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- [Narrator] The following content

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is provided under a

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- All right, hello.

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Welcome to a Godel, Escher, Bach:

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A Mental Space Odyssey.

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My name is Justin Curry

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and I'm a senior in

mathematics here at MIT.

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I've spent the last year

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at Cambridge University at UK,

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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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but I'm excited to teach

Godel, Escher, Bach again.

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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

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all in one go.

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And it's impossible.

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Most undergrads can't get

through it in 13 weeks.

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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

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but to get the essence of

Godel, Escher Bach out.

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But I want to make sure

we introduce everybody

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just to get people's names.

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This will help me take attendance

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and it will also...

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Just I also want you to say

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what is it when you read

the course catalogue

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that interested you most

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and why essentially why

you're sitting here today.

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I'm curious.

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So what is the idea behind this book?

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I interviewed a good

many of you this morning

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and just to make sure that you guys

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felt comfortable with mathematics.

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This course isn't directly

about mathematics.

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There's a lot of mathematics

being talked about.

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Yes you have a question?

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- [Student] What is the class about.

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- Ok so, that's what I'm

gonna go through right now.

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The idea here is that Douglas Hofstadter

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is interested in one primary question

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and that question is how does a self

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come out of things which have no selves?

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How is it that all these carbon atoms

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and molecules and proteins

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which make us up in the physical universe,

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how do they go from being meaningless

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to developing into an entity

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which can refer to itself.

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Like right now I'm saying I think this,

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I think you like this.

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I'm meeting all of you as individuals.

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Each one of you claim to have a self.

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You might remember Des

Cartes' famous quote,

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"I think therefore I am."

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So it seems like the I,

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when I say the I,

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I mean the things we call ourselves,

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as a real existent thing.

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But it's a complex question.

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How do we get I's out of non-I's.

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And that's that's kind of

gonna be the goal over here.

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So I'm just gonna call it I.

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But how do you get to an I.

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And get to an I by having

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a bunch of meaningless primitives.

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Things like atoms,

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proteins,

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molecules I should say if I want to, etc.

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Like this 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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Douglas Hofstadter,

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he wrote this book back in the 70s

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when he was doing

graduate school in physics

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and this was after him

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doing a math undergrad at Stanford.

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He believed that he saw

that he saw the answer.

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When he was playing

around with mathematics

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and the very formal systems we play with.

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Like when we write down things

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like two plus two equals four,

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these are just symbols.

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And as we go through today,

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I'll show you completely equivalent ways

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of doing addition,

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which will look like this and...

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And these are just logical primitives.

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Like if you've seen any set theory,

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and you know don't feel scared

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if you haven't seen any of these symbols,

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but like there exists an X for every.

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We give these interpretations

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but the idea is that mathematics

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can be reduced to a bunch

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of meaningless operations,

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just symbol shunting.

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But what's interesting is that

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within mathematics there exists

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an equivalent to self reference.

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This is a bunch of atoms and proteins

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referring to itself, calling itself an I.

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What happens here

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and this is going to be kind of underneath

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the name of Godel,

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is we're gonna get to some

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incompleteness theorems.

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Gonna get to some statements

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which 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

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how do we go from meaningless symbols

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to something which refers to itself

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and which has meaning.

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The claim then is

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is that these two systems are equivalent

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and this is really the profound idea.

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I'm gonna draw this symbol.

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And I'm going to use a

term called isomorphism.

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An isomorphism is basically an equals too,

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and equals in a different sense.

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But the idea here is in many ways

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we can link atoms

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and proteins to kind of logical

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symbolic primitives in mathematics

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and we understand how

we get self-reference

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in mathematics.

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So maybe we can use this to understand

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how we get I's,

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how self comes out of now non-self.

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This is a really tall order

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but we're gonna try to do it.

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And that's what this book attempts to do.

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And what I've done

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is isolate the chapters in this book,

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which I think are most pertinent

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to this string of thought.

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Basically what we're gonna do

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is we're gonna learn how

it works in mathematics.

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We're gonna go from logical primitives

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and workup self-reference.

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And talk about Zen

Buddhism, consciousness etc.

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But that's going to happen

as we leap over here

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because we're gonna work up, down,

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and then around.

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And we'll conclude the course

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with some interesting questions

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about artificial intelligence

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and how intelligent things

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come out of unintelligent things.

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So when I was teaching this course

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two years ago or two springs ago,

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I ran into kind of five things,

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which I viewed as really important tools

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for thinking.

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And this is kind of, I've

had to condense a little bit

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into my famous tools for thinking lecture.

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The idea here is that Godel, Escher, Bach

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has an incredible number

of conceptual tools

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for thinking about this complex problem

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of how do we go from a non-self to a self.

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And just outline these real quick.

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We're gonna have

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isomorphisms.

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And I'll explain all

these terms as we go on.

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Recursion.

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I'm going to leave this one mainly up

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to current on the second lecture.

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Paradox.

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And this is infinity,

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and all these concepts

are very closely linked.

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And finally,

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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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So first let me go through

kind of definitions

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of these terms.

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And isomorphism,

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I want you to all be

very careful with this

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because when you start

talking to mathematicians

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you know growing 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,

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the way it's going to

be used in this class

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is very loose.

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We're gonna make very kind of

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intuitive statements, like

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you know what's the isomorphism

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between a car,

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I'm not a great artist here.

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What's the isomorphism

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between a skateboard and a car?

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And you know you might say lots of things

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like it carries a person.

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It has four wheels.

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So what we do is we construct a map,

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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

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and preserve information,

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preserve kind of structure.

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If you really feel like following along,

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I've included actually a quote

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from Douglas Hofstadter.

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And on a page seven 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

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"when to complex structures

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"can be mapped onto each other

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"in a ways that to each

part of one structure

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"there's a corresponding part

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"and the other structure,

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"where corresponding means

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"that the two parts play

similar similar roles

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"in their respective structures."

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This is how we're going to

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always use the term

isomorphism in this class.

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If you're taking the

abstract algebra class,

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it's going to mean something

a lot more specific,

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and you're gonna have a lot more details.

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You might actually think of these

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as kind of,

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what I'll say but don't worry about it

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is a homomorphism.

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And the idea with the homomorphism

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is that there are a lot more details here

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than there are here.

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And for example,

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there's no steering well.

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There's a steering well in a car

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but there's no steering well

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specifically in a skateboard.

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So if you were to go,

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if you were to create a map

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from the car to the skateboard

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that detail would have

to go somewhere else.

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But don't worry about those necessities.

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But when I say the term isomorphism,

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think of equals.

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And then I'll often use

that symbol right there.

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So this is gonna be really important

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because it's going to be how we're

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gonna get meaning out of things.

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and you'll see it a lot coming up

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and over the book.

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But first I'm gonna hop on

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and talk about recursion.

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Recursion is basically...

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It's seen everywhere

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but it's kind of a list of instructions,

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which you follow,

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but then repeat until you've reached

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kind of a final case.

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So suppose you're cooking

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and you could have a recursive algorithm

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for stirring eggs.

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And that would be whirl

and then whirl again

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and keep whirling until,

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essentially everything looks mixed up.

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That's a very loose way

of understanding it.

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But another way

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which you are probably familiar with,

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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