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JUSTIN CURRY: All right, hello.

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Welcome back.

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Today's I think
the fifth lecture.

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So I want to start off
actually with some apologies.

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And it's going to be a
bit of a sobering note.

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I want to quickly recap
some of the things we talked

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about last lecture,
and kind of bring forth

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some words of caution.

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But before I do that, I
want to also apologize

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for this chapter.

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How many of you were
kind of out of your chair

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excited in reading
this past chapter?

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All right.

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No.

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In fact, I see the opposite
of shooting your hand up.

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So I take it that most people
didn't enjoy this last chapter.

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Yeah, I mean, neither did I.
And really the only reason

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I assigned it as reading
is so that you guys could

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get kind of a fundamental idea
of MIU, like the PQ system,

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we had a completely formal
typographical system

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for encoding statements.

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And we could play in
a very mechanical way

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with these statements.

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Of course, unlike PQ and MIU--

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PQ to an extent, TNT--

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typographical number theory--
has the advantage of coding

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something which we think
we know things about--

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and that being numbers, and the
property of natural numbers--

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0, 1, 2, 3, 4, and so on--

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and truths about them.

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So I wanted you mainly to become
familiar with the notation,

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00:01:57,466 --> 00:01:58,840
and become familiar
with the idea

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00:01:58,840 --> 00:02:04,390
that you can start with a kind
of couple of seeds of axioms.

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And just by applying these
kind of recursive rules

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of development, you
can produce kind

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of a whole web of
new strings, which

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happen to, when
interpreted, provide truths,

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or things which we think
we know about numbers.

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And that was really the
main point of this chapter.

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There are going to
be a few other things

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I'm going to highlight.

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But we're going to, once
again, do a mishmash of topics,

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00:02:30,430 --> 00:02:32,080
like we seem to
do every lecture.

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And I think we'll have
something exciting to show you.

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00:02:35,740 --> 00:02:39,370
So first off, recap of--

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not yesterday, but yesterday
in terms of learning--

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last week, where we
were interested in kind

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of fundamentally a
theory of meaning.

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And this is already something
I want to kind of offer

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00:03:07,310 --> 00:03:09,210
my apologies for
is, in many ways,

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I think we provide a pretty
convincing idea of what we

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think a theory of meaning
is, even though it's not

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at all agreed upon in
the academic linguistic

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philosophical community, what a
theory of meaning actually is.

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What we kind of advertised
is that the meaning

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of kind of "snow is white."

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And what we take meaning
to mean is that there's

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some sort of complex.

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And in the words of
Douglas Hofstadter,

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an exotic isomorphism.

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So I could have exotic, or E for
exotic, for the activity which

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goes on in your brain when
you, as your visual perceptual

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devices hone in
on the chalkboard,

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and we see these broken bits of
limestone against this shale,

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or whatever the
chalkboard's made out of.

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And as our brain undergoes
kind of a complex set of edge

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detection algorithms, we
hint this light and dark here

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and here and here and here.

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And then, as this
comes into focus,

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we then recognize we have
kind of global objects.

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I don't know if you care
about the quotations,

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but you probably noticed this
group here, and here and here.

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And then once you have
these broken down,

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you then say, OK,
I recognize these.

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You then perform edge detection
on each of the letters.

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And you say OK,
S-N-O-W spells "snow."

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Snow then feeds
to this whole kind

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of long and complicated
conceptual semantic network

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of the memory of when you
first saw snow falling.

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Let's see.

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00:05:02,480 --> 00:05:06,950
You might just associate it with
the sensory feeling of cold.

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You might also associate
snow with going

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skiing or snowboarding,
and all sorts of things.

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And this continuing
to branch out,

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and it becoming a
very individual thing.

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And I think that's a very
important point to notice

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is that really what
"snow is white"

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means to me is, in some
ways, fundamentally

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different than what
it means for you.

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And that's a
dangerous statement.

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That means that kind
of a theory of meaning

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00:05:40,970 --> 00:05:44,730
lacks almost a formal
mathematical single

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interpretation, which
we intend it to be

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What philosophers and
linguists always wanted is

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we want "snow" to point somehow
to this exterior world--

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the world which we
live in-- and refer it

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00:06:01,325 --> 00:06:04,230
to the actual crystallized
structure of water

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00:06:04,230 --> 00:06:05,860
we know as snow.

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00:06:05,860 --> 00:06:10,655
And that was the upstanding
citizen meaning of snow.

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00:06:10,655 --> 00:06:12,030
And there was
nothing else-- none

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of this personal
understanding of, well,

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there was that one
time where we got

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00:06:16,440 --> 00:06:19,980
caught in a blizzard
in Kansas, or whatever.

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Like we wanted to
get rid of all kind

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of individual interpretations.

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So I advertised this
isomorphism between what

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the perceptual input of
"snow is white" does,

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and what that lights
up in your brain.

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But then we kind of carried
forth to another idea.

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And we brought out this very,
very, very connected notion

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of information.

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So if you will cast
your mind backwards,

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00:07:02,380 --> 00:07:05,710
Hofstadter talks
about this record

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which we strap onto a
space shuttle or something

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00:07:09,990 --> 00:07:14,230
and send it rocketing
across the universe.

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00:07:14,230 --> 00:07:18,100
And he kind of poses
the problem of suppose

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00:07:18,100 --> 00:07:22,090
some alien civilization
stumbled upon this record

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00:07:22,090 --> 00:07:28,610
and then said, OK, what is this?

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00:07:28,610 --> 00:07:30,670
And there is this
kind of argument

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00:07:30,670 --> 00:07:33,610
of to what extent would
the record mean anything

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00:07:33,610 --> 00:07:35,560
when it was completely
out of context

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00:07:35,560 --> 00:07:39,520
of this kind of human cultural
sociological entity which

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00:07:39,520 --> 00:07:42,280
we support here on earth.

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I mean, what does a record
mean to someone living

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00:07:47,990 --> 00:07:49,180
on alpha centauri.

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00:07:49,180 --> 00:07:52,660
I don't know.

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00:07:52,660 --> 00:07:58,750
And one of the big
arguments right away

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00:07:58,750 --> 00:08:06,250
was that the
structure of a record,

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00:08:06,250 --> 00:08:09,960
with kind of these very
neat, concentric grooves--

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00:08:09,960 --> 00:08:17,890
or actually spiraling, to be
a little more appropriate.

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00:08:17,890 --> 00:08:23,290
And somehow these aliens
busted out their microscopes

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00:08:23,290 --> 00:08:25,690
and started analyzing
little sections

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00:08:25,690 --> 00:08:30,652
of this strange artifact.

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00:08:30,652 --> 00:08:32,860
And they would then notice
these kind of regularities

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00:08:32,860 --> 00:08:34,225
in sort of grooves.

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00:08:39,780 --> 00:08:43,860
And then they would kind
of ask, well, that's odd.

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00:08:43,860 --> 00:08:46,980
I wonder if that means anything?

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00:08:46,980 --> 00:08:49,920
And then this idea of them
somehow from the record

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00:08:49,920 --> 00:08:53,010
be able to reverse engineer
what a record player might be,

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00:08:53,010 --> 00:08:56,440
or look like, and then
actually play the music.

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So even assuming if they
had somehow figured out

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this incredible string
of ideas, even if they

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00:09:03,185 --> 00:09:04,810
were to play the
music, would the music

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00:09:04,810 --> 00:09:06,540
mean anything to them?

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00:09:06,540 --> 00:09:11,000
Would it just sound like a
bunch of garbled mishmash?

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00:09:11,000 --> 00:09:12,660
Latif, you have a comment?

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00:09:12,660 --> 00:09:14,760
AUDIENCE: [INAUDIBLE] noise.

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00:09:14,760 --> 00:09:17,365
JUSTIN CURRY: So Latif argues
that music would just be noise.

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00:09:17,365 --> 00:09:18,740
How many of you
agree with Latif?

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00:09:21,420 --> 00:09:24,120
You think Latif's right?

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00:09:24,120 --> 00:09:27,191
Does anyone have the cahoneys
to stand up to Latif?

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00:09:27,191 --> 00:09:31,956
AUDIENCE: [INAUDIBLE]

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00:09:31,956 --> 00:09:33,830
JUSTIN CURRY: So Max is
saying we're assuming

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00:09:33,830 --> 00:09:34,913
that they can hear at all.

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00:09:34,913 --> 00:09:36,440
OK.

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00:09:36,440 --> 00:09:38,820
So let's go ahead
and grant them that.

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And if we're going
to kind of engage

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in this philosophical dialogue,
let's grant them the ability

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to hear, and let's
even assume that they

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can hear in a similar
frequency range that we do.

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00:09:50,060 --> 00:09:52,100
Let's say if in
many ways somehow

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there is this universal
form known as a human,

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00:09:54,920 --> 00:09:57,440
and for planets with
similar gravitational pulls

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and things like that,
and just the fact

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that carbon happens to be
a really stable molecule,

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00:10:01,970 --> 00:10:05,030
that evolution carried
out in almost exact sense

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00:10:05,030 --> 00:10:07,020
on this other planet.

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00:10:07,020 --> 00:10:09,154
Would the music still
mean the same thing?

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00:10:09,154 --> 00:10:10,070
Would it have meaning?

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00:10:13,490 --> 00:10:14,236
Yes?

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00:10:14,236 --> 00:10:17,530
AUDIENCE: Well, the sounds
themselves would still

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00:10:17,530 --> 00:10:18,990
pretty much sound the same.

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00:10:18,990 --> 00:10:20,736
But if there was
somebody singing,

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00:10:20,736 --> 00:10:22,652
that probably wouldn't
make any sense to them.

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00:10:22,652 --> 00:10:26,537
It would be like listening to
any foreign music in a language

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00:10:26,537 --> 00:10:27,620
that you don't understand.

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00:10:27,620 --> 00:10:28,370
JUSTIN CURRY: OK.

185
00:10:28,370 --> 00:10:30,710
So let's say it was
a piece of Bach.

186
00:10:33,650 --> 00:10:35,540
No vocals.

187
00:10:35,540 --> 00:10:41,240
Just kind of triumphant
sounds of glory or whatever.

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00:10:45,190 --> 00:10:45,980
Felix.

189
00:10:45,980 --> 00:10:47,970
AUDIENCE: [INAUDIBLE]

190
00:10:47,970 --> 00:10:48,970
JUSTIN CURRY: All right.

191
00:10:48,970 --> 00:10:51,095
So it'd probably [? wouldn't ?]
know what piano is.

192
00:10:51,095 --> 00:10:52,750
But do you think
they would still

193
00:10:52,750 --> 00:10:56,710
have this kind of
trigger of beauty,

194
00:10:56,710 --> 00:11:02,500
or even kind of awe or
desire over the music,

195
00:11:02,500 --> 00:11:05,407
even if they didn't know
it, just from the sound?

196
00:11:05,407 --> 00:11:07,902
AUDIENCE: Where do
those desires come from?

197
00:11:07,902 --> 00:11:09,690
What triggers those desires?

198
00:11:09,690 --> 00:11:10,690
JUSTIN CURRY: So let's--

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00:11:10,690 --> 00:11:13,150
I should be careful--
not desires, but just

200
00:11:13,150 --> 00:11:14,441
a sense of beauty.

201
00:11:14,441 --> 00:11:15,940
How fundamental do
you think a sense

202
00:11:15,940 --> 00:11:21,044
of beauty and organization
is in this music?

203
00:11:21,044 --> 00:11:22,960
AUDIENCE: That's based
on your interpretation.

204
00:11:22,960 --> 00:11:26,519
If somebody else
[INAUDIBLE] says that OK,

205
00:11:26,519 --> 00:11:27,810
I can associate this with that.

206
00:11:27,810 --> 00:11:28,934
This sounds more like this.

207
00:11:28,934 --> 00:11:32,520
Or this makes me
feel a certain way.

208
00:11:32,520 --> 00:11:35,710
And that's what makes me
feel that it is beautiful

209
00:11:35,710 --> 00:11:36,460
[INAUDIBLE].

210
00:11:36,460 --> 00:11:38,001
JUSTIN CURRY: So
even for individuals

211
00:11:38,001 --> 00:11:39,990
on this earth, what
Bach's music means is

212
00:11:39,990 --> 00:11:41,790
different for everyone.

213
00:11:41,790 --> 00:11:42,588
OK.

214
00:11:42,588 --> 00:11:45,396
AUDIENCE: But there's
still sensation there.

215
00:11:45,396 --> 00:11:47,502
And I think that's one
of the coolest things

216
00:11:47,502 --> 00:11:49,150
that you can have.

217
00:11:49,150 --> 00:11:55,230
When you listen to foreign
music, you can in many ways

218
00:11:55,230 --> 00:11:58,760
decipher despair, and all
these different feelings.

219
00:11:58,760 --> 00:12:03,698
So maybe it really
depends on if these people

220
00:12:03,698 --> 00:12:07,360
get the same sort of
sensations [INAUDIBLE]

221
00:12:07,360 --> 00:12:09,690
from those different
frequencies.

222
00:12:09,690 --> 00:12:11,280
JUSTIN CURRY: Exactly.

223
00:12:11,280 --> 00:12:14,920
But even putting aside the
very complex semantic networks,

224
00:12:14,920 --> 00:12:16,920
which everybody has
internally when they trigger

225
00:12:16,920 --> 00:12:20,610
with this music, and assuming
that these aliens don't

226
00:12:20,610 --> 00:12:22,560
have it, I think
in many ways you

227
00:12:22,560 --> 00:12:24,330
can make a good
case that there's

228
00:12:24,330 --> 00:12:28,500
something about pattern
which is fundamentally

229
00:12:28,500 --> 00:12:30,540
beautiful in music.

230
00:12:30,540 --> 00:12:33,330
And that in many ways this is
mathematically describable.

231
00:12:36,035 --> 00:12:37,410
And that's kind
of a careful note

232
00:12:37,410 --> 00:12:40,470
to think about, just the
idea that pattern is itself

233
00:12:40,470 --> 00:12:44,430
beauty, and it's something
which is universal,

234
00:12:44,430 --> 00:12:47,760
that pattern is
detectable by anyone,

235
00:12:47,760 --> 00:12:50,780
and that it is, in some ways,
the only thing which there

236
00:12:50,780 --> 00:12:54,760
is to meaning, the
idea of pattern.

237
00:12:54,760 --> 00:12:59,490
So last time we really
connected pattern--

238
00:12:59,490 --> 00:13:00,785
or I tried to connect--

239
00:13:00,785 --> 00:13:03,480
to information.

240
00:13:03,480 --> 00:13:05,940
Now once again, before
we go running off

241
00:13:05,940 --> 00:13:09,270
in all sorts of
tangents, I wanted

242
00:13:09,270 --> 00:13:10,590
to provide a word of caution.

243
00:13:13,670 --> 00:13:16,310
Because what this means,
what information--

244
00:13:16,310 --> 00:13:18,990
and in particular I
at least mentioned

245
00:13:18,990 --> 00:13:26,490
the idea of this kind of
entropy version of information.

246
00:13:26,490 --> 00:13:30,820
And I'm not really going to
explain this formula that well.

247
00:13:30,820 --> 00:13:33,090
Sorry, there's
[? the sum ?] place.

248
00:13:38,000 --> 00:13:40,630
I forgot the minus
sign last time.

249
00:13:40,630 --> 00:13:41,360
Log px.

250
00:13:46,270 --> 00:13:49,960
But this is just this idea
of the probability assigned

251
00:13:49,960 --> 00:13:56,140
to something, and in some ways
the less probable something

252
00:13:56,140 --> 00:14:01,030
is, the more entropy
there is to it--

253
00:14:01,030 --> 00:14:02,380
or the other way around.

254
00:14:02,380 --> 00:14:05,050
But either way, the main
thing I want to point out

255
00:14:05,050 --> 00:14:07,480
is that there are these
mathematical ideas

256
00:14:07,480 --> 00:14:11,080
of how to measure
information in entropy form.

257
00:14:11,080 --> 00:14:14,020
And what this
fundamentally boils down to

258
00:14:14,020 --> 00:14:16,360
is if you were to describe
by either this record,

259
00:14:16,360 --> 00:14:19,540
or, as we talked about
previously, a picture

260
00:14:19,540 --> 00:14:21,640
or a piece of music,
and just feed it

261
00:14:21,640 --> 00:14:29,770
kind of as a string
of 0s and 1s,

262
00:14:29,770 --> 00:14:35,650
we could actually assign
certain values of entropy,

263
00:14:35,650 --> 00:14:39,694
and view this as a
measure of information.

264
00:14:39,694 --> 00:14:41,860
And it's interesting that
there are certain patterns

265
00:14:41,860 --> 00:14:44,305
that all languages and
certain symbols share.

266
00:14:46,880 --> 00:14:51,760
And some of these are closely
related to notions of entropy.

267
00:14:51,760 --> 00:14:57,559
And you can actually produce
an information entropic measure

268
00:14:57,559 --> 00:14:58,850
of the letters in the alphabet.

269
00:14:58,850 --> 00:15:00,599
And that's just based
on how commonly they

270
00:15:00,599 --> 00:15:01,990
occur in language.

271
00:15:01,990 --> 00:15:04,300
But there's yes-- sorry--

272
00:15:04,300 --> 00:15:05,290
go ahead Latif.

273
00:15:05,290 --> 00:15:09,285
AUDIENCE: [INAUDIBLE] because
that's the only sort of thing

274
00:15:09,285 --> 00:15:11,410
the human mind can handle.

275
00:15:11,410 --> 00:15:12,979
So it creates all
of these languages

276
00:15:12,979 --> 00:15:14,830
that have the same structure.

277
00:15:14,830 --> 00:15:18,652
JUSTIN CURRY: So Latif
argues that these patterns

278
00:15:18,652 --> 00:15:20,110
and languages emerge
because that's

279
00:15:20,110 --> 00:15:21,460
all the human mind can handle.

280
00:15:21,460 --> 00:15:23,140
But I would argue no.

281
00:15:23,140 --> 00:15:25,900
Because they've actually done
the same analysis on DNA.

282
00:15:25,900 --> 00:15:30,010
And DNA, as far as a language,
presents the same patterns.

283
00:15:30,010 --> 00:15:37,090
So unless you think the
human mind created DNA,

284
00:15:37,090 --> 00:15:39,170
then I would argue no.

285
00:15:41,980 --> 00:15:46,990
And once again, so then I talked
about kind of also a related

286
00:15:46,990 --> 00:15:47,507
phenomenon.

287
00:15:47,507 --> 00:15:49,090
And these are things
for you to Google

288
00:15:49,090 --> 00:15:54,710
and pursue in your own
time, is Zipf's law.

289
00:15:54,710 --> 00:15:57,340
And it's this idea that if
you were to take any language,

290
00:15:57,340 --> 00:16:02,110
and just to rank the
symbols based on frequency

291
00:16:02,110 --> 00:16:06,115
of occurrence, there's actually
a very nice power law behavior.

292
00:16:08,980 --> 00:16:10,750
The second most
common letter occurs

293
00:16:10,750 --> 00:16:13,060
exactly half the
time as the first,

294
00:16:13,060 --> 00:16:16,190
and the third most common
letter happens exactly a third

295
00:16:16,190 --> 00:16:17,350
of the time than the first.

296
00:16:17,350 --> 00:16:19,090
And there's this
power law behavior

297
00:16:19,090 --> 00:16:20,950
which comes out of
languages if you just

298
00:16:20,950 --> 00:16:26,037
sort them by frequently
appearing symbols.

299
00:16:26,037 --> 00:16:27,870
And this is kind of an
interesting argument.

300
00:16:27,870 --> 00:16:30,430
Zipf was, I believe,
a linguist at Harvard

301
00:16:30,430 --> 00:16:32,650
in the '40s, or '50s--

302
00:16:32,650 --> 00:16:33,220
I'm not sure.

303
00:16:35,810 --> 00:16:37,067
And he noticed this behavior.

304
00:16:37,067 --> 00:16:38,650
And it's interesting,
because then you

305
00:16:38,650 --> 00:16:41,800
can take something and
ask, is this a fake?

306
00:16:41,800 --> 00:16:43,360
Is this an actual language?

307
00:16:43,360 --> 00:16:44,990
I have no idea what it means.

308
00:16:44,990 --> 00:16:49,150
But I just noticed these certain
re-appearing symbols like that.

309
00:16:49,150 --> 00:16:54,940
And I also notice that, and
then start ranking them.

310
00:16:54,940 --> 00:16:57,160
And there's this idea
that you actually

311
00:16:57,160 --> 00:17:00,910
discern intelligible
languages based

312
00:17:00,910 --> 00:17:08,920
completely on the frequency of
occurrence of these symbols.

313
00:17:08,920 --> 00:17:13,804
Well, all this has
a rigorous footing.

314
00:17:13,804 --> 00:17:15,470
We talked about, then,
a related notion.

315
00:17:18,940 --> 00:17:21,670
Because fundamentally,
I said, this picture

316
00:17:21,670 --> 00:17:26,230
doesn't capture what
we mean by information.

317
00:17:26,230 --> 00:17:30,527
Because if you take a picture--

318
00:17:30,527 --> 00:17:31,360
where is my eraser--

319
00:17:40,162 --> 00:17:47,990
if you take a picture,
and you kind of

320
00:17:47,990 --> 00:17:52,160
have just a bunch of
seething dog barf.

321
00:17:55,520 --> 00:17:58,730
And you convert it
into these 0s and 1s,

322
00:17:58,730 --> 00:18:02,300
and then do this kind of
information entropy analysis

323
00:18:02,300 --> 00:18:06,740
on it, it might not
differ all that much

324
00:18:06,740 --> 00:18:15,920
from if you take a picture
which has something

325
00:18:15,920 --> 00:18:18,980
like the Sierpinski
gasket drawn on it,

326
00:18:18,980 --> 00:18:23,150
which has obvious
regularity, obvious pattern,

327
00:18:23,150 --> 00:18:29,420
and, it could be argued,
obvious meaning, and so on ad

328
00:18:29,420 --> 00:18:31,700
infinitum.

329
00:18:31,700 --> 00:18:36,850
So this measure,
this kind of Shannon

330
00:18:36,850 --> 00:18:40,160
and entropy information,
it's all just

331
00:18:40,160 --> 00:18:42,300
kind of a field of
information theory

332
00:18:42,300 --> 00:18:45,740
which if you guys want
to, you can come to MIT

333
00:18:45,740 --> 00:18:47,900
and audit a class
in if you want to.

334
00:18:51,620 --> 00:18:55,640
This kind of analysis on
this conversion to 0s and 1s

335
00:18:55,640 --> 00:18:58,880
of this versus the
eating dog barf

336
00:18:58,880 --> 00:19:03,149
would be not that
all distinguishable.

337
00:19:03,149 --> 00:19:04,690
So then we presented
this other idea.

338
00:19:04,690 --> 00:19:06,940
And we really got to
play around with this,

339
00:19:06,940 --> 00:19:09,110
is that, well, this
obviously has meaning,

340
00:19:09,110 --> 00:19:14,355
because I can write a short
little program with some syntax

341
00:19:14,355 --> 00:19:17,820
and et cetera, blah,
and then run it.

342
00:19:17,820 --> 00:19:19,830
And it's very short.

343
00:19:19,830 --> 00:19:22,640
And it presents
this entire picture.

344
00:19:22,640 --> 00:19:27,470
This sequence of 0s and 1s
that go on for a million,

345
00:19:27,470 --> 00:19:30,440
billion, whatever
digits, that's needed

346
00:19:30,440 --> 00:19:36,800
to encode the visual picture
of a Sierpinski gasket

347
00:19:36,800 --> 00:19:40,340
is actually an excessively
long description.

348
00:19:40,340 --> 00:19:43,490
It's kind of like taking
a pendulum, and then

349
00:19:43,490 --> 00:19:45,020
someone asking you
to describe it,

350
00:19:45,020 --> 00:19:47,150
and you going, well
it rocks there,

351
00:19:47,150 --> 00:19:50,670
and back, and there, and
back, and there, and back.

352
00:19:50,670 --> 00:19:51,920
And you just keep saying that.

353
00:19:51,920 --> 00:19:54,086
And then forever, however
long period of a time, you

354
00:19:54,086 --> 00:19:56,750
want to describe the pendulum,
you just keep saying that.

355
00:19:56,750 --> 00:19:58,790
You record your
voice saying that.

356
00:19:58,790 --> 00:20:03,230
It's a very inefficient way
to encode the regularities

357
00:20:03,230 --> 00:20:06,640
of the phenomenon.

358
00:20:06,640 --> 00:20:08,330
There's the idea that
this can actually

359
00:20:08,330 --> 00:20:11,630
be expressed
mathematically using

360
00:20:11,630 --> 00:20:16,280
just kind of a idea of
iteration of symbols.

361
00:20:16,280 --> 00:20:19,850
And we had this
Lindenmayer system.

362
00:20:24,210 --> 00:20:27,536
I have no idea if I'm
spelling this correctly.

363
00:20:27,536 --> 00:20:30,470
L-I-- linden.

364
00:20:30,470 --> 00:20:31,814
But there has to be a mayer.

365
00:20:31,814 --> 00:20:32,730
AUDIENCE: Lindenmayer.

366
00:20:32,730 --> 00:20:33,438
JUSTIN CURRY: Oh.

367
00:20:33,438 --> 00:20:40,260
So there's an N-M. N-M. I
never won the spelling bee.

368
00:20:40,260 --> 00:20:47,760
And was F minus R minus
or plus plus R minus F.

369
00:20:47,760 --> 00:20:49,410
And we could encode
these, and then we

370
00:20:49,410 --> 00:20:52,020
had this very nice program
for expanding this out

371
00:20:52,020 --> 00:20:53,040
in a recursive manner.

372
00:20:53,040 --> 00:20:56,710
And it just took a
few lines of code,

373
00:20:56,710 --> 00:20:58,830
which takes much
less information

374
00:20:58,830 --> 00:21:03,780
to encode the lines of code
than an actual picture of what's

375
00:21:03,780 --> 00:21:06,600
going on.

376
00:21:06,600 --> 00:21:09,510
So this leads to a very
important idea of kind

377
00:21:09,510 --> 00:21:11,590
of algorithmic information.

378
00:21:24,350 --> 00:21:26,750
And we then really,
really started

379
00:21:26,750 --> 00:21:30,925
playing around with
the idea of, well,

380
00:21:30,925 --> 00:21:35,930
all science ever does
is reverse engineering.

381
00:21:35,930 --> 00:21:39,830
We take phenomenon, like
the Sierpinski gasket,

382
00:21:39,830 --> 00:21:44,240
or the regularities
of a pendulum swinging

383
00:21:44,240 --> 00:21:49,190
back and forth with respect
to some angle theta,

384
00:21:49,190 --> 00:22:00,350
and we describe it very simply.

385
00:22:00,350 --> 00:22:03,350
So we make our algorithmic
content for the pendulum

386
00:22:03,350 --> 00:22:08,420
as low as possible by using
these symbols to encode

387
00:22:08,420 --> 00:22:10,670
the regularities of
the dynamics here.

388
00:22:10,670 --> 00:22:12,380
And I really apologize
to those of you

389
00:22:12,380 --> 00:22:13,790
who haven't had calculus.

390
00:22:13,790 --> 00:22:16,730
But these double
dots indicate a rate

391
00:22:16,730 --> 00:22:20,170
of change of the rate of
change of the angle theta.

392
00:22:25,570 --> 00:22:27,990
So we then started
playing around

393
00:22:27,990 --> 00:22:33,150
with looking at cellular
automaton, which is essentially

394
00:22:33,150 --> 00:22:35,860
this grid.

395
00:22:35,860 --> 00:22:37,020
And we divide it up.

396
00:22:39,600 --> 00:22:43,920
And the value of this grid,
whether it's black or white,

397
00:22:43,920 --> 00:22:46,230
is somehow governed
by the neighbors.

398
00:22:46,230 --> 00:22:48,560
And by just playing
around with those rules,

399
00:22:48,560 --> 00:22:56,700
we suddenly stumbled upon an
entire wealth of behavior.

400
00:22:56,700 --> 00:23:00,870
By changing the number
of colors it could have,

401
00:23:00,870 --> 00:23:03,180
and how quickly it
changed, we could actually

402
00:23:03,180 --> 00:23:04,830
emulate a wave equation.

403
00:23:04,830 --> 00:23:07,860
We could make
splashes in a puddle

404
00:23:07,860 --> 00:23:14,700
using this kind of very simple
finite and deterministic rule.

405
00:23:14,700 --> 00:23:18,670
We modeled voting patterns
by just saying, well,

406
00:23:18,670 --> 00:23:20,160
I'm going to make
my colors somehow

407
00:23:20,160 --> 00:23:22,130
dependent on what my
neighbors are doing,

408
00:23:22,130 --> 00:23:23,580
or what my neighbors
are thinking,

409
00:23:23,580 --> 00:23:27,097
kind of indicating the
dangers of groupthink.

410
00:23:27,097 --> 00:23:28,680
If you always let
yourself be affected

411
00:23:28,680 --> 00:23:30,750
by what other
people are thinking,

412
00:23:30,750 --> 00:23:32,700
we just become kind of
a homogeneous culture.

413
00:23:36,210 --> 00:23:39,060
So somehow, just by these
very, very simple rules,

414
00:23:39,060 --> 00:23:43,450
which Curran gracefully
coded up in just a few lines,

415
00:23:43,450 --> 00:23:46,590
we were suddenly
describing huge realms

416
00:23:46,590 --> 00:23:51,330
of complex behavior in the
social, physical, and cultural

417
00:23:51,330 --> 00:23:54,000
worlds.

418
00:23:54,000 --> 00:23:56,880
And that kind of presented
some interesting ideas.

419
00:23:56,880 --> 00:23:59,340
We started asking, well,
what about this and that?

420
00:23:59,340 --> 00:24:04,830
And I remember one line
we ended up stumbling upon

421
00:24:04,830 --> 00:24:10,870
was, well, because we see the
self-similarity of behavior,

422
00:24:10,870 --> 00:24:12,960
or the same voting
rule which describes

423
00:24:12,960 --> 00:24:17,760
how humans act in groups,
also describes gas droplet

424
00:24:17,760 --> 00:24:19,070
condensation.

425
00:24:19,070 --> 00:24:22,200
If you have a bunch of spraying
water on your shower curtain,

426
00:24:22,200 --> 00:24:25,980
the reason why it [? droplets ?]
[? up ?] is because each

427
00:24:25,980 --> 00:24:31,650
molecule of water is seeking
to reduce its overall energy,

428
00:24:31,650 --> 00:24:34,360
and it can do that by
grouping up with people.

429
00:24:34,360 --> 00:24:37,410
And suddenly you've got this
clustering, this coarsening

430
00:24:37,410 --> 00:24:40,260
property, which we
saw in water droplets.

431
00:24:40,260 --> 00:24:43,170
And then we started, oh, does
that mean that the universe is

432
00:24:43,170 --> 00:24:46,230
fractal, and that there's
this concepts which

433
00:24:46,230 --> 00:24:47,440
apply on all levels.

434
00:24:47,440 --> 00:24:51,030
And I've kind of felt
ourselves getting caught up

435
00:24:51,030 --> 00:24:53,070
in the moment, and
maybe leading you guys

436
00:24:53,070 --> 00:24:56,730
down paths which,
although exciting,

437
00:24:56,730 --> 00:24:58,865
aren't exactly well founded.

438
00:24:58,865 --> 00:25:00,990
It's not like everyone in
the scientific community,

439
00:25:00,990 --> 00:25:03,420
if you went to them and
said, is the universe

440
00:25:03,420 --> 00:25:06,750
a giant fractal, a lot of
people would probably say no,

441
00:25:06,750 --> 00:25:08,480
obviously not.

442
00:25:08,480 --> 00:25:11,490
And there would be some
various reasons for this.

443
00:25:11,490 --> 00:25:15,960
Because if you're talking about
self-similarity along scales,

444
00:25:15,960 --> 00:25:20,235
this view of, well, we have
the solar system, the sun,

445
00:25:20,235 --> 00:25:25,370
and these tiny planets
going around, et cetera.

446
00:25:25,370 --> 00:25:28,130
And like, a-ha, but it's the
same model that the atom is.

447
00:25:28,130 --> 00:25:29,520
Wrong.

448
00:25:29,520 --> 00:25:31,710
One of the fundamental
things in physics

449
00:25:31,710 --> 00:25:34,590
nowadays is patching
up the disagreement

450
00:25:34,590 --> 00:25:37,740
in mathematical description
and behavior of things

451
00:25:37,740 --> 00:25:41,490
at the atomic realm, which
is not at all like this.

452
00:25:41,490 --> 00:25:44,220
It's, in fact, if you guys
have done chemistry, and looked

453
00:25:44,220 --> 00:25:46,410
through the books, you
see these weird drawings

454
00:25:46,410 --> 00:25:49,830
of probability clouds, which
are governed by, fundamentally,

455
00:25:49,830 --> 00:25:52,710
quantum mechanics.

456
00:25:52,710 --> 00:25:56,610
And it doesn't at all match
the classical mechanical

457
00:25:56,610 --> 00:25:58,960
description of the solar system.

458
00:25:58,960 --> 00:26:01,094
So in that sense, no, at all.

459
00:26:01,094 --> 00:26:02,760
The universe is
obviously not a fractal,

460
00:26:02,760 --> 00:26:05,940
because the rules don't
apply on all the same levels.

461
00:26:08,460 --> 00:26:12,782
And then I did this very
exciting demonstration--

462
00:26:12,782 --> 00:26:15,240
or at least I kind of thought
it was exciting at the time--

463
00:26:15,240 --> 00:26:18,930
where I took a piece of
paper, and I crumpled it up,

464
00:26:18,930 --> 00:26:19,890
and then I unfolded it.

465
00:26:19,890 --> 00:26:26,580
And I said, well, the kind
of typology and morphology

466
00:26:26,580 --> 00:26:30,600
of the piece of paper is almost
identical to the kind of thing

467
00:26:30,600 --> 00:26:33,300
we see in mountains
and valleys and rivers.

468
00:26:33,300 --> 00:26:35,370
On this kind of pattern
formation along scales,

469
00:26:35,370 --> 00:26:38,700
the fact that what I do
here on a very local level

470
00:26:38,700 --> 00:26:43,440
is what we see on
a very large level.

471
00:26:43,440 --> 00:26:46,140
I suggested this idea of
the universe conceptually

472
00:26:46,140 --> 00:26:47,250
being a fractal.

473
00:26:47,250 --> 00:26:50,910
But what a mathematician or a
physicist might actually tell

474
00:26:50,910 --> 00:26:55,110
you is that my equations for
describing the phenomenon

475
00:26:55,110 --> 00:26:57,660
are scale-free.

476
00:26:57,660 --> 00:27:03,720
And there's this idea
of being scale-free.

477
00:27:03,720 --> 00:27:07,020
And we do this all the
time in fluid dynamics.

478
00:27:07,020 --> 00:27:10,680
I can drag my finger
through the water,

479
00:27:10,680 --> 00:27:15,180
and see little vortices
form behind my fingers.

480
00:27:15,180 --> 00:27:18,360
And then we've actually got
exact pictures of mountains

481
00:27:18,360 --> 00:27:19,650
which pierce the clouds.

482
00:27:19,650 --> 00:27:24,210
And the clouds are flowing
around this mountain tip.

483
00:27:24,210 --> 00:27:26,360
And you get these
von Karman sheets,

484
00:27:26,360 --> 00:27:29,700
where these vortices split off.

485
00:27:36,140 --> 00:27:38,010
I think it's von Karman.

486
00:27:40,617 --> 00:27:42,200
But there's so many
beautiful pictures

487
00:27:42,200 --> 00:27:44,324
you can see in fluid dynamics
and things like that.

488
00:27:44,324 --> 00:27:47,600
So this being mountain.

489
00:27:47,600 --> 00:27:50,285
Or I could equally erase
this and say finger.

490
00:27:54,140 --> 00:27:55,670
And really what I
should be trying

491
00:27:55,670 --> 00:27:59,890
to do here is
advocating the beauty

492
00:27:59,890 --> 00:28:04,040
and elegance in our formalism
with mathematical equations,

493
00:28:04,040 --> 00:28:07,550
and how we can make
these scale-free,

494
00:28:07,550 --> 00:28:12,360
and describe phenomenon
at all sorts of levels.

495
00:28:12,360 --> 00:28:16,250
So I just wanted to
be cautious, and not

496
00:28:16,250 --> 00:28:19,610
say things which will
later get me in trouble.

497
00:28:22,790 --> 00:28:26,150
But aside from that, I encourage
you all to get carried away

498
00:28:26,150 --> 00:28:27,650
and explore new ideas.

499
00:28:27,650 --> 00:28:30,149
And then the reason why
Curran and I fundamentally

500
00:28:30,149 --> 00:28:31,940
believe in computer
science and mathematics

501
00:28:31,940 --> 00:28:33,860
as a framework for
thinking is that you

502
00:28:33,860 --> 00:28:36,110
can be as lofty and
philosophical and out there as

503
00:28:36,110 --> 00:28:36,950
possible.

504
00:28:36,950 --> 00:28:37,940
But at the end of
the day, you have

505
00:28:37,940 --> 00:28:39,773
to either be able to
make it work rigorously

506
00:28:39,773 --> 00:28:42,890
with proof, or by execution
on a computer, which

507
00:28:42,890 --> 00:28:46,610
for most intents and purposes,
is just as good as proof.

508
00:28:46,610 --> 00:28:48,200
Because either it
works or it doesn't.

509
00:28:48,200 --> 00:28:50,990
And you're thinking about the
universe either works or it

510
00:28:50,990 --> 00:28:52,400
doesn't.

511
00:28:52,400 --> 00:28:55,910
And fundamentally you
need falsifiability.

512
00:28:55,910 --> 00:29:00,860
So there's a chance that
your thinking might not work.

513
00:29:00,860 --> 00:29:04,900
And it's fun to be
metaphysical and talk

514
00:29:04,900 --> 00:29:09,510
about things higher than the
universe and space and time,

515
00:29:09,510 --> 00:29:11,305
and what the structure
of these things are.

516
00:29:11,305 --> 00:29:12,680
But at the end of
the day we want

517
00:29:12,680 --> 00:29:14,690
to be able to test
our observations,

518
00:29:14,690 --> 00:29:19,440
or prove them mathematically,
or using computers.

519
00:29:19,440 --> 00:29:23,880
So that's kind of my
sobering note of caution.

520
00:29:23,880 --> 00:29:26,930
And I'm sorry for those of
you that had to go through it

521
00:29:26,930 --> 00:29:29,270
and didn't want to hear it.

522
00:29:29,270 --> 00:29:33,755
Now on to slightly
more fun things.

523
00:29:33,755 --> 00:29:37,130
First, let me check the time.

524
00:29:37,130 --> 00:29:38,320
So number theory.

525
00:29:41,860 --> 00:29:44,560
I really want to just
draw your attention

526
00:29:44,560 --> 00:29:48,830
to the idea of what we can
take as to be an axiom,

527
00:29:48,830 --> 00:29:53,140
and why we can't
prove these things.

528
00:29:53,140 --> 00:30:01,832
And if you'll cast your
eyes onto page 221,

529
00:30:01,832 --> 00:30:05,504
we saw kind of this pyramid of--

530
00:30:19,350 --> 00:30:21,820
And we had 0 plus SS.

531
00:30:21,820 --> 00:30:26,550
S0 equals SSS0, and so on.

532
00:30:29,619 --> 00:30:31,410
And, well, first of
all I want to highlight

533
00:30:31,410 --> 00:30:32,550
some things, some elegance.

534
00:30:32,550 --> 00:30:34,680
And this is really
what makes logicians

535
00:30:34,680 --> 00:30:37,290
the most anal creatures
on the planet,

536
00:30:37,290 --> 00:30:41,040
is that we don't
have numbers here.

537
00:30:41,040 --> 00:30:42,100
We just have one number.

538
00:30:42,100 --> 00:30:43,800
And that number is 0.

539
00:30:43,800 --> 00:30:46,410
And then every other number
is just the successor of 0,

540
00:30:46,410 --> 00:30:49,150
or the successor of the
successor of the successor

541
00:30:49,150 --> 00:30:52,620
of 0, or the
successor times what

542
00:30:52,620 --> 00:30:57,300
we like to say in our
metalanguage 1,729 S's,

543
00:30:57,300 --> 00:31:03,010
and then a 0, and that
representing the number 1,729.

544
00:31:03,010 --> 00:31:05,010
But there's some also
elegance in this, the fact

545
00:31:05,010 --> 00:31:07,030
that you only need one concept--

546
00:31:07,030 --> 00:31:08,250
well, two concepts.

547
00:31:08,250 --> 00:31:10,920
You need a 0, and you need
the concept of a successor.

548
00:31:10,920 --> 00:31:14,104
Then, suddenly you get
the whole thing for free.

549
00:31:14,104 --> 00:31:14,770
Which is pretty.

550
00:31:17,410 --> 00:31:21,000
But what was highlighted
by this example

551
00:31:21,000 --> 00:31:28,080
was that if we didn't
assume it, we couldn't prove

552
00:31:28,080 --> 00:31:32,270
this statement, which
says that for every a,

553
00:31:32,270 --> 00:31:42,415
where a is some
variable, 0 plus a is a.

554
00:31:42,415 --> 00:31:46,600
What a dull and obvious thing.

555
00:31:46,600 --> 00:31:54,570
But all we have at our disposal
are kind of these things.

556
00:31:54,570 --> 00:31:57,110
And in fact, we get
this whole mountain--

557
00:31:57,110 --> 00:32:00,600
this pyramid of true statements.

558
00:32:00,600 --> 00:32:03,020
And we want to the
leap to this generality

559
00:32:03,020 --> 00:32:05,810
that, well, this is
obviously the case.

560
00:32:05,810 --> 00:32:10,700
But fundamentally our desire
to leap to this conclusion

561
00:32:10,700 --> 00:32:13,280
comes from our understanding,
our mental models

562
00:32:13,280 --> 00:32:16,460
of how numbers, and how
specifically integers behave.

563
00:32:20,920 --> 00:32:24,180
And that's an important point.

564
00:32:24,180 --> 00:32:27,450
This idea of really all we
ever have at our disposal

565
00:32:27,450 --> 00:32:32,380
are our mental models, and
by making them rigorous,

566
00:32:32,380 --> 00:32:35,430
and trying to make them
rigorous through formal systems,

567
00:32:35,430 --> 00:32:37,450
we really kind of
see whether or not

568
00:32:37,450 --> 00:32:41,480
our definitions encompass
as much as we want.

569
00:32:41,480 --> 00:32:44,740
So we could never actually prove
this statement with the way

570
00:32:44,740 --> 00:32:47,260
that TNT was set up
without this axiom.

571
00:32:47,260 --> 00:32:49,570
So we eventually
had to assume it.

572
00:32:49,570 --> 00:32:51,250
And Hofstadter calls this--

573
00:32:54,280 --> 00:32:57,010
just to write it here--

574
00:32:57,010 --> 00:33:01,600
omega incomplete, where
omega is kind of the thing

575
00:33:01,600 --> 00:33:05,350
we used to refer to all
the integers at once.

576
00:33:05,350 --> 00:33:08,920
And it's just that idea
that even though we

577
00:33:08,920 --> 00:33:11,050
have this infinite
stack of true things,

578
00:33:11,050 --> 00:33:14,080
we don't have the thing which
describes them all as a truth--

579
00:33:14,080 --> 00:33:15,010
namely this.

580
00:33:18,012 --> 00:33:19,720
But you know, this
notion of mental model

581
00:33:19,720 --> 00:33:21,700
I think is really important.

582
00:33:21,700 --> 00:33:25,250
Because I remember first seeing
this proof in high school.

583
00:33:25,250 --> 00:33:28,450
And I remember just being kind
of utterly shocked and in some

584
00:33:28,450 --> 00:33:29,440
ways horrified by it.

585
00:33:32,500 --> 00:33:36,360
So, how many people
believe the following?

586
00:33:36,360 --> 00:33:44,563
0.9999 repeating is equal to 1?

587
00:33:44,563 --> 00:33:45,960
[? Ders ?] believes it.

588
00:33:45,960 --> 00:33:48,150
[? Devin ?] believes it.

589
00:33:48,150 --> 00:33:51,390
Can anyone come up and prove it?

590
00:33:51,390 --> 00:33:53,220
Felix, do you believe it?

591
00:33:53,220 --> 00:33:55,329
Can you prove it?

592
00:33:55,329 --> 00:33:57,120
AUDIENCE: I'd have to
think about it first.

593
00:33:57,120 --> 00:33:58,222
JUSTIN CURRY: Think
about it first.

594
00:33:58,222 --> 00:33:59,106
AUDIENCE: [INAUDIBLE]

595
00:33:59,106 --> 00:34:00,150
JUSTIN CURRY: Deep down.

596
00:34:02,670 --> 00:34:04,650
Does anyone want to
show off, and leap

597
00:34:04,650 --> 00:34:06,840
to the chalkboard
like a young Gauss,

598
00:34:06,840 --> 00:34:10,267
and just go ahead
and show this to me?

599
00:34:10,267 --> 00:34:10,980
[? Ders? ?]

600
00:34:10,980 --> 00:34:14,310
AUDIENCE: Is there something
like multiplying by 10?

601
00:34:14,310 --> 00:34:18,630
JUSTIN CURRY: So I'm going to go
ahead and take your suggestion,

602
00:34:18,630 --> 00:34:20,460
and kind of lead
you down the proof.

603
00:34:20,460 --> 00:34:22,739
So we're going to
call this thing x.

604
00:34:22,739 --> 00:34:25,824
And we'll temporarily
forget that we're 1.

605
00:34:25,824 --> 00:34:29,739
We're going to call
x this 0.9 repeating.

606
00:34:29,739 --> 00:34:32,610
So [? Ders ?]
recommends we consider

607
00:34:32,610 --> 00:34:41,422
the quantity 10x, which
I think is 9.9 repeating.

608
00:34:47,360 --> 00:34:49,459
A bunch of hands.

609
00:34:49,459 --> 00:34:50,929
AUDIENCE: [INAUDIBLE]
that infinity

610
00:34:50,929 --> 00:34:54,359
will be 1, because
it's approaching

611
00:34:54,359 --> 00:34:56,670
a number [INAUDIBLE].

612
00:34:56,670 --> 00:34:59,250
JUSTIN CURRY: Well, without
going to the idea of a limit,

613
00:34:59,250 --> 00:35:01,620
we can try to prove
it more fundamentally.

614
00:35:01,620 --> 00:35:02,520
I don't really know.

615
00:35:02,520 --> 00:35:03,587
Felix, do you know?

616
00:35:03,587 --> 00:35:05,100
AUDIENCE: [INAUDIBLE]

617
00:35:05,100 --> 00:35:07,620
JUSTIN CURRY: So Felix
says we should subtract x.

618
00:35:10,532 --> 00:35:16,695
So suddenly we get 9.0.

619
00:35:16,695 --> 00:35:17,520
Oh, sorry.

620
00:35:17,520 --> 00:35:21,000
I mean x itself
is 0.9 repeating.

621
00:35:21,000 --> 00:35:21,930
Dot, dot, dot.

622
00:35:21,930 --> 00:35:25,770
So then we perform the
subtraction, we get 9.

623
00:35:25,770 --> 00:35:31,490
So we have now this new
truth that 9x equals 9.

624
00:35:31,490 --> 00:35:35,420
And the only number which
satisfies that is x equals 1.

625
00:35:40,360 --> 00:35:44,650
And I actually will never forget
the girl sitting next to me

626
00:35:44,650 --> 00:35:47,630
in pre-calculus when my
teacher first did this.

627
00:35:47,630 --> 00:35:50,080
And she goes no!

628
00:35:50,080 --> 00:35:51,900
It can't be!

629
00:35:51,900 --> 00:35:53,490
But it's not 1!

630
00:35:53,490 --> 00:35:55,750
Look!

631
00:35:55,750 --> 00:35:59,320
Because if it were 1,
I would have written 1.

632
00:35:59,320 --> 00:36:02,510
But somehow I wrote down a
completely different number,

633
00:36:02,510 --> 00:36:04,990
which was the same thing.

634
00:36:04,990 --> 00:36:09,610
And this kind of shows the
really important thing which

635
00:36:09,610 --> 00:36:12,100
mental models have to tell us.

636
00:36:12,100 --> 00:36:14,440
We can have what appears to
be a correct understanding

637
00:36:14,440 --> 00:36:15,670
of an object--

638
00:36:15,670 --> 00:36:18,670
namely, real numbers.

639
00:36:18,670 --> 00:36:20,170
But then we're
continually surprised

640
00:36:20,170 --> 00:36:22,969
when they do things
like this to us, even

641
00:36:22,969 --> 00:36:24,010
at the most basic levels.

642
00:36:27,102 --> 00:36:28,810
So these are just kind
of meta statements

643
00:36:28,810 --> 00:36:33,130
about what mathematics does,
and what we do in mathematics.

644
00:36:33,130 --> 00:36:34,990
And in particular
what we're doing

645
00:36:34,990 --> 00:36:40,420
in TNT is we have this idea that
number theory should encompass-

646
00:36:40,420 --> 00:36:43,180
or at least TNT should
encompass all of our thoughts

647
00:36:43,180 --> 00:36:45,010
of number theory.

648
00:36:45,010 --> 00:36:47,800
And this goes back
to what Euclid said

649
00:36:47,800 --> 00:36:50,300
about his axioms of geometry.

650
00:36:50,300 --> 00:36:53,150
He said, we know these things
to be fundamentally true,

651
00:36:53,150 --> 00:36:55,720
or at least we
believe they are true,

652
00:36:55,720 --> 00:37:00,640
and that they encompass all
of our knowledge of geometry,

653
00:37:00,640 --> 00:37:04,990
and that everything
they produce as a result

654
00:37:04,990 --> 00:37:07,510
of those assumptions
should be true and certain.

655
00:37:10,060 --> 00:37:14,320
But then we got the whole case
with [? Sakura ?] and Gauss

656
00:37:14,320 --> 00:37:16,191
in hiding, because
of course Gauss,

657
00:37:16,191 --> 00:37:18,190
who discovered it 50 years
before everyone else,

658
00:37:18,190 --> 00:37:19,480
and then show his notebooks.

659
00:37:19,480 --> 00:37:21,940
He goes, oh, sorry,
I did that theorem

660
00:37:21,940 --> 00:37:24,640
in between my
first cup of coffee

661
00:37:24,640 --> 00:37:26,830
and my second cup of coffee.

662
00:37:26,830 --> 00:37:31,780
But, well, good, I'm glad
you discovered it, too.

663
00:37:31,780 --> 00:37:34,470
And just this concept of
non-Euclidean geometry,

664
00:37:34,470 --> 00:37:36,140
where you break that
fifth postulate,

665
00:37:36,140 --> 00:37:42,250
that if you have a
parallel line or a line,

666
00:37:42,250 --> 00:37:45,550
and a point not on it, that
there exists a unique line--

667
00:37:45,550 --> 00:37:51,400
unique exclamation point-- such
that it does not intersect.

668
00:37:51,400 --> 00:37:54,040
But if you're working on
the surface of a sphere,

669
00:37:54,040 --> 00:37:58,650
for example, and you define
your lines to be great circles,

670
00:37:58,650 --> 00:38:01,480
and then, in fact, any
possible line you draw,

671
00:38:01,480 --> 00:38:03,400
where "line" is great
circle, you actually

672
00:38:03,400 --> 00:38:07,000
have two intersection points.

673
00:38:07,000 --> 00:38:09,831
Something like that, although
that's a terrible picture.

674
00:38:09,831 --> 00:38:10,330
Sorry.

675
00:38:13,330 --> 00:38:16,160
So Hilbert, this very
guy who is trying

676
00:38:16,160 --> 00:38:19,520
to advocate all this stuff
involving number theory,

677
00:38:19,520 --> 00:38:22,970
said, well, we just have these
fundamental axiomatic notions

678
00:38:22,970 --> 00:38:24,286
of a point and line.

679
00:38:24,286 --> 00:38:25,660
And the most we
can get from them

680
00:38:25,660 --> 00:38:28,310
are their logical
relationships with each other.

681
00:38:28,310 --> 00:38:31,460
The second we try to
interpret these things,

682
00:38:31,460 --> 00:38:33,800
we then get ourselves
into trouble.

683
00:38:33,800 --> 00:38:36,290
When we interpret
the statement line,

684
00:38:36,290 --> 00:38:41,510
and we mean the straight
thing on this flat board,

685
00:38:41,510 --> 00:38:46,307
and not the curvy thing on
the surface of the Earth,

686
00:38:46,307 --> 00:38:48,140
we get ourselves into
trouble, because we're

687
00:38:48,140 --> 00:38:49,850
providing an
interpretation, and not

688
00:38:49,850 --> 00:38:52,395
sticking directly
to the formalism.

689
00:38:52,395 --> 00:38:54,770
Shows you also one of the kind
of dangers of getting away

690
00:38:54,770 --> 00:38:59,040
with your interpretations of
what your formalities tell you.

691
00:38:59,040 --> 00:39:01,220
And I'm kind of
proud of Hofstadter

692
00:39:01,220 --> 00:39:05,839
for doing that, and cautioning
us against interpretation.

693
00:39:05,839 --> 00:39:07,130
Because it gets you in trouble.

694
00:39:14,044 --> 00:39:16,460
First I'm going to field some
questions about this chapter

695
00:39:16,460 --> 00:39:20,840
before I kind of introduce
what Curran's going to do.

696
00:39:20,840 --> 00:39:23,750
And then move on to newer
and more exciting things.

697
00:39:26,450 --> 00:39:27,860
TNT.

698
00:39:27,860 --> 00:39:31,430
I mean, I remember sitting
in an undergraduate seminar

699
00:39:31,430 --> 00:39:33,140
on Godel/Escher/Bach,
and someone

700
00:39:33,140 --> 00:39:37,010
going, what the was he talking
about with these supernatural

701
00:39:37,010 --> 00:39:41,412
numbers, and omega
inconsistency?

702
00:39:41,412 --> 00:39:43,370
Because, I mean, these
aren't trivial concepts.

703
00:39:43,370 --> 00:39:44,870
I mean, entire
fields of mathematics

704
00:39:44,870 --> 00:39:47,160
have been devoted to them.

705
00:39:47,160 --> 00:39:50,180
So if you guys are
feeling completely,

706
00:39:50,180 --> 00:39:55,180
ahem, I understood
this in my sleep.

707
00:39:55,180 --> 00:39:56,285
Yeah, Sandra.

708
00:39:56,285 --> 00:39:59,135
AUDIENCE: I was confused with
how many different rules TNT

709
00:39:59,135 --> 00:40:02,935
had, because it talks
about seven categories.

710
00:40:02,935 --> 00:40:05,320
[INAUDIBLE]

711
00:40:05,320 --> 00:40:06,410
JUSTIN CURRY: Right.

712
00:40:06,410 --> 00:40:09,010
So he also talked
propositional calculus, and how

713
00:40:09,010 --> 00:40:11,369
he assumed that into TNT.

714
00:40:11,369 --> 00:40:13,535
Yeah, and I'm sorry, but I
didn't assign that really

715
00:40:13,535 --> 00:40:14,210
as reading.

716
00:40:14,210 --> 00:40:15,668
So that was kind
of out of context.

717
00:40:18,160 --> 00:40:21,340
But did any of you guys
try the little exercises

718
00:40:21,340 --> 00:40:23,320
in the book itself?

719
00:40:23,320 --> 00:40:25,360
There's one section
I want to highlight.

720
00:40:25,360 --> 00:40:29,159
And I give brownie
points to everybody

721
00:40:29,159 --> 00:40:30,325
who gives me a right answer.

722
00:40:34,300 --> 00:40:36,874
Because I think I have
my own view of answers.

723
00:40:36,874 --> 00:40:38,290
But I'm not sure
if they're right.

724
00:40:41,080 --> 00:40:44,630
Let's also test you guys'
knowledge of the notation.

725
00:40:44,630 --> 00:40:58,890
So we have tilde, upside down
A, C, colon, backwards E, b,

726
00:40:58,890 --> 00:41:13,530
colon, parentheses, SS0,
dot b, parallel lines, C.

727
00:41:13,530 --> 00:41:17,540
First of all, can someone tell
me what this means by giving it

728
00:41:17,540 --> 00:41:19,770
an interpretation, and
then tell me whether or not

729
00:41:19,770 --> 00:41:21,600
it's true or false.

730
00:41:21,600 --> 00:41:22,366
Latif.

731
00:41:22,366 --> 00:41:24,800
AUDIENCE: There
exists-- there is no C--

732
00:41:24,800 --> 00:41:28,880
or no, for all of C,
there's not [INAUDIBLE]..

733
00:41:28,880 --> 00:41:33,090
JUSTIN CURRY: Well, here, before
we do the tilde, let's do this.

734
00:41:33,090 --> 00:41:33,660
Say again.

735
00:41:33,660 --> 00:41:34,570
AUDIENCE: For all C--

736
00:41:34,570 --> 00:41:35,929
JUSTIN CURRY: For all C--

737
00:41:35,929 --> 00:41:37,096
AUDIENCE: There exists no b.

738
00:41:37,096 --> 00:41:38,720
JUSTIN CURRY: There
exists a b or no b?

739
00:41:38,720 --> 00:41:40,012
AUDIENCE: No, there exists a b.

740
00:41:40,012 --> 00:41:41,303
JUSTIN CURRY: There exists a b.

741
00:41:41,303 --> 00:41:43,390
AUDIENCE: Such that the
successor of the successor

742
00:41:43,390 --> 00:41:46,680
of 0 times b is C.

743
00:41:46,680 --> 00:41:47,860
JUSTIN CURRY: OK.

744
00:41:47,860 --> 00:41:52,424
So this without the tilde,
is it true or false?

745
00:41:52,424 --> 00:41:54,325
AUDIENCE: False.

746
00:41:54,325 --> 00:41:55,075
JUSTIN CURRY: Yes.

747
00:41:55,075 --> 00:41:55,575
It does.

748
00:41:55,575 --> 00:41:57,420
With the tilde it's true.

749
00:41:57,420 --> 00:41:58,570
And why?

750
00:41:58,570 --> 00:42:01,852
AUDIENCE: Because two times
anything is not [INAUDIBLE]..

751
00:42:06,150 --> 00:42:11,040
JUSTIN CURRY: So fundamentally,
what this statement is saying

752
00:42:11,040 --> 00:42:12,600
is that for all numbers--

753
00:42:12,600 --> 00:42:18,120
natural numbers--
that number is even,

754
00:42:18,120 --> 00:42:22,500
because it's divisible by
2, or it's a multiple of 2--

755
00:42:22,500 --> 00:42:24,970
Which is clearly not the case.

756
00:42:24,970 --> 00:42:26,640
And using our rule
of substitution,

757
00:42:26,640 --> 00:42:33,400
or specialization, we could
have picked, well, let's say 3,

758
00:42:33,400 --> 00:42:35,400
or, I'm sorry.

759
00:42:35,400 --> 00:42:36,910
3 doesn't exist in our system.

760
00:42:36,910 --> 00:42:39,420
It's SSS0.

761
00:42:44,700 --> 00:42:47,940
And through our
interpretation, we

762
00:42:47,940 --> 00:42:51,840
can see that, well,
there is no number b.

763
00:42:51,840 --> 00:42:56,640
There does not exist a b such
that 2 times that number is 3,

764
00:42:56,640 --> 00:42:57,830
namely because 3 is odd.

765
00:43:00,540 --> 00:43:05,100
So let's try a little
harder one real quick.

766
00:43:05,100 --> 00:43:07,740
And then with the
tilde, which means not,

767
00:43:07,740 --> 00:43:11,520
or false statement
is now made true.

768
00:43:11,520 --> 00:43:16,230
So this was round one.

769
00:43:16,230 --> 00:43:18,450
Round two.

770
00:43:18,450 --> 00:43:27,995
For every C that not
[INAUDIBLE] b colon, colon.

771
00:43:31,320 --> 00:43:41,790
SS0, dot b, equals C. Anyone
who is not Latif can answer.

772
00:43:41,790 --> 00:43:44,192
I'm going to successfully
knock you guys out.

773
00:43:44,192 --> 00:43:45,650
So the last man
standing, actually,

774
00:43:45,650 --> 00:43:46,733
gets the hardest question.

775
00:43:50,490 --> 00:43:54,150
So first, anybody willing to
field an interpretation who's

776
00:43:54,150 --> 00:43:55,364
not Latif?

777
00:43:55,364 --> 00:43:56,120
Max.

778
00:43:56,120 --> 00:44:02,054
AUDIENCE: For all C there
doesn't exist b such that--

779
00:44:02,054 --> 00:44:03,720
JUSTIN CURRY: [INAUDIBLE]
and call it 2.

780
00:44:03,720 --> 00:44:04,553
AUDIENCE: All right.

781
00:44:04,553 --> 00:44:08,190
2 times b is C.

782
00:44:08,190 --> 00:44:14,220
JUSTIN CURRY: So for
all C there does not

783
00:44:14,220 --> 00:44:24,530
exist a b such that
two times b is C.

784
00:44:24,530 --> 00:44:26,590
You have a 50/50 shot on this.

785
00:44:26,590 --> 00:44:27,631
AUDIENCE: Oh, it's false.

786
00:44:27,631 --> 00:44:28,060
JUSTIN CURRY:
Yeah, there you go.

787
00:44:28,060 --> 00:44:28,410
Exactly.

788
00:44:28,410 --> 00:44:28,910
It's false.

789
00:44:28,910 --> 00:44:32,860
Because we specify let C be 2.

790
00:44:32,860 --> 00:44:36,670
And clearly when there
exists a B-- not not exists--

791
00:44:36,670 --> 00:44:40,030
there exists a b, namely
[? one ?] that 2 times 1 is 1.

792
00:44:40,030 --> 00:44:45,010
So we now have two
people eliminated.

793
00:44:48,850 --> 00:44:50,020
Three.

794
00:44:50,020 --> 00:44:52,300
So now let's try this.

795
00:44:52,300 --> 00:44:56,620
For every C such that
there exists a b--

796
00:44:56,620 --> 00:45:01,740
dammit-- not a tilde, I mean--

797
00:45:01,740 --> 00:45:12,252
SS0 dot b equals C. [? Ders. ?]

798
00:45:12,252 --> 00:45:14,488
AUDIENCE: For every
C there exists a b

799
00:45:14,488 --> 00:45:19,820
that is not 2 times b equals C.

800
00:45:19,820 --> 00:45:24,128
JUSTIN CURRY: And is
that true or false?

801
00:45:24,128 --> 00:45:26,600
AUDIENCE: False.

802
00:45:26,600 --> 00:45:28,381
JUSTIN CURRY: So say again.

803
00:45:28,381 --> 00:45:28,880
We have a--

804
00:45:28,880 --> 00:45:29,661
AUDIENCE: There's not a C.

805
00:45:29,661 --> 00:45:30,643
JUSTIN CURRY: For every C--

806
00:45:30,643 --> 00:45:31,625
AUDIENCE: For every C--

807
00:45:31,625 --> 00:45:32,300
JUSTIN CURRY:
--there exists a b.

808
00:45:32,300 --> 00:45:33,710
AUDIENCE: --there
exists a b that is not--

809
00:45:33,710 --> 00:45:35,043
JUSTIN CURRY: --such that this--

810
00:45:35,043 --> 00:45:37,654
AUDIENCE: --such that
2 times b equals C.

811
00:45:37,654 --> 00:45:38,590
JUSTIN CURRY: Right.

812
00:45:38,590 --> 00:45:40,485
So actually, so specify.

813
00:45:46,400 --> 00:45:48,980
So what do you say this is?

814
00:45:48,980 --> 00:45:50,190
True or false?

815
00:45:50,190 --> 00:45:51,125
AUDIENCE: False.

816
00:45:51,125 --> 00:45:52,750
JUSTIN CURRY: Think
it's actually true.

817
00:45:58,620 --> 00:46:04,740
Because this statement
says that for every number

818
00:46:04,740 --> 00:46:13,120
there exists another number
such that this doesn't work.

819
00:46:13,120 --> 00:46:25,700
But if you have, say, 4, you
could pick anything-- like 3.

820
00:46:25,700 --> 00:46:31,880
And 2 times 3 is not equal to 4,
which is exactly what that is.

821
00:46:31,880 --> 00:46:35,000
So instead of the tildes,
which I find really confusing,

822
00:46:35,000 --> 00:46:36,710
because it's out
front, you can just

823
00:46:36,710 --> 00:46:41,100
consider it to be a not equals
here, which I think is easier.

824
00:46:41,100 --> 00:46:42,320
So this is true, I believe.

825
00:46:46,520 --> 00:46:47,360
Fourth round.

826
00:46:50,720 --> 00:46:51,220
Sorry.

827
00:46:51,220 --> 00:46:52,780
Even as I'm saying
these things, I'm

828
00:46:52,780 --> 00:46:53,988
getting tripped up on myself.

829
00:46:57,460 --> 00:47:01,390
Notation's kind of cumbersome.

830
00:47:01,390 --> 00:47:05,680
For all C, such that the
successor of successor

831
00:47:05,680 --> 00:47:17,280
of 0 times b equals C.

832
00:47:17,280 --> 00:47:17,780
All right.

833
00:47:17,780 --> 00:47:27,306
So anyone who is neither Max nor
Latif nor [? Ders. ?] Sandra.

834
00:47:27,306 --> 00:47:28,794
AUDIENCE: All right.

835
00:47:28,794 --> 00:47:33,754
There does not exist a b such
that all of C, all members,

836
00:47:33,754 --> 00:47:38,714
such that [INAUDIBLE].

837
00:47:38,714 --> 00:47:42,186
So that 2 times b equals C.

838
00:47:42,186 --> 00:47:45,180
JUSTIN CURRY: Yeah, exactly.

839
00:47:45,180 --> 00:47:47,800
First, it's easiest to
consider without the tilde,

840
00:47:47,800 --> 00:47:51,085
and just consider there
exists a b for all of C such

841
00:47:51,085 --> 00:47:54,590
that this is the case.

842
00:47:54,590 --> 00:47:58,170
So with the tilde, just
the way it's written,

843
00:47:58,170 --> 00:48:00,050
is this true or false?

844
00:48:00,050 --> 00:48:02,050
AUDIENCE: I think it's false.

845
00:48:02,050 --> 00:48:03,050
JUSTIN CURRY: So, wait.

846
00:48:03,050 --> 00:48:03,598
Hold on.

847
00:48:03,598 --> 00:48:04,973
Did I write down
the right thing?

848
00:48:08,540 --> 00:48:12,240
So first, let's consider
the case without the tilde.

849
00:48:12,240 --> 00:48:15,630
So we have this magical
b, such that any number

850
00:48:15,630 --> 00:48:19,650
we put in, this number is
the product of 2 in that.

851
00:48:22,530 --> 00:48:30,370
So is there any one number
such that any number

852
00:48:30,370 --> 00:48:32,090
is just equal to 2 times that?

853
00:48:32,090 --> 00:48:38,750
So as an example, we could
specify C to be 4, or 3-- yeah,

854
00:48:38,750 --> 00:48:39,250
exactly--

855
00:48:39,250 --> 00:48:44,410
3, and then b to be 1.

856
00:48:44,410 --> 00:48:46,720
And clearly we have
found a C, namely

857
00:48:46,720 --> 00:48:49,780
3, such that 2
times 1 is not that.

858
00:48:49,780 --> 00:48:51,370
So this is false.

859
00:48:51,370 --> 00:48:53,571
With the tilde it's true.

860
00:48:53,571 --> 00:48:54,070
Excellent.

861
00:48:54,070 --> 00:48:56,214
Good work.

862
00:48:56,214 --> 00:48:59,469
AUDIENCE: [INAUDIBLE]

863
00:48:59,469 --> 00:49:00,468
JUSTIN CURRY: Say again?

864
00:49:00,468 --> 00:49:01,304
AUDIENCE: You can't use minus 1?

865
00:49:01,304 --> 00:49:02,140
[INAUDIBLE]

866
00:49:02,140 --> 00:49:03,181
JUSTIN CURRY: No, no, no.

867
00:49:03,181 --> 00:49:05,080
Exactly.

868
00:49:05,080 --> 00:49:07,330
Well, you would express
1 in this notation

869
00:49:07,330 --> 00:49:10,870
as just the successor of 0.

870
00:49:10,870 --> 00:49:13,960
So it's not that you
don't have access to 1.

871
00:49:13,960 --> 00:49:19,380
It's just that in
our notation this

872
00:49:19,380 --> 00:49:22,110
is how we would
write 1, basically.

873
00:49:22,110 --> 00:49:22,950
All right.

874
00:49:22,950 --> 00:49:24,390
Well, good.

875
00:49:24,390 --> 00:49:26,550
I just want people to try these.

876
00:49:26,550 --> 00:49:31,710
So two more to go, I believe.

877
00:49:31,710 --> 00:49:34,410
So anyone who is not any
of the other four people

878
00:49:34,410 --> 00:49:40,050
who have gone, riddle
me this, Batman.

879
00:49:40,050 --> 00:49:49,475
So b such that not for all
C there exists SS0 times b

880
00:49:49,475 --> 00:49:57,110
equals C. Anybody.

881
00:49:57,110 --> 00:49:58,023
[? Navine. ?]

882
00:49:58,023 --> 00:50:07,200
AUDIENCE: [INAUDIBLE] such that
for [INAUDIBLE] such that 2

883
00:50:07,200 --> 00:50:09,742
times b equals C.

884
00:50:09,742 --> 00:50:10,450
JUSTIN CURRY: OK.

885
00:50:10,450 --> 00:50:12,945
Can you give me a
truth evaluation?

886
00:50:21,927 --> 00:50:25,440
AUDIENCE: [INAUDIBLE]

887
00:50:25,440 --> 00:50:27,440
JUSTIN CURRY: So we have this b.

888
00:50:30,470 --> 00:50:36,310
And we're saying that for
this specified value of b,

889
00:50:36,310 --> 00:50:44,890
regardless of what we put
in here, this will hold.

890
00:50:44,890 --> 00:50:47,780
Or to kind of say it a
little more clearly--

891
00:50:47,780 --> 00:50:50,050
and this involves some
of the symbols shunting

892
00:50:50,050 --> 00:50:52,840
you do in the chapter--

893
00:50:52,840 --> 00:50:59,240
there exists a b where
if you put in any C,

894
00:50:59,240 --> 00:51:00,240
it's not going to equal.

895
00:51:08,760 --> 00:51:11,780
So you're saying false?

896
00:51:11,780 --> 00:51:13,490
Or is it true?

897
00:51:13,490 --> 00:51:26,530
There exists b such that not for
every C. 2 times b is whatever.

898
00:51:26,530 --> 00:51:30,115
So let's pick 3.

899
00:51:34,030 --> 00:51:41,330
And I mean, yeah, I
mean, this is actually

900
00:51:41,330 --> 00:51:42,230
I'm pretty sure true.

901
00:51:45,580 --> 00:51:48,500
Because the second you fix b--

902
00:51:48,500 --> 00:51:51,640
let it be 4--

903
00:51:51,640 --> 00:51:56,530
it's clear that not for
every number, 4 times 2

904
00:51:56,530 --> 00:51:58,960
is that number.

905
00:51:58,960 --> 00:52:09,880
So in particular, for
6, this is not the case.

906
00:52:09,880 --> 00:52:12,760
I can't put in any
number and get it.

907
00:52:17,680 --> 00:52:20,029
Are there still questions here?

908
00:52:20,029 --> 00:52:22,100
AUDIENCE: [INAUDIBLE] last one.

909
00:52:22,100 --> 00:52:23,433
JUSTIN CURRY: Do this one again?

910
00:52:23,433 --> 00:52:24,320
AUDIENCE: Do the next one.

911
00:52:24,320 --> 00:52:24,670
JUSTIN CURRY: All right.

912
00:52:24,670 --> 00:52:25,336
Do the next one.

913
00:52:25,336 --> 00:52:27,490
So we're OK with
this being true?

914
00:52:27,490 --> 00:52:28,138
OK.

915
00:52:28,138 --> 00:52:29,789
AUDIENCE: Yeah, [INAUDIBLE].

916
00:52:29,789 --> 00:52:30,580
JUSTIN CURRY: Yeah.

917
00:52:30,580 --> 00:52:32,820
And it's really weird
complicated logical relations.

918
00:52:32,820 --> 00:52:34,570
That's why most of the
time mathematicians

919
00:52:34,570 --> 00:52:36,111
don't use these
things-- because they

920
00:52:36,111 --> 00:52:37,720
get tripped up on the symbols.

921
00:52:37,720 --> 00:52:39,910
Whereas they already know
up here what they're doing

922
00:52:39,910 --> 00:52:42,670
is right.

923
00:52:42,670 --> 00:52:44,280
So since we're
running out of space,

924
00:52:44,280 --> 00:52:46,995
I'm going to put the
sixth round up here.

925
00:52:46,995 --> 00:52:48,370
And let's try to
do this quickly,

926
00:52:48,370 --> 00:52:49,600
so we can pass off the show.

927
00:52:52,730 --> 00:52:54,750
So there exists a b--

928
00:52:54,750 --> 00:52:57,280
and I could do the
interpretation.

929
00:52:57,280 --> 00:53:04,780
Backwards E, b, colon, upside
down A, C, colon, tilde,

930
00:53:04,780 --> 00:53:11,210
parentheses, SS0, dot
b, parallel lines,

931
00:53:11,210 --> 00:53:18,240
C. So anyone who is none of
the previous five people--

932
00:53:18,240 --> 00:53:20,415
Mia.

933
00:53:20,415 --> 00:53:21,915
AUDIENCE: Do you
want me to read it?

934
00:53:21,915 --> 00:53:22,880
JUSTIN CURRY: Yeah,
give me interpretation,

935
00:53:22,880 --> 00:53:24,907
and truth value if you're brave.

936
00:53:24,907 --> 00:53:27,889
AUDIENCE: There
exists b for every C

937
00:53:27,889 --> 00:53:33,517
so that there is not
2 times b equals C.

938
00:53:33,517 --> 00:53:34,350
JUSTIN CURRY: Right.

939
00:53:34,350 --> 00:53:37,160
So the colon, I could
replace the tilde with what

940
00:53:37,160 --> 00:53:38,646
other symbol?

941
00:53:38,646 --> 00:53:40,910
A line here.

942
00:53:40,910 --> 00:53:42,260
So I think that's easier.

943
00:53:42,260 --> 00:53:46,630
So there exists a b
[INAUDIBLE] for all C

944
00:53:46,630 --> 00:53:52,716
2 times b is not equal to
C. So is that true or false?

945
00:53:52,716 --> 00:53:55,080
AUDIENCE: It's true.

946
00:53:55,080 --> 00:53:57,262
JUSTIN CURRY: OK.

947
00:53:57,262 --> 00:53:57,970
Think about this.

948
00:53:57,970 --> 00:54:05,420
So there exists a b
such that for all C--

949
00:54:05,420 --> 00:54:07,183
so regardless of what
you put in here--

950
00:54:07,183 --> 00:54:07,930
AUDIENCE: No.

951
00:54:07,930 --> 00:54:08,638
JUSTIN CURRY: OK.

952
00:54:08,638 --> 00:54:11,160
So then it's false.

953
00:54:11,160 --> 00:54:14,165
This flipping back and forth
between these existential

954
00:54:14,165 --> 00:54:17,310
and quantifier--
existential and quantifiers,

955
00:54:17,310 --> 00:54:20,140
and things like [INAUDIBLE]
can really make this confusing,

956
00:54:20,140 --> 00:54:26,310
because what you have here
is that there's a b such that

957
00:54:26,310 --> 00:54:29,540
regardless of what
you put in here,

958
00:54:29,540 --> 00:54:32,240
this equality never holds.

959
00:54:32,240 --> 00:54:38,520
But that's not true, because
you can take any C. Eventually

960
00:54:38,520 --> 00:54:44,130
it's going to work, because
if we specified b to be 2,

961
00:54:44,130 --> 00:54:46,470
that's the thing which exists.

962
00:54:46,470 --> 00:54:50,010
Then we can find a C. So
it's not true that for all C.

963
00:54:50,010 --> 00:54:52,890
And particularly
that C value being 4.

964
00:54:52,890 --> 00:54:55,980
So 2 times 2 is equal to
4, even though this thing

965
00:54:55,980 --> 00:55:00,270
claims that this value
of b which we picked,

966
00:55:00,270 --> 00:55:03,910
would never be equal, regardless
of what you put in here.

967
00:55:03,910 --> 00:55:07,440
So I think that's my answer key.

968
00:55:07,440 --> 00:55:10,750
True, false, true, true, false.

969
00:55:10,750 --> 00:55:13,770
And it fits the second hint,
when he says either there are

970
00:55:13,770 --> 00:55:17,200
four true and two false,
or four false and two true.

971
00:55:17,200 --> 00:55:20,640
And that's related to actually
how you shunt these tildes.

972
00:55:24,280 --> 00:55:27,660
But once again,
unless you're really

973
00:55:27,660 --> 00:55:29,490
planning on a life
as a logician,

974
00:55:29,490 --> 00:55:31,448
you're not going to have
to spend a lot of time

975
00:55:31,448 --> 00:55:34,350
manipulating formal systems.

976
00:55:34,350 --> 00:55:37,500
And as such, it's good to
get practice with this, just

977
00:55:37,500 --> 00:55:39,240
because the
difficulty, which you

978
00:55:39,240 --> 00:55:43,320
have of taking in these
new symbols, and forming--

979
00:55:43,320 --> 00:55:47,310
kind of creating more space
in your neural network

980
00:55:47,310 --> 00:55:50,030
for fitting these places in--

981
00:55:50,030 --> 00:55:52,387
is, I think, an
important exercise.

982
00:55:52,387 --> 00:55:53,970
And that really goes
back to that idea

983
00:55:53,970 --> 00:55:55,800
of a theory of meaning.

984
00:55:55,800 --> 00:55:57,450
And it's one of
the last things I

985
00:55:57,450 --> 00:55:59,940
want to apologize before
I hand off the lecture

986
00:55:59,940 --> 00:56:04,200
to Curran, is that when I
say things like recursion,

987
00:56:04,200 --> 00:56:09,900
and I say things like formal
systems, isomorphisms,

988
00:56:09,900 --> 00:56:13,890
algorithmic information,
Shannon entropy, et cetera, et

989
00:56:13,890 --> 00:56:18,990
cetera, neural nets--

990
00:56:18,990 --> 00:56:22,830
these terms don't really mean
the same thing for you people

991
00:56:22,830 --> 00:56:26,580
that it does to
a professor who's

992
00:56:26,580 --> 00:56:31,800
been spending years
working long nights

993
00:56:31,800 --> 00:56:34,770
torturing himself over
solving these problems late

994
00:56:34,770 --> 00:56:36,130
into the night.

995
00:56:36,130 --> 00:56:38,490
And that process of
thinking again and again,

996
00:56:38,490 --> 00:56:40,530
and making mistakes,
and then refining

997
00:56:40,530 --> 00:56:44,610
your understanding
of something, forces

998
00:56:44,610 --> 00:56:48,000
certain parts of your brain to
meet here and here and here.

999
00:56:48,000 --> 00:56:50,280
And just by talking to
it at you guys, what

1000
00:56:50,280 --> 00:56:52,410
I'm trying to really
do is just inspire

1001
00:56:52,410 --> 00:56:55,420
an interest in what I'm saying.

1002
00:56:55,420 --> 00:56:58,380
And it's not like I
can condense somehow

1003
00:56:58,380 --> 00:57:02,580
seven years of undergraduate
and postgraduate work

1004
00:57:02,580 --> 00:57:06,300
into a two-hour lecture,
and just immediately go

1005
00:57:06,300 --> 00:57:09,280
into your brain, and I want to
put this neuron here and there

1006
00:57:09,280 --> 00:57:11,670
and there, and suddenly
you have the same depth

1007
00:57:11,670 --> 00:57:13,620
of understanding
about these subjects

1008
00:57:13,620 --> 00:57:17,700
that Seth Lloyd or
anyone else who is

1009
00:57:17,700 --> 00:57:20,680
a specialist in these fields,
the kind of understanding they

1010
00:57:20,680 --> 00:57:21,180
have.

1011
00:57:21,180 --> 00:57:24,650
I can't endow that in just
a simple two-hour lecture,

1012
00:57:24,650 --> 00:57:28,350
unless I assign pages
and pages of problems,

1013
00:57:28,350 --> 00:57:32,400
and you're working
100 hours a week,

1014
00:57:32,400 --> 00:57:34,770
which I'm not going to
do, because I'm not evil.

1015
00:57:34,770 --> 00:57:37,590
But aside from that,
the fundamental thing

1016
00:57:37,590 --> 00:57:39,300
to pull out here--

1017
00:57:39,300 --> 00:57:41,310
and I've noticed that
I say fundamental--

1018
00:57:44,190 --> 00:57:46,980
the important idea
here is that you

1019
00:57:46,980 --> 00:57:50,250
can start with essentially a
basic set of statements which

1020
00:57:50,250 --> 00:57:52,380
you think can capture
something true,

1021
00:57:52,380 --> 00:57:55,290
and apply a recursive
rule, a recursive algorithm

1022
00:57:55,290 --> 00:57:59,280
to producing new strings,
which can produce new things.

1023
00:57:59,280 --> 00:58:03,600
So as we go into the next
part of today's lecture,

1024
00:58:03,600 --> 00:58:05,940
I want you to kind of
think of this truth

1025
00:58:05,940 --> 00:58:08,010
tree, which we try to grow.

1026
00:58:10,770 --> 00:58:15,280
And it all starts kind of from
Peano's axioms, number theory.

1027
00:58:15,280 --> 00:58:17,760
And we just apply
these rules, and create

1028
00:58:17,760 --> 00:58:18,720
different statements.

1029
00:58:18,720 --> 00:58:22,351
It's just like the MIU tree that
we made in the first lecture

1030
00:58:22,351 --> 00:58:22,850
to.

1031
00:58:22,850 --> 00:58:26,576
And I noticed no one's
challenged me on my $20.

1032
00:58:26,576 --> 00:58:28,891
AUDIENCE: [INAUDIBLE]
all these sums are true,

1033
00:58:28,891 --> 00:58:30,837
and they fall into
the [INAUDIBLE]..

1034
00:58:30,837 --> 00:58:31,670
JUSTIN CURRY: Right.

1035
00:58:31,670 --> 00:58:34,760
So what happens is that
some of these things,

1036
00:58:34,760 --> 00:58:38,280
we start with those
five basic axioms,

1037
00:58:38,280 --> 00:58:40,250
which Hofstadter outlines.

1038
00:58:40,250 --> 00:58:42,020
0 is not the successor
of any number.

1039
00:58:45,320 --> 00:58:46,300
Let's see.

1040
00:58:46,300 --> 00:58:49,120
0 is not the successor
of any number.

1041
00:58:49,120 --> 00:58:51,875
You have to assume that 0 plus
the number is just that number

1042
00:58:51,875 --> 00:58:52,375
itself.

1043
00:58:55,280 --> 00:58:56,560
And here you go.

1044
00:59:02,640 --> 00:59:05,684
Yes, he actually states
them in this form.

1045
00:59:05,684 --> 00:59:07,850
So you don't go ahead and
give it an interpretation.

1046
00:59:07,850 --> 00:59:09,800
Genie is a djinn.

1047
00:59:09,800 --> 00:59:12,680
Genie is a 0, and
djinn is a number.

1048
00:59:12,680 --> 00:59:15,300
Every djinn has a meta,
which is also a djinn.

1049
00:59:15,300 --> 00:59:19,230
So every number has a successor,
which is also a number.

1050
00:59:19,230 --> 00:59:22,770
So Genie is not the
meta of any djinn.

1051
00:59:22,770 --> 00:59:26,590
So 0 is not the successor
of any other number.

1052
00:59:26,590 --> 00:59:29,970
Different djinns--
sorry, this is page 216--

1053
00:59:29,970 --> 00:59:31,695
different djinns
have different metas.

1054
00:59:31,695 --> 00:59:35,190
So if two numbers are not
equal, the next guy after them

1055
00:59:35,190 --> 00:59:37,530
are also not going to be equal.

1056
00:59:37,530 --> 00:59:41,880
And then finally, if Genie
has x, and each djinn,

1057
00:59:41,880 --> 00:59:45,450
relays x to its meta,
then all djinns get x.

1058
00:59:45,450 --> 00:59:47,190
And that's the
principle of induction,

1059
00:59:47,190 --> 00:59:55,410
that if 0 has a property P, and
the successor and any number

1060
00:59:55,410 --> 00:59:58,110
relays that property
P to its successor,

1061
00:59:58,110 --> 01:00:00,300
then all numbers
have it, because you

1062
01:00:00,300 --> 01:00:03,690
can give from 0 to 1, and then
1 gives it to its successor,

1063
01:00:03,690 --> 01:00:04,530
and it's 2.

1064
01:00:04,530 --> 01:00:06,150
And it always has
that property P.

1065
01:00:06,150 --> 01:00:09,480
And that's what mathematical
induction relies on.

1066
01:00:09,480 --> 01:00:13,920
And from these basic
things, you can actually

1067
01:00:13,920 --> 01:00:16,290
derive most of number theory.

1068
01:00:16,290 --> 01:00:20,160
And you just apply these
rules of induction.

1069
01:00:20,160 --> 01:00:23,530
And you start with this trunk.

1070
01:00:23,530 --> 01:00:26,400
And you get a new theorem--

1071
01:00:26,400 --> 01:00:32,730
well, since 1 and
2 aren't equal,

1072
01:00:32,730 --> 01:00:34,500
then 2 and 3 aren't equal.

1073
01:00:34,500 --> 01:00:37,140
So you just kind of
add things to your tree

1074
01:00:37,140 --> 01:00:40,200
based on these rules.

1075
01:00:40,200 --> 01:00:41,850
And it's completely local.

1076
01:00:41,850 --> 01:00:46,630
It's completely based on what
you have at your given point,

1077
01:00:46,630 --> 01:00:50,700
and what rule you're
willing to apply.

1078
01:00:50,700 --> 01:00:52,830
And then it's amazing the
kind of emergent patterns

1079
01:00:52,830 --> 01:00:53,580
which you can get.

1080
01:00:53,580 --> 01:00:55,390
And we happen to call
that number theory.

1081
01:00:55,390 --> 01:00:58,190
Today, you'll shortly
see some things which

1082
01:00:58,190 --> 01:00:59,850
are a little more interesting.

1083
01:00:59,850 --> 01:01:01,350
But on that note I
think we're going

1084
01:01:01,350 --> 01:01:03,840
to take a two minute
break to de-stress,

1085
01:01:03,840 --> 01:01:06,020
and then get things set up
for Curran to take over.

1086
01:01:06,020 --> 01:01:06,787
Thank you.

1087
01:01:06,787 --> 01:01:07,620
CURRAN KELLEHER: OK.

1088
01:01:07,620 --> 01:01:11,460
So what he was
actually talking about

1089
01:01:11,460 --> 01:01:14,770
can also be called a
context-free grammar.

1090
01:01:14,770 --> 01:01:16,440
A context-free
grammar is something

1091
01:01:16,440 --> 01:01:22,050
that has symbols and
production rules.

1092
01:01:22,050 --> 01:01:23,460
And the symbols
here in this case

1093
01:01:23,460 --> 01:01:29,470
are S and 0, and their various
mathematical operators.

1094
01:01:29,470 --> 01:01:32,280
But they're just symbols
in terms of the grammar.

1095
01:01:32,280 --> 01:01:35,970
And the production
rules are the rules

1096
01:01:35,970 --> 01:01:38,610
of inference, where
you can take one string

1097
01:01:38,610 --> 01:01:41,310
and perform some
manipulation on a part of it

1098
01:01:41,310 --> 01:01:42,300
to get a new string.

1099
01:01:42,300 --> 01:01:44,070
It's a rule of inference.

1100
01:01:44,070 --> 01:01:49,170
So you can use a similar
system to define moving around

1101
01:01:49,170 --> 01:01:53,380
circles on the screen.

1102
01:01:53,380 --> 01:01:56,080
So I'm going to explain
what's going on here.

1103
01:01:56,080 --> 01:01:58,740
This is a program
called Context Free.

1104
01:01:58,740 --> 01:02:01,440
It's an open source project
that you can download and play

1105
01:02:01,440 --> 01:02:04,120
with yourself.

1106
01:02:04,120 --> 01:02:07,290
So what's going on here,
the code startshape

1107
01:02:07,290 --> 01:02:08,740
just is the entry point.

1108
01:02:08,740 --> 01:02:11,010
It's not going to change.

1109
01:02:11,010 --> 01:02:14,280
And we define a
rule called SPIRAL.

1110
01:02:14,280 --> 01:02:19,290
And inside this rule
we have CIRCLE, which

1111
01:02:19,290 --> 01:02:21,750
plots a circle on the screen.

1112
01:02:21,750 --> 01:02:24,270
And then we call SPIRAL again.

1113
01:02:24,270 --> 01:02:30,210
And y space 2 means that
we increment the y position

1114
01:02:30,210 --> 01:02:32,870
by 2 units.

1115
01:02:32,870 --> 01:02:39,320
And size space 0.9 means that
we multiply the size of this

1116
01:02:39,320 --> 01:02:42,860
by 0.9 every time we go up.

1117
01:02:42,860 --> 01:02:47,960
So this rule defines this image.

1118
01:02:47,960 --> 01:02:51,680
And what this
program does is, when

1119
01:02:51,680 --> 01:02:54,750
the thing gets too small to
see, it just stops doing that.

1120
01:02:54,750 --> 01:02:57,690
So this is actually
an infinite recursion.

1121
01:02:57,690 --> 01:03:01,310
But it stops at some point,
because it gets so small.

1122
01:03:01,310 --> 01:03:03,410
So this is our framework.

1123
01:03:03,410 --> 01:03:07,430
And by changing this
code little by little,

1124
01:03:07,430 --> 01:03:11,937
we're going to get
some amazing pictures.

1125
01:03:11,937 --> 01:03:13,520
So I'm going to just
do these changes,

1126
01:03:13,520 --> 01:03:15,740
and explain them as I go.

1127
01:03:15,740 --> 01:03:18,410
So I just had this spacing
to be 2, so it's clear

1128
01:03:18,410 --> 01:03:19,975
that we're just trying circles.

1129
01:03:19,975 --> 01:03:25,660
I'm going to decrease
the spacing to be 0.4.

1130
01:03:25,660 --> 01:03:29,300
And rerender so it
looks like that.

1131
01:03:32,270 --> 01:03:39,730
So I'm going to define another
rule, also called SPIRAL.

1132
01:03:39,730 --> 01:03:41,350
And so as I go, I'm
going to explain

1133
01:03:41,350 --> 01:03:45,510
the features of this language.

1134
01:03:45,510 --> 01:03:49,410
When you define a rule that
has the same name twice, what

1135
01:03:49,410 --> 01:03:54,260
happens is whenever
you call this rule,

1136
01:03:54,260 --> 01:03:58,630
it calls one or the other
with equal probability.

1137
01:03:58,630 --> 01:04:00,480
So what I'm going
to do here is say

1138
01:04:00,480 --> 01:04:07,470
flip 90, which means flip our
sort of frame of reference

1139
01:04:07,470 --> 01:04:08,130
by 90 degrees.

1140
01:04:11,610 --> 01:04:14,010
Actually, first of
all, before I do this--

1141
01:04:14,010 --> 01:04:16,670
sorry-- I'm going
to add a rotation.

1142
01:04:16,670 --> 01:04:20,010
So rotate 1.

1143
01:04:20,010 --> 01:04:22,365
Rotate 1 degree each time.

1144
01:04:22,365 --> 01:04:23,490
So it rotates a little bit.

1145
01:04:23,490 --> 01:04:24,560
You see that?

1146
01:04:24,560 --> 01:04:26,470
1 degree each time.

1147
01:04:26,470 --> 01:04:30,450
So if we decrease the
size by 0.99 every time,

1148
01:04:30,450 --> 01:04:34,780
it's going to get smaller
a little bit slower.

1149
01:04:34,780 --> 01:04:37,680
So we see this spiral happen.

1150
01:04:37,680 --> 01:04:43,530
So if we do 0.9999, it
will be even more spirally.

1151
01:04:43,530 --> 01:04:45,690
So that's what we get.

1152
01:04:45,690 --> 01:04:48,104
I don't know why it's
going off the screen.

1153
01:04:48,104 --> 01:04:50,620
There we go.

1154
01:04:50,620 --> 01:04:53,040
So now I'm going to define
another rule called SPIRAL,

1155
01:04:53,040 --> 01:04:55,330
and flip by 90 degrees.

1156
01:04:55,330 --> 01:05:00,300
So if I render this, what do you
guys think is going to happen?

1157
01:05:00,300 --> 01:05:01,534
What do you think?

1158
01:05:01,534 --> 01:05:03,720
AUDIENCE: It's going
to flip [INAUDIBLE]..

1159
01:05:03,720 --> 01:05:06,070
CURRAN KELLEHER:
It's going to flip?

1160
01:05:06,070 --> 01:05:07,680
So maybe I wasn't
clear about what it

1161
01:05:07,680 --> 01:05:09,910
means to call that flip thing.

1162
01:05:09,910 --> 01:05:15,360
So when we say rotate 1, we're
rotating 1 degree this way.

1163
01:05:15,360 --> 01:05:19,320
So after we call flip,
when we say rotate 1,

1164
01:05:19,320 --> 01:05:23,190
it's going to rotate
the opposite direction.

1165
01:05:23,190 --> 01:05:26,400
So say we call the
first rule five times,

1166
01:05:26,400 --> 01:05:27,880
draws five of these circles.

1167
01:05:27,880 --> 01:05:30,671
Then we call the second
rule once, it flips.

1168
01:05:30,671 --> 01:05:32,670
And then we call the first
real five more times,

1169
01:05:32,670 --> 01:05:36,139
it's going to just turn
the other way a little bit.

1170
01:05:36,139 --> 01:05:37,680
And so initially,
they're going to be

1171
01:05:37,680 --> 01:05:39,190
called with equal probability.

1172
01:05:39,190 --> 01:05:41,760
So if I render it,
this is what we get--

1173
01:05:41,760 --> 01:05:44,340
this sort of meandering thing.

1174
01:05:44,340 --> 01:05:47,040
So what's happening
is half the time

1175
01:05:47,040 --> 01:05:49,500
it's calling the first
rule, which moves

1176
01:05:49,500 --> 01:05:51,570
the circle up a little bit.

1177
01:05:51,570 --> 01:05:54,430
It rotates it, and decreases
its size, and draws the circle.

1178
01:05:54,430 --> 01:05:56,430
And half the time we're
calling this other rule,

1179
01:05:56,430 --> 01:05:59,340
which flips the direction
in which we're rotating.

1180
01:05:59,340 --> 01:06:01,920
So this is what we get.

1181
01:06:01,920 --> 01:06:04,350
Another feature
of the language is

1182
01:06:04,350 --> 01:06:07,110
we can change the probabilities
at which these things are

1183
01:06:07,110 --> 01:06:08,910
called.

1184
01:06:08,910 --> 01:06:14,560
So at, say, 0.1, I put 0.1
right next to the second SPIRAL

1185
01:06:14,560 --> 01:06:17,670
rule, the flipping rule.

1186
01:06:17,670 --> 01:06:20,190
If I don't specify
a number, it gets 1.

1187
01:06:20,190 --> 01:06:22,230
So that means that
the first rule

1188
01:06:22,230 --> 01:06:24,962
is going to get called
with a probability of 1.

1189
01:06:24,962 --> 01:06:26,670
And the second rule
is going to be called

1190
01:06:26,670 --> 01:06:29,460
with a probability of 0.1.

1191
01:06:29,460 --> 01:06:31,152
And this is whenever--

1192
01:06:31,152 --> 01:06:33,120
AUDIENCE: Maybe 1 minus that.

1193
01:06:33,120 --> 01:06:35,108
CURRAN KELLEHER: 1 minus that?

1194
01:06:35,108 --> 01:06:36,607
AUDIENCE: No, in
the first rule they

1195
01:06:36,607 --> 01:06:39,844
call the probability
of 1 minus 0.1,

1196
01:06:39,844 --> 01:06:41,356
because you're rolling a die.

1197
01:06:41,356 --> 01:06:43,532
You can never go beyond
a probability of 1.

1198
01:06:43,532 --> 01:06:44,490
CURRAN KELLEHER: Right.

1199
01:06:44,490 --> 01:06:50,150
So talking about probability,
what it actually does,

1200
01:06:50,150 --> 01:06:52,220
I think, is computes
the sum, and then

1201
01:06:52,220 --> 01:06:56,420
takes the fraction of that
sum that each of the rules

1202
01:06:56,420 --> 01:06:57,630
have for a probability.

1203
01:06:57,630 --> 01:07:02,181
Either way, this one happens
less probable than 50% now.

1204
01:07:02,181 --> 01:07:04,430
So we can see that it goes
for slightly longer periods

1205
01:07:04,430 --> 01:07:07,820
of iteration without
any flippage.

1206
01:07:07,820 --> 01:07:18,929
So if we do it again, decrease
it 0.01, it'll flip even less.

1207
01:07:18,929 --> 01:07:20,220
Flips with even less frequency.

1208
01:07:20,220 --> 01:07:24,830
So if we make it 0.001
it flips a lot less.

1209
01:07:28,080 --> 01:07:30,600
So already we're seeing
these really cool pictures

1210
01:07:30,600 --> 01:07:33,030
being generated by
such simple rules,

1211
01:07:33,030 --> 01:07:34,913
simple context-free grammars.

1212
01:07:34,913 --> 01:07:36,332
Any questions so far?

1213
01:07:40,600 --> 01:07:46,930
So what happens if I
add inside this rule

1214
01:07:46,930 --> 01:07:51,107
another instance of
spiral without the flip?

1215
01:07:51,107 --> 01:07:51,940
What does this mean?

1216
01:07:54,844 --> 01:08:00,974
AUDIENCE: [INAUDIBLE] First it
flips, and then it just goes on

1217
01:08:00,974 --> 01:08:02,004
like that for awhile.

1218
01:08:02,004 --> 01:08:02,920
CURRAN KELLEHER: Yeah.

1219
01:08:02,920 --> 01:08:03,586
So more or less.

1220
01:08:03,586 --> 01:08:05,860
You said first it
flips, and then it

1221
01:08:05,860 --> 01:08:07,570
goes on without flipping.

1222
01:08:07,570 --> 01:08:12,400
So yeah, with this rule,
instead of just flipping,

1223
01:08:12,400 --> 01:08:16,330
what it's going to do is start
this one off on its tangent,

1224
01:08:16,330 --> 01:08:21,290
and also just keep going its
original way without flipping.

1225
01:08:21,290 --> 01:08:23,040
So we'll render this
and see what it does.

1226
01:08:26,340 --> 01:08:28,267
So it does exactly that.

1227
01:08:28,267 --> 01:08:29,510
See?

1228
01:08:29,510 --> 01:08:32,180
Whenever it branches,
it also keeps

1229
01:08:32,180 --> 01:08:35,990
going its original direction.

1230
01:08:35,990 --> 01:08:40,160
So we can change some
parameters around,

1231
01:08:40,160 --> 01:08:42,950
and we're going to get
some organic-looking forms.

1232
01:08:42,950 --> 01:08:45,950
So if we decrease
the probability-- no,

1233
01:08:45,950 --> 01:08:50,430
increase the probability
of branching, we get this.

1234
01:08:50,430 --> 01:08:52,760
It just goes out of control.

1235
01:08:52,760 --> 01:08:55,890
So maybe that's not
what we want to do.

1236
01:08:55,890 --> 01:08:58,640
So now I put it back.

1237
01:08:58,640 --> 01:09:04,189
If we multiply a size by
0.99, yeah, there we go.

1238
01:09:04,189 --> 01:09:09,040
Now we increase the
probability of branching.

1239
01:09:09,040 --> 01:09:10,068
And we get trees.

1240
01:09:10,068 --> 01:09:10,609
Look at that.

1241
01:09:10,609 --> 01:09:11,609
It looks like a tree.

1242
01:09:15,840 --> 01:09:18,960
That's really wild.

1243
01:09:18,960 --> 01:09:20,430
So if we increase
the probability

1244
01:09:20,430 --> 01:09:25,420
of branching even more,
we get thicker trees,

1245
01:09:25,420 --> 01:09:26,468
because they branch more.

1246
01:09:29,819 --> 01:09:31,569
It's pretty wild, isn't it?

1247
01:09:31,569 --> 01:09:34,558
Isn't that cool?

1248
01:09:34,558 --> 01:09:38,050
So it makes you
wonder, does nature

1249
01:09:38,050 --> 01:09:41,170
use these sort of
context-free grammars

1250
01:09:41,170 --> 01:09:45,100
in the way it grows
plants, or is it

1251
01:09:45,100 --> 01:09:46,290
like a Lindenmayer system?

1252
01:09:46,290 --> 01:09:49,430
Is it fixed, these
global rules that

1253
01:09:49,430 --> 01:09:52,029
get applied at smaller
and smaller scales?

1254
01:09:52,029 --> 01:09:55,112
I think in nature, what we see
is sort of a mixture of both,

1255
01:09:55,112 --> 01:09:57,070
because some plants have
very regular features,

1256
01:09:57,070 --> 01:09:59,950
and some plants don't.

1257
01:09:59,950 --> 01:10:03,880
So it's still a mystery, plants.

1258
01:10:03,880 --> 01:10:06,730
But it looks pretty organic.

1259
01:10:06,730 --> 01:10:12,430
So I'm just amazed by
this sort of thing.

1260
01:10:12,430 --> 01:10:14,290
So we can play with
the parameters,

1261
01:10:14,290 --> 01:10:20,290
and get really cool growths.

1262
01:10:20,290 --> 01:10:25,860
So let's think about this as
a model of plant development.

1263
01:10:25,860 --> 01:10:28,410
We're modeling that
the size of the plant

1264
01:10:28,410 --> 01:10:30,140
is just constantly shrinking.

1265
01:10:30,140 --> 01:10:33,480
And when it branches,
it's mass sort

1266
01:10:33,480 --> 01:10:37,150
of doubles, which is
sort of a bad model.

1267
01:10:37,150 --> 01:10:38,130
Think about a tree.

1268
01:10:38,130 --> 01:10:40,470
Think about a tree
just going up.

1269
01:10:40,470 --> 01:10:44,550
And then, when it branches,
it still keeps going up,

1270
01:10:44,550 --> 01:10:47,530
but a branch goes
off to the side.

1271
01:10:47,530 --> 01:10:50,540
So we can change our
grammar to do this.

1272
01:10:50,540 --> 01:10:53,130
And we'll get some
tree-like things.

1273
01:10:55,690 --> 01:10:57,300
So this is the rule.

1274
01:10:57,300 --> 01:11:00,750
The rule on top is just the rule
that it does when it's going.

1275
01:11:03,916 --> 01:11:11,530
So I'm going to change
this to just increment y.

1276
01:11:11,530 --> 01:11:13,450
And I'm going to change this.

1277
01:11:18,690 --> 01:11:21,270
So one of these, it's
just going to go straight.

1278
01:11:21,270 --> 01:11:24,620
And the other one is going
to branch and get smaller.

1279
01:11:24,620 --> 01:11:34,910
So size, point 0.9, but
for just going straight.

1280
01:11:34,910 --> 01:11:37,960
And what flip does
in this context--

1281
01:11:37,960 --> 01:11:40,700
well, first, I'll
explain it later.

1282
01:11:40,700 --> 01:11:46,190
I'll say rotate 45 degrees.

1283
01:11:46,190 --> 01:11:53,840
And size is 0.3.

1284
01:11:53,840 --> 01:11:56,330
So what we get is this
very sparse structure.

1285
01:11:56,330 --> 01:11:59,160
So we can increase the
probability of branching

1286
01:11:59,160 --> 01:12:02,870
to 0.5 or something.

1287
01:12:02,870 --> 01:12:08,370
Maybe 0.9.

1288
01:12:08,370 --> 01:12:11,249
Maybe 0.2.

1289
01:12:11,249 --> 01:12:13,415
So we get these things that
sort of look like trees.

1290
01:12:16,354 --> 01:12:18,770
So I'll just explain the rules,
in case you didn't follow.

1291
01:12:18,770 --> 01:12:27,950
So the first spiral in this
rule, this part of the rule

1292
01:12:27,950 --> 01:12:31,550
encodes for the fact that
whenever it branches,

1293
01:12:31,550 --> 01:12:35,880
the size gets a little
smaller by a factor of 0.9.

1294
01:12:35,880 --> 01:12:37,310
And you form this branch.

1295
01:12:37,310 --> 01:12:40,910
And this, the second
rule here, says size 0.3.

1296
01:12:40,910 --> 01:12:43,120
That means that the
size of this branch

1297
01:12:43,120 --> 01:12:47,630
is 0.3 the size of
the original trunk.

1298
01:12:47,630 --> 01:12:51,860
And the flip part,
flip 90 in this rule,

1299
01:12:51,860 --> 01:12:54,620
means that next time
it branches it's going

1300
01:12:54,620 --> 01:12:56,140
to branch in that direction.

1301
01:12:56,140 --> 01:12:57,650
We can actually take it out.

1302
01:12:57,650 --> 01:13:01,640
And if we take out
the flip, the branches

1303
01:13:01,640 --> 01:13:06,680
will all just go in the
same direction all the time.

1304
01:13:06,680 --> 01:13:09,725
So if we take it out, see there?

1305
01:13:09,725 --> 01:13:14,400
They only go in one direction.

1306
01:13:14,400 --> 01:13:19,030
So we can sort of play
with the parameters here.

1307
01:13:19,030 --> 01:13:22,000
Let's say, 0.95.

1308
01:13:22,000 --> 01:13:23,500
Let's just see what happens.

1309
01:13:29,500 --> 01:13:33,890
Yeah, we got some pretty
interesting things.

1310
01:13:33,890 --> 01:13:35,900
If we make the branches
a little bit thicker--

1311
01:13:35,900 --> 01:13:39,384
maybe 0.4.

1312
01:13:39,384 --> 01:13:39,884
See?

1313
01:13:39,884 --> 01:13:43,188
Look at that.

1314
01:13:43,188 --> 01:13:46,520
It's like a tree or something.

1315
01:13:46,520 --> 01:13:48,374
Any questions so
far, and comments?

1316
01:13:52,310 --> 01:13:55,390
So if we add a little bit of
rotation to the main rule--

1317
01:13:55,390 --> 01:14:02,154
rotate 1 degree-- yeah, Latif.

1318
01:14:02,154 --> 01:14:06,337
AUDIENCE: [INAUDIBLE]

1319
01:14:06,337 --> 01:14:07,780
CURRAN KELLEHER: Say again?

1320
01:14:07,780 --> 01:14:12,430
AUDIENCE: [INAUDIBLE] does it
call itself the thing that it's

1321
01:14:12,430 --> 01:14:13,888
calling [INAUDIBLE].

1322
01:14:13,888 --> 01:14:15,346
CURRAN KELLEHER:
Ah, good question.

1323
01:14:18,270 --> 01:14:19,040
Yes.

1324
01:14:19,040 --> 01:14:22,710
When it calls
itself, like inside,

1325
01:14:22,710 --> 01:14:26,295
like this one, or this
one, or this one--

1326
01:14:26,295 --> 01:14:28,503
AUDIENCE: Which one does it
call, the [INAUDIBLE] one

1327
01:14:28,503 --> 01:14:29,530
or the [INAUDIBLE]?

1328
01:14:29,530 --> 01:14:30,488
CURRAN KELLEHER: Right.

1329
01:14:30,488 --> 01:14:33,170
So which one does it call
when you call SPIRAL?

1330
01:14:33,170 --> 01:14:35,130
So this is where the
probability comes in.

1331
01:14:35,130 --> 01:14:38,570
Anytime you call it, there
are these three instances

1332
01:14:38,570 --> 01:14:41,280
where SPIRAL is called
inside of SPIRAL.

1333
01:14:41,280 --> 01:14:43,020
[INTERPOSING VOICES]

1334
01:14:43,020 --> 01:14:46,960
CURRAN KELLEHER: So
it calls either one

1335
01:14:46,960 --> 01:14:49,790
with certain probabilities.

1336
01:14:49,790 --> 01:14:56,890
And the probability of the
first one is 1 out of 1.02.

1337
01:14:56,890 --> 01:15:04,510
And the probability of the
second one is 0.2 out of 1.2.

1338
01:15:04,510 --> 01:15:08,530
Sorry, the probability of
the first one is 1 of 1.2.

1339
01:15:08,530 --> 01:15:09,867
So it's a very high percentage.

1340
01:15:09,867 --> 01:15:11,950
And the probability of the
second one being called

1341
01:15:11,950 --> 01:15:14,818
is 0.2 out of 1.2.

1342
01:15:14,818 --> 01:15:20,440
So yeah, every time you call
it from within anything,

1343
01:15:20,440 --> 01:15:23,380
it goes into this
program and say, tell me

1344
01:15:23,380 --> 01:15:24,970
which one should I call.

1345
01:15:24,970 --> 01:15:27,400
And the program
assigns which one

1346
01:15:27,400 --> 01:15:30,009
it is based on
these probabilities.

1347
01:15:30,009 --> 01:15:31,550
AUDIENCE: So it
doesn't matter if you

1348
01:15:31,550 --> 01:15:35,996
put it [INAUDIBLE] it
will still [INAUDIBLE]..

1349
01:15:41,450 --> 01:15:43,480
CURRAN KELLEHER: So
you're saying maybe we

1350
01:15:43,480 --> 01:15:44,500
could put it outside?

1351
01:15:44,500 --> 01:15:46,131
AUDIENCE: [INAUDIBLE] put it
inside the function is so that

1352
01:15:46,131 --> 01:15:47,093
the function--

1353
01:15:47,093 --> 01:15:48,980
I mean, the thing
itself could be called.

1354
01:15:48,980 --> 01:15:51,480
CURRAN KELLEHER: The reason why
I put it inside the function

1355
01:15:51,480 --> 01:15:55,590
is so that it can call itself.

1356
01:15:55,590 --> 01:15:57,162
So like--

1357
01:15:57,162 --> 01:15:59,920
JUSTIN CURRY: What happens if
you take, say, SPIRAL flip 90,

1358
01:15:59,920 --> 01:16:03,350
and you put it up in
the top row for SPIRAL?

1359
01:16:03,350 --> 01:16:06,577
[INAUDIBLE]

1360
01:16:06,577 --> 01:16:08,160
CURRAN KELLEHER: So
you're saying what

1361
01:16:08,160 --> 01:16:10,687
if I take this out of
here, and put it up here?

1362
01:16:10,687 --> 01:16:11,720
JUSTIN CURRY: Sure.

1363
01:16:11,720 --> 01:16:12,636
CURRAN KELLEHER: Yeah.

1364
01:16:12,636 --> 01:16:14,322
I mean, I don't know.

1365
01:16:14,322 --> 01:16:17,813
[CHUCKLING]

1366
01:16:17,813 --> 01:16:19,510
I don't know.

1367
01:16:19,510 --> 01:16:21,030
But it has to be
inside a function.

1368
01:16:23,914 --> 01:16:25,608
JUSTIN CURRY: So
I think the idea

1369
01:16:25,608 --> 01:16:28,910
is that you have
kind of two options,

1370
01:16:28,910 --> 01:16:33,290
whether or not you're just
doing the simple SPIRAL circle

1371
01:16:33,290 --> 01:16:43,330
routine, or if you're then
doing this other spiral

1372
01:16:43,330 --> 01:16:45,550
routine, where they
rotate 45 degrees

1373
01:16:45,550 --> 01:16:48,970
and change the size by 0.4,
instead of this spiral where

1374
01:16:48,970 --> 01:16:53,680
you flip 90, and you
change the size by 0.5.

1375
01:16:53,680 --> 01:16:55,360
And see, this was
what I thought.

1376
01:16:55,360 --> 01:16:57,650
I'm not exactly sure how
the algorithm's implemented.

1377
01:16:57,650 --> 01:17:03,670
But I think by having 0.2 next
to the bottom spiral, that

1378
01:17:03,670 --> 01:17:07,030
means the top spiral is called
only with a probability of 0.8.

1379
01:17:07,030 --> 01:17:09,220
But that's obviously
much higher than 0.2.

1380
01:17:09,220 --> 01:17:12,470
So every time the computer rolls
its die, it's trying to decide,

1381
01:17:12,470 --> 01:17:15,370
do I either execute
the top spiral,

1382
01:17:15,370 --> 01:17:17,680
or do I execute
the bottom spiral.

1383
01:17:17,680 --> 01:17:22,050
And then the content of
each of those spirals

1384
01:17:22,050 --> 01:17:23,890
then governs the
behavior you see here.

1385
01:17:26,948 --> 01:17:32,600
CURRAN KELLEHER: So what we
did, when we moved this up,

1386
01:17:32,600 --> 01:17:35,210
whenever we call a spiral
a spiral, it branches.

1387
01:17:35,210 --> 01:17:37,400
We have two spirals now.

1388
01:17:37,400 --> 01:17:40,340
And that's the probability
of that one is really high.

1389
01:17:40,340 --> 01:17:42,950
So that means pretty much
every time we're branching.

1390
01:17:42,950 --> 01:17:45,080
So we each get this
mass of branches.

1391
01:17:45,080 --> 01:17:47,970
And we'll never finish.

1392
01:17:47,970 --> 01:17:50,270
So yeah, I mean
there are all kinds

1393
01:17:50,270 --> 01:17:58,100
of really interesting modes
that we can come to with this.

1394
01:17:58,100 --> 01:18:00,350
So I'll take it out, and
put it back where it was.

1395
01:18:02,980 --> 01:18:04,538
It's still computing.

1396
01:18:04,538 --> 01:18:05,038
OK.

1397
01:18:05,038 --> 01:18:05,538
There we go.

1398
01:18:05,538 --> 01:18:06,505
It stopped.

1399
01:18:14,830 --> 01:18:16,570
Ah, yes.

1400
01:18:16,570 --> 01:18:19,900
So it looks crazy, doesn't it?

1401
01:18:19,900 --> 01:18:22,780
So if I take out that rotate,
it's all very regular.

1402
01:18:22,780 --> 01:18:26,050
It's sort of a
regular structure.

1403
01:18:26,050 --> 01:18:29,330
And we can change the
angle maybe 60 degrees--

1404
01:18:34,230 --> 01:18:38,430
I don't know-- or 90 even.

1405
01:18:38,430 --> 01:18:42,350
And it looks sort of
like roads, maybe.

1406
01:18:45,770 --> 01:18:48,610
I don't know.

1407
01:18:48,610 --> 01:18:53,690
AUDIENCE: It's like a
tree near a railroad.

1408
01:18:53,690 --> 01:18:55,914
You can see that [INAUDIBLE].

1409
01:18:55,914 --> 01:18:56,830
CURRAN KELLEHER: Yeah.

1410
01:18:56,830 --> 01:18:58,300
It's like you're
saying maybe it's

1411
01:18:58,300 --> 01:19:02,770
like a river with these little
sub-rivers going off of it.

1412
01:19:02,770 --> 01:19:04,810
Yeah, I mean this
kind of structure

1413
01:19:04,810 --> 01:19:06,160
is found everywhere in nature.

1414
01:19:06,160 --> 01:19:08,750
It's really amazing.

1415
01:19:08,750 --> 01:19:10,500
So if we add a little
bit of rotation

1416
01:19:10,500 --> 01:19:18,080
to the main going forward
rule, or rotate 1,

1417
01:19:18,080 --> 01:19:19,520
we get this sort of veiny.

1418
01:19:22,134 --> 01:19:24,050
AUDIENCE: Maybe that's
when the wind is going.

1419
01:19:24,050 --> 01:19:27,160
CURRAN KELLEHER: Yeah, when the
wind is blowing on the tree.

1420
01:19:27,160 --> 01:19:30,570
It looks a lot like vines.

1421
01:19:30,570 --> 01:19:32,460
When a vine is
crawling up a wall.

1422
01:19:34,998 --> 01:19:38,451
Or a root, a root of a plant.

1423
01:19:38,451 --> 01:19:40,740
AUDIENCE: [INAUDIBLE]
underground root.

1424
01:19:40,740 --> 01:19:41,412
Underground.

1425
01:19:41,412 --> 01:19:42,870
CURRAN KELLEHER:
Yeah, underground,

1426
01:19:42,870 --> 01:19:45,330
a root sort of looks like this.

1427
01:19:48,630 --> 01:19:54,000
AUDIENCE: [INAUDIBLE] for
the special tree if you tried

1428
01:19:54,000 --> 01:19:55,840
[INAUDIBLE].

1429
01:19:55,840 --> 01:19:58,480
CURRAN KELLEHER: So can I
make the Koch snowflake,

1430
01:19:58,480 --> 01:20:02,800
or the branching tree
with this sort of thing.

1431
01:20:02,800 --> 01:20:04,480
I haven't tried.

1432
01:20:04,480 --> 01:20:06,850
Maybe I could do it somehow.

1433
01:20:06,850 --> 01:20:10,730
But I don't think so.

1434
01:20:10,730 --> 01:20:14,830
I don't think I can,
because how this program

1435
01:20:14,830 --> 01:20:16,914
acts is completely random.

1436
01:20:16,914 --> 01:20:17,580
It's stochastic.

1437
01:20:17,580 --> 01:20:21,052
It chooses which rule
to do on the fly.

1438
01:20:21,052 --> 01:20:23,395
AUDIENCE: [INAUDIBLE] It
wouldn't be recursive.

1439
01:20:23,395 --> 01:20:25,270
CURRAN KELLEHER: Well,
it is still recursive.

1440
01:20:25,270 --> 01:20:27,010
It's totally recursive.

1441
01:20:27,010 --> 01:20:30,050
But it's not deterministic.

1442
01:20:30,050 --> 01:20:32,170
So it's something
being deterministic

1443
01:20:32,170 --> 01:20:35,302
gives it absolute
rigid regularity

1444
01:20:35,302 --> 01:20:36,760
that we find in
the Koch snowflake,

1445
01:20:36,760 --> 01:20:40,810
or that branching tree
that I showed earlier.

1446
01:20:40,810 --> 01:20:42,570
But yeah, they're
executed randomly.

1447
01:20:42,570 --> 01:20:46,420
So you get these
irregular fractals.

1448
01:20:46,420 --> 01:20:48,900
JUSTIN CURRY: But the
Sierpinski gasket, you

1449
01:20:48,900 --> 01:20:50,466
can also generate
it stochastically

1450
01:20:50,466 --> 01:20:54,440
using just a random
random [? die ?] rule.

1451
01:20:54,440 --> 01:20:58,780
And you pick your
three points, and then

1452
01:20:58,780 --> 01:21:01,480
you just throw a random dart.

1453
01:21:01,480 --> 01:21:03,940
And then, what, you
then take the distance

1454
01:21:03,940 --> 01:21:09,460
to the closest edge, and
you fill that point in.

1455
01:21:09,460 --> 01:21:10,330
You can do that.

1456
01:21:10,330 --> 01:21:11,050
I forget.

1457
01:21:11,050 --> 01:21:12,400
So the chaos game.

1458
01:21:12,400 --> 01:21:16,720
CURRAN KELLEHER: The chaos game
with the Sierpinski triangle,

1459
01:21:16,720 --> 01:21:19,930
what you do is you
have these three dots.

1460
01:21:19,930 --> 01:21:22,340
I did explain this once before.

1461
01:21:22,340 --> 01:21:24,091
But I'll just do
it again quickly.

1462
01:21:24,091 --> 01:21:26,590
You have these three dots, and
you start at a certain point,

1463
01:21:26,590 --> 01:21:28,780
like say here.

1464
01:21:28,780 --> 01:21:31,060
And you take where
you are, and you

1465
01:21:31,060 --> 01:21:33,100
choose at random one
of the three dots,

1466
01:21:33,100 --> 01:21:35,870
and go half way from
where you are to that dot.

1467
01:21:35,870 --> 01:21:38,290
So let's say we choose this
dot, we go halfway there.

1468
01:21:38,290 --> 01:21:40,157
Say we choose it
again, we go halfway.

1469
01:21:40,157 --> 01:21:42,490
And then say we choose this
dot, we go halfway from here

1470
01:21:42,490 --> 01:21:43,880
to here.

1471
01:21:43,880 --> 01:21:48,790
So after doing this 1,000
times, we get all these points

1472
01:21:48,790 --> 01:21:50,564
that in the limit,
if you were to do it

1473
01:21:50,564 --> 01:21:51,980
in an infinite
number of times, it

1474
01:21:51,980 --> 01:21:54,730
would be the Sierpinski gasket.

1475
01:21:54,730 --> 01:21:56,860
And so it starts looking
like the Sierpinski gasket

1476
01:21:56,860 --> 01:21:58,660
after a little while.

1477
01:21:58,660 --> 01:22:00,340
So that's stochastic.

1478
01:22:00,340 --> 01:22:03,130
I mean, I don't
know, maybe I could

1479
01:22:03,130 --> 01:22:04,540
code that in this language.

1480
01:22:04,540 --> 01:22:05,710
Maybe it's possible.

1481
01:22:05,710 --> 01:22:08,320
I haven't tried.

1482
01:22:08,320 --> 01:22:12,040
So there's some
prepared examples

1483
01:22:12,040 --> 01:22:15,700
that I sort of have prepared.

1484
01:22:22,500 --> 01:22:24,620
And one of the things
that I want to talk about

1485
01:22:24,620 --> 01:22:34,510
is these regions of stability
and instability in this system.

1486
01:22:34,510 --> 01:22:37,990
It's a very dynamic system.

1487
01:22:37,990 --> 01:22:41,014
And you get these
different situations.

1488
01:22:44,990 --> 01:22:45,650
Yeah.

1489
01:22:45,650 --> 01:22:46,200
Look at this.

1490
01:22:50,550 --> 01:22:51,050
Oh, wait.

1491
01:22:51,050 --> 01:22:52,370
That's a PNG.

1492
01:22:52,370 --> 01:22:53,740
That's not the actual thing.

1493
01:22:53,740 --> 01:22:54,402
Oh, well.

1494
01:22:57,180 --> 01:22:59,371
Just consider this.

1495
01:22:59,371 --> 01:23:02,490
I'll open the actual one.

1496
01:23:02,490 --> 01:23:03,440
Here it is.

1497
01:23:06,290 --> 01:23:16,385
So what's going on here
is I have this one rule.

1498
01:23:20,750 --> 01:23:22,940
First of all, this
rule, these two rules

1499
01:23:22,940 --> 01:23:24,390
are similar to the ones that--

1500
01:23:24,390 --> 01:23:29,722
I'll just get rid of this one
for the sake of explanation.

1501
01:23:29,722 --> 01:23:30,680
So this is pretty cool.

1502
01:23:30,680 --> 01:23:31,620
It's a tree.

1503
01:23:31,620 --> 01:23:33,450
So I'll explain the rules.

1504
01:23:33,450 --> 01:23:38,420
The first rule, goes
forward by a unit of 1,

1505
01:23:38,420 --> 01:23:39,680
and the circles are at 1.

1506
01:23:39,680 --> 01:23:42,620
So we can actually see
all the little circles.

1507
01:23:42,620 --> 01:23:46,170
Goes forward by 1,
and decreases in size.

1508
01:23:46,170 --> 01:23:49,550
The size is multiplied by 0.99.

1509
01:23:49,550 --> 01:23:51,410
So that's the main rule.

1510
01:23:51,410 --> 01:23:56,240
And then we have this other rule
which has a probability of 0.02

1511
01:23:56,240 --> 01:24:00,430
over 1.02, the sum of them.

1512
01:24:00,430 --> 01:24:01,680
And that's the branching rule.

1513
01:24:01,680 --> 01:24:04,970
So it just calls tree again,
with a rotation of 20 degrees

1514
01:24:04,970 --> 01:24:06,230
or minus 20 degrees.

1515
01:24:06,230 --> 01:24:09,390
So it's 20 minus 20.

1516
01:24:09,390 --> 01:24:12,310
So this is our rule.

1517
01:24:12,310 --> 01:24:14,510
And it generates this tree.

1518
01:24:14,510 --> 01:24:18,020
And then we add this other rule.

1519
01:24:18,020 --> 01:24:21,500
So this is a rule
that multiplies

1520
01:24:21,500 --> 01:24:28,085
the size of the thing by
5, which is pretty extreme.

1521
01:24:30,680 --> 01:24:33,050
But it happens with a
very small probability.

1522
01:24:35,710 --> 01:24:38,040
So I'll decrease the
probability even more--

1523
01:24:38,040 --> 01:24:43,910
0.0003.

1524
01:24:43,910 --> 01:24:49,470
So maybe 0.0001.

1525
01:24:49,470 --> 01:24:49,970
OK.

1526
01:24:49,970 --> 01:24:51,290
It just happened a few times.

1527
01:24:51,290 --> 01:24:54,770
So we can see that
usually it doesn't happen.

1528
01:24:54,770 --> 01:24:57,870
All these branching and
all these iterations,

1529
01:24:57,870 --> 01:24:59,510
it didn't happen.

1530
01:24:59,510 --> 01:25:04,010
But on this one here, the
size of it multiplied by five.

1531
01:25:04,010 --> 01:25:06,900
So we had this bigger circle.

1532
01:25:06,900 --> 01:25:08,750
And then it propagated more.

1533
01:25:08,750 --> 01:25:12,470
And then it happened again
on this tip, this very end

1534
01:25:12,470 --> 01:25:13,870
branch of it.

1535
01:25:13,870 --> 01:25:16,860
And it happened again.

1536
01:25:16,860 --> 01:25:18,302
So I mean--

1537
01:25:18,302 --> 01:25:20,030
JUSTIN CURRY:
[? It's an evil ?] tree.

1538
01:25:20,030 --> 01:25:23,450
CURRAN KELLEHER: This pattern
appears in evolution, actually,

1539
01:25:23,450 --> 01:25:24,900
which is really fascinating.

1540
01:25:24,900 --> 01:25:28,400
So in evolution, you have
these species and whatnot,

1541
01:25:28,400 --> 01:25:31,010
or different strains
of genome, which

1542
01:25:31,010 --> 01:25:34,520
sort of diverge and branch out.

1543
01:25:34,520 --> 01:25:39,137
And say at this point in time--

1544
01:25:39,137 --> 01:25:41,720
and the point in time being the
number of iterations overall--

1545
01:25:41,720 --> 01:25:45,200
at this point in time, say,
some huge catastrophic event

1546
01:25:45,200 --> 01:25:46,610
happened on the earth.

1547
01:25:46,610 --> 01:25:50,690
And this one organism,
or small set of organisms

1548
01:25:50,690 --> 01:25:52,730
is the only one that survived.

1549
01:25:52,730 --> 01:25:55,490
So their weight, so
it suddenly increased.

1550
01:25:55,490 --> 01:25:58,960
And then they propagated
and spread themselves.

1551
01:25:58,960 --> 01:26:03,840
And this is what we get, this
new branching of evolution.

1552
01:26:03,840 --> 01:26:05,423
And then the same
thing happened here.

1553
01:26:05,423 --> 01:26:06,150
Bam.

1554
01:26:06,150 --> 01:26:09,710
And then it branched again,
and it branched again.

1555
01:26:09,710 --> 01:26:15,342
So it's a very unstable,
unpredictable system,

1556
01:26:15,342 --> 01:26:17,300
because you could have
these little events that

1557
01:26:17,300 --> 01:26:21,290
just completely change
the face of the system.

1558
01:26:23,947 --> 01:26:25,780
Without this new rule,
it's a stable system.

1559
01:26:25,780 --> 01:26:29,840
We know the size is
always going to get

1560
01:26:29,840 --> 01:26:32,730
really smaller and smaller and
smaller until it disappears.

1561
01:26:32,730 --> 01:26:34,856
That's guaranteed in the limit.

1562
01:26:34,856 --> 01:26:36,230
But with this
system we introduce

1563
01:26:36,230 --> 01:26:39,245
a rule which goes backwards, so
it makes the system unstable,

1564
01:26:39,245 --> 01:26:42,180
and much more unpredictable.

1565
01:26:42,180 --> 01:26:46,620
So if we increase the
probability of this rule

1566
01:26:46,620 --> 01:26:54,720
to 0.001, the system goes out
of control really quickly.

1567
01:26:54,720 --> 01:27:00,320
And we have 0.01, it just gets
completely out of control.

1568
01:27:00,320 --> 01:27:04,070
And it just gets bigger
and bigger and bigger.

1569
01:27:04,070 --> 01:27:05,380
See this message here?

1570
01:27:05,380 --> 01:27:06,380
"A shape got too big"?

1571
01:27:09,830 --> 01:27:12,200
So this is an error that
we get when the thing just

1572
01:27:12,200 --> 01:27:15,059
goes out of control.

1573
01:27:15,059 --> 01:27:16,850
JUSTIN CURRY: So this
could also correspond

1574
01:27:16,850 --> 01:27:18,755
to meteorites being
frequently thrown

1575
01:27:18,755 --> 01:27:20,870
in the face of the
earth, and only

1576
01:27:20,870 --> 01:27:25,460
very few strands of species
surviving at a time.

1577
01:27:25,460 --> 01:27:27,912
CURRAN KELLEHER:
Well, yeah, I guess

1578
01:27:27,912 --> 01:27:29,162
this would correspond to that.

1579
01:27:32,389 --> 01:27:34,430
JUSTIN CURRY: Versus
destroying genetic diversity

1580
01:27:34,430 --> 01:27:36,680
at more frequent intervals.

1581
01:27:36,680 --> 01:27:38,930
CURRAN KELLEHER: Yeah.

1582
01:27:38,930 --> 01:27:43,340
Think of the metaphor to
evolution again, I don't know,

1583
01:27:43,340 --> 01:27:46,250
I guess it would correspond to
just huge catastrophic events

1584
01:27:46,250 --> 01:27:49,550
happening all the time,
and miraculously every time

1585
01:27:49,550 --> 01:27:52,340
one species survives.

1586
01:27:52,340 --> 01:27:53,690
Sort of extreme metaphor.

1587
01:27:53,690 --> 01:27:54,600
Doesn't really hold.

1588
01:27:54,600 --> 01:28:00,664
But it's interesting,
this regions

1589
01:28:00,664 --> 01:28:03,540
of stability and
instability in this space

1590
01:28:03,540 --> 01:28:05,940
of parameters to the system.

1591
01:28:05,940 --> 01:28:09,569
So yeah, it's
pretty fascinating.

1592
01:28:09,569 --> 01:28:11,110
So I'll just show
some more examples,

1593
01:28:11,110 --> 01:28:12,026
and then I'll be done.

1594
01:28:15,490 --> 01:28:17,480
This thick tree.

1595
01:28:17,480 --> 01:28:17,980
Yeah.

1596
01:28:17,980 --> 01:28:22,390
When we have certain
parameters sets,

1597
01:28:22,390 --> 01:28:24,360
we get these really
organic-looking forms.

1598
01:28:24,360 --> 01:28:26,620
So if we increase
the probability

1599
01:28:26,620 --> 01:28:30,070
of branching even more,
you get these really nice,

1600
01:28:30,070 --> 01:28:35,021
thick organic-looking trees.

1601
01:28:35,021 --> 01:28:36,889
AUDIENCE: So do
organisms [INAUDIBLE]??

1602
01:28:40,082 --> 01:28:41,790
JUSTIN CURRY: That's
the question, right?

1603
01:28:41,790 --> 01:28:44,340
CURRAN KELLEHER: So she
asks, do organisms actually

1604
01:28:44,340 --> 01:28:46,845
use this sort of system to live?

1605
01:28:46,845 --> 01:28:50,240
AUDIENCE: And if they had ones
that didn't have that system,

1606
01:28:50,240 --> 01:28:56,016
and did have some system,
is that [INAUDIBLE] or not?

1607
01:28:56,016 --> 01:28:57,390
CURRAN KELLEHER:
Right, so you're

1608
01:28:57,390 --> 01:29:02,050
asking this question sort of
from a totally different angle.

1609
01:29:02,050 --> 01:29:07,110
She's saying, for a given
organism if it develops

1610
01:29:07,110 --> 01:29:10,920
this kind of recursive system,
is it evolutionarily more

1611
01:29:10,920 --> 01:29:14,840
advantageous, like trees?

1612
01:29:14,840 --> 01:29:18,360
I think for plants that's
one of the main things

1613
01:29:18,360 --> 01:29:21,120
that plants sort of
developed, and that made

1614
01:29:21,120 --> 01:29:24,120
them, evolutionarily speaking,
more feasible, more fit.

1615
01:29:24,120 --> 01:29:25,820
So that's why.

1616
01:29:25,820 --> 01:29:30,390
JUSTIN CURRY: So that's an
important point to consider.

1617
01:29:30,390 --> 01:29:33,540
However, what would actually
make a tree bend like that?

1618
01:29:37,109 --> 01:29:38,900
Let's say that it's a
relatively calm area,

1619
01:29:38,900 --> 01:29:41,600
and we're not having hurricane
force winds forcing our trees

1620
01:29:41,600 --> 01:29:44,090
to bend that way.

1621
01:29:44,090 --> 01:29:46,309
Why would a tree grow like that?

1622
01:29:46,309 --> 01:29:47,225
CURRAN KELLEHER: Yeah.

1623
01:29:47,225 --> 01:29:48,120
[INAUDIBLE]

1624
01:29:48,120 --> 01:29:48,915
AUDIENCE: Sunlight?

1625
01:29:48,915 --> 01:29:49,885
JUSTIN CURRY: Sunlight.

1626
01:29:49,885 --> 01:29:50,385
Exactly.

1627
01:29:50,385 --> 01:29:52,050
It's phototaxis.

1628
01:29:52,050 --> 01:29:54,380
And plants have this feedback
mechanism where they're

1629
01:29:54,380 --> 01:29:56,600
actually able to sense light.

1630
01:29:56,600 --> 01:29:59,480
And it's just like if were
to have a potted plant here,

1631
01:29:59,480 --> 01:30:03,220
and that light in the
corner illuminating us,

1632
01:30:03,220 --> 01:30:05,570
it would actually be able
to dynamically change

1633
01:30:05,570 --> 01:30:07,315
the probability of splitting.

1634
01:30:07,315 --> 01:30:08,690
And of course the
method for that

1635
01:30:08,690 --> 01:30:10,500
is actually a little different.

1636
01:30:10,500 --> 01:30:12,820
You have these [? oxygen ?]
chemicals being exchanged,

1637
01:30:12,820 --> 01:30:13,920
collapses cell walls.

1638
01:30:13,920 --> 01:30:18,470
And a tree is actually very
quickly able to grow and bend

1639
01:30:18,470 --> 01:30:19,847
in a different direction.

1640
01:30:23,670 --> 01:30:26,339
So it's not exactly
just a probabilistic--

1641
01:30:26,339 --> 01:30:28,130
what happened there--
it's not exactly just

1642
01:30:28,130 --> 01:30:30,960
a probabilistic,
context-free grammar.

1643
01:30:30,960 --> 01:30:33,800
It's got to have some sort
of feedback mechanism.

1644
01:30:33,800 --> 01:30:35,510
And then with the
level of evolution,

1645
01:30:35,510 --> 01:30:36,760
changing the rules even there.

1646
01:30:40,890 --> 01:30:45,770
So yeah, I mean look at all
the number of leaf branches.

1647
01:30:45,770 --> 01:30:49,020
And when I say leaf, I
mean the one on the end.

1648
01:30:51,937 --> 01:30:53,020
There are so many of them.

1649
01:30:53,020 --> 01:30:58,830
And it maximizes the amount
of sunlight that hits.

1650
01:30:58,830 --> 01:31:01,120
It maximizes the amount of
surface area on the tree.

1651
01:31:01,120 --> 01:31:06,180
So I think that's one of the
big reasons why it's more fit.

1652
01:31:06,180 --> 01:31:07,212
You had a question?

1653
01:31:07,212 --> 01:31:09,550
AUDIENCE: No, I
just had a comment.

1654
01:31:09,550 --> 01:31:12,595
It also makes sense, like
you said, the surface area.

1655
01:31:12,595 --> 01:31:15,590
And it wouldn't make any sense
to have a branch at the bottom,

1656
01:31:15,590 --> 01:31:19,662
because all the line
was already gone from--

1657
01:31:19,662 --> 01:31:20,620
CURRAN KELLEHER: Right.

1658
01:31:20,620 --> 01:31:21,820
It wouldn't make any sense--

1659
01:31:21,820 --> 01:31:26,387
AUDIENCE: Yeah, if you
had some mutation where

1660
01:31:26,387 --> 01:31:28,304
a branch did come at
the bottom, that tree

1661
01:31:28,304 --> 01:31:29,470
wouldn't have any advantage.

1662
01:31:29,470 --> 01:31:30,428
CURRAN KELLEHER: Right.

1663
01:31:30,428 --> 01:31:32,070
So you're getting at selection.

1664
01:31:32,070 --> 01:31:34,770
So he said, if there
were some trees that

1665
01:31:34,770 --> 01:31:37,080
had these set of rules that
made it start branching

1666
01:31:37,080 --> 01:31:40,567
at the bottom and on the top,
it would be selected against,

1667
01:31:40,567 --> 01:31:42,650
because it's not feasible,
because the light would

1668
01:31:42,650 --> 01:31:44,233
be blocked out by
the higher branches.

1669
01:31:44,233 --> 01:31:46,250
Yeah, that's the
nature of evolution.

1670
01:31:46,250 --> 01:31:46,886
Yeah.

1671
01:31:46,886 --> 01:31:48,677
AUDIENCE: And that also
looks like a brain,

1672
01:31:48,677 --> 01:31:50,930
because you've got this
cerebral cortex on the top.

1673
01:31:50,930 --> 01:31:52,400
That's [INAUDIBLE].

1674
01:31:52,400 --> 01:31:56,679
And you've got the connections
and the [INAUDIBLE]..

1675
01:31:56,679 --> 01:31:57,720
CURRAN KELLEHER: Exactly.

1676
01:31:57,720 --> 01:32:01,610
So he said it's sort
of like a brain,

1677
01:32:01,610 --> 01:32:03,290
like this is a cerebral cortex.

1678
01:32:03,290 --> 01:32:05,240
And you have all
these connections

1679
01:32:05,240 --> 01:32:08,150
that go out to the surface
of the brain, which

1680
01:32:08,150 --> 01:32:10,672
have all these brain cells.

1681
01:32:10,672 --> 01:32:12,762
JUSTIN CURRY: Or even
a vein structure.

1682
01:32:12,762 --> 01:32:13,970
CURRAN KELLEHER: Yeah, veins.

1683
01:32:13,970 --> 01:32:15,500
The structure of the
veins in your body

1684
01:32:15,500 --> 01:32:16,541
coming out of your heart.

1685
01:32:16,541 --> 01:32:19,580
It just fractals out
to all of your body.

1686
01:32:19,580 --> 01:32:23,150
So, yeah, it's this
sort of universal form

1687
01:32:23,150 --> 01:32:25,230
that appears everywhere.

1688
01:32:25,230 --> 01:32:28,330
It's really amazing.

1689
01:32:28,330 --> 01:32:31,380
So all we can do is sort
of awe at it say, wow,

1690
01:32:31,380 --> 01:32:33,570
"They're like the same, man!"

1691
01:32:33,570 --> 01:32:37,510
But where does it come from?

1692
01:32:37,510 --> 01:32:38,260
What does it mean?

1693
01:32:38,260 --> 01:32:38,850
I don't know.

1694
01:32:38,850 --> 01:32:42,821
It's something that needs
to be explored, I think.

1695
01:32:42,821 --> 01:32:45,510
AUDIENCE: [INAUDIBLE]
it had to be that way?

1696
01:32:45,510 --> 01:32:48,490
CURRAN KELLEHER: Could it be
that it had to be that way?

1697
01:32:48,490 --> 01:32:49,833
What do you mean by that?

1698
01:32:49,833 --> 01:32:55,720
AUDIENCE: You can see that it's
the most efficient algorithm,

1699
01:32:55,720 --> 01:32:59,790
and [INAUDIBLE] stress
towards efficiency.

1700
01:32:59,790 --> 01:33:02,750
So at some point we had
to hit that algorithm.

1701
01:33:02,750 --> 01:33:05,950
And when it gets something good,
it doesn't want to let it go.

1702
01:33:05,950 --> 01:33:06,870
CURRAN KELLEHER: Yeah.

1703
01:33:06,870 --> 01:33:13,295
So you're talking about it
being the whole system of all

1704
01:33:13,295 --> 01:33:14,960
of biology and things evolving.

1705
01:33:14,960 --> 01:33:18,080
He said maybe it's the
only thing that works.

1706
01:33:18,080 --> 01:33:20,900
It has to exist, because it's
the most efficient algorithm

1707
01:33:20,900 --> 01:33:24,585
for developing our
biology, our forms.

1708
01:33:24,585 --> 01:33:27,830
And so once it
appeared, it sort of

1709
01:33:27,830 --> 01:33:34,720
took hold in this big
evolutionary series of events.

1710
01:33:34,720 --> 01:33:35,890
And I think you're right.

1711
01:33:35,890 --> 01:33:42,430
And it also is coded in a
very simple set of rules.

1712
01:33:42,430 --> 01:33:44,980
Look at this.

1713
01:33:44,980 --> 01:33:48,760
It's very little text that
encodes for all of this.

1714
01:33:48,760 --> 01:33:53,070
And I think similarly,
in our genomes,

1715
01:33:53,070 --> 01:33:56,050
if the evolutionary
paradigm comes up

1716
01:33:56,050 --> 01:34:00,640
with an efficient way of
encoding a certain set of rules

1717
01:34:00,640 --> 01:34:05,950
to do something, which is
fit, which makes us more fit,

1718
01:34:05,950 --> 01:34:08,140
then it sticks.

1719
01:34:08,140 --> 01:34:12,760
Yeah, I think I think
this notion of fractals

1720
01:34:12,760 --> 01:34:16,490
and the recursive algorithm
being encoded into our genomes

1721
01:34:16,490 --> 01:34:21,980
is a reasonable
thing to hypothesize.

1722
01:34:21,980 --> 01:34:24,325
AUDIENCE: Doesn't that
also show up in [INAUDIBLE]

1723
01:34:24,325 --> 01:34:27,394
behaviors, [INAUDIBLE]?

1724
01:34:27,394 --> 01:34:28,310
CURRAN KELLEHER: Yeah.

1725
01:34:28,310 --> 01:34:29,140
Ants.

1726
01:34:29,140 --> 01:34:29,750
Ant colonies.

1727
01:34:29,750 --> 01:34:31,090
Oh, man.

1728
01:34:31,090 --> 01:34:32,810
Yeah.

1729
01:34:32,810 --> 01:34:34,044
It's everywhere.

1730
01:34:34,044 --> 01:34:35,460
JUSTIN CURRY:
Emergent properties.

1731
01:34:35,460 --> 01:34:37,001
CURRAN KELLEHER:
Emergent properties.

1732
01:34:40,810 --> 01:34:42,330
So I think I'm done.

1733
01:34:42,330 --> 01:34:43,530
I think this is my spiel.

1734
01:34:43,530 --> 01:34:45,310
And I'll give it back to Justin.

1735
01:34:45,310 --> 01:34:47,310
JUSTIN CURRY: I just want
to kind of wrap things

1736
01:34:47,310 --> 01:34:49,286
up, kind of give a
sense of conclusion,

1737
01:34:49,286 --> 01:34:50,910
and the direction of
where we're going.

1738
01:34:55,340 --> 01:34:58,460
Ah, if you want to
kill the projector.

1739
01:34:58,460 --> 01:35:01,720
So exactly, Curran's kind
of hinting at an idea,

1740
01:35:01,720 --> 01:35:04,960
and we're all kind of at
the brink of this concept

1741
01:35:04,960 --> 01:35:06,640
which I told you at
the very beginning

1742
01:35:06,640 --> 01:35:10,210
is the stated thesis of
Godel/Escher/Bach, which

1743
01:35:10,210 --> 01:35:12,970
is that the universe,
at a fundamental level,

1744
01:35:12,970 --> 01:35:16,450
is a formal system,
and that it obeys

1745
01:35:16,450 --> 01:35:20,610
certain deterministic rules,
or perhaps probabilistic rules,

1746
01:35:20,610 --> 01:35:23,350
but kind of formal
system nonetheless.

1747
01:35:23,350 --> 01:35:26,950
We have this kind of label which
we just stick on something,

1748
01:35:26,950 --> 01:35:29,050
and that being the
[? I ?] label, which

1749
01:35:29,050 --> 01:35:32,787
actually hides a lot of detail,
just like in the way that--

1750
01:35:32,787 --> 01:35:34,870
But trying to understand
them in a fundamental way

1751
01:35:34,870 --> 01:35:38,500
is the stated goal
of this course.

1752
01:35:38,500 --> 01:35:41,230
And I'm still
debating, because I'm

1753
01:35:41,230 --> 01:35:43,390
kind of continuously
and probabilistic

1754
01:35:43,390 --> 01:35:46,510
modifying the course
of this course.

1755
01:35:46,510 --> 01:35:50,500
And I'm trying to decide,
and the stated plan right now

1756
01:35:50,500 --> 01:35:53,770
is that after we do
Murmann and Godl,

1757
01:35:53,770 --> 01:35:57,250
and you'll get this weird
Asian kick from both Curran

1758
01:35:57,250 --> 01:36:00,610
and I combining zen and logic.

1759
01:36:00,610 --> 01:36:02,680
And we talk about Godel's
incompleteness theorem,

1760
01:36:02,680 --> 01:36:04,510
finally.

1761
01:36:04,510 --> 01:36:10,600
And then we're going to leap
forward chapters in the book 16

1762
01:36:10,600 --> 01:36:16,690
to a self ref and self
rep chapter, which will be

1763
01:36:16,690 --> 01:36:17,940
kind of a little over the top.

1764
01:36:17,940 --> 01:36:19,606
First of all, it's a
very long chapter--

1765
01:36:19,606 --> 01:36:21,640
57 pages.

1766
01:36:21,640 --> 01:36:24,250
But it's going to have
this idea of a kind

1767
01:36:24,250 --> 01:36:26,071
of typographical genetics.

1768
01:36:26,071 --> 01:36:28,570
And we're going to look at how
genetics and protein folding,

1769
01:36:28,570 --> 01:36:33,292
and kind of the processes which
kind of make us correspond

1770
01:36:33,292 --> 01:36:34,750
to some of the
formal systems we've

1771
01:36:34,750 --> 01:36:38,350
been talking about
in a prescribed way.

1772
01:36:38,350 --> 01:36:40,275
Then we're going
to leap backwards.

1773
01:36:40,275 --> 01:36:42,400
And since what I've realized
this course has become

1774
01:36:42,400 --> 01:36:46,510
is really a topics course
of a bunch of things,

1775
01:36:46,510 --> 01:36:48,970
then leap to essentially
brains and thoughts.

1776
01:36:48,970 --> 01:36:52,374
Because the bottom line
is Hofstadter's thinking

1777
01:36:52,374 --> 01:36:54,040
hasn't been patched
through all the way.

1778
01:36:54,040 --> 01:36:55,649
Otherwise we would
have it solved.

1779
01:36:55,649 --> 01:36:56,440
We'd be like, a-ha!

1780
01:36:56,440 --> 01:36:58,010
Good thing we solved
consciousness.

1781
01:36:58,010 --> 01:36:59,830
We can go on and
do other things.

1782
01:36:59,830 --> 01:37:01,560
It's not solved.

1783
01:37:01,560 --> 01:37:05,680
And there are these huge
kind of gaps missing between.

1784
01:37:05,680 --> 01:37:10,132
OK, maybe I buy it that the
universe is a formal system.

1785
01:37:10,132 --> 01:37:12,340
But what on earth does
Godel's incompleteness theorem

1786
01:37:12,340 --> 01:37:13,855
have to say about
physical systems?

1787
01:37:17,170 --> 01:37:19,074
And let's ignore all
that, and then start

1788
01:37:19,074 --> 01:37:21,490
talking about the brain, and
kind of these meta structures

1789
01:37:21,490 --> 01:37:23,114
in the brain and the
mind, and thinking

1790
01:37:23,114 --> 01:37:24,400
in artificial intelligence.

1791
01:37:24,400 --> 01:37:26,627
And that kind of
wrapping up the course.

1792
01:37:26,627 --> 01:37:28,210
But then, of course,
I'm also thinking

1793
01:37:28,210 --> 01:37:31,390
about possibly showing a movie.

1794
01:37:31,390 --> 01:37:32,710
And that being Waking Life.

1795
01:37:32,710 --> 01:37:35,350
I don't know if any
of you have seen it.

1796
01:37:35,350 --> 01:37:38,470
And that just being essentially
my gift to you guys for working

1797
01:37:38,470 --> 01:37:40,060
so hard in this class.

1798
01:37:40,060 --> 01:37:42,280
But I might need to
get permission slips.

1799
01:37:42,280 --> 01:37:44,650
So that's yet to come.

1800
01:37:44,650 --> 01:37:47,530
And I kind of
apologized for whatever

1801
01:37:47,530 --> 01:37:49,690
slow pace today's lecture was.

1802
01:37:49,690 --> 01:37:52,900
Hopefully next one will
be a little more exciting.

1803
01:37:52,900 --> 01:37:58,600
But other than that
read, Murmann and Godel.

1804
01:37:58,600 --> 01:38:02,050
I meant to do the dialogue that
precedes the chapter today.

1805
01:38:02,050 --> 01:38:03,550
But obviously we can't.

1806
01:38:03,550 --> 01:38:06,670
We can do it for next
lecture, if we want.

1807
01:38:06,670 --> 01:38:09,440
And then, yeah,
that should be fun.

1808
01:38:09,440 --> 01:38:12,160
And read that handout
from I am a Strange Loop.

1809
01:38:12,160 --> 01:38:13,750
Because I think
it will explicate.

1810
01:38:13,750 --> 01:38:16,291
And that's really what I'll be
lecturing from for a large bit

1811
01:38:16,291 --> 01:38:18,120
of next lecture.