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JUSTIN CURRY: All right.

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00:00:21,370 --> 00:00:21,869
Hello.

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00:00:21,869 --> 00:00:25,660
Welcome to Godel, Escher,
Bach-- a Mental Space Odyssey.

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00:00:25,660 --> 00:00:29,020
My name is Justin Curry, and
I'm a senior in mathematics

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00:00:29,020 --> 00:00:30,075
here at MIT.

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00:00:30,075 --> 00:00:32,075
I've spent the last year
at Cambridge University

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at UK and, the summer before
that, living in Germany.

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So it's kind of a reverse
culture shock coming back,

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00:00:38,140 --> 00:00:42,040
but I'm excited to teach
Godel, Escher, Bach again.

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00:00:42,040 --> 00:00:44,320
I taught this course
in spring, 2006.

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It was a 10-week course then.

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And we attempted
the impossible task

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of trying to get through this
thick monster all in one go.

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00:00:52,870 --> 00:00:56,290
And it's impossible.

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00:00:56,290 --> 00:00:59,260
Most undergrads can't get
through it in 13 weeks.

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00:00:59,260 --> 00:01:01,690
I got through it in
about seven years.

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So you're going to be
attempting a feat here,

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not to complete the entire
book but to get the essence

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00:01:07,420 --> 00:01:09,550
of Godel, Escher, Bach out.

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00:01:09,550 --> 00:01:11,740
But I want to make sure
we introduce everybody,

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00:01:11,740 --> 00:01:12,940
just to get people's names.

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This will help me
take attendance.

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00:01:14,870 --> 00:01:17,770
And it will also--

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I also want you
to say, what is it

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00:01:19,390 --> 00:01:22,410
when you read the course
catalog that interested you most

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and why, essentially, why
you're sitting here today?

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00:01:25,930 --> 00:01:28,780
I'm curious.

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00:01:28,780 --> 00:01:34,000
So what is the idea
behind this book?

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00:01:34,000 --> 00:01:39,550
I interviewed a good
many of you this morning,

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just to make sure that
you guys felt comfortable

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00:01:41,962 --> 00:01:42,670
with mathematics.

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00:01:42,670 --> 00:01:45,280
This course isn't directly
about mathematics.

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00:01:45,280 --> 00:01:47,972
There's a lot of mathematics
being talked about.

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Yes, do you have a question?

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00:01:49,138 --> 00:01:50,530
STUDENT: What's
this class about?

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00:01:50,530 --> 00:01:51,238
JUSTIN CURRY: OK.

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00:01:51,238 --> 00:01:54,910
So that's what I'm going
to go through right now.

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The idea here is that--

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00:01:58,910 --> 00:02:02,580
Douglas Hofstadter is interested
in one primary question.

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00:02:02,580 --> 00:02:05,540
And that question is,
how does a self come out

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00:02:05,540 --> 00:02:07,400
of things which have no selves?

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00:02:07,400 --> 00:02:11,630
How is it that all these
carbon atoms and molecules

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00:02:11,630 --> 00:02:14,930
and proteins which make us
up in the physical universe,

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00:02:14,930 --> 00:02:20,120
how do they go from being
meaningless to developing

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00:02:20,120 --> 00:02:23,752
into an entity which
can refer to itself?

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00:02:23,752 --> 00:02:26,320
Like, right now, I'm
saying, I think this.

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I think you like this.

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00:02:27,520 --> 00:02:29,730
I'm meeting all of
you, as individuals.

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00:02:29,730 --> 00:02:32,300
Each one of you
claim to have a self.

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00:02:32,300 --> 00:02:34,550
You might remember
Descartes' famous quote,

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"I think, therefore I am."

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00:02:35,750 --> 00:02:37,332
So it seems like the I--

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when I say the "I,"
I mean the things

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we call ourselves-- is
a real, existent thing.

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00:02:43,764 --> 00:02:44,930
But it's a complex question.

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00:02:44,930 --> 00:02:48,380
How do we get I's
out of non-I's?

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00:02:48,380 --> 00:02:52,170
And that's going to be
the goal, over here.

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So I'm just going to call it I.

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But how do you get to an I?

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You get to an I
by having a bunch

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of meaningless primitives,
things like atoms, proteins--

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molecules, I should say--

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et cetera.

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This is what you're made up of.

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But none of these
things mean anything.

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None of these things
have I's or selves.

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But you do.

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So what's the relationship here?

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00:03:25,020 --> 00:03:28,630
Douglas Hofstadter wrote
this book back in the 70s

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when he was doing graduate
school in physics.

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00:03:31,960 --> 00:03:34,630
And this was after him doing
a math undergrad at Stanford.

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He saw the answer when
he was playing around

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with mathematics in
the very formal systems

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we play with, like when we
write down things like 2

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plus 2 equals 4.

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These are just symbols.

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And as we go through
today, I'll show you

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completely equivalent
ways of doing addition,

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which will look like this.

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And these are just
logical primitives.

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And if you've seen
any set theory--

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and don't feel
scared if you haven't

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00:04:09,936 --> 00:04:11,060
seen any of these symbols--

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but there exists
an x for every--

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we give these interpretations.

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But the idea is
that mathematics can

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be reduced to a bunch of
meaningless operations, just

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00:04:19,730 --> 00:04:20,540
symbol shunting.

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But what's interesting is
that, within mathematics, there

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exists an equivalent
to self-reference.

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00:04:31,540 --> 00:04:34,130
This is a bunch of atoms and
proteins referring to itself,

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00:04:34,130 --> 00:04:37,310
calling itself an I.
What happens here--

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00:04:37,310 --> 00:04:41,500
and this is going to be
underneath the name of Godel--

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00:04:49,231 --> 00:04:51,480
is we're going to get to
some incompleteness theorems.

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00:04:51,480 --> 00:04:53,313
We're going to get to
some statements which,

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in mathematics,
refer to themselves.

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And the question of
how this happens,

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we understand this rigorously.

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Mathematicians have
worked out, how

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do we go from
meaningless symbols

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00:05:02,610 --> 00:05:06,030
to something which refers to
itself and which has meaning?

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00:05:06,030 --> 00:05:10,090
The claim, then, is that these
two systems are equivalent.

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And this is, really,
the profound idea.

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I'm going to draw
this symbol, and I'm

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00:05:14,460 --> 00:05:16,280
going to use a term
called isomorphism.

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00:05:16,280 --> 00:05:19,320
And isomorphism is,
basically, an equals to--

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and equals in a different sense.

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00:05:21,210 --> 00:05:25,560
But the idea here
is, in many ways,

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we can link atoms and proteins
to logical symbolic primitives

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in mathematics.

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And we understand how we get
self-reference in mathematics.

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So maybe we can use
this to understand

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how we get I's, how self
comes out of non-self.

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This is a really tall order,
but we're going to try to do it.

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And that's what this
book attempts to do.

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00:05:48,720 --> 00:05:53,790
And what I've done is
isolate the chapters

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00:05:53,790 --> 00:05:56,460
in this book which I
think are most pertinent

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to this stream of thought.

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00:05:57,576 --> 00:05:58,950
Basically, what
we're going to do

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is we're going to learn how
it works in mathematics.

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00:06:00,660 --> 00:06:02,700
We're going to go from
logical primitives

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00:06:02,700 --> 00:06:06,210
and work up to self-reference
and talk about Zen Buddhism

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00:06:06,210 --> 00:06:08,020
consciousness, et cetera.

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00:06:08,020 --> 00:06:10,020
But that's going to happen
as we leap over here.

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Because we're going to work
up, down, and then around.

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And we'll conclude the course
with some interesting questions

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about artificial intelligence
and how intelligent things come

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out of unintelligent things.

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So when I was teaching this
course two years ago, or two

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springs ago, I ran
into five things

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which I viewed as really
important tools for thinking.

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And I've had to condense a
little bit into my famous Tools

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for Thinking lecture.

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The idea here is that
Godel, Escher, Bach has

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00:06:58,120 --> 00:07:01,420
an incredible number of
conceptual tools for thinking

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00:07:01,420 --> 00:07:05,050
about this complex problem of,
how do we go from a non-self

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00:07:05,050 --> 00:07:06,130
to a self?

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00:07:06,130 --> 00:07:09,542
And just to outline
these real quick,

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00:07:09,542 --> 00:07:19,680
I'm going to have isomorphisms--

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00:07:19,680 --> 00:07:24,719
and I'll explain all these terms
as we go along; recursion--

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00:07:24,719 --> 00:07:26,260
I'm going to leave
this one mainly up

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00:07:26,260 --> 00:07:42,090
to Curran on the second lecture;
paradox; and this is infinity--

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and all these concepts
are very closely linked.

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And finally, the main
subject for today's lecture

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is going to be formal systems.

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All righty.

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00:08:04,442 --> 00:08:13,040
So first, let me go through
definitions of these terms.

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An isomorphism-- I want you all
to be very careful with this.

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Because when you start
talking to mathematicians--

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00:08:21,980 --> 00:08:24,500
grown-up, professional
mathematicians--

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they're going to use
the term isomorphism

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to mean something
very, very specific.

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The way it's used in Godel,
Escher, Bach, the way

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it's going to be used in
this class, is very loose.

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We're going to make very
intuitive statements,

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like what's the
isomorphism between a car--

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00:08:48,040 --> 00:08:49,350
I'm not a great artist here--

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what's the isomorphism between
a skateboard and a car?

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And you might say
lots of things,

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00:09:01,190 --> 00:09:06,450
like it carries a person,
it has four wheels.

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00:09:06,450 --> 00:09:10,940
So what we do is we construct a
map which also has an inverse.

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And that's the way you
think of an isomorphism.

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You can go either way
and preserve information,

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preserve structure.

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If you really feel
like following along,

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I've included, actually, a quote
from Douglas Hofstadter on page

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7 of your lecture notes.

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He says-- and this is in
the middle of the page--

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"The word isomorphism applies
when two complex structures

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can be mapped onto
each other in ways

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that, to each part
of one structure,

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00:09:47,960 --> 00:09:50,720
there's a corresponding part
in the other structure, where

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00:09:50,720 --> 00:09:53,070
corresponding means
that the two parts play

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00:09:53,070 --> 00:09:55,917
similar roles in their
respective structures."

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This is how we're going
to always use the term

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00:10:02,750 --> 00:10:04,747
isomorphism in this class.

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00:10:04,747 --> 00:10:06,580
If you're taking the
abstract algebra class,

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00:10:06,580 --> 00:10:09,290
it's going to mean something
a lot more specific,

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00:10:09,290 --> 00:10:12,770
and you're going to
have a lot more details.

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00:10:12,770 --> 00:10:16,520
You might, actually, think
of these as kind of a--

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00:10:16,520 --> 00:10:18,920
I'll say it, but
don't worry about it--

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00:10:18,920 --> 00:10:20,864
is a homomorphism.

190
00:10:20,864 --> 00:10:22,280
And the idea with
the homomorphism

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is that there are a
lot more details here

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00:10:25,010 --> 00:10:27,560
than there are here.

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00:10:27,560 --> 00:10:31,204
And for example, there's
no steering wheel.

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00:10:31,204 --> 00:10:32,620
There's a steering
wheel in a car,

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00:10:32,620 --> 00:10:36,030
but there's no steering wheel,
specifically, in a skateboard.

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00:10:36,030 --> 00:10:43,220
So if you were to create a map
from the car to the skateboard,

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00:10:43,220 --> 00:10:47,360
that detail would have
to go somewhere else.

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00:10:47,360 --> 00:10:49,520
But don't worry about
those necessities.

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00:10:49,520 --> 00:10:52,019
But when I say the term
isomorphism, think of equals.

200
00:10:52,019 --> 00:10:53,810
And I'll often use that
symbol right there.

201
00:10:58,300 --> 00:10:59,800
This is going to
be really important

202
00:10:59,800 --> 00:11:01,500
because it's going
to be how we're going

203
00:11:01,500 --> 00:11:03,130
to get meaning out of things.

204
00:11:03,130 --> 00:11:06,840
And you'll see it a lot
coming up over the book.

205
00:11:06,840 --> 00:11:11,350
But first I want to hop on
and talk about recursion.

206
00:11:11,350 --> 00:11:16,710
Recursion is, basically--
it's seen everywhere.

207
00:11:16,710 --> 00:11:20,490
But it's a list of
instructions which

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00:11:20,490 --> 00:11:24,940
you follow but then repeat until
you've reached a final case.

209
00:11:24,940 --> 00:11:27,630
So suppose you were cooking.

210
00:11:27,630 --> 00:11:32,760
And you could have a recursive
algorithm for stirring eggs.

211
00:11:32,760 --> 00:11:37,510
And that would be whirl,
and then, whirl again.

212
00:11:37,510 --> 00:11:39,450
Keep whirling until,
essentially, everything

213
00:11:39,450 --> 00:11:40,690
looks mixed up.

214
00:11:40,690 --> 00:11:42,524
That's a very loose way
of understanding it.

215
00:11:42,524 --> 00:11:44,314
But another way, which
you all are probably

216
00:11:44,314 --> 00:11:45,960
familiar with, and
much more rigorous,

217
00:11:45,960 --> 00:11:50,020
in term of mathematics,
is the Fibonacci sequence.

218
00:11:50,020 --> 00:11:51,770
This is where you start
with two numbers--

219
00:11:51,770 --> 00:11:55,320
1 and 1-- and then, you
construct the next number

220
00:11:55,320 --> 00:11:57,690
by summing the previous two.

221
00:11:57,690 --> 00:12:03,240
So you have that, and you
have 3, and you have 5,

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00:12:03,240 --> 00:12:06,220
and you have 8, and so on.

223
00:12:06,220 --> 00:12:09,870
And you can create what's called
a recursive definition where

224
00:12:09,870 --> 00:12:13,110
you define the n-th
Fibonacci number--

225
00:12:20,950 --> 00:12:22,870
this is for n greater
than or equal to 2.

226
00:12:25,380 --> 00:12:30,940
And here, you define the
thing in terms of itself.

227
00:12:30,940 --> 00:12:34,320
And this is a classic
example of recursion.

228
00:12:34,320 --> 00:12:38,700
What it is is, really,
itself on a smaller level.

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00:12:38,700 --> 00:12:41,130
I think one of the most exciting
applications of recursion

230
00:12:41,130 --> 00:12:42,930
are fractals.

231
00:12:42,930 --> 00:12:46,491
Because the way we create
fractals is through recursion.

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00:12:46,491 --> 00:12:48,240
So I don't know if you
all have seen this,

233
00:12:48,240 --> 00:12:51,420
but the Sierpinksi triangle,
or the Sierpinksi gasket,

234
00:12:51,420 --> 00:12:53,340
is a classic fractal.

235
00:12:53,340 --> 00:12:58,650
Here, you divide a
triangle up into three.

236
00:12:58,650 --> 00:13:01,280
And then, you just
repeat the process

237
00:13:01,280 --> 00:13:06,080
for an infinite number of times
on each remaining triangle.

238
00:13:06,080 --> 00:13:09,541
And you create these very
beautiful mosaic forms.

239
00:13:09,541 --> 00:13:11,040
But the nice thing
about mathematics

240
00:13:11,040 --> 00:13:13,830
is that we can be very
precise and do things that we

241
00:13:13,830 --> 00:13:15,780
can't do in the real world.

242
00:13:15,780 --> 00:13:18,060
And that's repeat
this infinitely--

243
00:13:18,060 --> 00:13:20,610
and so on.

244
00:13:20,610 --> 00:13:22,470
Just for a quick
digression, and I really

245
00:13:22,470 --> 00:13:24,095
don't want to spend
too much time on it

246
00:13:24,095 --> 00:13:29,850
because Curran will do more,
why is it called a fractal?

247
00:13:29,850 --> 00:13:32,205
Does anyone know?

248
00:13:32,205 --> 00:13:35,769
STUDENT: I think it's like
a fragment of something.

249
00:13:35,769 --> 00:13:36,560
JUSTIN CURRY: Sure.

250
00:13:39,380 --> 00:13:44,290
It was a term coined by Benoit
Mandelbrot in 1977, I believe.

251
00:13:44,290 --> 00:13:47,070
It, actually, refers to
its number of dimensions.

252
00:13:47,070 --> 00:13:49,940
So this might be kind of a
mind-bending concept for most

253
00:13:49,940 --> 00:13:52,910
of you, but we like to think
we live in one, two, or three,

254
00:13:52,910 --> 00:13:55,380
or four dimensions--

255
00:13:55,380 --> 00:13:56,540
all integers, right?

256
00:13:56,540 --> 00:14:00,470
But my claim is that the
Sierpinski gasket actually

257
00:14:00,470 --> 00:14:04,430
lives in between one
and two dimensions.

258
00:14:04,430 --> 00:14:09,350
It lives in, like,
1.63-something dimensions.

259
00:14:09,350 --> 00:14:11,660
And I want to help
you think about that.

260
00:14:11,660 --> 00:14:14,780
And if you want to
hop along to page 9,

261
00:14:14,780 --> 00:14:19,787
I've got a recipe for helping
you think about dimension.

262
00:14:19,787 --> 00:14:20,370
You know what?

263
00:14:20,370 --> 00:14:22,670
It's weird because only
mathematicians would ever

264
00:14:22,670 --> 00:14:25,310
worry about rigorously
understanding the concept

265
00:14:25,310 --> 00:14:27,090
of what a dimension means.

266
00:14:27,090 --> 00:14:30,500
So here's one way
to think about it.

267
00:14:30,500 --> 00:14:36,530
If you take a line,
and you double it,

268
00:14:36,530 --> 00:14:40,770
you have two copies of the
line that you started with.

269
00:14:40,770 --> 00:14:42,990
This guy's here and there.

270
00:14:42,990 --> 00:14:51,130
If you have a square, and you
double the sides of the square,

271
00:14:51,130 --> 00:14:54,430
you have four copies
of the original square.

272
00:14:54,430 --> 00:14:56,930
Similarly-- and I'm not going
to try to draw this because it

273
00:14:56,930 --> 00:14:59,375
will get too complicated
way too fast--

274
00:14:59,375 --> 00:15:08,060
if you take a cube, and you
double each of the sides,

275
00:15:08,060 --> 00:15:11,600
you get, if you think
about it, eight copies

276
00:15:11,600 --> 00:15:14,100
of the original cube.

277
00:15:14,100 --> 00:15:16,100
So if you're
perceptive enough, you

278
00:15:16,100 --> 00:15:20,580
might realize this action
of powers going on here.

279
00:15:20,580 --> 00:15:24,110
So here, we had, after our
doubling process, two copies.

280
00:15:24,110 --> 00:15:25,850
We had 2 to the 1.

281
00:15:25,850 --> 00:15:31,100
Here, after our doubling
process, we had 2 to the 2.

282
00:15:31,100 --> 00:15:35,650
After our doubling process
here, we had 2 to the 3--

283
00:15:35,650 --> 00:15:37,370
8.

284
00:15:37,370 --> 00:15:38,780
So this is weird.

285
00:15:38,780 --> 00:15:44,400
Because notice that the cube
lives in three dimensions.

286
00:15:44,400 --> 00:15:47,102
And the square lives
in two dimensions.

287
00:15:47,102 --> 00:15:50,840
And the line lives
in one dimension.

288
00:15:50,840 --> 00:15:57,920
So this might suggest to you the
relationship that 2 to the d,

289
00:15:57,920 --> 00:16:01,340
where d is the dimension of
the space you're living in,

290
00:16:01,340 --> 00:16:04,370
equals the number
of copies you have

291
00:16:04,370 --> 00:16:06,630
after the doubling process.

292
00:16:06,630 --> 00:16:09,860
So let's return to our
friend, the Sierpinski gasket.

293
00:16:09,860 --> 00:16:13,100
If we start here, and
we imagine doubling

294
00:16:13,100 --> 00:16:16,070
each of the sides of the
Sierpinski gasket, here

295
00:16:16,070 --> 00:16:20,330
and here, we're very strangely
led to the conclusion

296
00:16:20,330 --> 00:16:26,692
that whatever dimension the
Sierpinski gasket lives in,

297
00:16:26,692 --> 00:16:30,230
it obeys this rule.

298
00:16:30,230 --> 00:16:41,410
So take the logarithms,
and d times--

299
00:16:41,410 --> 00:16:43,050
sorry, this is getting crowded.

300
00:16:49,329 --> 00:16:52,420
if you take the logarithm of
both sides and solve for d,

301
00:16:52,420 --> 00:16:56,760
you'll see that the dimension
of the Sierpinski gasket

302
00:16:56,760 --> 00:17:02,810
is log 3 over log 2, which
is approximately 1.585

303
00:17:02,810 --> 00:17:05,030
on to infinity.

304
00:17:05,030 --> 00:17:08,060
So here's an exact example of
something which lives somewhere

305
00:17:08,060 --> 00:17:09,540
between one and two dimensions.

306
00:17:09,540 --> 00:17:11,248
And I think that's a
really cool concept.

307
00:17:14,540 --> 00:17:19,560
Moving on for other tools for
thinking, we have paradoxes.

308
00:17:19,560 --> 00:17:22,130
Paradoxes come in all
sorts of different flavors.

309
00:17:22,130 --> 00:17:24,380
I don't know if some of you
have heard of the birthday

310
00:17:24,380 --> 00:17:28,040
paradox, where it's
the idea of, OK, what's

311
00:17:28,040 --> 00:17:30,230
the probability that
someone else in the room

312
00:17:30,230 --> 00:17:31,970
has your same birthday?

313
00:17:31,970 --> 00:17:33,560
Everybody thinks
it's really small.

314
00:17:33,560 --> 00:17:35,434
But if you actually work
out the mathematics,

315
00:17:35,434 --> 00:17:37,414
it turns out you actually
have a good chance.

316
00:17:37,414 --> 00:17:39,080
If you're in a room
with over 40 people,

317
00:17:39,080 --> 00:17:40,455
you have an
extremely high chance

318
00:17:40,455 --> 00:17:44,420
of finding someone else
with your same birthday.

319
00:17:44,420 --> 00:17:49,240
So I've actually
list listed out--

320
00:17:49,240 --> 00:17:52,875
this is courtesy of
Wikipedia and Mr. Quine--

321
00:18:04,130 --> 00:18:05,860
we have three
variants of paradoxes.

322
00:18:09,430 --> 00:18:13,030
This is veridical.

323
00:18:13,030 --> 00:18:17,920
And these are things
which are true but may

324
00:18:17,920 --> 00:18:19,210
seem paradoxical at first.

325
00:18:21,780 --> 00:18:24,100
There's falsidical.

326
00:18:28,700 --> 00:18:30,510
And I'll give an example
of each of these.

327
00:18:30,510 --> 00:18:33,290
And then, the classic, the
one which we're going to be

328
00:18:33,290 --> 00:18:35,740
interested in-- and these
are real paradoxes--

329
00:18:35,740 --> 00:18:36,975
are antinomies.

330
00:18:40,090 --> 00:18:43,030
To give you an example of
another classic paradox,

331
00:18:43,030 --> 00:18:45,820
and one which is visited in
Godel, Escher, Bach very early

332
00:18:45,820 --> 00:18:49,420
on, it's called Zeno's paradox.

333
00:18:49,420 --> 00:18:54,190
And the idea is if I want to
get from here to my laptop,

334
00:18:54,190 --> 00:18:59,440
I first need to walk
halfway across the distance.

335
00:18:59,440 --> 00:19:02,650
And then, if I want to walk
the remaining distance,

336
00:19:02,650 --> 00:19:04,975
I need to walk half of that.

337
00:19:04,975 --> 00:19:06,850
And if I want to walk
the remaining distance,

338
00:19:06,850 --> 00:19:11,949
I need to walk half of that, and
then half that, half of that.

339
00:19:11,949 --> 00:19:13,990
And eventually, I get
stuck in this infinite loop

340
00:19:13,990 --> 00:19:18,880
where it seems like I'm
not getting to my laptop.

341
00:19:18,880 --> 00:19:20,500
A variant of this
paradox is the idea

342
00:19:20,500 --> 00:19:23,620
that, if I even want to move
at all, if my atoms want

343
00:19:23,620 --> 00:19:29,720
to pass in space, first,
they have to go halfway.

344
00:19:29,720 --> 00:19:31,870
But before it can
go halfway, it's

345
00:19:31,870 --> 00:19:34,390
got to go halfway by half
and halfway of that half

346
00:19:34,390 --> 00:19:35,790
and a half of that half.

347
00:19:35,790 --> 00:19:37,960
So Zeno, back in
Greece, actually

348
00:19:37,960 --> 00:19:40,480
used this to prove that
motion was impossible

349
00:19:40,480 --> 00:19:43,610
and that any motion we saw in
the universe was an illusion.

350
00:19:43,610 --> 00:19:47,390
So it's weird.

351
00:19:47,390 --> 00:19:49,330
Why?

352
00:19:49,330 --> 00:19:52,310
And nobody really could answer
Zeno for the longest time.

353
00:19:52,310 --> 00:19:56,060
But then it took,
essentially, the understanding

354
00:19:56,060 --> 00:19:58,640
of limits and
calculus to really get

355
00:19:58,640 --> 00:20:01,550
an idea of why this
wasn't paradoxical.

356
00:20:01,550 --> 00:20:05,300
What, rigorously, did we mean
by an infinite number of steps?

357
00:20:05,300 --> 00:20:08,222
How could we actually
get across the room?

358
00:20:08,222 --> 00:20:10,430
It seemed paradoxical, but
we knew it had to be true.

359
00:20:10,430 --> 00:20:11,846
We knew motion had
to be possible.

360
00:20:14,480 --> 00:20:16,580
I'm sure when you
all were younger,

361
00:20:16,580 --> 00:20:20,810
or even now, you've seen all
sorts of falsidical paradoxes

362
00:20:20,810 --> 00:20:22,730
where somebody will
write out a string

363
00:20:22,730 --> 00:20:30,760
of, if you take 1 minus 1
plus 1 minus 1, dot, dot, dot.

364
00:20:30,760 --> 00:20:33,170
And the person convinces
you, well, look,

365
00:20:33,170 --> 00:20:36,950
if you look in groups of
this, these are all zeros.

366
00:20:36,950 --> 00:20:39,230
So if you just add a
bunch of zeros together,

367
00:20:39,230 --> 00:20:42,064
this is necessarily 0.

368
00:20:42,064 --> 00:20:43,480
This is an infinite
string, right?

369
00:20:43,480 --> 00:20:45,512
And we can repeat the pattern.

370
00:20:45,512 --> 00:20:46,720
What happens if we add a one?

371
00:20:49,850 --> 00:20:54,380
So suddenly, we get these weird
conclusions where 0 equals 1.

372
00:20:54,380 --> 00:20:56,930
And they're usually built
on doing something illegal

373
00:20:56,930 --> 00:20:58,630
involving infinities.

374
00:20:58,630 --> 00:21:01,010
And infinity is going to
be a very important concept

375
00:21:01,010 --> 00:21:03,650
that we'll encounter
again and again.

376
00:21:03,650 --> 00:21:07,190
Finally, the antinomy.

377
00:21:07,190 --> 00:21:11,270
These are the important
paradoxes to think about.

378
00:21:11,270 --> 00:21:13,916
I once went out to dinner with
a bunch of mathematicians.

379
00:21:13,916 --> 00:21:16,290
I don't know how I ended up
in that but, let me tell you,

380
00:21:16,290 --> 00:21:18,500
it was kind of frightening.

381
00:21:18,500 --> 00:21:21,230
And there was this
Korean mathematician

382
00:21:21,230 --> 00:21:23,921
who said, well, you know what?

383
00:21:23,921 --> 00:21:25,670
Most of these questions
don't even matter.

384
00:21:25,670 --> 00:21:28,680
We don't understand some of
the most fundamental things.

385
00:21:28,680 --> 00:21:33,290
And the thing he was most
interested in and, I think,

386
00:21:33,290 --> 00:21:35,000
which bothers
mathematicians the most,

387
00:21:35,000 --> 00:21:39,935
is the antimony of the
liar and Russell's paradox.

388
00:21:42,830 --> 00:21:47,730
So the Liar's Paradox you
probably have heard before.

389
00:21:47,730 --> 00:21:50,990
And it's based on, actually,
a biblical reference.

390
00:21:50,990 --> 00:22:02,200
But it, essentially, says
"This sentence is not true."

391
00:22:07,050 --> 00:22:10,840
So is it true or is it not true?

392
00:22:10,840 --> 00:22:17,350
Well, if it's true, then it says
of itself that it's not true.

393
00:22:17,350 --> 00:22:19,390
So true implies not true--

394
00:22:19,390 --> 00:22:20,740
contradiction.

395
00:22:20,740 --> 00:22:22,740
So if it's not
true, then we know

396
00:22:22,740 --> 00:22:25,500
that, if we believe in the law
of the excluded middle, which

397
00:22:25,500 --> 00:22:28,540
means that things have to
either be true or not true,

398
00:22:28,540 --> 00:22:30,210
that it's negation is true.

399
00:22:30,210 --> 00:22:34,930
So if it's not true, then
the sentence is true.

400
00:22:34,930 --> 00:22:36,860
So not true implies true.

401
00:22:36,860 --> 00:22:39,800
So we're stuck.

402
00:22:39,800 --> 00:22:42,470
The liar paradox
still hounds us today.

403
00:22:42,470 --> 00:22:50,750
Unlike Zeno's paradox,
it hasn't been solved.

404
00:22:50,750 --> 00:22:53,130
We still don't know
how to deal with it.

405
00:22:53,130 --> 00:22:56,690
And when we talk
about Godel's theorem,

406
00:22:56,690 --> 00:22:58,560
the way he proves his
result is actually

407
00:22:58,560 --> 00:23:02,225
going to be intimately linked
with a variant on this.

408
00:23:02,225 --> 00:23:04,610
So instead of saying, I'm
not true, it's going to say,

409
00:23:04,610 --> 00:23:07,410
I'm not provable.

410
00:23:07,410 --> 00:23:09,780
And that's going to be
a very interesting idea.

411
00:23:09,780 --> 00:23:12,180
And we'll explore that
a little bit later.

412
00:23:12,180 --> 00:23:14,710
The other antinomy
I want to look at

413
00:23:14,710 --> 00:23:19,730
is Russell's paradox, also
known as the barber's paradox.

414
00:23:19,730 --> 00:23:22,371
And that's how I'm going to tell
it, as the barber's paradox.

415
00:23:22,371 --> 00:23:23,870
I think it's a
little more friendly.

416
00:23:29,740 --> 00:23:31,890
So you have a town.

417
00:23:31,890 --> 00:23:36,240
And there's this male barber.

418
00:23:36,240 --> 00:23:38,640
And he abides by
the rule that he

419
00:23:38,640 --> 00:23:41,880
shaves all people
and only people who

420
00:23:41,880 --> 00:23:44,980
don't shave themselves.

421
00:23:44,980 --> 00:23:49,410
So what does the barber
do when his beard

422
00:23:49,410 --> 00:23:51,300
is getting as thick as mine?

423
00:23:51,300 --> 00:23:54,420
Does he shave himself,
or does he not?

424
00:23:54,420 --> 00:23:55,560
Well, let's see.

425
00:23:55,560 --> 00:23:57,570
So by definition,
the barber only

426
00:23:57,570 --> 00:24:00,940
shaves those people who
don't shave themselves.

427
00:24:00,940 --> 00:24:03,722
So if he shaves himself,
then he doesn't.

428
00:24:03,722 --> 00:24:06,720
And if he doesn't shave
himself, then, by definition,

429
00:24:06,720 --> 00:24:09,720
he must shave himself.

430
00:24:09,720 --> 00:24:13,920
A variant of this, which
was coined by both Bertrand

431
00:24:13,920 --> 00:24:18,060
Russell, Cambridge mathematician
and philosopher, and Zermelo,

432
00:24:18,060 --> 00:24:22,800
a great German
magician, is the idea

433
00:24:22,800 --> 00:24:27,210
that you can consider the set--

434
00:24:27,210 --> 00:24:31,290
let's call it omega--

435
00:24:31,290 --> 00:24:39,190
which contains all sets that
aren't members of themselves.

436
00:24:51,930 --> 00:24:55,310
So remember, a set is just
a collection of objects.

437
00:24:55,310 --> 00:24:56,750
And mathematicians
really believed

438
00:24:56,750 --> 00:25:00,770
that set theory was going
to be what gave mathematics

439
00:25:00,770 --> 00:25:04,190
its ultimate sure and
logical foundation.

440
00:25:04,190 --> 00:25:09,990
So let's give an example of
a set which contains itself.

441
00:25:09,990 --> 00:25:12,320
So let's think of the
set of all things which

442
00:25:12,320 --> 00:25:15,020
aren't Joan of Arc.

443
00:25:15,020 --> 00:25:17,760
Well, sets aren't people.

444
00:25:17,760 --> 00:25:19,610
They're people, not sets.

445
00:25:19,610 --> 00:25:23,600
So that set of all things
which aren't Joan of Arc

446
00:25:23,600 --> 00:25:24,380
includes itself.

447
00:25:24,380 --> 00:25:26,550
Because a set can
never be a person.

448
00:25:26,550 --> 00:25:31,230
So that set is
contained in itself.

449
00:25:31,230 --> 00:25:33,620
So we have a bunch
of things in here

450
00:25:33,620 --> 00:25:37,140
which are sets which aren't
members of themselves.

451
00:25:37,140 --> 00:25:44,240
And then, we ask the question,
is omega an element of itself?

452
00:25:44,240 --> 00:25:45,800
And this means "is in."

453
00:25:51,740 --> 00:25:57,000
Well, if omega contains itself--

454
00:25:57,000 --> 00:25:59,659
but omega, by definition,
only contains things

455
00:25:59,659 --> 00:26:00,950
which don't contain themselves.

456
00:26:00,950 --> 00:26:03,180
So it can't contain itself.

457
00:26:03,180 --> 00:26:05,360
Well, if it can't
contain itself,

458
00:26:05,360 --> 00:26:08,050
it doesn't contain
itself, and that means

459
00:26:08,050 --> 00:26:09,780
it should contain itself--

460
00:26:09,780 --> 00:26:12,450
contradiction.

461
00:26:12,450 --> 00:26:14,900
This really, really bothered
a lot of mathematicians

462
00:26:14,900 --> 00:26:17,450
for a long time.

463
00:26:17,450 --> 00:26:21,080
And it's an exact variant
on the barber's paradox.

464
00:26:21,080 --> 00:26:25,640
So this is kind of interesting
things to play around with.

465
00:26:25,640 --> 00:26:27,110
Finally is the
concept of infinity.

466
00:26:27,110 --> 00:26:28,776
I can't, really, talk
too much about it.

467
00:26:28,776 --> 00:26:30,440
We're going to look at it more.

468
00:26:30,440 --> 00:26:32,330
But I want to introduce
you guys to the idea

469
00:26:32,330 --> 00:26:34,770
that there are multiple
types of infinity.

470
00:26:34,770 --> 00:26:40,010
So you have the integers, and
you also have the real numbers.

471
00:26:40,010 --> 00:26:45,410
And it is true that you
cannot create a direct link.

472
00:26:45,410 --> 00:26:50,070
You can't match every real
number, like 0.333333--

473
00:26:50,070 --> 00:26:53,180
well, 0.35-something
random-- pi.

474
00:26:53,180 --> 00:26:54,110
Let's pick pi.

475
00:26:54,110 --> 00:26:57,980
You can't put pi
directly in connection

476
00:26:57,980 --> 00:27:01,820
with a natural
number, an integer.

477
00:27:01,820 --> 00:27:03,200
And this is kind of famous--

478
00:27:03,200 --> 00:27:04,820
Cantor's diagonalization
argument.

479
00:27:04,820 --> 00:27:07,520
So somehow, there are
different degrees of infinity.

480
00:27:07,520 --> 00:27:12,360
And the real numbers is a
higher degree of infinity.

481
00:27:12,360 --> 00:27:16,280
So that's an important
thing to think about.

482
00:27:16,280 --> 00:27:23,190
Now, we're going to jump ahead
to our last tool for thinking.

483
00:27:23,190 --> 00:27:26,160
And this is going
to be the reason why

484
00:27:26,160 --> 00:27:31,170
we ignore the first three
chapters of Godel, Escher Bach.

485
00:27:31,170 --> 00:27:36,501
And it's the idea
of a formal system.

486
00:27:36,501 --> 00:27:40,150
The problem is is formal
systems are boring.

487
00:27:40,150 --> 00:27:43,800
And Douglas Hofstadter
takes his sweet, sweet time

488
00:27:43,800 --> 00:27:48,702
in introducing you to the
concept of a formal system.

489
00:27:48,702 --> 00:27:50,160
So I want to try
to speed things up

490
00:27:50,160 --> 00:27:52,034
because I know you all
are smarter than that,

491
00:27:52,034 --> 00:27:54,600
and you can get through
these concepts very quickly.

492
00:27:57,360 --> 00:27:58,990
We're going to play a game.

493
00:27:58,990 --> 00:28:03,930
It's called the
mu puzzle, or M-U.

494
00:28:03,930 --> 00:28:06,540
And the way you
play it is you start

495
00:28:06,540 --> 00:28:11,000
with a bag of three letters.

496
00:28:14,060 --> 00:28:21,160
And you're going to have a
rule you're going to start with

497
00:28:21,160 --> 00:28:22,380
You pull two letters out.

498
00:28:22,380 --> 00:28:26,670
And you get M, I. And we're
going to have four rules.

499
00:28:26,670 --> 00:28:34,290
And these are completely
strict typographical rules

500
00:28:34,290 --> 00:28:36,890
for deriving new things that
we can pull from our bag.

501
00:28:40,130 --> 00:28:45,908
Our first rule is
that if we have an I--

502
00:28:45,908 --> 00:28:53,097
so suppose we have MI, or we
could have anything and then an

503
00:28:53,097 --> 00:28:54,071
I--

504
00:28:54,071 --> 00:29:01,410
we can tack a U on, so IU.

505
00:29:01,410 --> 00:29:03,440
So right away, we know
that we can create MIU.

506
00:29:07,600 --> 00:29:14,680
Our second rule is,
suppose we have M and then

507
00:29:14,680 --> 00:29:18,050
a string of letters that are
I's and U's, since they're

508
00:29:18,050 --> 00:29:22,170
in our bag of alphabet,
our alphabet here.

509
00:29:22,170 --> 00:29:26,210
Then, you're going to
get, for free, Mxx.

510
00:29:26,210 --> 00:29:28,490
So just as an example,
suppose, somehow, you

511
00:29:28,490 --> 00:29:34,220
had MI, which we do.

512
00:29:34,220 --> 00:29:37,360
You're going to
get MII for free.

513
00:29:40,410 --> 00:29:48,510
Third rule-- suppose
somewhere along the way,

514
00:29:48,510 --> 00:29:50,755
you end up with a
cluster of three I's.

515
00:29:50,755 --> 00:29:52,130
They don't have
to be at the end.

516
00:29:52,130 --> 00:29:53,630
They can be anywhere--

517
00:29:53,630 --> 00:29:57,210
just needs to be three
I's all together.

518
00:29:57,210 --> 00:29:59,410
And you can replace
all three of those I's.

519
00:29:59,410 --> 00:30:10,740
They're equal to a U.

520
00:30:10,740 --> 00:30:18,180
And our final rule is that if
we have a double pair of U's, we

521
00:30:18,180 --> 00:30:20,490
can drop them, and
they just go away.

522
00:30:23,010 --> 00:30:29,271
So somehow, if we had
MUU, we could just have M.

523
00:30:29,271 --> 00:30:31,770
Now, you have these rules.

524
00:30:31,770 --> 00:30:33,570
You have these letters.

525
00:30:33,570 --> 00:30:37,260
You start with one guy.

526
00:30:37,260 --> 00:30:39,020
He's going to be our axiom.

527
00:30:39,020 --> 00:30:41,580
An axiom is a starting
point for reasoning

528
00:30:41,580 --> 00:30:42,720
for applying these rules.

529
00:30:48,120 --> 00:30:51,210
And the game is, can you get MU?

530
00:30:51,210 --> 00:30:55,700
Starting from MI, and using
only these four rules,

531
00:30:55,700 --> 00:30:58,640
can you get MU?

532
00:30:58,640 --> 00:31:05,850
I will give $20 to the first
person who can derive MU--

533
00:31:05,850 --> 00:31:08,710
that's in this room--

534
00:31:08,710 --> 00:31:11,320
only applying these four rules
and starting directly from MI.

535
00:31:13,992 --> 00:31:16,450
Just to give you an idea of
where you might be going, where

536
00:31:16,450 --> 00:31:20,440
you might be playing, just
going off of our rules,

537
00:31:20,440 --> 00:31:22,510
we already saw that if we
had MI, we can get MIU.

538
00:31:25,840 --> 00:31:29,762
We also saw that, using rule
two-- that's using rule one--

539
00:31:29,762 --> 00:31:30,920
we can get MII.

540
00:31:37,750 --> 00:31:43,240
We saw if we have anything like
that, we can repeat it twice.

541
00:31:43,240 --> 00:31:45,220
So we can get MIUIU--

542
00:31:45,220 --> 00:31:48,640
that's applying rule two again--

543
00:31:48,640 --> 00:31:49,270
and so on.

544
00:31:51,870 --> 00:31:53,180
Leave this as a puzzle.

545
00:31:53,180 --> 00:31:55,040
Take your time with it.

546
00:31:55,040 --> 00:31:56,960
You'll be working on
it for a few hours.

547
00:31:56,960 --> 00:32:01,430
But first person that's in
this room, derive MU from this,

548
00:32:01,430 --> 00:32:03,771
gets $20.

549
00:32:03,771 --> 00:32:04,270
Yes?

550
00:32:04,270 --> 00:32:06,960
STUDENT: Fourth rule
only applies to U?

551
00:32:06,960 --> 00:32:07,820
JUSTIN CURRY: Yes.

552
00:32:07,820 --> 00:32:10,950
Fourth rule only
applies to two U's.

553
00:32:10,950 --> 00:32:15,635
So yes, if you have two
U's, you can remove them.

554
00:32:15,635 --> 00:32:17,790
You can subtract them.

555
00:32:17,790 --> 00:32:18,510
All right.

556
00:32:18,510 --> 00:32:22,230
And once again, I do urge
everyone to buy the book.

557
00:32:22,230 --> 00:32:25,530
These rules are listed
explicitly in the chapter.

558
00:32:25,530 --> 00:32:27,870
And you might gain
some insight on how

559
00:32:27,870 --> 00:32:30,095
to derive what you want here.

560
00:32:38,340 --> 00:32:41,530
So why is this interesting?

561
00:32:41,530 --> 00:32:44,030
We're just playing with
letters and strings and things

562
00:32:44,030 --> 00:32:46,070
like that.

563
00:32:46,070 --> 00:32:53,420
Well, although this seems pretty
meaningless and kind of dumb,

564
00:32:53,420 --> 00:32:56,117
does anybody feel
like when they're just

565
00:32:56,117 --> 00:32:57,950
looking at this game,
looking at this rules,

566
00:32:57,950 --> 00:33:00,530
that they're just,
essentially, playing around

567
00:33:00,530 --> 00:33:03,860
with algebra that they
learned in middle school

568
00:33:03,860 --> 00:33:05,930
or high school?

569
00:33:05,930 --> 00:33:07,610
Really, what we're
doing here is we've

570
00:33:07,610 --> 00:33:13,490
got some statements
like 2 plus 2 equals 4

571
00:33:13,490 --> 00:33:14,840
And we all learned that.

572
00:33:14,840 --> 00:33:19,226
We have a typographical
rule for when

573
00:33:19,226 --> 00:33:20,600
we have an equals
sign like that,

574
00:33:20,600 --> 00:33:26,040
we can add 1 to both sides
and preserve equality.

575
00:33:26,040 --> 00:33:33,290
So suddenly, we have
2 plus 3 equals 5.

576
00:33:33,290 --> 00:33:35,870
So really, what
mathematics reduces

577
00:33:35,870 --> 00:33:40,940
to is just playing around
with systems of this form

578
00:33:40,940 --> 00:33:43,525
and applying these rigorous
typographical rules.

579
00:33:43,525 --> 00:33:45,650
Except, here, there doesn't
seem to be any meaning.

580
00:33:45,650 --> 00:33:47,050
It's just meaningless.

581
00:33:47,050 --> 00:33:48,550
One of the important
questions we're

582
00:33:48,550 --> 00:33:51,860
going to address in this class
is, how do things gain meaning?

583
00:33:51,860 --> 00:33:55,696
How do we go from
meaningless to meaning?

584
00:33:55,696 --> 00:33:57,320
This, obviously,
seems to have meaning,

585
00:33:57,320 --> 00:33:59,060
but I want you to
ask yourself why.

586
00:34:03,580 --> 00:34:06,770
Before we proceed,
it's necessary--

587
00:34:06,770 --> 00:34:12,699
it's my duty-- to do the
boring task of writing down

588
00:34:12,699 --> 00:34:15,489
just a few definitions
of things, what you can

589
00:34:15,489 --> 00:34:16,739
call these, so you have words.

590
00:34:16,739 --> 00:34:18,330
So we already saw axiom.

591
00:34:18,330 --> 00:34:20,050
That's a definition.

592
00:34:20,050 --> 00:34:22,674
You call any of
these guys a string.

593
00:34:26,210 --> 00:34:39,280
So a string is just any ordered
sequence of, in this case, M,

594
00:34:39,280 --> 00:34:40,280
I's, and U's.

595
00:34:48,239 --> 00:34:49,840
We already met an axiom.

596
00:34:53,170 --> 00:34:54,880
An axiom is a starting point.

597
00:34:57,680 --> 00:35:02,475
It's your first thing that
you can apply the rules to.

598
00:35:02,475 --> 00:35:05,440
And this, actually, has a lot
to do with mathematical logic.

599
00:35:05,440 --> 00:35:07,040
Because in math
logic, the idea is

600
00:35:07,040 --> 00:35:09,040
that we start from really
primitive things which

601
00:35:09,040 --> 00:35:15,760
seem obvious, like the
successor of 0 is 1,

602
00:35:15,760 --> 00:35:17,680
and then we work
from that concept,

603
00:35:17,680 --> 00:35:19,930
and we derive all these
truths of number theory

604
00:35:19,930 --> 00:35:22,570
and mathematics.

605
00:35:22,570 --> 00:35:25,360
Here, your axiom
is MI, and you're

606
00:35:25,360 --> 00:35:27,570
trying to prove the theorem--

607
00:35:27,570 --> 00:35:31,940
and that's our next guy here--

608
00:35:31,940 --> 00:35:33,740
we are trying to prove
the theorem of MU.

609
00:35:38,630 --> 00:35:44,050
So a theorem is, basically,
a string which results

610
00:35:44,050 --> 00:35:45,340
at the end of a derivation.

611
00:36:00,040 --> 00:36:02,260
And a derivation
is like a proof.

612
00:36:02,260 --> 00:36:04,760
For those of you who have done
geometry, when you're saying,

613
00:36:04,760 --> 00:36:06,410
OK, well, this
triangle's congruent

614
00:36:06,410 --> 00:36:08,900
to this triangle because of
side, angle, side and things

615
00:36:08,900 --> 00:36:14,240
like that, you're making
rigorous justifications

616
00:36:14,240 --> 00:36:16,200
for your leaps in logic.

617
00:36:16,200 --> 00:36:17,990
So here, our rigorous
justification

618
00:36:17,990 --> 00:36:21,600
that MIU was the
theorem, well, we

619
00:36:21,600 --> 00:36:24,020
applied typographical
rule number one.

620
00:36:24,020 --> 00:36:29,420
That's a rigorous leap in logic,
and we got to this theorem.

621
00:36:29,420 --> 00:36:35,580
And you can just call
these four rules here,

622
00:36:35,580 --> 00:36:37,311
these are rules of inference.

623
00:36:45,802 --> 00:36:47,510
And logic and a lot
of things that you'll

624
00:36:47,510 --> 00:36:50,360
play around with eventually,
on SATs and things

625
00:36:50,360 --> 00:36:55,370
like that, if you have
the statement that p

626
00:36:55,370 --> 00:36:58,790
implies a statement q--

627
00:36:58,790 --> 00:37:02,100
if it's cloudy,
then it will rain--

628
00:37:02,100 --> 00:37:05,330
you have that this
is equivalent to--

629
00:37:05,330 --> 00:37:07,590
I should use a
different arrow here--

630
00:37:11,120 --> 00:37:16,610
to not q implies not p.

631
00:37:16,610 --> 00:37:20,060
And these are really
nice because they're just

632
00:37:20,060 --> 00:37:21,080
typographical rules.

633
00:37:21,080 --> 00:37:23,750
When you see something,
like when you have--

634
00:37:23,750 --> 00:37:26,664
well, I've got M followed
by any string of letters--

635
00:37:26,664 --> 00:37:27,830
well, then, I can double it.

636
00:37:27,830 --> 00:37:31,102
That's a rule of
inference, just like this

637
00:37:31,102 --> 00:37:32,060
is a rule of inference.

638
00:37:32,060 --> 00:37:35,150
If I have p implies q, I
can always replace that.

639
00:37:35,150 --> 00:37:38,420
It's completely equivalent
to not q implies not p.

640
00:37:42,647 --> 00:37:44,480
But for those of you
who are scrambling away

641
00:37:44,480 --> 00:37:45,770
because you want
$20 really fast,

642
00:37:45,770 --> 00:37:46,894
I want you to take a break.

643
00:37:46,894 --> 00:37:48,440
Because once again,
we should focus

644
00:37:48,440 --> 00:37:49,940
on what we're saying right now.

645
00:37:52,177 --> 00:37:54,260
And we're going to talk a
little bit about jumping

646
00:37:54,260 --> 00:37:55,180
outside the system.

647
00:37:55,180 --> 00:37:58,610
This is the cool renegade stuff
that Hofstadter fills his book

648
00:37:58,610 --> 00:37:59,650
with.

649
00:37:59,650 --> 00:38:02,360
And it's the idea
that, as you're

650
00:38:02,360 --> 00:38:03,950
playing around with
this, right now,

651
00:38:03,950 --> 00:38:05,780
you're just playing a game.

652
00:38:05,780 --> 00:38:09,230
And what mathematicians,
and what anybody human,

653
00:38:09,230 --> 00:38:13,010
does is when they feel like
they're caught in loops, just

654
00:38:13,010 --> 00:38:14,570
cranking through
pages of algebra,

655
00:38:14,570 --> 00:38:16,610
and they're not
getting anywhere,

656
00:38:16,610 --> 00:38:19,730
humans are intelligent
enough to stop.

657
00:38:19,730 --> 00:38:24,599
They exit the system, and
they say, I don't know.

658
00:38:24,599 --> 00:38:26,390
I don't think this is
going to go anywhere.

659
00:38:26,390 --> 00:38:31,520
Or well, let me think
about why I'm not getting,

660
00:38:31,520 --> 00:38:35,240
or how might I get, MU?

661
00:38:35,240 --> 00:38:40,190
Maybe it has something to do
with numbers of I's and U's

662
00:38:40,190 --> 00:38:42,950
or things like that.

663
00:38:42,950 --> 00:38:45,740
You start doing what I
like to call meta-thinking.

664
00:38:45,740 --> 00:38:48,420
You're not thinking
in the system,

665
00:38:48,420 --> 00:38:51,740
applying typographical rules,
applying rules of inference

666
00:38:51,740 --> 00:38:53,800
to existing strings--

667
00:38:53,800 --> 00:38:56,834
axioms-- and getting theorems.

668
00:38:56,834 --> 00:38:58,250
That's thinking
inside the system.

669
00:38:58,250 --> 00:38:59,770
That's just thinking.

670
00:38:59,770 --> 00:39:02,660
Meta-thinking involves you
leaping outside the system

671
00:39:02,660 --> 00:39:05,750
and making judgments about
it, thoughts which cannot be

672
00:39:05,750 --> 00:39:09,830
expressed as any just normal
typographical role within

673
00:39:09,830 --> 00:39:10,460
the system.

674
00:39:10,460 --> 00:39:13,940
You're doing meta-thinking.

675
00:39:13,940 --> 00:39:21,230
One of my favorite parts of
this section in Godel, Escher,

676
00:39:21,230 --> 00:39:25,040
Bach is when Hofstadter says--

677
00:39:25,040 --> 00:39:29,270
and, once again, stop
the drive in you--

678
00:39:29,270 --> 00:39:34,560
try to turn to page 24
in your lecture notes.

679
00:39:34,560 --> 00:39:35,640
Oops.

680
00:39:35,640 --> 00:39:37,472
Somebody's syllabus.

681
00:39:37,472 --> 00:39:38,960
Let me get that.

682
00:39:38,960 --> 00:39:40,880
No worries.

683
00:39:40,880 --> 00:39:46,280
Page 24-- Hofstadter kind of
uses this as a life lesson.

684
00:39:46,280 --> 00:39:49,100
He says, look, "Of
course, there are

685
00:39:49,100 --> 00:39:52,580
cases when only a rare
individual will have the vision

686
00:39:52,580 --> 00:39:54,950
to perceive a system which
governs many people's

687
00:39:54,950 --> 00:39:57,710
lives, a system which
had never before even

688
00:39:57,710 --> 00:39:59,870
been recognized as a system.

689
00:39:59,870 --> 00:40:01,610
Then such people often
devote their lives

690
00:40:01,610 --> 00:40:05,480
to convincing other people
that the system really is there

691
00:40:05,480 --> 00:40:08,350
and that it ought
to be exited from."

692
00:40:08,350 --> 00:40:12,770
It's as if our social
customs and our cultures

693
00:40:12,770 --> 00:40:15,062
are really just formal games.

694
00:40:15,062 --> 00:40:16,020
You know, we say hello.

695
00:40:16,020 --> 00:40:17,240
We shake your hand.

696
00:40:17,240 --> 00:40:21,200
That's an instance of a formal
rule, which we all follow.

697
00:40:21,200 --> 00:40:22,700
But you know, every
once in a while,

698
00:40:22,700 --> 00:40:26,060
you get somebody who says, ah,
I don't want to shake your hand.

699
00:40:26,060 --> 00:40:30,475
I'm going to exit the
hand-shaking formal system.

700
00:40:30,475 --> 00:40:32,600
But of course, there are
much more radical examples

701
00:40:32,600 --> 00:40:36,620
of this-- like, I said
Karl Marx and communism.

702
00:40:36,620 --> 00:40:39,590
He viewed this idea
of, look, you've

703
00:40:39,590 --> 00:40:42,830
got these people who are
collecting money and property.

704
00:40:42,830 --> 00:40:46,010
And they're getting someone
else to do all the work,

705
00:40:46,010 --> 00:40:49,280
and they're oppressing
this whole class of people.

706
00:40:49,280 --> 00:40:51,350
Can't people
recognize the system?

707
00:40:51,350 --> 00:40:53,870
So then, people like
Karl Marx and Fred Engels

708
00:40:53,870 --> 00:40:56,540
start writing in pamphlets,
encouraging people

709
00:40:56,540 --> 00:40:58,880
to overthrow
governments, et cetera,

710
00:40:58,880 --> 00:41:00,120
because they viewed a system.

711
00:41:00,120 --> 00:41:02,640
They said, look, we
need to exit the system.

712
00:41:02,640 --> 00:41:06,530
For intelligent beings, we
can think on a higher level.

713
00:41:06,530 --> 00:41:08,700
Of course, I'm not trying
to promote communism here.

714
00:41:08,700 --> 00:41:11,720
I'm just showing you an
example of historical interest.

715
00:41:14,570 --> 00:41:17,570
You know, anarchism, socialism
today, working people,

716
00:41:17,570 --> 00:41:18,962
the media.

717
00:41:18,962 --> 00:41:21,170
Nowadays, I think it's one
of the most popular things

718
00:41:21,170 --> 00:41:22,910
for people to say
is, well, you know,

719
00:41:22,910 --> 00:41:24,650
it's just the media
trying to do this.

720
00:41:24,650 --> 00:41:27,410
Before, we used to never
just refer to this entity

721
00:41:27,410 --> 00:41:28,190
as "the media."

722
00:41:28,190 --> 00:41:30,565
The media is trying to obscure
our understanding of this.

723
00:41:30,565 --> 00:41:33,560
The media is trying to scare us.

724
00:41:33,560 --> 00:41:34,910
Also, the government.

725
00:41:34,910 --> 00:41:38,540
The government's responsible!

726
00:41:38,540 --> 00:41:42,590
Of course, a classic example
is also what Karl Marx said,

727
00:41:42,590 --> 00:41:43,811
the church.

728
00:41:43,811 --> 00:41:45,060
It's the opiate of the masses.

729
00:41:45,060 --> 00:41:46,980
That's what he said.

730
00:41:46,980 --> 00:41:47,840
And also, school.

731
00:41:47,840 --> 00:41:50,890
School is my favorite example
of a system which people have

732
00:41:50,890 --> 00:41:52,570
encouraged you to exit from.

733
00:41:52,570 --> 00:41:55,310
It's like, well, it's just
a daycare that we have.

734
00:41:55,310 --> 00:41:58,870
And we don't actually want
kids to learn and grow up.

735
00:41:58,870 --> 00:42:01,420
And this inspired a lot of
new free-thinking educational

736
00:42:01,420 --> 00:42:05,200
movements like the Montessoris
and things like that.

737
00:42:05,200 --> 00:42:07,000
And I really want you
guys to think about,

738
00:42:07,000 --> 00:42:09,050
in your daily actions,
am I living, perhaps,

739
00:42:09,050 --> 00:42:12,840
in a kind of formal system which
is acting in a similar way?

740
00:42:12,840 --> 00:42:16,140
Try to do some meta-thinking,
thinking on a higher level.

741
00:42:16,140 --> 00:42:20,500
And is it worth
exiting that system?

742
00:42:20,500 --> 00:42:24,160
Hofstadter classifies these
three levels of thinking.

743
00:42:24,160 --> 00:42:27,520
And he likes to call it a
mechanical mode, when you're

744
00:42:27,520 --> 00:42:32,700
doing the normal games of the
system, an intelligent mode,

745
00:42:32,700 --> 00:42:34,630
and, then, just an unmode.

746
00:42:34,630 --> 00:42:36,990
Unmode is when you
just reject the system.

747
00:42:36,990 --> 00:42:39,340
He calls it the zen way
of approaching things.

748
00:42:39,340 --> 00:42:41,920
And this is something we like
to talk about a little more.

749
00:42:46,390 --> 00:42:48,490
I want to quickly
introduce you to another--

750
00:42:52,710 --> 00:42:56,520
well, first of all, I want to
talk about a concept of what

751
00:42:56,520 --> 00:42:58,050
we've previously mentioned.

752
00:42:58,050 --> 00:42:59,550
We're eventually
going to be talking

753
00:42:59,550 --> 00:43:00,870
about artificial intelligence.

754
00:43:00,870 --> 00:43:04,860
And it's weird
because humans really

755
00:43:04,860 --> 00:43:08,070
like to say that their
thoughts are logical.

756
00:43:08,070 --> 00:43:13,030
We like to say that we
do think in this manner.

757
00:43:13,030 --> 00:43:15,090
But a lot of times, we don't.

758
00:43:15,090 --> 00:43:21,900
We like to use just inference
about collective events.

759
00:43:21,900 --> 00:43:25,050
One of our favorite tools
of thinking is induction.

760
00:43:25,050 --> 00:43:27,330
Well, the sun has rised
all these previous days.

761
00:43:27,330 --> 00:43:29,750
I'm sure it'll rise tomorrow.

762
00:43:29,750 --> 00:43:33,060
And there's no real
formal line of logic

763
00:43:33,060 --> 00:43:35,160
that's saying that,
well, sun rised yesterday

764
00:43:35,160 --> 00:43:37,260
and that, thus, it
will rise tomorrow.

765
00:43:37,260 --> 00:43:39,360
And I want you to
think of whether or not

766
00:43:39,360 --> 00:43:42,690
our thoughts are actually
just computations

767
00:43:42,690 --> 00:43:45,780
in a formal system,
much like MIU,

768
00:43:45,780 --> 00:43:47,220
p implies q, and
things like that.

769
00:43:50,170 --> 00:43:54,540
And that's going to bring me
to another formal system which

770
00:43:54,540 --> 00:43:57,647
I have to mention just
because, in chapter 4,

771
00:43:57,647 --> 00:43:58,730
he's going to refer to it.

772
00:44:03,000 --> 00:44:05,360
And it's going to lead us
to this interesting line

773
00:44:05,360 --> 00:44:11,510
of dialogue of when a formal
system with meaningless symbols

774
00:44:11,510 --> 00:44:13,550
gains meaning.

775
00:44:13,550 --> 00:44:15,270
And it's called the pq system.

776
00:44:18,970 --> 00:44:21,290
We're going to have three
new letters-- well, three

777
00:44:21,290 --> 00:44:23,020
new characters.

778
00:44:23,020 --> 00:44:26,210
It's now going to
be p, q, and hyphen.

779
00:44:29,840 --> 00:44:32,750
And you've, actually, got an
infinite number of axioms here.

780
00:44:32,750 --> 00:44:48,620
And you've got a definition,
and that's that if xp hyphen--

781
00:44:48,620 --> 00:44:54,216
I'm going to make sure I
have, just, an underlined p--

782
00:44:54,216 --> 00:44:54,716
qx.

783
00:44:59,190 --> 00:45:08,240
And this is going to
be an axiom whenever

784
00:45:08,240 --> 00:45:10,227
x is just a string of hyphens.

785
00:45:10,227 --> 00:45:15,950
So it's just some
string of hyphens.

786
00:45:15,950 --> 00:45:17,000
So what's this saying?

787
00:45:17,000 --> 00:45:23,940
It's saying that, well, if you
have something like this, well,

788
00:45:23,940 --> 00:45:29,100
x here was two hyphens, so
we know that that's an axiom.

789
00:45:32,981 --> 00:45:33,480
All right.

790
00:45:33,480 --> 00:45:34,854
It's a little
different than MIU.

791
00:45:34,854 --> 00:45:37,710
It seems just as meaningless.

792
00:45:37,710 --> 00:45:40,920
And we're going to
have different forms

793
00:45:40,920 --> 00:45:44,340
for manipulating and
playing around with this.

794
00:45:44,340 --> 00:45:53,760
And one rule is that if
you have x, y, and z,

795
00:45:53,760 --> 00:45:56,250
which are just hyphen strings--

796
00:45:56,250 --> 00:46:07,040
xpyqz-- then you
can derive, you're

797
00:46:07,040 --> 00:46:20,280
given for free, the statement
xpy hyphen qz hyphen.

798
00:46:23,180 --> 00:46:25,970
Seems meaningless.

799
00:46:25,970 --> 00:46:30,420
But what does it remind you of?

800
00:46:33,450 --> 00:46:34,680
We've got this axiom.

801
00:46:34,680 --> 00:46:37,620
We, in fact, have a whole
infinite list of axioms.

802
00:46:37,620 --> 00:46:45,054
And maybe you've noticed that
they've got two hyphens here,

803
00:46:45,054 --> 00:46:51,180
one hyphen here, got
three hyphens here.

804
00:46:54,162 --> 00:46:55,450
Now, what does this do?

805
00:46:55,450 --> 00:46:57,910
STUDENT: [INAUDIBLE] .

806
00:46:57,910 --> 00:46:59,200
JUSTIN CURRY: Yeah, exactly.

807
00:46:59,200 --> 00:47:02,380
And what it does is
it says that, well,

808
00:47:02,380 --> 00:47:04,000
if this works, right?

809
00:47:04,000 --> 00:47:07,540
So let's apply this rule here.

810
00:47:07,540 --> 00:47:10,270
And we'll apply this rule here.

811
00:47:10,270 --> 00:47:12,670
So we can take this
and get for free

812
00:47:12,670 --> 00:47:18,260
that hyphen hyphen p hyphen--

813
00:47:18,260 --> 00:47:20,440
we can add another hyphen--

814
00:47:20,440 --> 00:47:22,290
q.

815
00:47:22,290 --> 00:47:24,850
Now, we had three hyphens
here, but this rule

816
00:47:24,850 --> 00:47:27,380
says we can tack
on another hyphen.

817
00:47:27,380 --> 00:47:30,230
What does that say?

818
00:47:30,230 --> 00:47:38,409
Well, this seems to say
that 2 plus 2 equals 4.

819
00:47:38,409 --> 00:47:40,450
So I want you to realize
that the symbolism which

820
00:47:40,450 --> 00:47:43,120
mathematicians have been
using, and what you've grown up

821
00:47:43,120 --> 00:47:45,350
learning, is just shorthand.

822
00:47:45,350 --> 00:47:47,740
It's meaningless notation.

823
00:47:47,740 --> 00:47:48,464
Yeah?

824
00:47:48,464 --> 00:47:49,432
STUDENT: Did you do
those two backwards--

825
00:47:49,432 --> 00:47:49,932
1 and 2?

826
00:47:49,932 --> 00:47:51,370
Should it be 2 and 1?

827
00:47:51,370 --> 00:47:52,578
JUSTIN CURRY: Well, yeah, no.

828
00:47:52,578 --> 00:47:54,040
What I meant to
say here is that we

829
00:47:54,040 --> 00:47:59,320
seem to be inferring this
rule that hyphen string 1 plus

830
00:47:59,320 --> 00:48:06,030
hyphen string 2 always
equals hyphen string 3.

831
00:48:06,030 --> 00:48:10,150
And so, just 1 here refers
to a whole string of hyphens.

832
00:48:10,150 --> 00:48:13,150
And 2 refers to a string
of hyphens, like y here.

833
00:48:13,150 --> 00:48:16,750
Or better yet, I could say
x plus y equals c here.

834
00:48:16,750 --> 00:48:23,620
And what makes this
system different than MIU?

835
00:48:23,620 --> 00:48:26,027
Does anyone have any ideas?

836
00:48:26,027 --> 00:48:27,610
Why do you suddenly
care a little more

837
00:48:27,610 --> 00:48:29,674
about this system than
MIU, other than the fact

838
00:48:29,674 --> 00:48:31,840
that you have $20 going on
the line for deriving MU?

839
00:48:37,708 --> 00:48:38,686
Anybody?

840
00:48:43,580 --> 00:48:45,890
What about this fact
that I've just showed

841
00:48:45,890 --> 00:48:48,760
you this equivalence here?

842
00:48:48,760 --> 00:48:53,660
Now, instead of applying
these typographical rules,

843
00:48:53,660 --> 00:48:56,780
I've showed you that, well, you
can also take this as 2 plus 2

844
00:48:56,780 --> 00:48:58,460
equals 4?

845
00:48:58,460 --> 00:49:00,740
And then, you're
going to say, aha!

846
00:49:00,740 --> 00:49:04,140
Well, now, I can do
all sorts of things.

847
00:49:04,140 --> 00:49:07,580
Now that I've discovered
the meaning of the pq hyphen

848
00:49:07,580 --> 00:49:10,430
system, I can go
ahead and just create

849
00:49:10,430 --> 00:49:14,600
all sorts of new
theorems, starting

850
00:49:14,600 --> 00:49:17,840
from any of our axioms.

851
00:49:17,840 --> 00:49:22,310
And you might even be tempted to
say, well, I know it's obvious.

852
00:49:22,310 --> 00:49:28,040
I know that 2 plus
2 plus 2 equals 6.

853
00:49:28,040 --> 00:49:29,960
And I've discovered
this isomorphism

854
00:49:29,960 --> 00:49:34,670
between p's and q's and
pluses and equal signs.

855
00:49:37,260 --> 00:49:44,420
So I'm tempted to say that
hyphen hyphen p hyphen hyphen p

856
00:49:44,420 --> 00:49:50,570
hyphen hyphen q hyphen hyphen
hyphen hyphen hyphen hyphen--

857
00:49:50,570 --> 00:49:53,730
that's a lot of hyphens.

858
00:49:53,730 --> 00:49:55,176
What's wrong with this?

859
00:49:59,963 --> 00:50:01,046
Does anyone see a problem?

860
00:50:06,002 --> 00:50:06,502
Yes?

861
00:50:06,502 --> 00:50:08,379
STUDENT: [INAUDIBLE] .

862
00:50:08,379 --> 00:50:09,670
JUSTIN CURRY: Exactly, exactly.

863
00:50:09,670 --> 00:50:11,440
It doesn't follow the rule.

864
00:50:11,440 --> 00:50:14,440
The rules I told you in the
axioms, which you start from,

865
00:50:14,440 --> 00:50:17,650
you only ever have
one p and one q.

866
00:50:17,650 --> 00:50:19,840
This is not even what we call--

867
00:50:19,840 --> 00:50:23,710
so this is not
what we will refer

868
00:50:23,710 --> 00:50:29,890
to as a well-formed formula.

869
00:50:34,890 --> 00:50:37,470
So you have to be
really careful with what

870
00:50:37,470 --> 00:50:40,680
meaning means and
when you try to create

871
00:50:40,680 --> 00:50:44,160
an isomorphism between what
you know about addition

872
00:50:44,160 --> 00:50:47,130
and the formal systems you play.

873
00:50:47,130 --> 00:50:50,550
Try to come up with an
alternative interpretation.

874
00:50:50,550 --> 00:50:56,700
We could have just interpreted
these p's, q's and hyphens as,

875
00:50:56,700 --> 00:51:03,270
we're going to call p, we're
going to say that's horse.

876
00:51:03,270 --> 00:51:07,305
And q, that's apple.

877
00:51:10,130 --> 00:51:17,890
And one hyphen is
happy, and two hyphens

878
00:51:17,890 --> 00:51:26,030
is happy happy, and so on.

879
00:51:26,030 --> 00:51:29,210
So suddenly, we have an
interpretation for this string.

880
00:51:29,210 --> 00:51:33,710
It's not 2 plus 3
equals 4, but it's

881
00:51:33,710 --> 00:51:37,490
happy happy horse, happy
happy apple, happy happy

882
00:51:37,490 --> 00:51:38,420
happy happy happy.

883
00:51:40,970 --> 00:51:46,070
Doesn't mean anything, but
it's an interpretation.

884
00:51:46,070 --> 00:51:50,360
And there's no reason not
to make that interpretation.

885
00:51:50,360 --> 00:51:54,170
Perhaps to horses, this
is, actually, more sensible

886
00:51:54,170 --> 00:51:55,640
than addition.

887
00:51:55,640 --> 00:51:58,040
First of all, when
we do addition,

888
00:51:58,040 --> 00:52:00,680
we're representing
these numbers in base 10

889
00:52:00,680 --> 00:52:02,930
because we have 10 fingers.

890
00:52:02,930 --> 00:52:04,700
But horses don't
have 10 fingers.

891
00:52:04,700 --> 00:52:07,910
And numbers written in base 10
don't mean anything to horses.

892
00:52:07,910 --> 00:52:11,570
But perhaps happy, horse, apple
really makes much more sense

893
00:52:11,570 --> 00:52:13,610
to a horse.

894
00:52:13,610 --> 00:52:16,250
So we're going to throw out--
and I have to be a little

895
00:52:16,250 --> 00:52:18,580
rushed about this--

896
00:52:18,580 --> 00:52:21,180
be thinking about where
does meaning come from?

897
00:52:21,180 --> 00:52:24,960
How do we actually assign
meaning to meaningless symbols?

898
00:52:24,960 --> 00:52:26,210
Because that's the goal here.

899
00:52:26,210 --> 00:52:27,960
We're going to go from
meaningless symbols

900
00:52:27,960 --> 00:52:29,967
in mathematics to meaning.

901
00:52:29,967 --> 00:52:31,550
And then, we're going
to try to create

902
00:52:31,550 --> 00:52:33,620
an isomorphism
between the universe

903
00:52:33,620 --> 00:52:35,450
and our formal systems.

904
00:52:35,450 --> 00:52:40,570
And this leads me
perfectly into this idea of

905
00:52:40,570 --> 00:52:42,515
is reality a formal system?

906
00:52:45,440 --> 00:52:50,240
And if you go to page
29 in your notes,

907
00:52:50,240 --> 00:52:52,820
you've got this long quote.

908
00:52:52,820 --> 00:52:55,267
It stretches on to 30.

909
00:52:55,267 --> 00:52:56,350
I'll go and start reading.

910
00:52:56,350 --> 00:52:58,046
It's at the bottom.

911
00:52:58,046 --> 00:53:02,180
It says, "Can all of reality
be turned into a formal system?

912
00:53:02,180 --> 00:53:05,196
In a very broad sense, the
answer might appear to be yes.

913
00:53:05,196 --> 00:53:07,070
One could suggest, for
instance, that reality

914
00:53:07,070 --> 00:53:11,110
is itself nothing but one very
complicated formal system.

915
00:53:11,110 --> 00:53:13,730
Its symbols do not move
around on paper but, rather,

916
00:53:13,730 --> 00:53:15,360
in a three-dimensional vacuum--

917
00:53:15,360 --> 00:53:16,090
space.

918
00:53:16,090 --> 00:53:17,840
They are the elementary
particles of which

919
00:53:17,840 --> 00:53:19,690
everything is composed--

920
00:53:19,690 --> 00:53:22,160
tacit assumption that there
is an end to the descending

921
00:53:22,160 --> 00:53:23,810
chain of matter,
that the expression

922
00:53:23,810 --> 00:53:27,950
'elementary particles'
makes sense.

923
00:53:27,950 --> 00:53:29,720
The typographical
rules are the laws

924
00:53:29,720 --> 00:53:32,750
of physics, which tell
how--" we're on page 29,

925
00:53:32,750 --> 00:53:34,299
if you just want to catch up--

926
00:53:34,299 --> 00:53:35,840
"The typographical
rules are the laws

927
00:53:35,840 --> 00:53:37,880
of physics, which tell
how, given the positions

928
00:53:37,880 --> 00:53:40,630
and velocities of all the
particles at a given instant,

929
00:53:40,630 --> 00:53:43,160
to modify them,
resulting in a new set

930
00:53:43,160 --> 00:53:46,160
of positions and velocities
belonging to the next instant.

931
00:53:46,160 --> 00:53:49,970
So the theorems of this
grand formal system

932
00:53:49,970 --> 00:53:52,880
are the possible configurations
of particles at different times

933
00:53:52,880 --> 00:53:54,806
in the universe.

934
00:53:54,806 --> 00:53:57,230
The sole axiom is,
or perhaps was,

935
00:53:57,230 --> 00:53:58,970
the original
configuration of all the

936
00:53:58,970 --> 00:54:01,160
particles at the
beginning of time.

937
00:54:01,160 --> 00:54:03,670
This is so grandiose
a conception, however,

938
00:54:03,670 --> 00:54:05,570
it has only the most
theoretical interest.

939
00:54:05,570 --> 00:54:09,170
And besides, quantum mechanics
and other parts of physics

940
00:54:09,170 --> 00:54:11,420
cast at least some doubt on
even the theoretical worth

941
00:54:11,420 --> 00:54:12,890
of this idea.

942
00:54:12,890 --> 00:54:14,660
Basically, we are
asking if the universe

943
00:54:14,660 --> 00:54:20,900
operates deterministically,
which is an open question."

944
00:54:20,900 --> 00:54:23,090
I think it was Laplace
who said, well,

945
00:54:23,090 --> 00:54:25,520
look, if you were to
give me the position

946
00:54:25,520 --> 00:54:28,940
and momentum of every
particle in the universe,

947
00:54:28,940 --> 00:54:31,340
I could tell you the
rest of the future.

948
00:54:31,340 --> 00:54:34,850
And this leads to one of the
grand philosophical questions

949
00:54:34,850 --> 00:54:40,380
which we'll be investigating
as part of this class, as well,

950
00:54:40,380 --> 00:54:44,150
which is, if the universe
operates deterministically,

951
00:54:44,150 --> 00:54:48,200
if Newton's laws
govern how my arm falls

952
00:54:48,200 --> 00:54:52,840
and how all the atoms in my
body interact, where does free

953
00:54:52,840 --> 00:54:54,200
will creep into?

954
00:54:54,200 --> 00:54:56,900
How do I know I have
control over these actions,

955
00:54:56,900 --> 00:54:59,060
and it's not the fact
that, at the Big Bang,

956
00:54:59,060 --> 00:55:02,150
there was a denser
cluster of atoms over here

957
00:55:02,150 --> 00:55:05,420
and a less dense over here,
and things evolved according

958
00:55:05,420 --> 00:55:08,600
to deterministic laws, much
like the formal systems

959
00:55:08,600 --> 00:55:10,640
we're playing with here?

960
00:55:10,640 --> 00:55:14,100
So this question, you can
really think of on two levels--

961
00:55:14,100 --> 00:55:19,370
one, can the universe be
thought of as being modeled

962
00:55:19,370 --> 00:55:22,970
by a formal system,
having forces, and solving

963
00:55:22,970 --> 00:55:25,490
equations for the
particles here,

964
00:55:25,490 --> 00:55:28,500
and it collides with another
particle at this angle,

965
00:55:28,500 --> 00:55:32,450
they go off like this,
and things like this?

966
00:55:32,450 --> 00:55:36,800
but also, I think, likes to
ask another question, which

967
00:55:36,800 --> 00:55:41,240
is version 2, for those of
you who are Matrix fans.

968
00:55:41,240 --> 00:55:45,320
To what extent is the universe
a formal system proper,

969
00:55:45,320 --> 00:55:46,760
in a sense?

970
00:55:46,760 --> 00:55:48,830
Is it a program running
in the background

971
00:55:48,830 --> 00:55:51,920
of some hyperdimensional
alien who's playing WoW,

972
00:55:51,920 --> 00:55:55,910
and he's just
running our universe

973
00:55:55,910 --> 00:56:00,180
as a simulation on his
supercomputer cluster

974
00:56:00,180 --> 00:56:01,430
that he's got in his basement?

975
00:56:04,760 --> 00:56:05,340
Who knows?

976
00:56:05,340 --> 00:56:07,048
I mean, if the universe
is deterministic,

977
00:56:07,048 --> 00:56:09,140
or he's just coded up--

978
00:56:09,140 --> 00:56:10,520
hacking away in Python--

979
00:56:10,520 --> 00:56:12,380
all of our rules
of our universe,

980
00:56:12,380 --> 00:56:15,110
and he said, all right,
let's let this simulation go.

981
00:56:15,110 --> 00:56:16,820
And here we are,
in his computer,

982
00:56:16,820 --> 00:56:19,410
having all these dramatic
interactions with people,

983
00:56:19,410 --> 00:56:22,662
et cetera, et cetera,
and he's just, oh, well,

984
00:56:22,662 --> 00:56:25,730
a bug came up, et cetera.

985
00:56:25,730 --> 00:56:30,390
It's kind of interesting
to think about.

986
00:56:30,390 --> 00:56:38,980
So we've, now, really hit home
these five tools for thinking.

987
00:56:38,980 --> 00:56:42,040
And we're going to be
revisiting all of these ideas

988
00:56:42,040 --> 00:56:43,460
throughout the entire book.

989
00:56:43,460 --> 00:56:48,610
And one of the things that
Douglas Hofstadter does

990
00:56:48,610 --> 00:56:53,560
is he structures his book in its
own kind of recursive fashion.

991
00:56:53,560 --> 00:56:55,750
And I only gave you a
few specific instances

992
00:56:55,750 --> 00:56:57,410
of where recursion shows up.

993
00:56:57,410 --> 00:57:00,430
And this represents my bias.

994
00:57:00,430 --> 00:57:02,920
For me, I'm very much an art
person and a math person.

995
00:57:02,920 --> 00:57:05,320
But I'm not so much
of a music person.

996
00:57:05,320 --> 00:57:08,530
And I really encourage you guys
to bring in different elements.

997
00:57:08,530 --> 00:57:12,160
Because GEB has such
high-dimensional structure

998
00:57:12,160 --> 00:57:15,320
to it, everybody contributes
their own slice to it.

999
00:57:15,320 --> 00:57:18,750
And one thing which I would
hate to deny you guys from

1000
00:57:18,750 --> 00:57:21,560
is the music aspect
of this book.

1001
00:57:21,560 --> 00:57:24,934
Each one of Douglas
Hofstadter's dialogues

1002
00:57:24,934 --> 00:57:26,350
is, actually,
structured and based

1003
00:57:26,350 --> 00:57:28,570
upon a piece of Bach's music.

1004
00:57:28,570 --> 00:57:30,824
And if you listen
to Bach's music,

1005
00:57:30,824 --> 00:57:32,740
and you read the dialogue,
he might, actually,

1006
00:57:32,740 --> 00:57:35,890
hint at some of the connections,
some of the isomorphism

1007
00:57:35,890 --> 00:57:38,740
that Hofstadter's alluding to.

1008
00:57:38,740 --> 00:57:41,740
But first of all,
you should know

1009
00:57:41,740 --> 00:57:45,940
why he chose Bach, how
recursion acts in music.

1010
00:57:45,940 --> 00:57:49,420
And that's why I have this
whole speaker setup, here.

1011
00:57:49,420 --> 00:57:56,060
So allow me to play.

1012
00:57:56,060 --> 00:57:57,811
So this is Bach's
Little Fugue in G minor.

1013
00:57:57,811 --> 00:57:59,560
[MUSIC - BACH, "LITTLE FUGUE IN
 G MINOR"]

1014
00:57:59,560 --> 00:58:01,730
Just as a nice anecdote,
who here has seen

1015
00:58:01,730 --> 00:58:04,090
A Beautiful Mind, the movie?

1016
00:58:04,090 --> 00:58:04,590
All right.

1017
00:58:04,590 --> 00:58:06,230
So John Nash, the
mathematician who

1018
00:58:06,230 --> 00:58:08,830
went crazy, Princeton,
et cetera, the story

1019
00:58:08,830 --> 00:58:10,580
goes that he used to
actually stalk around

1020
00:58:10,580 --> 00:58:12,290
the halls of the
math department,

1021
00:58:12,290 --> 00:58:15,740
smoking cigarettes and
whistling this song constantly.

1022
00:58:15,740 --> 00:58:18,930
And what were some
of the things which

1023
00:58:18,930 --> 00:58:21,500
you noticed about this piece?

1024
00:58:21,500 --> 00:58:24,390
For those of you with
good auditory abilities,

1025
00:58:24,390 --> 00:58:25,652
what did you notice?

1026
00:58:25,652 --> 00:58:27,620
STUDENT: They're,
sort of, patterns.

1027
00:58:27,620 --> 00:58:28,570
JUSTIN CURRY: OK.

1028
00:58:28,570 --> 00:58:31,734
Elaborate a little
bit on these patterns.

1029
00:58:31,734 --> 00:58:32,650
STUDENT: I don't know.

1030
00:58:32,650 --> 00:58:34,499
I'm not a music person, either.

1031
00:58:34,499 --> 00:58:35,040
I don't know.

1032
00:58:35,040 --> 00:58:38,214
Maybe, just, after a certain
number of notes, it repeats.

1033
00:58:38,214 --> 00:58:39,130
JUSTIN CURRY: Exactly.

1034
00:58:39,130 --> 00:58:41,330
So you heard it come
in at a different tone,

1035
00:58:41,330 --> 00:58:43,360
at a different volume.

1036
00:58:43,360 --> 00:58:45,490
And you noticed it
was the same theme.

1037
00:58:45,490 --> 00:58:47,530
It's the same theme
that he played--

1038
00:58:47,530 --> 00:58:50,680
stretched, inverted,
backwards, on higher levels,

1039
00:58:50,680 --> 00:58:51,790
on lower levels.

1040
00:58:51,790 --> 00:58:55,619
So GEB is, actually, very
much structured like a fugue.

1041
00:58:55,619 --> 00:58:57,160
Hofstadter lays out
for us-- and what

1042
00:58:57,160 --> 00:58:59,590
I did in this first
lecture was I'm

1043
00:58:59,590 --> 00:59:01,260
laying out the
entire book for you,

1044
00:59:01,260 --> 00:59:03,520
all in one go, so that
way, you understand it

1045
00:59:03,520 --> 00:59:06,040
when I play it stretched
out, inverted, backwards,

1046
00:59:06,040 --> 00:59:08,150
and at different volumes.

1047
00:59:08,150 --> 00:59:09,747
So this is nice.

1048
00:59:09,747 --> 00:59:11,080
You have a musical illustration.

1049
00:59:11,080 --> 00:59:12,913
You have artistic
illustrations of the ideas

1050
00:59:12,913 --> 00:59:14,090
we're talking about.

1051
00:59:14,090 --> 00:59:22,480
But we need to, actually,
settle into the book itself.

1052
00:59:22,480 --> 00:59:26,140
So Curran Kelleher and
I, or anyone else who's

1053
00:59:26,140 --> 00:59:28,152
really excited about reading--

1054
00:59:28,152 --> 00:59:29,860
anybody really excited
about volunteering

1055
00:59:29,860 --> 00:59:32,650
for reading the dialogue?

1056
00:59:32,650 --> 00:59:35,740
Anybody have the book
with them right now?

1057
00:59:35,740 --> 00:59:38,050
Oh, good job.

1058
00:59:38,050 --> 00:59:39,550
Would you like to read?

1059
00:59:39,550 --> 00:59:42,650
You don't have to.

1060
00:59:42,650 --> 00:59:45,507
STUDENT: [INAUDIBLE] Yeah, sure.

1061
00:59:45,507 --> 00:59:46,590
JUSTIN CURRY: You want to?

1062
00:59:46,590 --> 00:59:48,450
OK.

1063
00:59:48,450 --> 00:59:51,000
So we're going to spend
the last 15 minutes going

1064
00:59:51,000 --> 00:59:51,960
through a dialogue.

1065
00:59:51,960 --> 00:59:55,008
I, actually, have another copy.

1066
00:59:55,008 --> 00:59:56,490
Good.

1067
00:59:56,490 --> 01:00:01,650
And so I need two characters--
one to be Achilles and one

1068
01:00:01,650 --> 01:00:02,700
to be Tortoise.

1069
01:00:02,700 --> 01:00:05,480
These are two characters we're
going to meet in this dialogue.

1070
01:00:05,480 --> 01:00:07,063
They're going to
play a prominent role

1071
01:00:07,063 --> 01:00:08,710
throughout the entire book.

1072
01:00:08,710 --> 01:00:13,665
So does anyone else want to be--

1073
01:00:13,665 --> 01:00:16,290
well, see, I like the tortoise,
so I'd like to be the tortoise.

1074
01:00:16,290 --> 01:00:18,623
But someone else can be the
tortoise if they want to be.

1075
01:00:20,790 --> 01:00:21,780
OK.

1076
01:00:21,780 --> 01:00:24,500
So we only have one soul
that's brave enough to do it.

1077
01:00:24,500 --> 01:00:26,880
All right.

1078
01:00:26,880 --> 01:00:28,710
All righty.

1079
01:00:28,710 --> 01:00:29,720
So page 79.

1080
01:00:33,569 --> 01:00:34,563
Yeah, sorry.

1081
01:00:40,540 --> 01:00:44,020
So I'm going to give you
some quick background

1082
01:00:44,020 --> 01:00:46,210
on this dialogue.

1083
01:00:46,210 --> 01:00:48,310
So Hofstadter, like
me, believes that it's

1084
01:00:48,310 --> 01:00:52,780
important to introduce the idea
of a topic conceptually first,

1085
01:00:52,780 --> 01:00:55,570
before you start
really diving into it.

1086
01:00:55,570 --> 01:00:58,820
So he prefaces every
chapter with a dialogue.

1087
01:00:58,820 --> 01:01:01,660
And the dialogue is kind of
a conceptual introduction

1088
01:01:01,660 --> 01:01:03,487
to the ideas we're
talking about.

1089
01:01:03,487 --> 01:01:06,070
To go ahead and give you an idea
of what this dialogue's based

1090
01:01:06,070 --> 01:01:11,050
on, it's going to be the
conflict of two mathematicians,

1091
01:01:11,050 --> 01:01:15,010
Kurt Godel and David Hilbert.

1092
01:01:15,010 --> 01:01:16,750
David Hilbert believed
that mathematics

1093
01:01:16,750 --> 01:01:20,800
could be put into a formal
system very rigorously,

1094
01:01:20,800 --> 01:01:23,440
and it could also be proved
to be consistent and complete.

1095
01:01:23,440 --> 01:01:25,045
Those are two words
which I'm going

1096
01:01:25,045 --> 01:01:28,649
to have to define at the
end of this dialogue.

1097
01:01:28,649 --> 01:01:30,190
But let's go and
start it off and try

1098
01:01:30,190 --> 01:01:31,939
to work quickly through this.

1099
01:01:31,939 --> 01:01:34,480
I'm going to ask that when you
have the italics, you go ahead

1100
01:01:34,480 --> 01:01:36,221
and read it as part
of your section,

1101
01:01:36,221 --> 01:01:38,470
so people have an idea of
what's going on in the book.

1102
01:01:50,430 --> 01:01:51,880
All right, excellent.

1103
01:01:51,880 --> 01:01:56,080
So we don't have,
really, any time left.

1104
01:01:56,080 --> 01:01:58,900
But I want to say one thing.

1105
01:01:58,900 --> 01:02:01,590
It's a challenge.

1106
01:02:01,590 --> 01:02:06,580
Pay attention to
Tortoise's quote on page 81

1107
01:02:06,580 --> 01:02:09,140
when she talks about acrostics.

1108
01:02:09,140 --> 01:02:13,080
if you can find two
acrostics in this dialogue,

1109
01:02:13,080 --> 01:02:15,530
I'll [AUDIO OUT].