09,455 --> 00:21:14,765 of the early modern period also reflect the Euclidian nature of space. 353 00:21:14,765 --> 00:21:18,915 Art and architecture reinforce the Euclidian intuition of space in the minds 354 00:21:18,915 --> 00:21:22,215 of everybody who look at the paintings, and lived and worked in the buildings and 355 00:21:22,215 --> 00:21:25,930 public squares, from the Renaissance to well into the 19th century. 356 00:21:25,930 --> 00:21:30,480 Perspective in art which is based on Euclid's geometry 357 00:21:30,480 --> 00:21:33,830 helped make people literally see space as Euclidian. 358 00:21:33,830 --> 00:21:37,030 And let me illustrate a bit. 359 00:21:37,030 --> 00:21:40,240 One of the most important early perspective paintings of the Renaissance 360 00:21:40,240 --> 00:21:42,860 here, the Trinity by Masaccio, 361 00:21:42,860 --> 00:21:47,940 We are used to two dimensional pictures that look three dimensional. 362 00:21:47,940 --> 00:21:50,660 Cuz we've got photography and television and iPhones and 363 00:21:50,660 --> 00:21:52,040 all sorts of wonderful things like that. 364 00:21:52,040 --> 00:21:53,840 In the Renaissance they didn't. 365 00:21:53,840 --> 00:21:57,330 So a painting like this as the quotation from Vasari makes clear, 366 00:21:57,330 --> 00:22:00,150 was incredibly exciting to them and the realistic 367 00:22:00,150 --> 00:22:05,220 illusion of depth in Renaissance art comes explicitly from geometry. 368 00:22:05,220 --> 00:22:08,890 Useful briefly to look at a couple Medieval works of art to appreciate 369 00:22:08,890 --> 00:22:11,750 the difference between Medieval and Renaissance. 370 00:22:11,750 --> 00:22:14,840 Picked a couple things I like. 371 00:22:14,840 --> 00:22:20,450 The Bayotuss tapestry, the people are the same size as the castle. 372 00:22:20,450 --> 00:22:24,010 This is Nicholas Verdun, The Crossing of the Red Sea, 373 00:22:24,010 --> 00:22:28,820 well the people are as big as the whole Red Sea Which is also red. 374 00:22:28,820 --> 00:22:33,960 This is wonderful art, but there's no convincing three-dimensionality. 375 00:22:33,960 --> 00:22:39,130 Now on the Renaissance we're going to move into a different kind of space. 376 00:22:39,130 --> 00:22:44,150 A pretty dramatic contrast. 377 00:22:44,150 --> 00:22:48,780 Okay, the geometry used in creating Renaissance art is literally Euclidian. 378 00:22:48,780 --> 00:22:52,500 It comes from Euclid's elements of geometry and Euclid's optics. 379 00:22:52,500 --> 00:22:57,630 Here's a theorem from Euclid's optics not Euclid's diagram. 380 00:22:57,630 --> 00:23:00,737 I think you see the truth of the result and 381 00:23:00,737 --> 00:23:06,760 here is something from Euclid's elements. 382 00:23:06,760 --> 00:23:10,450 If a straight line is drawn parallel to the base of a triangle it cuts the other 383 00:23:10,450 --> 00:23:11,990 two sides proportionally. 384 00:23:11,990 --> 00:23:14,910 That theorem is just essential 385 00:23:14,910 --> 00:23:19,660 to the Renaissance geometric theory of perspective. 386 00:23:19,660 --> 00:23:22,970 Here's how the geometry lets you see three dimensional reality on a flat canvas. 387 00:23:22,970 --> 00:23:26,060 This is known as the Alberti construction. 388 00:23:26,060 --> 00:23:30,860 What you've got there is a square floor divided into squares and 389 00:23:30,860 --> 00:23:38,790 the plane of that floor is perpendicular to the plane of, in this case, the screen. 390 00:23:38,790 --> 00:23:41,750 This photograph does the Alberti construction for 391 00:23:41,750 --> 00:23:45,320 two parallel railroad tracks, perpendicular to the plane of the screen. 392 00:23:45,320 --> 00:23:47,760 Now the checker board construction of Alberti's. 393 00:23:47,760 --> 00:23:51,390 Let's take a look at it again, checker board, okay. 394 00:23:51,390 --> 00:23:56,687 Was used by many Renaissance painters because you have to square it off 395 00:23:56,687 --> 00:24:02,900 pavement and it emphasizes the perspective and also helps the artist's draw it. 396 00:24:02,900 --> 00:24:08,737 So, this is supposedly a religious picture about Christ but 397 00:24:08,737 --> 00:24:13,438 I look at it as an application or a checker board. 398 00:24:13,438 --> 00:24:18,306 Here's another artist you've probably heard of Doing the same thing, 399 00:24:18,306 --> 00:24:21,690 checkerboard here is on top. 400 00:24:21,690 --> 00:24:25,453 These are artists who are also mathematicians, Piero, and 401 00:24:25,453 --> 00:24:27,530 Leonardo DaVinci. 402 00:24:27,530 --> 00:24:32,273 Moving north from Italy into Germany, here's another mathematician artist, 403 00:24:32,273 --> 00:24:35,841 Arbuteur showing you precisely how to construct the image. 404 00:24:35,841 --> 00:24:38,600 Well okay, you're a renaissance artist. 405 00:24:38,600 --> 00:24:41,740 Do you have to master all this mathematical theory to draw on 406 00:24:41,740 --> 00:24:42,780 three dimensions? 407 00:24:42,780 --> 00:24:47,600 Not at all, just find the right books. 408 00:24:47,600 --> 00:24:50,870 Schern is a student of Durer's and 409 00:24:50,870 --> 00:24:55,690 he says, just look at my drawing and you'll know what to do. 410 00:24:55,690 --> 00:24:59,890 Here's another German I like this picture by Count Johann 411 00:24:59,890 --> 00:25:04,960 because it literally embodies Euclid's theorem about 412 00:25:04,960 --> 00:25:08,710 disproportional sides of similar triangles. 413 00:25:08,710 --> 00:25:13,270 One more Renaissance painting. 414 00:25:13,270 --> 00:25:14,118 This is so great. 415 00:25:14,118 --> 00:25:19,040 Okay, in some the objects in 416 00:25:19,040 --> 00:25:23,730 the painting look the way an objective observer would actually see them. 417 00:25:23,730 --> 00:25:26,770 In a rational order created by Euclid's geometry. 418 00:25:26,770 --> 00:25:31,670 The art helps to teach to see our world as Euclidian. 419 00:25:31,670 --> 00:25:35,630 Also architecture, teaches us to see our world as Euclidian. 420 00:25:35,630 --> 00:25:40,350 Okay, whenever I talk about this I'm always in a room like this one. 421 00:25:40,350 --> 00:25:41,510 Full of parallel lines. 422 00:25:41,510 --> 00:25:44,002 Look at the sides of this auditorium. 423 00:25:44,002 --> 00:25:46,786 Full of parallel lines which are everywhere equidistant, 424 00:25:46,786 --> 00:25:48,964 which make equal right angles with the floor. 425 00:25:48,964 --> 00:25:52,659 The banister kindly let's us look at the making equal 426 00:25:52,659 --> 00:25:55,452 non-right angles with a transversal, 427 00:25:55,452 --> 00:26:00,520 all with the properties that Euclid used to prove his fifth postulate. 428 00:26:00,520 --> 00:26:03,950 This is the kind of room that you would design if you wanted to brainwash people 429 00:26:03,950 --> 00:26:07,010 into believing that space has to be Euclidian. 430 00:26:07,010 --> 00:26:10,435 >> [LAUGH] >> So, here's the 18th century world, 431 00:26:10,435 --> 00:26:12,815 the world of sufficient reason. 432 00:26:12,815 --> 00:26:16,195 It's symmetric, it's balanced, it's based on self evident and 433 00:26:16,195 --> 00:26:19,525 necessary truths, it's embedded in Euclidian space. 434 00:26:19,525 --> 00:26:22,940 We can figure it all out rationally by ourselves. 435 00:26:22,940 --> 00:26:27,740 Euclid's geometry is the universally agreed upon model of 436 00:26:27,740 --> 00:26:31,760 perfect intellectual authority. 437 00:26:31,760 --> 00:26:36,045 That is the end of part one of this talk, now we'll blow it all out of the water. 438 00:26:36,045 --> 00:26:41,920 >> [LAUGH] >> It's really amazing that 439 00:26:41,920 --> 00:26:46,180 in the early 19th century, Gauss, Bolyai, and Lobachevsky, the three independent 440 00:26:46,180 --> 00:26:50,300 inventors of the first non-Euclidean geometries, were able, in spite of all 441 00:26:50,300 --> 00:26:56,070 this Euclidean brainwashing, to imagine that space could be other than Euclidean. 442 00:26:56,070 --> 00:27:00,750 Now what they did was to realize that the absurd consequences. 443 00:27:00,750 --> 00:27:05,800 The absurd consequences of denying Euclid's postulate. 444 00:27:05,800 --> 00:27:08,650 Consequences like parallels are not unique. 445 00:27:08,650 --> 00:27:11,370 Parallel lines that is lines that never meet do not 446 00:27:11,370 --> 00:27:13,660 have to be everywhere equidistant. 447 00:27:13,660 --> 00:27:18,070 These consequences are not absurd at all, but are truths. 448 00:27:18,070 --> 00:27:23,380 Truths in some alternate counter intuitive reality. 449 00:27:23,380 --> 00:27:27,540 Well let's stick for the moment to two dimensions, a surface. 450 00:27:27,540 --> 00:27:32,450 Here's a modern non-Euclidean surface that obeys Euclid's first four postulates, but 451 00:27:32,450 --> 00:27:34,200 disobeys his fifth. 452 00:27:34,200 --> 00:27:40,840 Notice that the red and yellow shortest paths never meet their parallel. 453 00:27:40,840 --> 00:27:42,800 They're both parallel to the blue one. 454 00:27:42,800 --> 00:27:44,990 Notice that they intersect. 455 00:27:44,990 --> 00:27:48,380 So we got two parallels through the same point. 456 00:27:48,380 --> 00:27:51,630 And notice that those three parallel lines, the red, the yellow, and 457 00:27:51,630 --> 00:27:54,620 the blue, are not everywhere equidistant. 458 00:27:54,620 --> 00:27:58,060 The idea of non-Euclidean geometry was so revolutionary, 459 00:27:58,060 --> 00:28:01,715 that Gauss never even got up the nerve to publish his work on the subject. 460 00:28:01,715 --> 00:28:06,570 Non-Euclidean geometry required a paradigm change. 461 00:28:06,570 --> 00:28:10,350 Realizing that the logical implications of denying Euclid's postulate 462 00:28:10,350 --> 00:28:14,520 were not intellectual absurdities, as Lagrange had thought. 463 00:28:14,520 --> 00:28:16,950 But part of an alternate reality. 464 00:28:16,950 --> 00:28:19,450 And of course, the Lobachevsky inversion of non-Euclidean 465 00:28:19,450 --> 00:28:22,640 geometry with space of negative curvature was not the only one. 466 00:28:22,640 --> 00:28:28,930 There are lots more non-Euclidean manifolds, including some 467 00:28:28,930 --> 00:28:36,070 of positive curvature as was explained to mathematicians notably by Riemann. 468 00:28:36,070 --> 00:28:40,670 But, what about all that stuff that Kant said. 469 00:28:40,670 --> 00:28:43,980 That space is in the mind, and that the properties of space are the same for 470 00:28:43,980 --> 00:28:45,350 all human beings. 471 00:28:45,350 --> 00:28:47,800 And that we order all our perceptions in space. 472 00:28:47,800 --> 00:28:49,834 Kant's space was Euclidean, all right. 473 00:28:49,834 --> 00:28:56,669 We can't possible order our perceptions in a non-Euclidean space, can we? 474 00:28:56,669 --> 00:29:01,420 Yes said Hermann von Helmholtz. 475 00:29:01,420 --> 00:29:02,570 Of course, in order to do this, 476 00:29:02,570 --> 00:29:07,080 we have to divest ourselves of 2,000 years of Euclidian experience. 477 00:29:07,080 --> 00:29:11,890 But if we can order our perceptions in a three dimensional non-Euclidian space, 478 00:29:11,890 --> 00:29:13,375 Kant is wrong. 479 00:29:13,375 --> 00:29:17,730 And if Kant is wrong says Helmholtz, then the postulates of geometry 480 00:29:17,730 --> 00:29:22,620 are not dictated by the nature of the human intellect, or by logical necessity 481 00:29:22,620 --> 00:29:27,340 whether space is Euclidian or not is a question for experience. 482 00:29:27,340 --> 00:29:31,960 So I challenge, you let's see if through new experiences 483 00:29:31,960 --> 00:29:35,630 you can learn to order your perceptions in a non-Euclidian three dimensional space. 484 00:29:35,630 --> 00:29:40,174 Here we go. 485 00:29:40,174 --> 00:29:43,374 You probably have a convex mirror on your car. 486 00:29:43,374 --> 00:29:45,740 And there's a warning on it. 487 00:29:45,740 --> 00:29:47,070 Warning! 488 00:29:47,070 --> 00:29:49,285 The space you see in this mirror is not Eucledian. 489 00:29:49,285 --> 00:29:55,390 >> [LAUGH] >> See the parallel lines on the top and 490 00:29:55,390 --> 00:29:59,790 the bottom of the bookcase not being everywhere equidistant? 491 00:29:59,790 --> 00:30:02,060 Now, yes, you can learn to order your perceptions and 492 00:30:02,060 --> 00:30:05,950 touch a space if you have made it so far with your car. 493 00:30:05,950 --> 00:30:07,580 I salute you for having done so. 494 00:30:07,580 --> 00:30:10,270 But you might say, look, it's just a mirror. 495 00:30:10,270 --> 00:30:12,320 It's an illusion, the world in the mirror. 496 00:30:12,320 --> 00:30:16,890 Only our own world, the Euclidean world, can possibly be real. 497 00:30:16,890 --> 00:30:20,130 Oh yeah, says Helmholtz? 498 00:30:20,130 --> 00:30:23,570 Look imagine you're having a dialogue with the guy in the mirror. 499 00:30:23,570 --> 00:30:26,480 You say my world is real, yours is distorted. 500 00:30:26,480 --> 00:30:27,980 He says yeah how can you tell? 501 00:30:27,980 --> 00:30:30,810 You say, well in your world when you get closer, and 502 00:30:30,810 --> 00:30:33,180 closer to the mirror you grow bigger. 503 00:30:33,180 --> 00:30:36,310 When you get farther and farther away from the mirror you get smaller. 504 00:30:36,310 --> 00:30:40,540 This violates the obvious Euclidian fact that when you move something 505 00:30:40,540 --> 00:30:44,110 around through space it stays the same size and shape. 506 00:30:44,110 --> 00:30:45,200 Guy in the mirror says, really? 507 00:30:45,200 --> 00:30:46,980 Yeah, let me try that. 508 00:30:46,980 --> 00:30:49,054 So he takes a yard stick. 509 00:30:49,054 --> 00:30:52,869 He comes up to the interface between his world and ours, and he measures himself. 510 00:30:52,869 --> 00:30:54,825 I'm six feet tall. 511 00:30:54,825 --> 00:30:57,980 And he goes to the back of the room, away from the mirror, and 512 00:30:57,980 --> 00:30:59,890 he measures himself again. 513 00:30:59,890 --> 00:31:03,000 No, you're wrong, he says, I'm still six feet tall. 514 00:31:03,000 --> 00:31:07,534 And you say, well your yardstick changed its size also. 515 00:31:07,534 --> 00:31:11,699 And he says, come on, Jack. 516 00:31:11,699 --> 00:31:15,368 >> [LAUGH] >> Helmholtz says there is no geometrical 517 00:31:15,368 --> 00:31:19,239 experiment that you can do that will decide the question of which one of 518 00:31:19,239 --> 00:31:22,710 these worlds is the real one. 519 00:31:22,710 --> 00:31:26,680 By the way, if that sounds like relativity theory, it's no accident. 520 00:31:26,680 --> 00:31:30,090 There's a, I was gonna say a straight line but I shouldn't use that word. 521 00:31:30,090 --> 00:31:32,820 There is a line of historical influence in German 522 00:31:32,820 --> 00:31:38,260 philosophy from Helmholtz to Reinsma and then on to Einstein. 523 00:31:38,260 --> 00:31:42,060 The British mathematician, philosopher W.K. Clifford 524 00:31:42,060 --> 00:31:46,910 argued that the new geometry was a revolution like the Copernican revolution. 525 00:31:46,910 --> 00:31:52,290 He said, it used to be that the aim of every scientific student of every subject 526 00:31:52,290 --> 00:31:56,460 was to bring his knowledge of that subject into a form as perfect as that 527 00:31:56,460 --> 00:31:59,440 which geometry had attained. 528 00:31:59,440 --> 00:32:01,634 But no more. 529 00:32:01,634 --> 00:32:02,724 Before Copernicus and 530 00:32:02,724 --> 00:32:05,888 Clifford people thought they knew all about the whole universe. 531 00:32:05,888 --> 00:32:08,136 Now we only know one small piece of it. 532 00:32:08,136 --> 00:32:12,886 Likewise before non-Euclidian geometry the laws of space and 533 00:32:12,886 --> 00:32:18,720 motion implied an infinite space, who's properties were always the same. 534 00:32:18,720 --> 00:32:20,983 No more said Clifford. 535 00:32:20,983 --> 00:32:25,867 Okay, space appears flat and continuous, but 536 00:32:25,867 --> 00:32:30,640 only as far as we can explore, and no farther. 537 00:32:30,640 --> 00:32:33,590 Well here is an even more radical view. 538 00:32:33,590 --> 00:32:36,780 Also saw the new geometries as revolutionary, but 539 00:32:36,780 --> 00:32:39,820 he disagreed with both Kant and Helmholtz. 540 00:32:39,820 --> 00:32:43,980 If, as Helmholtz said, geometry comes from experience, 541 00:32:43,980 --> 00:32:48,130 then geometry would not be an exact science. 542 00:32:48,130 --> 00:32:52,200 Still, knows that we have more than one kind of space in our minds. 543 00:32:52,200 --> 00:32:54,910 He's on the other side of the non-Euclidean revolution. 544 00:32:54,910 --> 00:32:58,720 So the postulates of geometry are not synthetic a priori intuitions, 545 00:32:58,720 --> 00:32:59,980 as Kant said. 546 00:32:59,980 --> 00:33:02,600 They're not experimental facts as Hemholtz said, 547 00:33:02,600 --> 00:33:07,068 and they aren't necessary self-evident truths as Voltaire and LaGrange said. 548 00:33:07,068 --> 00:33:11,810 Geometrical axioms says are conventions. 549 00:33:11,810 --> 00:33:15,190 Okay, so how should we decide which set of axioms to use, 550 00:33:15,190 --> 00:33:18,780 those of Euclid, Lobochesky, or Riemann? 551 00:33:18,780 --> 00:33:23,400 Says our choice among these conventions can be guided by experience, but 552 00:33:23,400 --> 00:33:27,470 as long as we avoid contradictions, our choice remains free. 553 00:33:27,470 --> 00:33:30,860 What are we to think of the question is Euclidean geometry true? 554 00:33:30,860 --> 00:33:33,260 The question has no meaning. 555 00:33:33,260 --> 00:33:37,430 We might just as well ask if the metric system is true and the old weights and 556 00:33:37,430 --> 00:33:39,740 measures false. 557 00:33:39,740 --> 00:33:46,620 One geometry cannot be more true than another, it can only be more convenient. 558 00:33:46,620 --> 00:33:51,130 In the 20th century, it turned out that non-Euclidean geometries of the Riemannian 559 00:33:51,130 --> 00:33:55,605 type, that's positively curved space were in fact more convenient because 560 00:33:55,605 --> 00:34:02,400 Reman's type of geometry is what Einstein needed to do general relativity. 561 00:34:02,400 --> 00:34:04,070 According to Einstein's theory of gravity, 562 00:34:04,070 --> 00:34:08,240 the path of planets moving under gravity are shortest distances. 563 00:34:08,240 --> 00:34:12,757 In a non non-euclidean manifold, light no longer travels in Euclidean straight 564 00:34:12,757 --> 00:34:18,930 lines, but shortest distances as again in a non-euclidean space time manifold. 565 00:34:18,930 --> 00:34:20,760 Is real space really Riemannian? 566 00:34:20,760 --> 00:34:24,254 Would say that's what works. 567 00:34:24,254 --> 00:34:28,810 You know though, if we're really interested in what works, 568 00:34:28,810 --> 00:34:33,200 we also ought to study empirically how people actually perceive space and 569 00:34:33,200 --> 00:34:35,190 even before non-euclidean geometry, 570 00:34:35,190 --> 00:34:40,030 philosophers like Bishop Barkley pointed out that we don't see distance at all. 571 00:34:40,030 --> 00:34:41,790 What we see are angles, and 572 00:34:41,790 --> 00:34:45,620 we infer the geometry of what's out there from the angles we actually see. 573 00:34:45,620 --> 00:34:49,200 I'm gonna give you a really easy example. 574 00:34:49,200 --> 00:34:50,530 If I asked you about this, 575 00:34:50,530 --> 00:34:54,710 you'd say oh look, three 90 degree angles are coming together, but 576 00:34:54,710 --> 00:35:00,380 look at what you actually see, three 120 degree angles coming together. 577 00:35:00,380 --> 00:35:05,380 Our visual space isn't the same as the space that we claim to see, and 578 00:35:05,380 --> 00:35:09,080 the invention of non-euclidean geometry really made psychologists 579 00:35:09,080 --> 00:35:11,601 think a lot about things like that, like Helmholtz. 580 00:35:11,601 --> 00:35:14,930 Helmholtz did an experiment where he asked people in a dark room 581 00:35:14,930 --> 00:35:20,260 to arrange little points of light along a table. 582 00:35:20,260 --> 00:35:24,690 He wanted them to do two parallel lines going away from themselves. 583 00:35:24,690 --> 00:35:28,100 So the people made the lines by the little points of light, and then he turned 584 00:35:28,100 --> 00:35:32,050 the lights back on and you could see that the lines weren't parallel at all. 585 00:35:32,050 --> 00:35:37,526 They curved away from the observer and so Helmholtz concluded, ha ha! 586 00:35:37,526 --> 00:35:42,940 Space, our perception of space is Lobachevskian, but 587 00:35:42,940 --> 00:35:46,660 there are psychologists who have done many more experiments, and it turns out 588 00:35:46,660 --> 00:35:51,340 the space of visual perception isn't represented by any consistent geometry. 589 00:35:51,340 --> 00:35:53,590 That's depressing. 590 00:35:53,590 --> 00:35:56,150 Now this is all European material. 591 00:35:56,150 --> 00:35:59,090 The cultural linguist Stephen Levinson has shown 592 00:35:59,090 --> 00:36:02,200 that people in many different cultures have other ways of ordering their 593 00:36:02,200 --> 00:36:06,770 perceptions than an external Euclidean space. 594 00:36:06,770 --> 00:36:10,370 Some cultures do use the idea of a fixed coordinate system in space with 595 00:36:10,370 --> 00:36:15,810 four cardinal directions as when you say the car is to the south of the building. 596 00:36:15,810 --> 00:36:17,530 That's like Newton's framework. 597 00:36:17,530 --> 00:36:22,255 But other cultures order their perceptions in space more in line with Levinson's 598 00:36:22,255 --> 00:36:24,388 ideas, a relation between bodies. 599 00:36:24,388 --> 00:36:25,990 Some from the individual point of view, 600 00:36:25,990 --> 00:36:29,210 as if you say the car is to the right of the building. 601 00:36:29,210 --> 00:36:31,300 You're in that okay. 602 00:36:31,300 --> 00:36:35,320 Or, you can get the individual out of it entirely and just talk about the intrinsic 603 00:36:35,320 --> 00:36:41,670 properties of the objects as if you say car is in front of the building. 604 00:36:41,670 --> 00:36:46,230 In our own culture, GPS navigational systems are changing people's supposedly 605 00:36:46,230 --> 00:36:50,190 innate intuitions of space from the Newtonian to the. 606 00:36:50,190 --> 00:36:52,850 Once I asked a taxi drive in Maryland 607 00:36:52,850 --> 00:36:56,350 if his GPS system had changed how he thought about space. 608 00:36:56,350 --> 00:36:57,160 He was a smart guy. 609 00:36:57,160 --> 00:37:01,040 He thought that was a really cool question, and I wrote down what he said. 610 00:37:01,040 --> 00:37:02,130 Here's what he said. 611 00:37:02,130 --> 00:37:05,370 I used to have the whole geography of Greater Baltimore in my head, 612 00:37:05,370 --> 00:37:06,810 I don't any more. 613 00:37:06,810 --> 00:37:09,150 Like when I take you somewhere, I'll turn left out of the airport, 614 00:37:09,150 --> 00:37:11,980 I'll take the expressway to your exit, then I'll turn right. 615 00:37:11,980 --> 00:37:14,000 When I leave you off, I'll just reverse that. 616 00:37:14,000 --> 00:37:15,710 I'll turn left, I'll get on the express way, 617 00:37:15,710 --> 00:37:20,385 I'll turn right at the airport, I'll get back, but I won't know where I've been. 618 00:37:20,385 --> 00:37:23,410 >> [LAUGH] >> Well okay, let's get back to 619 00:37:23,410 --> 00:37:27,090 the paradigm shift from Euclidean geometry to non-euclidean geometry. 620 00:37:27,090 --> 00:37:28,608 I'll start with a classroom anecdote. 621 00:37:28,608 --> 00:37:30,128 Got you! 622 00:37:30,128 --> 00:37:31,140 Okay. 623 00:37:31,140 --> 00:37:33,670 Once, I passed a convex mirror around in class and 624 00:37:33,670 --> 00:37:35,613 one of my students looked at himself in the mirror and 625 00:37:35,613 --> 00:37:40,545 he said this is distorted and somebody else said, you're a Euclidean chauvinist. 626 00:37:40,545 --> 00:37:43,310 >> [LAUGH] >> So 627 00:37:43,310 --> 00:37:48,130 what does modern theorists of culture have to say about this paradigm shift? 628 00:37:48,130 --> 00:37:54,150 One especially interesting thinker on this topic was Jose Ortega y Gasset. 629 00:37:54,150 --> 00:37:57,430 Ortega used the new geometries to argue that provincialism, 630 00:37:57,430 --> 00:38:01,660 assuming that our own experience or values are universal, is wrong. 631 00:38:01,660 --> 00:38:03,270 He sounds like Clifford here. 632 00:38:03,270 --> 00:38:05,710 Euclidean geometry's provincial. 633 00:38:05,710 --> 00:38:11,060 It was an unwarranted extrapolation of what was locally observed to the entire 634 00:38:11,060 --> 00:38:11,835 universe. 635 00:38:11,835 --> 00:38:17,078 Instead, Ortega says reality organizes itself to be visible from all view points. 636 00:38:17,078 --> 00:38:21,924 Einstein's Theory of Relativity promotes the harmonious multiplicity 637 00:38:21,924 --> 00:38:25,906 of all possible points of view, not just for Mathematics and 638 00:38:25,906 --> 00:38:29,700 Physics, but also for Politics and Culture. 639 00:38:29,700 --> 00:38:37,360 There's a Chinese perspective Ortega says, that is fully as justified as the Western. 640 00:38:37,360 --> 00:38:39,683 Okay, now briefly to architecture. 641 00:38:39,683 --> 00:38:44,200 Take Zaha Hadid, the first woman to win the Pritzker Architecture prize. 642 00:38:44,200 --> 00:38:47,360 As an under graduate, she specialized in mathematics. 643 00:38:47,360 --> 00:38:52,600 Notice we aren't in the Renaissance ideal city any more. 644 00:38:52,600 --> 00:38:53,900 Smore. 645 00:38:53,900 --> 00:38:56,510 Here's what she says about the world of the 21st century, 646 00:38:56,510 --> 00:38:58,880 the most important thing is motion, the flux of things, 647 00:38:58,880 --> 00:39:04,570 a non-euclidean geometry in which nothing repeats itself, a new order of space 648 00:39:04,570 --> 00:39:08,390 and the new geometries inspire many other types of artistic freedom. 649 00:39:08,390 --> 00:39:12,433 I'm gonna run by rapidly three examples from cubist art, 650 00:39:12,433 --> 00:39:20,780 of what Ortega called reality organizing itself to be seen from all points of view. 651 00:39:20,780 --> 00:39:22,050 Tea time. 652 00:39:22,050 --> 00:39:23,360 I'm in England you know. 653 00:39:23,360 --> 00:39:30,990 Look at the tea cup, the guitar, a portrait by Picasso. 654 00:39:30,990 --> 00:39:32,890 Planes and angles from multiple points of views. 655 00:39:32,890 --> 00:39:37,150 Notice the guy in the picture is not in a visually graspable 656 00:39:37,150 --> 00:39:39,260 three dimensional space at all. 657 00:39:39,260 --> 00:39:44,060 A different kind of geometrical influence on art, plaster models like 658 00:39:44,060 --> 00:39:49,210 the ones in your building, this is all we got in Claremont, but you got lots more. 659 00:39:49,210 --> 00:39:50,890 Mathematicians use these to teach and 660 00:39:50,890 --> 00:39:54,760 study the geometry of non-euclidean surfaces. 661 00:39:54,760 --> 00:40:00,500 Let me show you some art inspired by such models. 662 00:40:00,500 --> 00:40:07,390 That's the world, the name of this one makes clear that it comes from geometry. 663 00:40:07,390 --> 00:40:11,390 Remember the helix, the thing I just showed you? 664 00:40:11,390 --> 00:40:15,100 It's just a lovely piece by Man Ray, it's just beautiful. 665 00:40:15,100 --> 00:40:17,665 The plaster models that Man Ray studied 666 00:40:17,665 --> 00:40:20,200 were the ones in the Pohanka Ray Institute. 667 00:40:20,200 --> 00:40:21,140 Man gets his credit. 668 00:40:21,140 --> 00:40:22,750 Pohanka Ray in Paris. 669 00:40:22,750 --> 00:40:24,580 Man Ray was just blown away by them. 670 00:40:24,580 --> 00:40:27,031 He said these shapes were so unusual, 671 00:40:27,031 --> 00:40:34,769 as revolutionary as anything that is being done today in painting or in sculpture. 672 00:40:34,769 --> 00:40:39,560 One more of many examples, I find this charming. 673 00:40:39,560 --> 00:40:42,304 So, artists are influenced by the new geometry. 674 00:40:42,304 --> 00:40:46,850 They cite it's authority and it's prestige on their own behalf, 675 00:40:46,850 --> 00:40:50,790 they use it as part of their creation of self consciously modern art and 676 00:40:50,790 --> 00:40:54,220 they help us see a different world. 677 00:40:54,220 --> 00:40:58,860 Coming now to my last pair of slides, here's the first. 678 00:40:58,860 --> 00:41:02,085 I found this searching Google image for hyperbolic paraboloid. 679 00:41:02,085 --> 00:41:05,770 >> [LAUGH] >> And I like this picture a lot 680 00:41:05,770 --> 00:41:09,680 because it shows us how much non-euclidean geometrically objects 681 00:41:09,680 --> 00:41:14,580 had become part of the general culture already more than half a century ago. 682 00:41:14,580 --> 00:41:20,590 And here's one that became famous much more recently in your own country. 683 00:41:20,590 --> 00:41:24,620 Maroff is another hyperbolic paraboloid. 684 00:41:24,620 --> 00:41:29,960 Okay so you're entitled to conclusion, conclusion. 685 00:41:29,960 --> 00:41:35,980 Twice in history, first in Euclid's world and then in the non-euclidean world, 686 00:41:35,980 --> 00:41:41,250 ideas about geometry have helped shaped art and architecture, and science, 687 00:41:41,250 --> 00:41:47,610 and philosophy, and have helped shape the way people see and think about the world. 688 00:41:47,610 --> 00:41:49,300 I hope that this talk, and 689 00:41:49,300 --> 00:41:54,530 in fact this entire conference, help a wider public better understand 690 00:41:54,530 --> 00:41:58,980 the intimate relationship between Mathematics and culture. 691 00:41:58,980 --> 00:41:59,605 Thank you very much. 692 00:41:59,605 --> 00:42:07,360 >> [APPLAUSE]