1
00:00:00,580 --> 00:00:01,590
- [Narrator] The following content

2
00:00:01,590 --> 00:00:04,580
is provided under a
Creative Commons license.

3
00:00:04,580 --> 00:00:06,740
Your support will help MIT OpenCourseWare

4
00:00:06,740 --> 00:00:08,590
continue to offer high quality

5
00:00:08,590 --> 00:00:11,150
educational resources for free.

6
00:00:11,150 --> 00:00:12,150
To make a donation

7
00:00:12,150 --> 00:00:13,730
or view additional materials

8
00:00:13,730 --> 00:00:15,320
from hundreds of MIT courses

9
00:00:15,320 --> 00:00:18,653
visit MIT opencourseware at ocw.mit.edu.

10
00:00:20,349 --> 00:00:21,930
- All right, hello.

11
00:00:21,930 --> 00:00:23,590
Welcome to a Godel, Escher, Bach:

12
00:00:23,590 --> 00:00:24,980
A Mental Space Odyssey.

13
00:00:24,980 --> 00:00:26,610
My name is Justin Curry

14
00:00:26,610 --> 00:00:30,170
and I'm a senior in
mathematics here at MIT.

15
00:00:30,170 --> 00:00:31,240
I've spent the last year

16
00:00:31,240 --> 00:00:34,220
at Cambridge University at UK,

17
00:00:34,220 --> 00:00:36,220
and the summer before
that living in Germany.

18
00:00:36,220 --> 00:00:38,110
So it's kind of a reverse
culture shock coming back

19
00:00:38,110 --> 00:00:41,630
but I'm excited to teach
Godel, Escher, Bach again.

20
00:00:41,630 --> 00:00:44,620
I taught this course in Spring 2006.

21
00:00:44,620 --> 00:00:46,290
It was a 10-week course then.

22
00:00:46,290 --> 00:00:48,410
And we attempted the impossible task

23
00:00:48,410 --> 00:00:51,410
of trying to get through
this thick monster

24
00:00:51,410 --> 00:00:53,230
all in one go.

25
00:00:53,230 --> 00:00:56,650
And it's impossible.

26
00:00:56,650 --> 00:00:59,030
Most undergrads can't get
through it in 13 weeks.

27
00:00:59,030 --> 00:01:01,930
I got through it in about seven years.

28
00:01:01,930 --> 00:01:05,260
So you're going to be
attempting a feat here

29
00:01:05,260 --> 00:01:06,360
not to complete the entire book

30
00:01:06,360 --> 00:01:09,830
but to get the essence of
Godel, Escher Bach out.

31
00:01:09,830 --> 00:01:12,000
But I want to make sure
we introduce everybody

32
00:01:12,000 --> 00:01:13,000
just to get people's names.

33
00:01:13,000 --> 00:01:14,830
This will help me take attendance

34
00:01:14,830 --> 00:01:17,750
and it will also...

35
00:01:17,750 --> 00:01:19,030
Just I also want you to say

36
00:01:19,030 --> 00:01:20,630
what is it when you read
the course catalogue

37
00:01:20,630 --> 00:01:22,640
that interested you most

38
00:01:22,640 --> 00:01:26,300
and why essentially why
you're sitting here today.

39
00:01:26,300 --> 00:01:29,020
I'm curious.

40
00:01:29,020 --> 00:01:33,770
So what is the idea behind this book?

41
00:01:33,770 --> 00:01:38,510
I interviewed a good
many of you this morning

42
00:01:38,510 --> 00:01:41,340
and just to make sure that you guys

43
00:01:41,340 --> 00:01:42,750
felt comfortable with mathematics.

44
00:01:42,750 --> 00:01:44,750
This course isn't directly
about mathematics.

45
00:01:44,750 --> 00:01:47,460
There's a lot of mathematics
being talked about.

46
00:01:47,460 --> 00:01:48,780
Yes you have a question?

47
00:01:48,780 --> 00:01:51,220
- [Student] What is the class about.

48
00:01:51,220 --> 00:01:55,070
- Ok so, that's what I'm
gonna go through right now.

49
00:01:55,070 --> 00:01:59,781
The idea here is that Douglas Hofstadter

50
00:01:59,781 --> 00:02:02,520
is interested in one primary question

51
00:02:02,520 --> 00:02:05,270
and that question is how does a self

52
00:02:05,270 --> 00:02:07,560
come out of things which have no selves?

53
00:02:07,560 --> 00:02:10,590
How is it that all these carbon atoms

54
00:02:10,590 --> 00:02:12,390
and molecules and proteins

55
00:02:12,390 --> 00:02:14,720
which make us up in the physical universe,

56
00:02:14,720 --> 00:02:17,810
how do they go from being meaningless

57
00:02:17,810 --> 00:02:20,760
to developing into an entity

58
00:02:20,760 --> 00:02:24,120
which can refer to itself.

59
00:02:24,120 --> 00:02:26,390
Like right now I'm saying I think this,

60
00:02:26,390 --> 00:02:27,530
I think you like this.

61
00:02:27,530 --> 00:02:29,800
I'm meeting all of you as individuals.

62
00:02:29,800 --> 00:02:32,510
Each one of you claim to have a self.

63
00:02:32,510 --> 00:02:34,590
You might remember Des
Cartes' famous quote,

64
00:02:34,590 --> 00:02:35,900
"I think therefore I am."

65
00:02:35,900 --> 00:02:36,900
So it seems like the I,

66
00:02:36,900 --> 00:02:38,150
when I say the I,

67
00:02:38,150 --> 00:02:40,540
I mean the things we call ourselves,

68
00:02:40,540 --> 00:02:44,060
as a real existent thing.

69
00:02:44,060 --> 00:02:45,130
But it's a complex question.

70
00:02:45,130 --> 00:02:48,790
How do we get I's out of non-I's.

71
00:02:48,790 --> 00:02:52,130
And that's that's kind of
gonna be the goal over here.

72
00:02:52,130 --> 00:02:56,470
So I'm just gonna call it I.

73
00:02:56,470 --> 00:02:57,860
But how do you get to an I.

74
00:02:57,860 --> 00:02:59,550
And get to an I by having

75
00:02:59,550 --> 00:03:03,250
a bunch of meaningless primitives.

76
00:03:03,250 --> 00:03:06,280
Things like atoms,

77
00:03:06,280 --> 00:03:08,740
proteins,

78
00:03:08,740 --> 00:03:11,590
molecules I should say if I want to, etc.

79
00:03:11,590 --> 00:03:13,050
Like this this is what you're made up of

80
00:03:13,050 --> 00:03:14,440
but none of these things mean anything.

81
00:03:14,440 --> 00:03:16,590
None of these things have I's or selves

82
00:03:16,590 --> 00:03:18,050
but you do.

83
00:03:18,050 --> 00:03:22,050
So what's the relationship here?

84
00:03:22,920 --> 00:03:25,120
Douglas Hofstadter,

85
00:03:25,120 --> 00:03:28,830
he wrote this book back in the 70s

86
00:03:28,830 --> 00:03:32,000
when he was doing
graduate school in physics

87
00:03:32,000 --> 00:03:33,310
and this was after him

88
00:03:33,310 --> 00:03:36,090
doing a math undergrad at Stanford.

89
00:03:36,090 --> 00:03:39,370
He believed that he saw
that he saw the answer.

90
00:03:39,370 --> 00:03:41,980
When he was playing
around with mathematics

91
00:03:41,980 --> 00:03:44,710
and the very formal systems we play with.

92
00:03:44,710 --> 00:03:45,870
Like when we write down things

93
00:03:45,870 --> 00:03:48,700
like two plus two equals four,

94
00:03:48,700 --> 00:03:50,550
these are just symbols.

95
00:03:50,550 --> 00:03:52,560
And as we go through today,

96
00:03:52,560 --> 00:03:54,120
I'll show you completely equivalent ways

97
00:03:54,120 --> 00:03:56,060
of doing addition,

98
00:03:56,060 --> 00:04:00,060
which will look like this and...

99
00:04:03,459 --> 00:04:04,910
And these are just logical primitives.

100
00:04:04,910 --> 00:04:07,940
Like if you've seen any set theory,

101
00:04:07,940 --> 00:04:09,760
and you know don't feel scared

102
00:04:09,760 --> 00:04:11,280
if you haven't seen any of these symbols,

103
00:04:11,280 --> 00:04:14,370
but like there exists an X for every.

104
00:04:14,370 --> 00:04:15,380
We give these interpretations

105
00:04:15,380 --> 00:04:17,280
but the idea is that mathematics

106
00:04:17,280 --> 00:04:18,280
can be reduced to a bunch

107
00:04:18,280 --> 00:04:19,590
of meaningless operations,

108
00:04:19,590 --> 00:04:23,630
just symbol shunting.

109
00:04:23,630 --> 00:04:25,050
But what's interesting is that

110
00:04:25,050 --> 00:04:28,800
within mathematics there exists

111
00:04:28,800 --> 00:04:31,900
an equivalent to self reference.

112
00:04:31,900 --> 00:04:33,250
This is a bunch of atoms and proteins

113
00:04:33,250 --> 00:04:36,710
referring to itself, calling itself an I.

114
00:04:36,710 --> 00:04:37,710
What happens here

115
00:04:37,710 --> 00:04:39,520
and this is going to be kind of underneath

116
00:04:39,520 --> 00:04:43,520
the name of Godel,

117
00:04:50,069 --> 00:04:51,060
is we're gonna get to some

118
00:04:51,060 --> 00:04:52,060
incompleteness theorems.

119
00:04:52,060 --> 00:04:53,060
Gonna get to some statements

120
00:04:53,060 --> 00:04:55,610
which in mathematics refer to themselves.

121
00:04:55,610 --> 00:04:58,030
And the question of how this happens,

122
00:04:58,030 --> 00:05:00,270
we understand this rigorously.

123
00:05:00,270 --> 00:05:01,270
Mathematicians have worked out

124
00:05:01,270 --> 00:05:02,660
how do we go from meaningless symbols

125
00:05:02,660 --> 00:05:04,490
to something which refers to itself

126
00:05:04,490 --> 00:05:06,150
and which has meaning.

127
00:05:06,150 --> 00:05:07,150
The claim then is

128
00:05:07,150 --> 00:05:10,030
is that these two systems are equivalent

129
00:05:10,030 --> 00:05:12,620
and this is really the profound idea.

130
00:05:12,620 --> 00:05:13,740
I'm gonna draw this symbol.

131
00:05:13,740 --> 00:05:15,990
And I'm going to use a
term called isomorphism.

132
00:05:15,990 --> 00:05:19,530
An isomorphism is basically an equals too,

133
00:05:19,530 --> 00:05:21,520
and equals in a different sense.

134
00:05:21,520 --> 00:05:25,589
But the idea here is in many ways

135
00:05:25,589 --> 00:05:27,160
we can link atoms

136
00:05:27,160 --> 00:05:29,550
and proteins to kind of logical

137
00:05:29,550 --> 00:05:32,110
symbolic primitives in mathematics

138
00:05:32,110 --> 00:05:33,930
and we understand how
we get self-reference

139
00:05:33,930 --> 00:05:35,490
in mathematics.

140
00:05:35,490 --> 00:05:37,090
So maybe we can use this to understand

141
00:05:37,090 --> 00:05:39,130
how we get I's,

142
00:05:39,130 --> 00:05:41,620
how self comes out of now non-self.

143
00:05:41,620 --> 00:05:43,460
This is a really tall order

144
00:05:43,460 --> 00:05:46,130
but we're gonna try to do it.

145
00:05:46,130 --> 00:05:49,090
And that's what this book attempts to do.

146
00:05:49,090 --> 00:05:50,940
And what I've done

147
00:05:50,940 --> 00:05:54,360
is isolate the chapters in this book,

148
00:05:54,360 --> 00:05:55,470
which I think are most pertinent

149
00:05:55,470 --> 00:05:57,710
to this string of thought.

150
00:05:57,710 --> 00:05:59,020
Basically what we're gonna do

151
00:05:59,020 --> 00:06:00,580
is we're gonna learn how
it works in mathematics.

152
00:06:00,580 --> 00:06:02,740
We're gonna go from logical primitives

153
00:06:02,740 --> 00:06:05,140
and workup self-reference.

154
00:06:05,140 --> 00:06:08,270
And talk about Zen
Buddhism, consciousness etc.

155
00:06:08,270 --> 00:06:10,080
But that's going to happen
as we leap over here

156
00:06:10,080 --> 00:06:13,060
because we're gonna work up, down,

157
00:06:13,060 --> 00:06:14,380
and then around.

158
00:06:14,380 --> 00:06:15,380
And we'll conclude the course

159
00:06:15,380 --> 00:06:16,380
with some interesting questions

160
00:06:16,380 --> 00:06:17,520
about artificial intelligence

161
00:06:17,520 --> 00:06:19,090
and how intelligent things

162
00:06:19,090 --> 00:06:22,110
come out of unintelligent things.

163
00:06:22,110 --> 00:06:26,259
So when I was teaching this course

164
00:06:26,259 --> 00:06:28,250
two years ago or two springs ago,

165
00:06:28,250 --> 00:06:32,110
I ran into kind of five things,

166
00:06:32,110 --> 00:06:36,030
which I viewed as really important tools

167
00:06:36,030 --> 00:06:38,900
for thinking.

168
00:06:38,900 --> 00:06:42,900
And this is kind of, I've
had to condense a little bit

169
00:06:43,819 --> 00:06:47,810
into my famous tools for thinking lecture.

170
00:06:52,539 --> 00:06:56,530
The idea here is that Godel, Escher, Bach

171
00:06:58,149 --> 00:07:01,050
has an incredible number
of conceptual tools

172
00:07:01,050 --> 00:07:03,050
for thinking about this complex problem

173
00:07:03,050 --> 00:07:06,460
of how do we go from a non-self to a self.

174
00:07:06,460 --> 00:07:09,490
And just outline these real quick.

175
00:07:09,490 --> 00:07:10,660
We're gonna have

176
00:07:10,660 --> 00:07:14,660
isomorphisms.

177
00:07:19,860 --> 00:07:22,180
And I'll explain all
these terms as we go on.

178
00:07:22,180 --> 00:07:23,180
Recursion.

179
00:07:23,180 --> 00:07:26,280
I'm going to leave this one mainly up

180
00:07:26,280 --> 00:07:30,280
to current on the second lecture.

181
00:07:33,720 --> 00:07:37,720
Paradox.

182
00:07:39,699 --> 00:07:41,880
And this is infinity,

183
00:07:41,880 --> 00:07:45,190
and all these concepts
are very closely linked.

184
00:07:45,190 --> 00:07:49,190
And finally,

185
00:07:50,360 --> 00:07:52,270
the main subject for today's lecture

186
00:07:52,270 --> 00:07:56,270
is going to be formal systems.

187
00:07:58,529 --> 00:08:02,520
All righty.

188
00:08:04,729 --> 00:08:08,720
So first let me go through
kind of definitions

189
00:08:12,020 --> 00:08:14,490
of these terms.

190
00:08:14,490 --> 00:08:18,210
And isomorphism,

191
00:08:18,210 --> 00:08:19,770
I want you to all be
very careful with this

192
00:08:19,770 --> 00:08:21,710
because when you start
talking to mathematicians

193
00:08:21,710 --> 00:08:24,720
you know growing up
professional mathematicians

194
00:08:24,720 --> 00:08:26,330
they're going to use the term isomorphism

195
00:08:26,330 --> 00:08:29,080
to mean something very, very specific.

196
00:08:29,080 --> 00:08:30,710
The way it's used in Godel, Escher, Bach,

197
00:08:30,710 --> 00:08:31,930
the way it's going to
be used in this class

198
00:08:31,930 --> 00:08:32,930
is very loose.

199
00:08:32,930 --> 00:08:34,810
We're gonna make very kind of

200
00:08:34,810 --> 00:08:38,810
intuitive statements, like

201
00:08:40,760 --> 00:08:43,460
you know what's the isomorphism

202
00:08:43,460 --> 00:08:48,100
between a car,

203
00:08:48,100 --> 00:08:52,100
I'm not a great artist here.

204
00:08:56,140 --> 00:08:57,140
What's the isomorphism

205
00:08:57,140 --> 00:08:59,850
between a skateboard and a car?

206
00:08:59,850 --> 00:09:01,200
And you know you might say lots of things

207
00:09:01,200 --> 00:09:03,890
like it carries a person.

208
00:09:03,890 --> 00:09:06,530
It has four wheels.

209
00:09:06,530 --> 00:09:09,050
So what we do is we construct a map,

210
00:09:09,050 --> 00:09:11,080
which also has an inverse.

211
00:09:11,080 --> 00:09:13,070
And that's the way you
think of an isomorphism.

212
00:09:13,070 --> 00:09:15,210
You can go either way

213
00:09:15,210 --> 00:09:17,830
and preserve information,

214
00:09:17,830 --> 00:09:21,550
preserve kind of structure.

215
00:09:21,550 --> 00:09:25,000
If you really feel like following along,

216
00:09:25,000 --> 00:09:27,190
I've included actually a quote

217
00:09:27,190 --> 00:09:29,190
from Douglas Hofstadter.

218
00:09:29,190 --> 00:09:33,190
And on a page seven of your lecture notes.

219
00:09:37,640 --> 00:09:40,240
He says, and this is in
the middle of the page,

220
00:09:40,240 --> 00:09:42,240
"The word isomorphism applies

221
00:09:42,240 --> 00:09:43,620
"when to complex structures

222
00:09:43,620 --> 00:09:45,460
"can be mapped onto each other

223
00:09:45,460 --> 00:09:47,760
"in a ways that to each
part of one structure

224
00:09:47,760 --> 00:09:49,180
"there's a corresponding part

225
00:09:49,180 --> 00:09:50,550
"and the other structure,

226
00:09:50,550 --> 00:09:51,740
"where corresponding means

227
00:09:51,740 --> 00:09:54,260
"that the two parts play
similar similar roles

228
00:09:54,260 --> 00:09:58,260
"in their respective structures."

229
00:09:59,620 --> 00:10:01,900
This is how we're going to

230
00:10:01,900 --> 00:10:05,030
always use the term
isomorphism in this class.

231
00:10:05,030 --> 00:10:06,680
If you're taking the
abstract algebra class,

232
00:10:06,680 --> 00:10:09,280
it's going to mean something
a lot more specific,

233
00:10:09,280 --> 00:10:12,930
and you're gonna have a lot more details.

234
00:10:12,930 --> 00:10:13,990
You might actually think of these

235
00:10:13,990 --> 00:10:16,160
as kind of,

236
00:10:16,160 --> 00:10:19,020
what I'll say but don't worry about it

237
00:10:19,020 --> 00:10:20,340
is a homomorphism.

238
00:10:20,340 --> 00:10:22,310
And the idea with the homomorphism

239
00:10:22,310 --> 00:10:25,050
is that there are a lot more details here

240
00:10:25,050 --> 00:10:27,880
than there are here.

241
00:10:27,880 --> 00:10:29,270
And for example,

242
00:10:29,270 --> 00:10:30,270
there's no steering well.

243
00:10:30,270 --> 00:10:32,640
There's a steering well in a car

244
00:10:32,640 --> 00:10:33,960
but there's no steering well

245
00:10:33,960 --> 00:10:36,220
specifically in a skateboard.

246
00:10:36,220 --> 00:10:38,330
So if you were to go,

247
00:10:38,330 --> 00:10:41,100
if you were to create a map

248
00:10:41,100 --> 00:10:43,240
from the car to the skateboard

249
00:10:43,240 --> 00:10:47,580
that detail would have
to go somewhere else.

250
00:10:47,580 --> 00:10:49,750
But don't worry about those necessities.

251
00:10:49,750 --> 00:10:51,270
But when I say the term isomorphism,

252
00:10:51,270 --> 00:10:52,470
think of equals.

253
00:10:52,470 --> 00:10:55,640
And then I'll often use
that symbol right there.

254
00:10:55,640 --> 00:10:59,760
So this is gonna be really important

255
00:10:59,760 --> 00:11:01,480
because it's going to be how we're

256
00:11:01,480 --> 00:11:03,320
gonna get meaning out of things.

257
00:11:03,320 --> 00:11:05,950
and you'll see it a lot coming up

258
00:11:05,950 --> 00:11:07,130
and over the book.

259
00:11:07,130 --> 00:11:08,870
But first I'm gonna hop on

260
00:11:08,870 --> 00:11:11,820
and talk about recursion.

261
00:11:11,820 --> 00:11:14,880
Recursion is basically...

262
00:11:14,880 --> 00:11:16,980
It's seen everywhere

263
00:11:16,980 --> 00:11:20,480
but it's kind of a list of instructions,

264
00:11:20,480 --> 00:11:21,480
which you follow,

265
00:11:21,480 --> 00:11:23,000
but then repeat until you've reached

266
00:11:23,000 --> 00:11:25,040
kind of a final case.

267
00:11:25,040 --> 00:11:27,670
So suppose you're cooking

268
00:11:27,670 --> 00:11:31,360
and you could have a recursive algorithm

269
00:11:31,360 --> 00:11:33,150
for stirring eggs.

270
00:11:33,150 --> 00:11:37,480
And that would be whirl
and then whirl again

271
00:11:37,480 --> 00:11:38,790
and keep whirling until,

272
00:11:38,790 --> 00:11:40,790
essentially everything looks mixed up.

273
00:11:40,790 --> 00:11:42,920
That's a very loose way
of understanding it.

274
00:11:42,920 --> 00:11:43,920
But another way

275
00:11:43,920 --> 00:11:45,200
which you are probably familiar with,

276
00:11:45,200 --> 00:11:47,540
a much more rigorous in
the term of mathematics

277
00:11:47,540 --> 00:11:50,200
is the Fibonacci sequence.

278
00:11:50,200 --> 00:11:52,030
This is where you start with two numbers,

279
00:11:52,030 --> 00:11:53,300
one and one,

280
00:11:53,300 --> 00:11:55,380
and then you construct the next number

281
00:11:55,380 --> 00:11:57,880
by summing the previous two.

282
00:11:57,880 --> 00:11:59,850
So you have that.

283
00:11:59,850 --> 00:12:00,850
And you have three.

284
00:12:00,850 --> 00:12:01,850
And you have five.

285
00:12:01,850 --> 00:12:04,220
And you have eight.

286
00:12:04,220 --> 00:12:06,290
And so on.

287
00:12:06,290 --> 00:12:07,700
And you can create what's called

288
00:12:07,700 --> 00:12:09,760
a recursive definition

289
00:12:09,760 --> 00:12:13,760
where you define the nth Fibonacci number,

290
00:12:21,130 --> 00:12:25,550
this is for n greater than or equal to two

291
00:12:25,550 --> 00:12:29,550
and here you define the
thing in terms of itself.

292
00:12:31,050 --> 00:12:34,530
And this is a classic
example of recursion.

293
00:12:34,530 --> 00:12:38,740
What it is is really
itself on a smaller level.

294
00:12:38,740 --> 00:12:40,340
I think one of the most
exciting applications

295
00:12:40,340 --> 00:12:43,050
of recursion are fractals

296
00:12:43,050 --> 00:12:44,340
because the way we create fractals

297
00:12:44,340 --> 00:12:46,810
is through recursion.

298
00:12:46,810 --> 00:12:48,420
So I don't know if you all have seen this

299
00:12:48,420 --> 00:12:51,510
but the Sierpinski Triangle
or the Sierpinski Gasket.

300
00:12:51,510 --> 00:12:52,910
It's kind of a classic fractal.

301
00:12:52,910 --> 00:12:56,910
Here you divide a triangle up into three.

302
00:12:58,870 --> 00:13:00,300
And then you just repeat the process

303
00:13:00,300 --> 00:13:04,110
for an infinite number of times

304
00:13:04,110 --> 00:13:06,090
on each remaining triangle.

305
00:13:06,090 --> 00:13:07,850
You create these very beautiful

306
00:13:07,850 --> 00:13:09,370
kind of mosaic forms.

307
00:13:09,370 --> 00:13:11,070
But the nice thing about mathematics

308
00:13:11,070 --> 00:13:12,460
is that we can be very precise

309
00:13:12,460 --> 00:13:15,840
and do things that we
can't do in the real world.

310
00:13:15,840 --> 00:13:16,990
And that's repeat this infinitely

311
00:13:16,990 --> 00:13:20,640
and so on.

312
00:13:20,640 --> 00:13:21,830
Just for a quick digression

313
00:13:21,830 --> 00:13:23,620
and I really don't spend
too much time on it

314
00:13:23,620 --> 00:13:25,770
cause Kurin will do more.

315
00:13:25,770 --> 00:13:29,990
Why is it called a fractal?

316
00:13:29,990 --> 00:13:31,060
Does anyone know?

317
00:13:31,060 --> 00:13:35,060
- [Stundent] I think it's
like a fragment of something.

318
00:13:37,990 --> 00:13:38,990
- Sure.

319
00:13:38,990 --> 00:13:41,770
That, it was a term coined
by Benoit Mandelbrot

320
00:13:41,770 --> 00:13:44,220
in 1977 I believe.

321
00:13:44,220 --> 00:13:47,170
It actually refers to
its number of dimensions.

322
00:13:47,170 --> 00:13:49,610
So this might be kind of
a mind-bending concept

323
00:13:49,610 --> 00:13:50,610
for most of you

324
00:13:50,610 --> 00:13:51,980
but we like to think we live in

325
00:13:51,980 --> 00:13:55,460
one two or three or four dimensions.

326
00:13:55,460 --> 00:13:56,840
All integers right?

327
00:13:56,840 --> 00:14:00,230
But my claim is that the Sierpinski Gasket

328
00:14:00,230 --> 00:14:04,230
actually lives in between
one and two dimensions,

329
00:14:05,080 --> 00:14:07,910
lives in like 1.63 something dimensions.

330
00:14:07,910 --> 00:14:11,710
But I'm gonna help you
kind of think about that

331
00:14:11,710 --> 00:14:13,140
and if you want to hop along

332
00:14:13,140 --> 00:14:15,740
to a page nine,

333
00:14:15,740 --> 00:14:16,740
kind of got a recipe

334
00:14:16,740 --> 00:14:20,240
for helping you think about dimension.

335
00:14:20,240 --> 00:14:22,400
You know it's weird
because only mathematicians

336
00:14:22,400 --> 00:14:23,940
would ever worry about

337
00:14:23,940 --> 00:14:25,320
rigorously understanding the concept

338
00:14:25,320 --> 00:14:27,150
of what a dimension means.

339
00:14:27,150 --> 00:14:29,790
So here's one way to think about it.

340
00:14:29,790 --> 00:14:32,670
If you take a line

341
00:14:32,670 --> 00:14:34,530
and you double it,

342
00:14:34,530 --> 00:14:38,410
you have two copies of the line

343
00:14:38,410 --> 00:14:40,960
that you started with.

344
00:14:40,960 --> 00:14:42,550
This guy's here and there.

345
00:14:42,550 --> 00:14:46,180
If you have a square

346
00:14:46,180 --> 00:14:50,060
and you double the sides of a square,

347
00:14:50,060 --> 00:14:54,280
you have four copies
of the original square.

348
00:14:54,280 --> 00:14:56,860
Similarly, and I'm not
going to try to draw this

349
00:14:56,860 --> 00:14:59,260
because we will get too
complicated way too fast,

350
00:14:59,260 --> 00:15:00,320
if you take a cube

351
00:15:00,320 --> 00:15:04,320
and you double each of the sides,

352
00:15:08,230 --> 00:15:11,610
you get, if you think
about it, eight copies

353
00:15:11,610 --> 00:15:14,170
of the original cube.

354
00:15:14,170 --> 00:15:16,160
So if you're perceptive enough,

355
00:15:16,160 --> 00:15:18,610
you might kind of realize this action

356
00:15:18,610 --> 00:15:20,670
of powers going on here.

357
00:15:20,670 --> 00:15:23,480
So here we had after our doubling process

358
00:15:23,480 --> 00:15:24,480
two copies.

359
00:15:24,480 --> 00:15:26,170
We had two to the one.

360
00:15:26,170 --> 00:15:27,490
Here after our doubling process,

361
00:15:27,490 --> 00:15:31,390
we had to the two.

362
00:15:31,390 --> 00:15:32,940
After our doubling process here,

363
00:15:32,940 --> 00:15:37,550
we had two to the three, eight.

364
00:15:37,550 --> 00:15:39,480
So this is weird because notice

365
00:15:39,480 --> 00:15:43,480
that the cube lives in three dimensions,

366
00:15:44,620 --> 00:15:47,110
and the square lives in two dimensions,

367
00:15:47,110 --> 00:15:51,020
and the line lives in one dimension.

368
00:15:51,020 --> 00:15:53,580
So this might suggest to you

369
00:15:53,580 --> 00:15:57,980
the relationship that two to the d

370
00:15:57,980 --> 00:15:58,980
where d is the dimension

371
00:15:58,980 --> 00:16:01,740
of the space you're living in

372
00:16:01,740 --> 00:16:04,440
equals the number of copies you have

373
00:16:04,440 --> 00:16:06,710
after the doubling process.

374
00:16:06,710 --> 00:16:10,020
So let's return to our
friend the Sierpinski Gasket.

375
00:16:10,020 --> 00:16:11,910
If we start here

376
00:16:11,910 --> 00:16:13,880
and we imagine doubling each of the sides

377
00:16:13,880 --> 00:16:15,940
of the Sierpinski Gasket,

378
00:16:15,940 --> 00:16:17,520
here and here,

379
00:16:17,520 --> 00:16:20,350
we're very strangely led to the conclusion

380
00:16:20,350 --> 00:16:24,350
that whatever dimension the
sierpinski gasket lives in,

381
00:16:26,290 --> 00:16:30,480
it obeys this rule.

382
00:16:30,480 --> 00:16:34,480
So take the logarithms

383
00:16:37,279 --> 00:16:40,230
and d times...

384
00:16:40,230 --> 00:16:45,000
Sorry this is getting crowded.

385
00:16:45,000 --> 00:16:49,000
If you take the logarithm of both sides

386
00:16:51,190 --> 00:16:52,260
and solve for d,

387
00:16:52,260 --> 00:16:53,660
you'll see that the dimension

388
00:16:53,660 --> 00:16:56,090
of the Sierpinski Gasket,

389
00:16:56,090 --> 00:16:59,510
it's log 3 over log 2,

390
00:16:59,510 --> 00:17:03,510
which it's approximately
1.585 on to infinity.

391
00:17:05,220 --> 00:17:07,010
So here's an exact example

392
00:17:07,010 --> 00:17:08,110
of something which lives somewhere

393
00:17:08,110 --> 00:17:09,480
between one and two dimensions,

394
00:17:09,480 --> 00:17:13,480
and I think that's a really cool concept.

395
00:17:14,779 --> 00:17:17,000
Moving on for other tools for thinking

396
00:17:17,000 --> 00:17:20,170
we have paradoxes.

397
00:17:20,170 --> 00:17:21,520
Paradoxes come in all
sorts of different flavors.

398
00:17:21,520 --> 00:17:22,990
I don't know if some of you

399
00:17:22,990 --> 00:17:24,680
have heard of the birthday paradox.

400
00:17:24,680 --> 00:17:27,920
Or it's the idea of okay

401
00:17:27,920 --> 00:17:29,800
what's the probability that someone else

402
00:17:29,800 --> 00:17:31,830
in the room has your same birthday?

403
00:17:31,830 --> 00:17:33,550
Everybody thinks it's really small

404
00:17:33,550 --> 00:17:35,540
but if you actually work
out the mathematics,

405
00:17:35,540 --> 00:17:37,460
it turns out you actually
have a good chance.

406
00:17:37,460 --> 00:17:39,220
If you're in a room with over 40 people

407
00:17:39,220 --> 00:17:40,290
you have an extremely high chance

408
00:17:40,290 --> 00:17:44,619
of finding someone else
with your same birthday.

409
00:17:44,619 --> 00:17:49,200
So I've actually listed out.

410
00:17:49,200 --> 00:17:51,910
This is courtesy of of a Wikipedia

411
00:17:51,910 --> 00:17:55,330
and Mr. Quine.

412
00:17:55,330 --> 00:17:59,419
We have sort of three variants of...

413
00:17:59,419 --> 00:18:01,140
Oops.

414
00:18:01,140 --> 00:18:05,140
You have three variants of paradoxes.

415
00:18:09,580 --> 00:18:12,140
This is a veridical.

416
00:18:12,140 --> 00:18:16,140
And these are things which are true

417
00:18:17,070 --> 00:18:21,070
but they seem paradoxical at first.

418
00:18:22,039 --> 00:18:26,030
There's falsidical.

419
00:18:28,919 --> 00:18:30,730
And I'll give an example of each of these.

420
00:18:30,730 --> 00:18:32,250
And then kind of the classic,

421
00:18:32,250 --> 00:18:34,000
the one which we are
gonna be interested in,

422
00:18:34,000 --> 00:18:38,090
and these are real
paradoxes, are antinomies.

423
00:18:38,090 --> 00:18:42,090
Give me an example of
another classic paradox

424
00:18:43,070 --> 00:18:45,360
and one which is visited
in a Godel, Escher, Bach

425
00:18:45,360 --> 00:18:48,040
very early on.

426
00:18:48,040 --> 00:18:49,460
It's called Zeno's paradox

427
00:18:49,460 --> 00:18:51,840
and the idea is if I want to get

428
00:18:51,840 --> 00:18:53,700
from here to my laptop,

429
00:18:53,700 --> 00:18:57,700
I first need to walk
halfway across the distance.

430
00:18:59,649 --> 00:19:02,750
And then if I want to walk
the remaining distance,

431
00:19:02,750 --> 00:19:04,550
I need to walk half of that.

432
00:19:04,550 --> 00:19:07,550
If I want to walk the remaining distance,

433
00:19:07,550 --> 00:19:08,990
I need to walk half of that.

434
00:19:08,990 --> 00:19:11,260
And then half of that, half of that,

435
00:19:11,260 --> 00:19:14,010
and eventually I get stuck
in this infinite loop

436
00:19:14,010 --> 00:19:17,900
where it seems like I'm
not getting to my laptop.

437
00:19:17,900 --> 00:19:20,520
A variant of this paradox is the idea

438
00:19:20,520 --> 00:19:22,420
that if I even want to move at all,

439
00:19:22,420 --> 00:19:26,529
if my atoms want to pass in space,

440
00:19:26,529 --> 00:19:29,920
first they have to go half way,

441
00:19:29,920 --> 00:19:31,160
but before it can go half way,

442
00:19:31,160 --> 00:19:33,110
it's got to go half way of that half,

443
00:19:33,110 --> 00:19:34,550
and half way to that half

444
00:19:34,550 --> 00:19:35,970
and then half of that half.

445
00:19:35,970 --> 00:19:38,370
So Zeno back in Greece actually used this

446
00:19:38,370 --> 00:19:40,530
to prove that motion was impossible

447
00:19:40,530 --> 00:19:41,960
and that any motion we saw in the universe

448
00:19:41,960 --> 00:19:43,880
was an illusion.

449
00:19:43,880 --> 00:19:48,250
So it's weird.

450
00:19:48,250 --> 00:19:49,530
Why?

451
00:19:49,530 --> 00:19:50,710
And nobody really could answer Zeno

452
00:19:50,710 --> 00:19:52,520
for the longest time.

453
00:19:52,520 --> 00:19:54,890
But then it took
essentially the development

454
00:19:54,890 --> 00:19:58,160
of the understanding
of limits and calculus

455
00:19:58,160 --> 00:19:59,970
to really get an idea of why

456
00:19:59,970 --> 00:20:01,770
this wasn't paradoxical,

457
00:20:01,770 --> 00:20:02,830
what rigorously did we mean

458
00:20:02,830 --> 00:20:05,530
by an infinite number of steps.

459
00:20:05,530 --> 00:20:08,550
How could we actually get across the room?

460
00:20:08,550 --> 00:20:09,550
it seemed paradoxical

461
00:20:09,550 --> 00:20:10,590
but we knew it had to be true.

462
00:20:10,590 --> 00:20:14,530
We knew motion had to be possible.

463
00:20:14,530 --> 00:20:16,610
I'm sure when you're all were younger

464
00:20:16,610 --> 00:20:18,200
or even now you've seen all sorts

465
00:20:18,200 --> 00:20:20,830
of kind of falsidical paradoxes

466
00:20:20,830 --> 00:20:24,570
where somebody will
write out a string of...

467
00:20:24,570 --> 00:20:28,570
If you take one minus
one plus one minus one,

468
00:20:29,409 --> 00:20:31,110
dot, dot, dot.

469
00:20:31,110 --> 00:20:33,000
And the person convinces you,

470
00:20:33,000 --> 00:20:36,110
well look if you look in groups of this,

471
00:20:36,110 --> 00:20:37,110
these are all zeros.

472
00:20:37,110 --> 00:20:39,790
So if you just add a
bunch of zeros together,

473
00:20:39,790 --> 00:20:42,100
this is necessarily zero.

474
00:20:42,100 --> 00:20:43,480
But I mean this is an
infinite string right

475
00:20:43,480 --> 00:20:45,710
and we can repeat the pattern.

476
00:20:45,710 --> 00:20:50,000
What happens if we add a one right?

477
00:20:50,000 --> 00:20:52,280
So sometimes we get
these weird conclusions

478
00:20:52,280 --> 00:20:54,400
where zero equals one.

479
00:20:54,400 --> 00:20:55,850
And they're usually built on kind of

480
00:20:55,850 --> 00:20:58,660
doing something illegal
involving infinities

481
00:20:58,660 --> 00:21:01,050
and infinity is going to
be a very important concept

482
00:21:01,050 --> 00:21:03,950
that we'll encounter again and again.

483
00:21:03,950 --> 00:21:08,350
Finally, the antinomy.

484
00:21:08,350 --> 00:21:11,340
These are the important
paradoxes to think about.

485
00:21:11,340 --> 00:21:12,340
I once went out to dinner

486
00:21:12,340 --> 00:21:13,350
with a bunch of mathematicians.

487
00:21:13,350 --> 00:21:15,490
I don't know how I ended up in that,

488
00:21:15,490 --> 00:21:18,760
but let me tell you it
was kind of frightening.

489
00:21:18,760 --> 00:21:21,260
And there was this Korean mathematician

490
00:21:21,260 --> 00:21:25,110
who said well you know
most these questions

491
00:21:25,110 --> 00:21:26,110
don't even matter.

492
00:21:26,110 --> 00:21:27,110
We don't we don't understand

493
00:21:27,110 --> 00:21:29,450
some of the most fundamental things.

494
00:21:29,450 --> 00:21:32,610
And the thing he was most interested in

495
00:21:32,610 --> 00:21:35,080
and I think which bothers
mathematicians the most

496
00:21:35,080 --> 00:21:39,100
is the antinomy of the liar

497
00:21:39,100 --> 00:21:43,000
and Russell's paradox.

498
00:21:43,000 --> 00:21:44,630
So the liar's paradox,

499
00:21:44,630 --> 00:21:48,010
you probably have heard before.

500
00:21:48,010 --> 00:21:50,980
And it's based on actually
a biblical reference

501
00:21:50,980 --> 00:21:54,280
but it essentially says

502
00:21:54,280 --> 00:21:58,280
this sentence is not true.

503
00:22:07,200 --> 00:22:10,970
So is it true or is it not true?

504
00:22:10,970 --> 00:22:14,970
Well if it's true then it says of itself

505
00:22:15,840 --> 00:22:17,510
that it's not true.

506
00:22:17,510 --> 00:22:19,830
So true implies not true.

507
00:22:19,830 --> 00:22:20,890
Contradiction.

508
00:22:20,890 --> 00:22:21,890
So if it's not true,

509
00:22:21,890 --> 00:22:24,080
then we know that we believe

510
00:22:24,080 --> 00:22:25,470
in the law of the excluded middle,

511
00:22:25,470 --> 00:22:26,610
which means that things have to either

512
00:22:26,610 --> 00:22:30,150
be true or not true that
it's a negation is true.

513
00:22:30,150 --> 00:22:31,200
So if it's not true,

514
00:22:31,200 --> 00:22:34,970
then the sentence is true.

515
00:22:34,970 --> 00:22:37,030
So not true implies true.

516
00:22:37,030 --> 00:22:39,910
So we're stuck.

517
00:22:39,910 --> 00:22:42,620
The liar paradox still hounds us today.

518
00:22:42,620 --> 00:22:44,530
Unlike Zeno's paradox,

519
00:22:44,530 --> 00:22:48,530
it hasn't been solved.

520
00:22:49,700 --> 00:22:53,420
We still don't know how to deal with it.

521
00:22:53,420 --> 00:22:56,820
And when we talk about Godel's theorem,

522
00:22:56,820 --> 00:22:58,230
the way he proves his result

523
00:22:58,230 --> 00:22:59,910
is actually going to be intimately linked

524
00:22:59,910 --> 00:23:02,300
with a variant on this.

525
00:23:02,300 --> 00:23:04,220
So instead of saying I'm not true,

526
00:23:04,220 --> 00:23:07,530
it's gonna say I'm not provable.

527
00:23:07,530 --> 00:23:09,870
And that's going to be
a very interesting idea.

528
00:23:09,870 --> 00:23:12,210
And we'll explore that a little bit later.

529
00:23:12,210 --> 00:23:14,860
The other antinomy I want to look at

530
00:23:14,860 --> 00:23:16,850
is Russell's paradox,

531
00:23:16,850 --> 00:23:19,880
also known as the barber's paradox.

532
00:23:19,880 --> 00:23:21,250
And that's how I'm gonna tell it.

533
00:23:21,250 --> 00:23:22,800
It's the barber's paradox.

534
00:23:22,800 --> 00:23:26,800
I think it's a little more friendly.

535
00:23:29,809 --> 00:23:32,480
So you have a town

536
00:23:32,480 --> 00:23:36,659
and there's this male barber.

537
00:23:36,659 --> 00:23:38,120
And he abides by the rule

538
00:23:38,120 --> 00:23:40,120
that he shaves all people

539
00:23:40,120 --> 00:23:44,120
and only people who
don't shave themselves.

540
00:23:45,139 --> 00:23:48,610
So what does the barber do

541
00:23:48,610 --> 00:23:51,330
when his beard is
getting as thick as mine?

542
00:23:51,330 --> 00:23:54,670
Does he shave himself or does he not?

543
00:23:54,670 --> 00:23:55,730
Well let's see,

544
00:23:55,730 --> 00:23:57,350
so by definition the barber

545
00:23:57,350 --> 00:23:58,610
only shaves those people

546
00:23:58,610 --> 00:24:01,030
who don't shave themselves.

547
00:24:01,030 --> 00:24:02,520
So if he shaves himself,

548
00:24:02,520 --> 00:24:03,770
then he doesn't.

549
00:24:03,770 --> 00:24:05,710
And if he doesn't shave himself,

550
00:24:05,710 --> 00:24:08,540
then by definition he must shave himself.

551
00:24:08,540 --> 00:24:12,250
A variant of this is which,

552
00:24:12,250 --> 00:24:15,070
which was coined by both Bertrand Russell,

553
00:24:15,070 --> 00:24:17,500
Cambridge mathematician and philosopher,

554
00:24:17,500 --> 00:24:21,850
and Zermelo, great German logician,

555
00:24:21,850 --> 00:24:25,850
is the idea that you can consider the set,

556
00:24:27,369 --> 00:24:31,460
let's call it omega,

557
00:24:31,460 --> 00:24:35,460
which contains all sets

558
00:24:36,869 --> 00:24:40,860
that are members of themselves.

559
00:24:52,389 --> 00:24:55,340
So remember a set is just
a collection of objects.

560
00:24:55,340 --> 00:24:56,760
And mathematicians really believe

561
00:24:56,760 --> 00:24:58,970
that set theory was going to be

562
00:24:58,970 --> 00:25:01,300
what gave mathematics its ultimate,

563
00:25:01,300 --> 00:25:04,360
sure and logical foundation.

564
00:25:04,360 --> 00:25:07,260
So let's give an example of a set,

565
00:25:07,260 --> 00:25:10,120
which contains itself.

566
00:25:10,120 --> 00:25:12,100
So let's think of the set of all things,

567
00:25:12,100 --> 00:25:15,270
which aren't Joan of Arc.

568
00:25:15,270 --> 00:25:17,580
Well sets aren't people.

569
00:25:17,580 --> 00:25:19,820
I mean they're people not sets.

570
00:25:19,820 --> 00:25:22,390
So that set of all things,

571
00:25:22,390 --> 00:25:23,650
which aren't Joan of Arc

572
00:25:23,650 --> 00:25:26,640
includes itself because a
set can never be a person.

573
00:25:26,640 --> 00:25:31,279
So that set is contained in itself.

574
00:25:31,279 --> 00:25:33,760
So we have a bunch of things in here,

575
00:25:33,760 --> 00:25:35,130
which are sets,

576
00:25:35,130 --> 00:25:36,610
which aren't members of themselves.

577
00:25:36,610 --> 00:25:40,610
And then we ask the question is omega

578
00:25:43,489 --> 00:25:44,480
an element of itself?

579
00:25:44,480 --> 00:25:48,480
And this means is in...

580
00:25:52,059 --> 00:25:56,050
Well if omega contains itself,

581
00:25:57,929 --> 00:25:59,280
but Omega by definition

582
00:25:59,280 --> 00:26:00,280
only contains things,

583
00:26:00,280 --> 00:26:01,280
which don't contain themselves.

584
00:26:01,280 --> 00:26:02,870
So it can't contain itself.

585
00:26:02,870 --> 00:26:05,480
Well, if it can't contain itself,

586
00:26:05,480 --> 00:26:07,030
it doesn't contain itself

587
00:26:07,030 --> 00:26:10,510
and that means it should contain itself.

588
00:26:10,510 --> 00:26:12,400
Contradiction.

589
00:26:12,400 --> 00:26:14,900
This really, really bothered
a lot of mathematicians

590
00:26:14,900 --> 00:26:17,710
for a long time.

591
00:26:17,710 --> 00:26:19,140
And it's an exact variant

592
00:26:19,140 --> 00:26:21,250
on the barbarous paradox.

593
00:26:21,250 --> 00:26:23,620
So this isn't kind of a interesting things

594
00:26:23,620 --> 00:26:25,140
to play around with.

595
00:26:25,140 --> 00:26:27,170
Finally, is the concept of infinity.

596
00:26:27,170 --> 00:26:28,630
I can't really talk too much about it.

597
00:26:28,630 --> 00:26:30,730
We're gonna look at it more.

598
00:26:30,730 --> 00:26:32,010
But I want to introduce you guys

599
00:26:32,010 --> 00:26:34,840
to the idea that there are
multiple types of infinity.

600
00:26:34,840 --> 00:26:37,070
So you have the integers.

601
00:26:37,070 --> 00:26:40,030
And you also have the real numbers .

602
00:26:40,030 --> 00:26:43,640
And it is true that you cannot

603
00:26:43,640 --> 00:26:45,510
create a direct link.

604
00:26:45,510 --> 00:26:47,170
You can't match every real number

605
00:26:47,170 --> 00:26:48,170
like 0.33333.

606
00:26:48,170 --> 00:26:51,450
Well 0.35 something, random, pi.

607
00:26:51,450 --> 00:26:54,230
That's the big pi.

608
00:26:54,230 --> 00:26:56,370
You can't put pi,

609
00:26:56,370 --> 00:27:00,320
directly in connection
with a natural number,

610
00:27:00,320 --> 00:27:02,110
and integer.

611
00:27:02,110 --> 00:27:03,130
And this is kind of famous

612
00:27:03,130 --> 00:27:05,000
Cantor's diagonalization argument.

613
00:27:05,000 --> 00:27:07,550
So somehow there are
different degrees of infinity

614
00:27:07,550 --> 00:27:11,580
and the real numbers is a
higher degree of infinity.

615
00:27:11,580 --> 00:27:15,580
So that's that's an important
thing to think about.

616
00:27:16,470 --> 00:27:20,470
Now we're going to jump ahead

617
00:27:21,350 --> 00:27:23,550
to our last tool for thinking.

618
00:27:23,550 --> 00:27:24,880
And this is going to be the reason

619
00:27:24,880 --> 00:27:29,109
why we ignore the first three chapters

620
00:27:29,109 --> 00:27:31,510
of Godel, Escher, Bach.

621
00:27:31,510 --> 00:27:35,510
And it's the idea of a formal system.

622
00:27:37,210 --> 00:27:40,530
Problem is is that formal
systems are boring.

623
00:27:40,530 --> 00:27:42,040
And Douglas Hofstadter

624
00:27:42,040 --> 00:27:43,830
takes his sweet. Sweet time

625
00:27:43,830 --> 00:27:45,240
in introducing you to the concept

626
00:27:45,240 --> 00:27:48,970
of a formal system.

627
00:27:48,970 --> 00:27:50,160
So I'm going to try to speed things up

628
00:27:50,160 --> 00:27:52,240
because I know you all
are smarter than that.

629
00:27:52,240 --> 00:27:53,300
And you can get through

630
00:27:53,300 --> 00:27:55,510
these concepts very quickly.

631
00:27:55,510 --> 00:27:59,150
We're gonna play a game.

632
00:27:59,150 --> 00:28:01,730
It's called the mu puzzle or M-U.

633
00:28:01,730 --> 00:28:06,080
And the way you play it

634
00:28:06,080 --> 00:28:07,720
is you start with,

635
00:28:07,720 --> 00:28:11,720
you have a bag of three letters.

636
00:28:14,460 --> 00:28:15,900
And you're gonna have a rule.

637
00:28:15,900 --> 00:28:19,710
You're gonna start with...

638
00:28:19,710 --> 00:28:22,440
You pull two letters out

639
00:28:22,440 --> 00:28:24,600
and get M-I.

640
00:28:24,600 --> 00:28:26,710
And we're gonna have four rules.

641
00:28:26,710 --> 00:28:30,500
And these are completely
strict typographical rules

642
00:28:30,500 --> 00:28:34,500
for deriving new things

643
00:28:35,919 --> 00:28:38,410
that we can pull from our bag.

644
00:28:38,410 --> 00:28:42,759
Our first rule

645
00:28:42,759 --> 00:28:45,630
is that if we have an I

646
00:28:45,630 --> 00:28:49,300
So, suppose we have M-I

647
00:28:49,300 --> 00:28:52,650
or we could have anything,

648
00:28:52,650 --> 00:28:54,530
and then an I,

649
00:28:54,530 --> 00:28:57,150
we can tack a U on.

650
00:28:57,150 --> 00:28:58,150
So,

651
00:28:58,150 --> 00:28:59,150
I-U.

652
00:28:59,150 --> 00:29:02,240
So right away we know

653
00:29:02,240 --> 00:29:05,550
that we can create M-I-U.

654
00:29:05,550 --> 00:29:09,550
Our second rule is

655
00:29:12,649 --> 00:29:14,350
suppose we have M

656
00:29:14,350 --> 00:29:17,750
and then a string of
letters that are I's and U's

657
00:29:17,750 --> 00:29:19,720
since they're in our bag of alphabet,

658
00:29:19,720 --> 00:29:22,280
our alphabet here,

659
00:29:22,280 --> 00:29:26,230
then you're gonna get for free M-X-X.

660
00:29:26,230 --> 00:29:27,860
So just as an example,

661
00:29:27,860 --> 00:29:31,640
suppose somehow you had M-I,

662
00:29:31,640 --> 00:29:34,420
which we do.

663
00:29:34,420 --> 00:29:39,200
You're gonna get M-I-I for free.

664
00:29:39,200 --> 00:29:43,200
Third rule suppose you have...

665
00:29:47,860 --> 00:29:49,140
Somewhere along the way you end up

666
00:29:49,140 --> 00:29:50,400
with a cluster of three I's.

667
00:29:50,400 --> 00:29:52,110
They don't have to be at the end.

668
00:29:52,110 --> 00:29:53,950
They can be anywhere.

669
00:29:53,950 --> 00:29:57,480
Just needs to be three I's all together.

670
00:29:57,480 --> 00:29:59,590
And you can replace
all three of those I's.

671
00:29:59,590 --> 00:30:03,590
They're equal to a U.

672
00:30:10,909 --> 00:30:12,480
And our final rule

673
00:30:12,480 --> 00:30:16,480
is that if we have a double pair of U's,

674
00:30:18,190 --> 00:30:19,720
we can drop them

675
00:30:19,720 --> 00:30:23,170
and they just go away.

676
00:30:23,170 --> 00:30:27,539
So somehow if we had M-U-U,

677
00:30:27,539 --> 00:30:29,510
we could just have M.

678
00:30:29,510 --> 00:30:31,870
Now you have these rules.

679
00:30:31,870 --> 00:30:33,670
You have these letters.

680
00:30:33,670 --> 00:30:37,420
You start with one guy.

681
00:30:37,420 --> 00:30:39,140
He's gonna be our axiom.

682
00:30:39,140 --> 00:30:41,640
An axiom is the starting
point for reasoning

683
00:30:41,640 --> 00:30:45,490
for applying these rules.

684
00:30:45,490 --> 00:30:49,820
And the game is can you get M-U?

685
00:30:49,820 --> 00:30:53,100
You starting from M-I

686
00:30:53,100 --> 00:30:55,880
and using only these two four rules,

687
00:30:55,880 --> 00:30:56,880
can you get M-U?

688
00:30:56,880 --> 00:31:00,880
I will give $20 to the first person

689
00:31:03,659 --> 00:31:07,650
who can derive M-U that's in this room,

690
00:31:08,870 --> 00:31:09,940
only applying these four rules

691
00:31:09,940 --> 00:31:13,940
and starting directly from M-I.

692
00:31:14,840 --> 00:31:15,840
Just to give you an idea

693
00:31:15,840 --> 00:31:16,840
of where you might be going,

694
00:31:16,840 --> 00:31:18,780
where you might be playing,

695
00:31:18,780 --> 00:31:20,330
just going off of our rules,

696
00:31:20,330 --> 00:31:21,690
we already saw that if we had M-I,

697
00:31:21,690 --> 00:31:25,950
we can get M-I-U.

698
00:31:25,950 --> 00:31:29,720
We also saw that using
rule two, using rule one,

699
00:31:29,720 --> 00:31:33,720
we can get M-I-I.

700
00:31:37,889 --> 00:31:41,720
We saw if we have anything like that,

701
00:31:41,720 --> 00:31:43,230
we can repeat it twice.

702
00:31:43,230 --> 00:31:45,340
So we get M-I-U-I-U.

703
00:31:45,340 --> 00:31:47,050
That's applying rule two again,

704
00:31:47,050 --> 00:31:50,830
and so on.

705
00:31:50,830 --> 00:31:53,450
Leave this as a puzzle.

706
00:31:53,450 --> 00:31:54,550
Take your time with it.

707
00:31:54,550 --> 00:31:57,240
You'll be working on it for a few hours.

708
00:31:57,240 --> 00:31:59,780
But first person that's in this room

709
00:31:59,780 --> 00:32:02,530
derive M-U from this gets $20.

710
00:32:02,530 --> 00:32:03,530
Yes.

711
00:32:03,530 --> 00:32:07,559
- [Student] Fourth rule
only applies to two U's?

712
00:32:07,559 --> 00:32:11,120
- Yes, fourth rule only
applies to two U's.

713
00:32:11,120 --> 00:32:15,539
So yes if you have to
U's you can remove them.

714
00:32:15,539 --> 00:32:17,390
You can subtract them.

715
00:32:17,390 --> 00:32:19,820
All right and once again,

716
00:32:19,820 --> 00:32:22,500
I do urge everyone to buy the book.

717
00:32:22,500 --> 00:32:25,820
These rules are listed
explicitly in the chapter.

718
00:32:25,820 --> 00:32:27,590
And you might gain some insight

719
00:32:27,590 --> 00:32:31,590
on how to derive what you want here.

720
00:32:38,429 --> 00:32:41,320
So why is this interesting?

721
00:32:41,320 --> 00:32:43,200
I mean we're just playing with letters

722
00:32:43,200 --> 00:32:46,690
and strings and things like that.

723
00:32:46,690 --> 00:32:49,210
Well,

724
00:32:49,210 --> 00:32:50,450
although this seems pretty meaningless

725
00:32:50,450 --> 00:32:53,580
and kind of dumb,

726
00:32:53,580 --> 00:32:55,500
does anybody feel like

727
00:32:55,500 --> 00:32:57,280
when they're just looking at this game,

728
00:32:57,280 --> 00:32:58,280
looking at these rules,

729
00:32:58,280 --> 00:33:00,580
that they're just
essentially playing around

730
00:33:00,580 --> 00:33:03,260
with algebra that they learned you know

731
00:33:03,260 --> 00:33:05,760
in middle school or high school?

732
00:33:05,760 --> 00:33:07,360
That really what we're doing here

733
00:33:07,360 --> 00:33:10,970
is we've got some statements like

734
00:33:10,970 --> 00:33:13,670
two plus two equals four.

735
00:33:13,670 --> 00:33:18,330
And we all learned that we
have a typographical rule

736
00:33:18,330 --> 00:33:20,730
for when we have an equal sign like that,

737
00:33:20,730 --> 00:33:24,250
we can add one to both sides

738
00:33:24,250 --> 00:33:26,140
and preserve equality.

739
00:33:26,140 --> 00:33:29,190
So something, we have

740
00:33:29,190 --> 00:33:33,470
two plus three equals five.

741
00:33:33,470 --> 00:33:36,610
So really what mathematics reduces to

742
00:33:36,610 --> 00:33:40,979
is just playing around
with systems of this form

743
00:33:40,979 --> 00:33:42,400
and applying these rigorous kind of

744
00:33:42,400 --> 00:33:43,570
typographical rules.

745
00:33:43,570 --> 00:33:44,820
Except here there doesn't seem

746
00:33:44,820 --> 00:33:45,830
to be any meaning.

747
00:33:45,830 --> 00:33:47,620
It's just meaningless.

748
00:33:47,620 --> 00:33:48,660
One of the important questions

749
00:33:48,660 --> 00:33:49,660
we're going to address in this class

750
00:33:49,660 --> 00:33:52,000
is how do things gain meaning?

751
00:33:52,000 --> 00:33:55,980
How do we go from meaningless to meaning?

752
00:33:55,980 --> 00:33:57,420
This obviously seems to have meaning

753
00:33:57,420 --> 00:33:59,400
but I want you to ask yourself why.

754
00:33:59,400 --> 00:34:03,400
Kind of before we proceed

755
00:34:05,620 --> 00:34:08,370
it's necessary, it's my duty

756
00:34:08,370 --> 00:34:12,770
to do the boring task of writing down

757
00:34:12,770 --> 00:34:15,590
just a few definitions of things,

758
00:34:15,590 --> 00:34:16,860
which you can call
these so you have words.

759
00:34:16,860 --> 00:34:18,550
So we already saw axiom.

760
00:34:18,550 --> 00:34:20,190
That's a definition.

761
00:34:20,190 --> 00:34:24,030
You call any of these guys a string.

762
00:34:24,030 --> 00:34:28,030
So a string is just any ordered

763
00:34:35,640 --> 00:34:38,470
sequence of

764
00:34:38,470 --> 00:34:42,350
in this case M, I's and U's.

765
00:34:42,350 --> 00:34:46,350
We already met an axiom.

766
00:34:51,670 --> 00:34:55,670
An axiom is a starting point.

767
00:34:57,790 --> 00:34:58,790
It's your first thing

768
00:34:58,790 --> 00:35:01,120
that you can apply the rules to.

769
00:35:01,120 --> 00:35:04,280
So and this actually has a lot to do

770
00:35:04,280 --> 00:35:06,780
with mathematical logic
because in math logic,

771
00:35:06,780 --> 00:35:08,990
the idea is that we start
from really primitive things,

772
00:35:08,990 --> 00:35:10,580
which seem obvious,

773
00:35:10,580 --> 00:35:14,580
like the successor of 0 is 1,

774
00:35:15,970 --> 00:35:17,790
and then we work from that concept,

775
00:35:17,790 --> 00:35:19,150
and we derive all these truths

776
00:35:19,150 --> 00:35:22,780
of number theory and mathematics.

777
00:35:22,780 --> 00:35:25,180
Here your axiom is M-I

778
00:35:25,180 --> 00:35:27,690
and you're trying to prove the theorem

779
00:35:27,690 --> 00:35:31,090
and that's kind of our next guy here.

780
00:35:31,090 --> 00:35:35,090
Well, you're trying to
prove the theorem of M-U.

781
00:35:38,820 --> 00:35:43,540
So a theorem is basically a string

782
00:35:43,540 --> 00:35:47,540
which results at the end of a derivation.

783
00:36:00,660 --> 00:36:02,490
And a derivation is like a proof.

784
00:36:02,490 --> 00:36:04,390
For those of you have done geometry,

785
00:36:04,390 --> 00:36:06,310
when you're saying okay well this triangle

786
00:36:06,310 --> 00:36:07,310
is congruent to this triangle

787
00:36:07,310 --> 00:36:10,670
because of side-angle-side
and things like that,

788
00:36:10,670 --> 00:36:14,230
you're deriving, you're
making rigorous justifications

789
00:36:14,230 --> 00:36:16,280
for your leaps in logic.

790
00:36:16,280 --> 00:36:18,030
So here our rigorous justification

791
00:36:18,030 --> 00:36:21,520
that M-I-U was the theorem was that well

792
00:36:21,520 --> 00:36:24,290
we applied typographical rule number one.

793
00:36:24,290 --> 00:36:26,450
That's a rigorous leap in logic

794
00:36:26,450 --> 00:36:29,870
when we got to this theorem.

795
00:36:29,870 --> 00:36:33,870
And you can just call
these four rules here...

796
00:36:35,630 --> 00:36:39,690
These are rules of inference.

797
00:36:39,690 --> 00:36:43,690
And logic and a lot of things

798
00:36:47,331 --> 00:36:48,530
that you'll play around with

799
00:36:48,530 --> 00:36:50,180
you know eventually on SATs

800
00:36:50,180 --> 00:36:51,180
and things like that

801
00:36:51,180 --> 00:36:53,270
or you know if you have
if you have the statement

802
00:36:53,270 --> 00:36:57,270
that p implies a statement q.

803
00:36:58,220 --> 00:37:02,250
So if it's cloudy then it will rain.

804
00:37:02,250 --> 00:37:05,620
You have that this is
kind of equivalent to,

805
00:37:05,620 --> 00:37:09,620
I should use a different arrow here,

806
00:37:11,140 --> 00:37:15,140
to not q implies not p.

807
00:37:16,950 --> 00:37:19,410
And these are really nice

808
00:37:19,410 --> 00:37:21,220
because they're just
type of graphical rules.

809
00:37:21,220 --> 00:37:24,000
When you see something like when you have,

810
00:37:24,000 --> 00:37:27,050
well I've got M followed
by any string of letters,

811
00:37:27,050 --> 00:37:28,050
well then I can double it.

812
00:37:28,050 --> 00:37:29,460
That's a rule of inference.

813
00:37:29,460 --> 00:37:32,050
Just like this is a rule of inference.

814
00:37:32,050 --> 00:37:33,540
If I have P implies Q,

815
00:37:33,540 --> 00:37:35,770
I can always replace that.

816
00:37:35,770 --> 00:37:39,770
It's completely equivalent
to not q implies not p.

817
00:37:43,220 --> 00:37:44,620
But for those of you
who are scrambling away

818
00:37:44,620 --> 00:37:45,730
because you want $20 really fast,

819
00:37:45,730 --> 00:37:46,770
I want you to take a break

820
00:37:46,770 --> 00:37:48,590
because once again you should focus on

821
00:37:48,590 --> 00:37:53,080
what we're saying right now.

822
00:37:53,080 --> 00:37:54,080
And we're gonna talk a little bit

823
00:37:54,080 --> 00:37:55,320
about jumping outside the system.

824
00:37:55,320 --> 00:37:56,880
This is kind of the cool renegade stuff

825
00:37:56,880 --> 00:37:59,930
that Hofstadter fills his book with.

826
00:37:59,930 --> 00:38:02,160
And it's the idea that

827
00:38:02,160 --> 00:38:03,830
as you're playing around with this,

828
00:38:03,830 --> 00:38:06,000
right now you're just playing a game.

829
00:38:06,000 --> 00:38:07,310
And what mathematicians

830
00:38:07,310 --> 00:38:10,510
and what anybody human does

831
00:38:10,510 --> 00:38:12,430
is when they feel like
they're caught in loops,

832
00:38:12,430 --> 00:38:14,550
just cranking through pages of algebra,

833
00:38:14,550 --> 00:38:16,340
and not getting anywhere,

834
00:38:16,340 --> 00:38:19,870
humans are intelligent enough to stop.

835
00:38:19,870 --> 00:38:21,900
They exit the system

836
00:38:21,900 --> 00:38:24,510
and they say,

837
00:38:24,510 --> 00:38:26,670
I don't know I don't think
this is gonna go anywhere.

838
00:38:26,670 --> 00:38:30,670
Or well let me think
about why I'm not getting

839
00:38:31,580 --> 00:38:35,150
or how might I get M-U.

840
00:38:35,150 --> 00:38:36,430
You know maybe it has something to do

841
00:38:36,430 --> 00:38:40,430
with numbers of I's U's
or things like that.

842
00:38:42,170 --> 00:38:43,170
I'm not sure.

843
00:38:43,170 --> 00:38:45,160
You start doing what I
like to call meta-thinking.

844
00:38:45,160 --> 00:38:48,620
You're not thinking in the system,

845
00:38:48,620 --> 00:38:50,000
applying typographical rules,

846
00:38:50,000 --> 00:38:51,800
applying rules of inference

847
00:38:51,800 --> 00:38:54,570
to existing strings, axioms,

848
00:38:54,570 --> 00:38:57,240
and getting theorems.

849
00:38:57,240 --> 00:38:58,340
That's thinking inside the system.

850
00:38:58,340 --> 00:39:00,050
That's just thinking.

851
00:39:00,050 --> 00:39:01,880
Meta-thinking involves you leaping

852
00:39:01,880 --> 00:39:02,880
outside the system

853
00:39:02,880 --> 00:39:04,680
and making judgments about it,

854
00:39:04,680 --> 00:39:07,930
thoughts which cannot be expressed

855
00:39:07,930 --> 00:39:09,530
as any just normal typographical role

856
00:39:09,530 --> 00:39:10,610
within the system.

857
00:39:10,610 --> 00:39:12,090
You're doing meta-thinking.

858
00:39:12,090 --> 00:39:16,090
One of my favorite parts of this section

859
00:39:20,780 --> 00:39:24,330
in Godel, Escher, Bach is when

860
00:39:24,330 --> 00:39:25,330
Hofstadter says,

861
00:39:25,330 --> 00:39:29,490
and, once again stop trying to derive M-U.

862
00:39:29,490 --> 00:39:33,490
Try to turn the page 24
in your lecture notes.

863
00:39:34,410 --> 00:39:36,640
Oops.

864
00:39:36,640 --> 00:39:41,100
No worries.

865
00:39:41,100 --> 00:39:42,100
Page 24.

866
00:39:42,100 --> 00:39:44,580
Hofstadter kind of uses this

867
00:39:44,580 --> 00:39:47,720
as like as a life lesson he says look,

868
00:39:47,720 --> 00:39:49,750
"Of course there are cases

869
00:39:49,750 --> 00:39:51,150
"when only a rare individual

870
00:39:51,150 --> 00:39:53,880
"will have the vision to perceive a system

871
00:39:53,880 --> 00:39:55,940
"which governs many people's lives,

872
00:39:55,940 --> 00:39:57,630
"a system which had never before

873
00:39:57,630 --> 00:39:59,560
"even been recognized as a system.

874
00:39:59,560 --> 00:40:01,720
"then such people often devote their lives

875
00:40:01,720 --> 00:40:02,800
"to convincing other people

876
00:40:02,800 --> 00:40:04,580
"that the system really is there

877
00:40:04,580 --> 00:40:07,390
"and that it ought to be exited from."

878
00:40:07,390 --> 00:40:11,040
It's as if our social customs

879
00:40:11,040 --> 00:40:12,800
and our kind of cultures

880
00:40:12,800 --> 00:40:15,260
are really just formal games.

881
00:40:15,260 --> 00:40:16,260
You know we say hello.

882
00:40:16,260 --> 00:40:17,380
We shake your hand.

883
00:40:17,380 --> 00:40:19,760
That's an instance of a formal rule,

884
00:40:19,760 --> 00:40:21,120
which we all follow.

885
00:40:21,120 --> 00:40:22,870
But you know every once in a while

886
00:40:22,870 --> 00:40:24,160
you get somebody who says,

887
00:40:24,160 --> 00:40:25,910
ah I want to shake your hand.

888
00:40:25,910 --> 00:40:29,910
I'm gonna exit the
handshaking formal system.

889
00:40:31,200 --> 00:40:32,200
But of course, they're much more

890
00:40:32,200 --> 00:40:33,210
radical examples of this,

891
00:40:33,210 --> 00:40:36,480
like I said Karl Marx and communism.

892
00:40:36,480 --> 00:40:38,090
You know he viewed this idea

893
00:40:38,090 --> 00:40:40,250
of like well look you've got these people

894
00:40:40,250 --> 00:40:43,150
who are collecting money and property.

895
00:40:43,150 --> 00:40:45,250
And you know they're getting someone else

896
00:40:45,250 --> 00:40:46,250
to do all the work.

897
00:40:46,250 --> 00:40:48,880
And they're pressing this
whole class of people.

898
00:40:48,880 --> 00:40:51,530
Can't people recognize the system?

899
00:40:51,530 --> 00:40:52,990
So then people like Karl Marx

900
00:40:52,990 --> 00:40:54,960
and Fred Engles like start writing

901
00:40:54,960 --> 00:40:56,520
and pamphlets encouraging people

902
00:40:56,520 --> 00:40:58,990
to overthrow governments etc.,

903
00:40:58,990 --> 00:41:00,240
because they viewed a system.

904
00:41:00,240 --> 00:41:02,780
They said look we need to
exit the thinking, system.

905
00:41:02,780 --> 00:41:03,960
We're intelligent beings.

906
00:41:03,960 --> 00:41:05,710
We can think on a higher level.

907
00:41:05,710 --> 00:41:07,160
Of course, I'm not trying

908
00:41:07,160 --> 00:41:08,780
to promote communism here.

909
00:41:08,780 --> 00:41:10,140
I'm just showing you an example

910
00:41:10,140 --> 00:41:14,750
of historical interest.

911
00:41:14,750 --> 00:41:16,860
You know anarchism, socialism today.

912
00:41:16,860 --> 00:41:17,860
Working peoples.

913
00:41:17,860 --> 00:41:19,160
The media.

914
00:41:19,160 --> 00:41:20,160
Nowadays I think it's one

915
00:41:20,160 --> 00:41:22,230
of most popular things for people to say

916
00:41:22,230 --> 00:41:23,720
is like well you know it's just the media

917
00:41:23,720 --> 00:41:24,990
trying to do this.

918
00:41:24,990 --> 00:41:26,180
Before we used to never like

919
00:41:26,180 --> 00:41:28,330
just refer to this entity as the media.

920
00:41:28,330 --> 00:41:29,330
The media is trying

921
00:41:29,330 --> 00:41:30,330
to obscure our understanding of this.

922
00:41:30,330 --> 00:41:33,890
The media is trying to scare us.

923
00:41:33,890 --> 00:41:35,040
Also you know the government.

924
00:41:35,040 --> 00:41:38,650
The government's responsible.

925
00:41:38,650 --> 00:41:40,040
Of course a classic example

926
00:41:40,040 --> 00:41:42,470
is also what Karl Marx said.

927
00:41:42,470 --> 00:41:46,140
The church it's the opiate
of the masses he said.

928
00:41:46,140 --> 00:41:48,120
And also school.

929
00:41:48,120 --> 00:41:49,550
School's my favorite example

930
00:41:49,550 --> 00:41:50,550
of you know a system

931
00:41:50,550 --> 00:41:52,450
which people have
encouraged you to exit from.

932
00:41:52,450 --> 00:41:54,540
It's like well you know it's just daycare

933
00:41:54,540 --> 00:41:55,540
that we have.

934
00:41:55,540 --> 00:41:56,990
And we don't actually want kids

935
00:41:56,990 --> 00:41:59,180
to learn and grow up.

936
00:41:59,180 --> 00:42:00,180
And this has inspired a lot

937
00:42:00,180 --> 00:42:01,890
of new free thinking educational movements

938
00:42:01,890 --> 00:42:02,990
like the Montessoris

939
00:42:02,990 --> 00:42:05,390
and things like that.

940
00:42:05,390 --> 00:42:07,430
And I really want you guys to think about

941
00:42:07,430 --> 00:42:08,430
in your daily actions

942
00:42:08,430 --> 00:42:09,430
and my living perhaps you know

943
00:42:09,430 --> 00:42:10,530
in a kind of formal system

944
00:42:10,530 --> 00:42:13,060
which is acting in a similar way.

945
00:42:13,060 --> 00:42:14,320
Try to do some meta-thinking,

946
00:42:14,320 --> 00:42:15,820
thinking on a higher level

947
00:42:15,820 --> 00:42:19,790
and is it worth exiting that system?

948
00:42:19,790 --> 00:42:21,820
Hofstadter kind of classifies

949
00:42:21,820 --> 00:42:24,470
these three levels of thinking.

950
00:42:24,470 --> 00:42:27,430
And he likes to call it a mechanical mode,

951
00:42:27,430 --> 00:42:29,220
when you're doing the normal

952
00:42:29,220 --> 00:42:32,890
games of the system, an intelligent mode,

953
00:42:32,890 --> 00:42:33,890
and then just an un-mode.

954
00:42:33,890 --> 00:42:36,940
Un-mode's when you just
kind of reject the system.

955
00:42:36,940 --> 00:42:39,660
He calls it the Zen way
of approaching things.

956
00:42:39,660 --> 00:42:41,120
And this is something we like

957
00:42:41,120 --> 00:42:45,120
to talk about a little more.

958
00:42:46,420 --> 00:42:48,000
I want to quickly introduce you

959
00:42:48,000 --> 00:42:50,530
to another...

960
00:42:50,530 --> 00:42:53,800
Well, first of all I want to talk about

961
00:42:53,800 --> 00:42:57,670
a concept of what we've
previously mentioned

962
00:42:57,670 --> 00:42:59,000
is you know we're eventually
going to be talking

963
00:42:59,000 --> 00:43:01,230
about artificial intelligence.

964
00:43:01,230 --> 00:43:04,770
And it's weird because humans

965
00:43:04,770 --> 00:43:08,210
really like to say that
their thoughts are logical.

966
00:43:08,210 --> 00:43:10,870
We like to say that

967
00:43:10,870 --> 00:43:13,150
we do think in this manner

968
00:43:13,150 --> 00:43:16,420
but a lot of times we don't.

969
00:43:16,420 --> 00:43:18,530
We like to use kind of just inference

970
00:43:18,530 --> 00:43:21,450
about just collective events.

971
00:43:21,450 --> 00:43:25,220
Like one of our favorite tools
of thinking is induction.

972
00:43:25,220 --> 00:43:26,250
Well you know the sun has rised

973
00:43:26,250 --> 00:43:27,680
all these previous days.

974
00:43:27,680 --> 00:43:29,930
Sure it'll rise tomorrow.

975
00:43:29,930 --> 00:43:32,340
And there's no real formal

976
00:43:32,340 --> 00:43:33,610
line of logic that's saying

977
00:43:33,610 --> 00:43:35,040
that well sun rised yesterday

978
00:43:35,040 --> 00:43:37,570
and thus it will rise tomorrow.

979
00:43:37,570 --> 00:43:38,570
And I want you to think

980
00:43:38,570 --> 00:43:40,490
of whether or not human, our thoughts,

981
00:43:40,490 --> 00:43:42,730
are actually just computations

982
00:43:42,730 --> 00:43:44,810
in a formal system much like M-I-U

983
00:43:44,810 --> 00:43:48,810
or p implies q and things like that.

984
00:43:50,220 --> 00:43:52,740
And that's going to bring me

985
00:43:52,740 --> 00:43:54,130
to another formal system

986
00:43:54,130 --> 00:43:56,850
which I have to mention just because

987
00:43:56,850 --> 00:43:59,770
in chapter four, he's
going to refer to it.

988
00:43:59,770 --> 00:44:03,770
And it's going to lead us to this

989
00:44:04,660 --> 00:44:06,660
kind of interesting line of dialogue

990
00:44:06,660 --> 00:44:08,750
of when a formal system

991
00:44:08,750 --> 00:44:11,860
with meaningless symbols

992
00:44:11,860 --> 00:44:13,820
gains meaning.

993
00:44:13,820 --> 00:44:16,670
And it's called the p q system.

994
00:44:16,670 --> 00:44:20,860
We're gonna have three new letters.

995
00:44:20,860 --> 00:44:22,240
Well, three new characters.

996
00:44:22,240 --> 00:44:26,230
It's now going to be p q and hyphen.

997
00:44:26,230 --> 00:44:30,230
And you've actually got an infinite number

998
00:44:31,380 --> 00:44:32,800
of axioms here.

999
00:44:32,800 --> 00:44:36,800
And you've got a definition.

1000
00:44:41,360 --> 00:44:43,250
And that's that

1001
00:44:43,250 --> 00:44:46,980
if you know xp hyphen,

1002
00:44:46,980 --> 00:44:50,410
I'm going to kind of make sure

1003
00:44:50,410 --> 00:44:53,640
I have just an underlined p,

1004
00:44:53,640 --> 00:44:57,780
q x.

1005
00:44:57,780 --> 00:45:01,780
And this is going to be an axiom.

1006
00:45:08,020 --> 00:45:10,790
Whenever x is just a string of hyphens.

1007
00:45:10,790 --> 00:45:14,790
So it's just some string of hyphens.

1008
00:45:16,120 --> 00:45:17,160
So what's this saying?

1009
00:45:17,160 --> 00:45:21,160
It's saying that well if you
have something like this,

1010
00:45:23,900 --> 00:45:26,010
well x here was two hyphens.

1011
00:45:26,010 --> 00:45:28,510
So we know that

1012
00:45:28,510 --> 00:45:29,510
that's an axiom.

1013
00:45:29,510 --> 00:45:33,510
All right it's a little
different than M-I-U.

1014
00:45:34,860 --> 00:45:38,030
Seems just as meaningless.

1015
00:45:38,030 --> 00:45:39,780
And we're gonna have

1016
00:45:39,780 --> 00:45:41,770
different forms for manipulating

1017
00:45:41,770 --> 00:45:44,650
and playing around with this.

1018
00:45:44,650 --> 00:45:48,650
And one rule

1019
00:45:50,370 --> 00:45:53,880
is that if you have x, y, and z,

1020
00:45:53,880 --> 00:45:58,170
which are just hyphen strings,

1021
00:45:58,170 --> 00:46:01,820
xpyqz,

1022
00:46:01,820 --> 00:46:06,490
then you can derive,

1023
00:46:06,490 --> 00:46:10,080
you're given for free,

1024
00:46:10,080 --> 00:46:14,080
the statement xpy-qz-.

1025
00:46:23,590 --> 00:46:26,250
Seems meaningless

1026
00:46:26,250 --> 00:46:30,250
but what does it remind you of?

1027
00:46:33,610 --> 00:46:34,620
We've got this axiom.

1028
00:46:34,620 --> 00:46:37,670
We in fact have a whole
infinite list of axioms

1029
00:46:37,670 --> 00:46:39,900
and maybe you've noticed

1030
00:46:39,900 --> 00:46:43,140
that we've got two hyphens here

1031
00:46:43,140 --> 00:46:47,140
and one hyphen here.

1032
00:46:48,260 --> 00:46:52,820
Got three hyphens here.

1033
00:46:52,820 --> 00:46:56,160
And what does this do?

1034
00:46:56,160 --> 00:46:58,120
(student speaking)

1035
00:46:58,120 --> 00:47:01,870
Yeah exactly I mean
what it does is it says

1036
00:47:01,870 --> 00:47:04,150
that well if this works right,

1037
00:47:04,150 --> 00:47:05,330
so, let's let's apply.

1038
00:47:05,330 --> 00:47:09,330
This rule here and we'll
apply this rule here.

1039
00:47:10,420 --> 00:47:11,420
So we can take this

1040
00:47:11,420 --> 00:47:15,420
and get for free that --p-,

1041
00:47:18,450 --> 00:47:20,370
we can add another hyphen,

1042
00:47:20,370 --> 00:47:21,450
q.

1043
00:47:21,450 --> 00:47:23,930
And we had three hyphens here.

1044
00:47:23,930 --> 00:47:26,660
But this rule says we can
tack on another hyphen.

1045
00:47:26,660 --> 00:47:29,360
What does that say?

1046
00:47:29,360 --> 00:47:33,360
This seems to say that
two plus two equals four.

1047
00:47:38,670 --> 00:47:40,360
So I want you to realize
that the symbolism

1048
00:47:40,360 --> 00:47:42,230
which mathematicians have been using

1049
00:47:42,230 --> 00:47:43,510
and what you've grown up learning

1050
00:47:43,510 --> 00:47:45,630
is just short hand.

1051
00:47:45,630 --> 00:47:47,320
It's meaningless notation.

1052
00:47:47,320 --> 00:47:49,270
(student speaking)

1053
00:47:49,270 --> 00:47:52,190
well, yeah no.

1054
00:47:52,190 --> 00:47:53,190
What I meant to say here

1055
00:47:53,190 --> 00:47:55,540
is that we seem to be inferring this rule

1056
00:47:55,540 --> 00:47:59,540
that hyphen string one
plus hyphen string two

1057
00:48:01,820 --> 00:48:06,280
always equals hyphen string of three.

1058
00:48:06,280 --> 00:48:08,010
And then see just one here refers

1059
00:48:08,010 --> 00:48:10,180
to a whole string of hyphens

1060
00:48:10,180 --> 00:48:12,120
and two refers to a string of hyphens,

1061
00:48:12,120 --> 00:48:13,300
like y here.

1062
00:48:13,300 --> 00:48:16,810
Or better yet, I could say
x plus y equals z here.

1063
00:48:16,810 --> 00:48:21,070
And what makes this system different

1064
00:48:21,070 --> 00:48:23,800
than M-I-U?

1065
00:48:23,800 --> 00:48:25,110
Does anyone have any ideas?

1066
00:48:25,110 --> 00:48:27,650
Why do you suddenly care a little more

1067
00:48:27,650 --> 00:48:29,290
about this system than M-I-U,

1068
00:48:29,290 --> 00:48:30,700
other than the fact that you have 20 bucks

1069
00:48:30,700 --> 00:48:34,700
going online for deriving M-I-U?

1070
00:48:37,890 --> 00:48:41,890
anybody?

1071
00:48:43,770 --> 00:48:45,130
What about this fact that

1072
00:48:45,130 --> 00:48:48,700
I've just kind of showed
you this equivalence here?

1073
00:48:48,700 --> 00:48:49,700
And you know instead

1074
00:48:49,700 --> 00:48:53,720
of applying these kind
of typographical rules,

1075
00:48:53,720 --> 00:48:54,920
I've showed you that

1076
00:48:54,920 --> 00:48:56,260
well you can also take this

1077
00:48:56,260 --> 00:48:58,680
as two plus two equals four.

1078
00:48:58,680 --> 00:49:00,880
And then you're gonna say aha,

1079
00:49:00,880 --> 00:49:02,240
well now I can do all sorts of things

1080
00:49:02,240 --> 00:49:06,420
like now that I discovered the meaning

1081
00:49:06,420 --> 00:49:09,810
of the p, q, hyphen system, I can go ahead

1082
00:49:09,810 --> 00:49:12,450
and just create all sorts of new theorems

1083
00:49:12,450 --> 00:49:17,220
and starting from any of our axioms.

1084
00:49:17,220 --> 00:49:19,720
And you might even be tempted to say,

1085
00:49:19,720 --> 00:49:22,290
well I know what's obvious.

1086
00:49:22,290 --> 00:49:26,290
I know that two plus
two plus two equals six.

1087
00:49:28,230 --> 00:49:30,050
And I've discovered this isomorphism

1088
00:49:30,050 --> 00:49:32,910
between p's and q's

1089
00:49:32,910 --> 00:49:37,350
and pluses and equal signs.

1090
00:49:37,350 --> 00:49:41,350
So I'm tempted to say that -
- p - - p - - q - - - - - -,

1091
00:49:47,620 --> 00:49:51,620
that's a lot of hyphens.

1092
00:49:54,040 --> 00:49:56,050
What's wrong with this?

1093
00:49:56,050 --> 00:50:00,050
Does anyone see a problem?

1094
00:50:03,800 --> 00:50:05,350
Yes?

1095
00:50:05,350 --> 00:50:06,900
Exactly.

1096
00:50:06,900 --> 00:50:11,540
Exactly, it doesn't follow the rule.

1097
00:50:11,540 --> 00:50:13,160
The rules I told you in the axioms

1098
00:50:13,160 --> 00:50:14,580
which you start from,

1099
00:50:14,580 --> 00:50:17,950
you only ever have one p and one q.

1100
00:50:17,950 --> 00:50:19,980
This is not even what we call,

1101
00:50:19,980 --> 00:50:21,060
so this is not

1102
00:50:21,060 --> 00:50:25,160
what we'll refer to as a well-formed

1103
00:50:25,160 --> 00:50:29,160
formula.

1104
00:50:35,060 --> 00:50:36,790
So you have to be really careful

1105
00:50:36,790 --> 00:50:39,570
with what meaning means,

1106
00:50:39,570 --> 00:50:42,000
and when you try to create an isomorphism

1107
00:50:42,000 --> 00:50:44,200
between what you know about addition

1108
00:50:44,200 --> 00:50:47,310
and the formal systems you play.

1109
00:50:47,310 --> 00:50:50,690
Try to come up with a
alternative interpretation.

1110
00:50:50,690 --> 00:50:51,940
We could have just interpreted

1111
00:50:51,940 --> 00:50:55,330
these p's, q's and hyphens as

1112
00:50:55,330 --> 00:50:57,730
you know we're gonna call p,

1113
00:50:57,730 --> 00:51:01,730
we're gonna say that's horse

1114
00:51:03,680 --> 00:51:05,480
and q.

1115
00:51:05,480 --> 00:51:09,480
And that's Apple.

1116
00:51:10,640 --> 00:51:14,640
And you know one hyphen is happy.

1117
00:51:16,620 --> 00:51:20,620
And you know two hyphens is happy, happy.

1118
00:51:24,660 --> 00:51:26,190
And so on.

1119
00:51:26,190 --> 00:51:27,770
So suddenly we have an interpretation

1120
00:51:27,770 --> 00:51:29,230
for for this string.

1121
00:51:29,230 --> 00:51:32,110
It's not two plus two equals four.

1122
00:51:32,110 --> 00:51:36,300
But it's happy, happy,
horse, happy, happy,

1123
00:51:36,300 --> 00:51:40,350
Apple, happy, happy, happy, happy.

1124
00:51:40,350 --> 00:51:43,290
It doesn't mean anything,

1125
00:51:43,290 --> 00:51:46,450
but it's an interpretation.

1126
00:51:46,450 --> 00:51:49,100
And there's no reason

1127
00:51:49,100 --> 00:51:50,720
not to make that interpretation.

1128
00:51:50,720 --> 00:51:52,460
Perhaps to horses,

1129
00:51:52,460 --> 00:51:54,980
this is actually more
sensible than addition.

1130
00:51:54,980 --> 00:51:58,250
I mean first of all, when we do addition,

1131
00:51:58,250 --> 00:51:59,440
we're representing these numbers

1132
00:51:59,440 --> 00:52:03,230
in base 10 because we have 10 fingers.

1133
00:52:03,230 --> 00:52:04,700
But horses don't have ten fingers

1134
00:52:04,700 --> 00:52:06,980
and numbers written in base 10

1135
00:52:06,980 --> 00:52:08,090
don't mean anything to horses.

1136
00:52:08,090 --> 00:52:10,670
But perhaps happy, horse, apple

1137
00:52:10,670 --> 00:52:13,790
really makes much more sense to a horse.

1138
00:52:13,790 --> 00:52:15,580
So we're gonna kind of throughout,

1139
00:52:15,580 --> 00:52:18,070
and I have to be a
little rushed about this,

1140
00:52:18,070 --> 00:52:19,640
be thinking about

1141
00:52:19,640 --> 00:52:21,250
where does the meaning come from?

1142
00:52:21,250 --> 00:52:23,390
How do we actually assign meaning

1143
00:52:23,390 --> 00:52:24,990
to meaningless symbols

1144
00:52:24,990 --> 00:52:26,080
because that's the goal here.

1145
00:52:26,080 --> 00:52:27,730
We're gonna go from meaningless symbols

1146
00:52:27,730 --> 00:52:30,640
in mathematics to meaning.

1147
00:52:30,640 --> 00:52:32,640
And then we're gonna try
to create an isomorphism

1148
00:52:32,640 --> 00:52:35,600
between the universe
and our formal systems.

1149
00:52:35,600 --> 00:52:37,400
And this leads me you know perfectly

1150
00:52:37,400 --> 00:52:40,950
into this idea of you know

1151
00:52:40,950 --> 00:52:44,950
is reality a formal system?

1152
00:52:45,800 --> 00:52:47,690
And if you go to page 29

1153
00:52:47,690 --> 00:52:51,120
and your notes you've got this

1154
00:52:51,120 --> 00:52:52,870
kind of long quote.

1155
00:52:52,870 --> 00:52:54,410
It stretches on to 30.

1156
00:52:54,410 --> 00:52:56,770
I'll go ahead and start reading.

1157
00:52:56,770 --> 00:52:58,100
It's at the bottom.

1158
00:52:58,100 --> 00:53:00,350
It says, "Can all of reality be turned

1159
00:53:00,350 --> 00:53:01,790
"into a formal system?

1160
00:53:01,790 --> 00:53:03,400
"In a very broad sense,

1161
00:53:03,400 --> 00:53:05,130
"the answer might appear to be yes.

1162
00:53:05,130 --> 00:53:06,650
"One could suggest for instance

1163
00:53:06,650 --> 00:53:08,130
"that reality is itself nothing

1164
00:53:08,130 --> 00:53:10,830
"but one very complicated formal system.

1165
00:53:10,830 --> 00:53:13,140
"Its symbols do not move around on paper,

1166
00:53:13,140 --> 00:53:16,020
"but rather in a
three-dimensional vacuum, space.

1167
00:53:16,020 --> 00:53:17,800
"They are the elementary particles

1168
00:53:17,800 --> 00:53:19,010
"of which everything is composed,"

1169
00:53:19,010 --> 00:53:20,450
Tacit assumption.

1170
00:53:20,450 --> 00:53:21,640
"That there is an end

1171
00:53:21,640 --> 00:53:22,860
"to the descending chain of matter,

1172
00:53:22,860 --> 00:53:24,880
"that the expression elementary particles

1173
00:53:24,880 --> 00:53:27,140
"make sense."

1174
00:53:27,140 --> 00:53:30,960
"The typographical rules
are the laws of physics,

1175
00:53:30,960 --> 00:53:31,960
which tell how,"

1176
00:53:31,960 --> 00:53:34,040
we're on page 29 if you
just want to catch up,

1177
00:53:34,040 --> 00:53:36,440
"The typographical rules
are the laws of physics,

1178
00:53:36,440 --> 00:53:38,020
"which tell how, given the positions

1179
00:53:38,020 --> 00:53:39,370
"and velocities of all the particles

1180
00:53:39,370 --> 00:53:41,990
"at a given instant, to modify them,

1181
00:53:41,990 --> 00:53:43,830
"resulting in a new set of positions

1182
00:53:43,830 --> 00:53:45,930
"and velocities belonging
to the next instant.

1183
00:53:45,930 --> 00:53:50,110
"So the theorems of
this grand formal system

1184
00:53:50,110 --> 00:53:51,310
"are the possible configurations

1185
00:53:51,310 --> 00:53:54,200
"of particles at different
times in the universe.

1186
00:53:54,200 --> 00:53:57,170
"The sole axiom is, or perhaps was,

1187
00:53:57,170 --> 00:53:59,420
"the original configuration
of all the particles

1188
00:53:59,420 --> 00:54:01,140
"at the beginning of time.

1189
00:54:01,140 --> 00:54:03,290
"this is so grandiose a conception,

1190
00:54:03,290 --> 00:54:05,710
"however, that is only the
most theoretical interest.

1191
00:54:05,710 --> 00:54:06,850
"And besides quantum mechanics,

1192
00:54:06,850 --> 00:54:09,900
"and other parts of
physics, can at least cast

1193
00:54:09,900 --> 00:54:11,350
"some doubt on even the theoretical

1194
00:54:11,350 --> 00:54:12,770
"worth of this idea.

1195
00:54:12,770 --> 00:54:14,310
"basically, we are asking

1196
00:54:14,310 --> 00:54:16,390
"if the universe operates
deterministically,

1197
00:54:16,390 --> 00:54:20,300
"which is an open question."

1198
00:54:20,300 --> 00:54:22,750
You know, I think it was Laplace

1199
00:54:22,750 --> 00:54:24,380
who said well look,

1200
00:54:24,380 --> 00:54:25,550
if you were to give me the position

1201
00:54:25,550 --> 00:54:28,440
and momentum of every
particle in the universe,

1202
00:54:28,440 --> 00:54:31,970
I could tell you the rest of the future.

1203
00:54:31,970 --> 00:54:33,210
And this is leads

1204
00:54:33,210 --> 00:54:35,020
to kind of the grand
philosophical questions,

1205
00:54:35,020 --> 00:54:38,430
which you know we'll be investigating

1206
00:54:38,430 --> 00:54:40,410
as part of this class as well,

1207
00:54:40,410 --> 00:54:44,360
which is if the universe
operates deterministically,

1208
00:54:44,360 --> 00:54:48,590
if Newton's laws govern how my arm falls,

1209
00:54:48,590 --> 00:54:52,030
and how all the atoms of my body interact,

1210
00:54:52,030 --> 00:54:54,240
where does free will creep into?

1211
00:54:54,240 --> 00:54:56,930
How do I know I have
control over these actions

1212
00:54:56,930 --> 00:54:58,580
and it's not the fact
that at the Big Bang,

1213
00:54:58,580 --> 00:55:02,490
there was a denser
cluster of atoms over here

1214
00:55:02,490 --> 00:55:04,150
and a less dense over here

1215
00:55:04,150 --> 00:55:05,420
and things evolved according

1216
00:55:05,420 --> 00:55:07,360
to deterministic laws,

1217
00:55:07,360 --> 00:55:08,620
much like the formal systems

1218
00:55:08,620 --> 00:55:10,840
we're playing with here?

1219
00:55:10,840 --> 00:55:12,670
So this question you can really think of

1220
00:55:12,670 --> 00:55:14,540
on two levels.

1221
00:55:14,540 --> 00:55:16,920
One, can the universe be thought of

1222
00:55:16,920 --> 00:55:20,880
as being modeled by formal system,

1223
00:55:20,880 --> 00:55:22,290
having forces

1224
00:55:22,290 --> 00:55:25,640
and solving equations
for the particles here

1225
00:55:25,640 --> 00:55:27,490
and it collides with another particle

1226
00:55:27,490 --> 00:55:30,520
at this angle, they go off like this,

1227
00:55:30,520 --> 00:55:31,520
and things like this?

1228
00:55:31,520 --> 00:55:35,450
But it also, I think, likes
to ask another question,

1229
00:55:35,450 --> 00:55:38,430
which is version two,

1230
00:55:38,430 --> 00:55:40,810
for those of you who
are kind of matrix fans,

1231
00:55:40,810 --> 00:55:44,660
to what extent is the
universe a formal system

1232
00:55:44,660 --> 00:55:47,610
proper in the sense is it a program

1233
00:55:47,610 --> 00:55:48,860
you know running in the background

1234
00:55:48,860 --> 00:55:50,550
of some hyperdimensional alien

1235
00:55:50,550 --> 00:55:52,200
who's playing WOW?

1236
00:55:52,200 --> 00:55:55,980
And you know he's just
running our universe

1237
00:55:55,980 --> 00:55:59,240
as a simulation on his you know

1238
00:55:59,240 --> 00:56:00,440
supercomputer cluster

1239
00:56:00,440 --> 00:56:02,840
that he's got in his basement.

1240
00:56:02,840 --> 00:56:04,070
Who knows?

1241
00:56:04,070 --> 00:56:06,740
I mean if the universe is deterministic

1242
00:56:06,740 --> 00:56:09,280
or he just coded up you know,

1243
00:56:09,280 --> 00:56:10,350
hacking away in Python,

1244
00:56:10,350 --> 00:56:11,830
all of our rules of our universe

1245
00:56:11,830 --> 00:56:15,180
and he said all right let's
let the simulation go.

1246
00:56:15,180 --> 00:56:16,780
And here we are in his computer

1247
00:56:16,780 --> 00:56:18,860
having all these kind
of dramatic interactions

1248
00:56:18,860 --> 00:56:20,290
with people, etc. Etc.

1249
00:56:20,290 --> 00:56:22,150
and he's just kind of interested,

1250
00:56:22,150 --> 00:56:25,290
in what bug came up etc.

1251
00:56:25,290 --> 00:56:29,290
It's kind of interesting to think about.

1252
00:56:30,570 --> 00:56:34,570
So we've now really kind of hit home

1253
00:56:35,770 --> 00:56:39,160
these five tools for thinking.

1254
00:56:39,160 --> 00:56:42,050
And we're gonna be
revisiting all of these ideas

1255
00:56:42,050 --> 00:56:43,650
throughout the entire book.

1256
00:56:43,650 --> 00:56:46,650
And one of the things that Bach does,

1257
00:56:46,650 --> 00:56:48,480
one of the things that
Douglas Hofstadter does

1258
00:56:48,480 --> 00:56:50,420
is he structures his book

1259
00:56:50,420 --> 00:56:53,780
in its own kind of recursive fashion.

1260
00:56:53,780 --> 00:56:54,780
And you know I only gave you

1261
00:56:54,780 --> 00:56:55,780
a few specific instances

1262
00:56:55,780 --> 00:56:57,470
of where recursion shows up.

1263
00:56:57,470 --> 00:57:00,570
And this represents kind of my bias.

1264
00:57:00,570 --> 00:57:01,950
For me I'm very much an art person

1265
00:57:01,950 --> 00:57:02,950
and a math person

1266
00:57:02,950 --> 00:57:05,460
but I'm not so much of a music person.

1267
00:57:05,460 --> 00:57:06,490
And I really encourage you guys

1268
00:57:06,490 --> 00:57:08,590
to bring in different elements

1269
00:57:08,590 --> 00:57:10,000
because GEB has such like

1270
00:57:10,000 --> 00:57:12,850
a high dimensional structure to it.

1271
00:57:12,850 --> 00:57:15,460
Everybody contributes
their own slice to it.

1272
00:57:15,460 --> 00:57:16,960
And one thing which I would hate

1273
00:57:16,960 --> 00:57:19,770
to deny you guys from is the music

1274
00:57:19,770 --> 00:57:21,730
aspect of this book.

1275
00:57:21,730 --> 00:57:25,050
Each one of Douglas Hofstadter's dialogues

1276
00:57:25,050 --> 00:57:26,050
is actually structured

1277
00:57:26,050 --> 00:57:28,980
and based upon a piece of Bach's music.

1278
00:57:28,980 --> 00:57:30,200
If you listen to Bach's music

1279
00:57:30,200 --> 00:57:32,000
and you read the dialogue,

1280
00:57:32,000 --> 00:57:34,350
he might actually hint at some of the

1281
00:57:34,350 --> 00:57:36,070
connections some, of the isomorphisms,

1282
00:57:36,070 --> 00:57:39,040
that Hofstadter is alluding to.

1283
00:57:39,040 --> 00:57:41,960
But first of all you should know

1284
00:57:41,960 --> 00:57:43,870
why he chose Bach,

1285
00:57:43,870 --> 00:57:45,990
how recursion acts in music

1286
00:57:45,990 --> 00:57:49,030
and that's why I have
this whole speaker setup

1287
00:57:49,030 --> 00:57:50,830
here so allow me to

1288
00:57:50,830 --> 00:57:54,830
play.

1289
00:57:56,240 --> 00:57:59,130
So this is Bach's Little Fugue in G minor

1290
00:57:59,130 --> 00:58:00,640
just as a nice anecdote.

1291
00:58:00,640 --> 00:58:03,050
Who here has seen A Beautiful Mind?

1292
00:58:03,050 --> 00:58:05,790
The movie, alright, so John Nash,

1293
00:58:05,790 --> 00:58:07,140
the mathematician who went crazy,

1294
00:58:07,140 --> 00:58:08,180
Princeton etc.

1295
00:58:08,180 --> 00:58:09,520
The story goes he used

1296
00:58:09,520 --> 00:58:11,460
to actually stalk around the halls

1297
00:58:11,460 --> 00:58:13,220
of the math department smoking cigarettes

1298
00:58:13,220 --> 00:58:15,770
and whistling this song constantly.

1299
00:58:15,770 --> 00:58:18,870
And what were some of the things

1300
00:58:18,870 --> 00:58:21,310
which you noticed about this piece?

1301
00:58:21,310 --> 00:58:24,600
For those of you with
good auditory abilities

1302
00:58:24,600 --> 00:58:25,970
what did you notice?

1303
00:58:25,970 --> 00:58:28,130
(student speaking)

1304
00:58:28,130 --> 00:58:29,510
Okay elaborate a little bit

1305
00:58:29,510 --> 00:58:32,520
on these patterns exactly.

1306
00:58:32,520 --> 00:58:37,020
(student speaking)

1307
00:58:37,020 --> 00:58:40,250
exactly, so you heard it come in

1308
00:58:40,250 --> 00:58:42,280
at a different tone,
at a different volume.

1309
00:58:42,280 --> 00:58:45,690
And you notice it was the same theme.

1310
00:58:45,690 --> 00:58:47,820
It's the same theme that he played,

1311
00:58:47,820 --> 00:58:50,070
stretched, inverted, backwards,

1312
00:58:50,070 --> 00:58:51,950
on higher levels on lower levels.

1313
00:58:51,950 --> 00:58:53,950
So GEB is actually very much
structured like a fugue.

1314
00:58:53,950 --> 00:58:56,970
Hofstadter lays out for us

1315
00:58:56,970 --> 00:58:58,240
and what I did in this first lecture

1316
00:58:58,240 --> 00:59:01,340
is I'm laying out the entire book for you

1317
00:59:01,340 --> 00:59:02,560
all in one go.

1318
00:59:02,560 --> 00:59:03,570
So that way you understand it

1319
00:59:03,570 --> 00:59:06,070
when I play it stretched
out inverted backwards

1320
00:59:06,070 --> 00:59:08,350
and at different volumes.

1321
00:59:08,350 --> 00:59:10,190
So this is nice.

1322
00:59:10,190 --> 00:59:11,190
You have a musical illustration.

1323
00:59:11,190 --> 00:59:12,300
You have artistic illustrations

1324
00:59:12,300 --> 00:59:14,280
of the ideas we're talking about.

1325
00:59:14,280 --> 00:59:18,280
But we need to actually
kind of settle into

1326
00:59:21,410 --> 00:59:22,690
the book itself.

1327
00:59:22,690 --> 00:59:25,220
So Kurin Kelleher and I

1328
00:59:25,220 --> 00:59:27,770
or anyone else who's really
excited about reading,

1329
00:59:27,770 --> 00:59:29,840
anybody really excited about volunteering

1330
00:59:29,840 --> 00:59:33,050
for reading a dialogue?

1331
00:59:33,050 --> 00:59:34,880
Anybody have the book with them right?

1332
00:59:34,880 --> 00:59:38,230
Oh good job.

1333
00:59:38,230 --> 00:59:39,680
Would you like to read?

1334
00:59:39,680 --> 00:59:43,680
You don't have to.

1335
00:59:46,230 --> 00:59:47,480
You want to?

1336
00:59:47,480 --> 00:59:48,620
Okay.

1337
00:59:48,620 --> 00:59:49,760
So we're gonna spend the last

1338
00:59:49,760 --> 00:59:52,040
kind of 15 minutes going through a dialog.

1339
00:59:52,040 --> 00:59:56,850
I actually have another copy.

1340
00:59:56,850 --> 01:00:00,300
And so I need two characters.

1341
01:00:00,300 --> 01:00:01,410
One to be Achilles

1342
01:00:01,410 --> 01:00:02,870
and one to be tortoise.

1343
01:00:02,870 --> 01:00:03,870
These are two characters

1344
01:00:03,870 --> 01:00:05,780
we're gonna meet in this dialogue.

1345
01:00:05,780 --> 01:00:06,840
And they're gonna play a prominent role

1346
01:00:06,840 --> 01:00:08,840
throughout the entire book.

1347
01:00:08,840 --> 01:00:11,710
So let's does anyone else

1348
01:00:11,710 --> 01:00:15,250
want to be, well see I like the tortoise

1349
01:00:15,250 --> 01:00:16,500
so I'd like to be the tortoise.

1350
01:00:16,500 --> 01:00:17,530
But someone else can be the tortoise

1351
01:00:17,530 --> 01:00:19,880
if they want to be.

1352
01:00:19,880 --> 01:00:22,480
Okay, so we only have one

1353
01:00:22,480 --> 01:00:25,420
so let's brave enough to do it.

1354
01:00:25,420 --> 01:00:27,160
All right.

1355
01:00:27,160 --> 01:00:28,900
All righty.

1356
01:00:28,900 --> 01:00:32,180
So page 79.

1357
01:00:32,180 --> 01:00:36,180
So yeah sorry.

1358
01:00:40,720 --> 01:00:44,460
So I'm gonna give you
some quick background

1359
01:00:44,460 --> 01:00:46,400
on this dialogue.

1360
01:00:46,400 --> 01:00:48,040
So Hofstadter, like me,

1361
01:00:48,040 --> 01:00:51,310
believes that it's important
to introduce the idea

1362
01:00:51,310 --> 01:00:52,810
of a topic conceptually first

1363
01:00:52,810 --> 01:00:55,600
before we start really diving into it

1364
01:00:55,600 --> 01:00:58,380
so he prefaces every
chapter with a dialogue.

1365
01:00:58,380 --> 01:01:01,690
And the dialogue is kind
of conceptual introduction

1366
01:01:01,690 --> 01:01:03,410
to the ideas we're talking about.

1367
01:01:03,410 --> 01:01:04,970
To go ahead and give you an idea

1368
01:01:04,970 --> 01:01:07,020
of what this dialogue is based on.

1369
01:01:07,020 --> 01:01:08,520
It's going to be the conflict

1370
01:01:08,520 --> 01:01:11,340
of two mathematicians,

1371
01:01:11,340 --> 01:01:13,640
Kurt Godel and David Hilbert.

1372
01:01:13,640 --> 01:01:16,860
David Hilbert believed that mathematics

1373
01:01:16,860 --> 01:01:20,850
could be put into a formal
system very rigorously

1374
01:01:20,850 --> 01:01:21,950
and it could also be proved

1375
01:01:21,950 --> 01:01:23,310
to be consistent and complete.

1376
01:01:23,310 --> 01:01:24,840
Those are two words

1377
01:01:24,840 --> 01:01:25,840
which I'm gonna have to define

1378
01:01:25,840 --> 01:01:29,090
kind of at the end of this dialogue,

1379
01:01:29,090 --> 01:01:30,090
but let's go ahead

1380
01:01:30,090 --> 01:01:32,010
and start it off and try to
work quickly through this.

1381
01:01:32,010 --> 01:01:33,010
I'm gonna ask that

1382
01:01:33,010 --> 01:01:34,010
when you have the italics,

1383
01:01:34,010 --> 01:01:36,470
you go ahead and read it
as part of your section.

1384
01:01:36,470 --> 01:01:37,470
So people have an idea

1385
01:01:37,470 --> 01:01:41,770
what's going on in the book.

1386
01:01:41,770 --> 01:01:45,770
All right, excellent.

1387
01:01:52,079 --> 01:01:55,290
So we don't have really any time left.

1388
01:01:55,290 --> 01:01:59,530
But I want to say one thing

1389
01:01:59,530 --> 01:02:01,000
that's a challenge.

1390
01:02:01,000 --> 01:02:05,000
Pay attention to
tortoise's quote on page 81

1391
01:02:06,740 --> 01:02:08,460
when she talks about acrostics.

1392
01:02:08,460 --> 01:02:12,460
If you can find two
acrostics in this dialogue...