[MUSIC PLAYING]
Stanford University.
And we are not going
to work our way
through the behavior on the
right and march to the left.
And instead, we'll
be trying to come up
with some ideas that are
going to apply to everything
we hear about in here.
And overall, these are probably
the most difficult lectures
of the course, the most
difficult material,
in part because I'm not sure
if I completely understand
what I'm talking about.
But also because
this is intrinsically
some really different ways
of thinking about things
in the realm of science.
And that's one of
the reasons why
I forced you guys to
read this Chaos book.
And again, as I think I
mentioned in the first lecture,
this incites a subset of
people to passionate enthusiasm
about the book.
It incites another small subset
into just the most greatest
level of irritation
that this was assigned.
And everybody else
is just vaguely
puzzled and kind of sort of
sees the point, but how come?
This book, when I first read
it, and so my first introduction
to the whole field, this
was like the first book
I had read where I finished it
and immediately started over
again, the first
one since, like,
Where the Wild Things Are
in terms of influence.
This was an incredibly
challenging book
in terms of questioning
all the ways in which I
think about sort of
reductive science.
And hopefully, it will
do the same for you.
And as part of it, posted, there
is the one and only homework
assignment in the
entire course, which,
just to make things easier, will
not be collected or looked at.
But what you should do is there
is a whole exercise up there
in generating something that
are called cellular automata.
Do not panic yet.
There's plenty of time to panic.
Where you will be making
some of those on your own.
And all I ask from
you guys in terms
of making sense of
these exercises is
do not sleep between
now and Friday.
Spend all of your time working
on them, do nothing else.
Take occasional breaks
to eat or the bathroom.
But other than that,
do nothing but this
between now and Friday.
And let's see how everybody
feels about it on Friday.
OK.
So picking up, where we
need to start off with
is trying to get a framework for
the standard Western approach
to understanding complicated
systems with scientific bases.
400 or so AD, Rome falls,
collapse of the Roman Empire,
entering an unbelievably dark
period in terms of ignorance,
in terms of the Dark
Ages throughout Europe,
the level at which
people did not
understand how the
world worked, the level
at which people had lost
knowledge from previous times.
Books were gone.
Philosophers were gone.
The level of just isolation,
intellectual isolation,
was phenomenal.
It was during that
period, it was
as if 500 years before people
had known the cure for cancer,
and for AIDS, and for being
able to fly on your own,
and all of that, and somehow
in the aftermath of the empire
falling apart, all of
that knowledge was lost.
Literacy went down the tubes.
And this was a period that
gave rise to words like,
having an audit, having an
audit about your finances,
making an oral argument
before the judges,
having hearings about something
or other before the court.
Because all of these
were about speaking,
about auditory transmission
of information because nobody
could read anymore.
This was a period
where there was
no Western European language
that had the word progress
or ambition in it.
These were non-existent
concepts at the time.
Utter intellectual isolation,
utter social isolation,
during a time the vast
majority of people
lived in small villages, where
you would go 50 miles away
and people spoke a dialect
that you could not understand,
that degree of isolation.
Estimates are the
average person never went
more than 12 to 15
miles away from where
they were born in their entire
life, incredible isolation.
And incredible
ignorance ultimately
about how you explain causality
in the world because of all
of the information was gone.
Then something dramatic
changed in the year
1085, which is the first
European Christian conquest
of a major Islamic city, a
major Moorish one in this case,
Toledo in Spain.
Spain at the time, which was
Moorish and known as Alhambra
at the time, this
was the first city
to fall to Christian troops
since Islam swept in there.
And this was basically sort
of a second rate kind of city.
This was not some
major center, Toledo.
But simply by
European forces having
captured that city, something
extraordinary happened, which
was within that city was a
library, with more books than
existed in all of Christian
Europe put together.
This simple library in this
sort out of the way podunk sort
of the city there
out in the boondocks,
one run-of-the-mill library
there had more cumulative
information than was available
to all of Europe at the time.
And suddenly, Europe got
to rediscover philosophers,
Aristotle, Plato, all of that.
They got to rediscover logic.
They got to rediscover
all the great works.
Suddenly all of
those flooded back
into Europe and the
first beginnings of sort
of a modern mindset about
complexity started to emerge.
People suddenly began to do
things, like able to think
transitive and
with transitivity,
in a transitive manner, where
you would see a is bigger
than b, b is bigger than c.
And you had this startling
revolutionary notion,
which is you now
can know something
about the relationship
between and a and c
without having to directly
compare them with each other.
This was an astonishing sort
of logical breakthrough.
Syllogism, syllogism
suddenly appeared in Europe
for the first time in
centuries, the ability
of people to do
things like, say,
if all things that
glow have fire, then
stars glow and thus stars have
fire, syllogistic thinking.
That had been utterly gone.
And suddenly, people
began to think
about what is, where that
had been lost for centuries.
And all of that culminated
in a certain sort
of emergence of what we
would now call science.
Thomas Aquinas coming
up with an amazing,
an amazing quote, that
summarizes all of what
was happening at the time.
He listed three things
that God could not do.
The first two were just
sort of theological stuff.
God cannot sin.
God cannot make a
copy of himself.
It's the third one that
was just earth-shaking.
Third, even God cannot make
a triangle with more than 180
degrees.
And in that one concept,
Aquinas had just
said if it comes up against
sort of the old knowledge
and science, science wins.
And that was an absolutely
landmark moment.
God could do anything, but
still can't make a triangle
with more than 180 degrees.
This was the beginning of the
transformation of the world.
And this immediately had
impact, all sorts of domains,
not just this very
pedestrian one,
which is if something
broke you could fix it.
That was a concept that was
very, very rare around then.
But the ability to
construct events
by looking at
overlapping fragments.
A crime has occurred
and there is
no individual who has watched
the entire crime happen.
But one person saw what
happened from point A to C. One
saw from B to D. One
for all from C to--
and suddenly, there
was this realization
that you can figure
out what went on
by putting these various
overlapping bits of data
together, a completely
revolutionary idea
and completely transformed
the notion that how
do you figure out if
somebody did a crime prior
to this period,
what you would do
is to throw them in
the river, for example.
And if they sank and
drowned, obviously they
had done the crime.
Good luck there.
In terms of having figured
that out, good detective work.
That's how you think
figured if somebody had
done something wrong or not.
You set fire to them.
And if they burned, oh,
they were obviously guilty.
And suddenly this
concept instead,
not only using facts, not
only using observational data,
but that you could derive
what occurred without any one
individual having
seen the whole thing.
You can reconstruct
things with overlapping.
This was just landmark.
This just transformed
everything.
And somewhere around
this time began
to emerge what we
would recognize
to be sort of the
proto-baby steps of what
would be modern science.
And in the aftermath of this
period came what was basically
the single most
important concept
in all of science in
the last 500 years,
which is the idea
of reductionism.
To find, very
simply, if you want
to understand a complex
system, you break it down
into its component parts.
And when you understand
the individual parts,
you will be able to
understand the complex system.
Reductionism, this is at
the core of everything
that we do in science, in
modern science, centuries worth.
The notion that
complicated things
can be explained by
looking at their component
parts, the smaller
pieces that make them up.
And what's intrinsic in that
is the concept of linearity,
of additivity.
You got something complex
and you break it down
into its component parts.
And once you figure out how
those component parts work,
all you need to do
is add them together
and they will increase in their
complexity in a linear manner
and you will produce the
whole complex system.
This is Westernized
reductionism.
And it came with a
bunch of corollaries,
that we take for granted by now.
The first one being,
in a reductive system
where the component parts
and how they worked,
just add them together.
And in a straightforward
way, that
will produce your
complex system.
One sort of
consequence of that is
if you know the starting
state of a system,
as defined here, if you know
what the little component
parts all are, if you
know the starting state,
you will have 100%
predictability
of what the full complex
mature system will look like.
So starting state allows you
to predict what comes later.
And related to that, if you
know the complex system,
you can figure out what
was the starting state.
That there is point
for point relationship
between the simple
building blocks
and the complex systems
that come out the other end.
And this gave rise to
an extraordinary thing,
which was the ability
to extrapolate,
to be able to see the
answer to something
in different iterations,
and to use the same rules
and apply them over and over.
OK, what do I mean by this?
Suddenly this amazing notion
that, OK, if x plus y equals z,
you will then know
that x plus 1 plus y
is going to equal z plus 1.
And x plus 2, all of that.
And the same exact
principle would
hold for some, like, bizarre
idiot equation or whatever,
like that.
It doesn't really matter
what all of this is.
You know absolutely beforehand
simply by this business
of additivity of component
parts that whatever this is,
it's going to equal z plus 1.
You could come up with
an answer to something
without having to go through
the calculations all over again.
You could go through
x plus whatever
and you know it's going to be
z plus whatever without having
to sit there and measure it.
You could extrapolate.
You can use reductive
knowledge, the linearity,
going from this to this.
You're still using
the same rules.
That allows you to
go from this to this.
Applications of it are
purely reductive linear set
of systems.
And this was revolutionary.
So that's great.
You don't need to go through all
the calculations at every step
of the way to be able to
know the starting state
and thus know what the
mature state is about.
Look at the mature
state, you know
what the starting state was
point for point to relatedness.
That's great.
Finally, another feature
of reductive systems
like these is the really
fancy ones require blueprints.
What do I mean by blueprints in
this case that requires already
a notion of what the mature
state is supposed to look like?
Which is intrinsic
in what I just said,
if you know the starting state,
you will know the mature state.
But the belief that in
terms of quality control,
you have to have some
representation of what
the mature state is
supposed to look like,
a blueprint; in order to know
if you're doing the right thing,
feedback along those lines.
If you're really going to
do something hard and fancy
in this reductive
world, you've got
to have a road map
at the beginning.
You have to have a blueprint.
You have to have something
that already shows you
what the outcome is going
to be when you apply
these linear additive rules.
And the way you
go about doing it
is shaped by the blueprint,
the instructions that
are intrinsic in this.
This is everything about
Westernized reductionism.
One important additional
component to it, which is you
go measure something or other.
And what's the normal
temperature in humans, 98.6.
There's no way you
take a whole bunch
of perfectly healthy humans and
they're all going to have 98.6.
There is going to
be 98.6 and there
will be variability around it.
And you could express
that so that you wind up
having an average of 98.6
and some sort of term
that denotes variability.
There's variability.
There's different
values for something
that winds up averaging
out to something like this.
And thus this critical
question, in a reductive world
of thinking about science and
the way fancy things work,
what do you make of variability?
What is variability about?
An intrinsic in this whole world
of know the starting state,
you know the mature state;
you know the mature state,
you knew how the
starting state was.
The rules allow
you to extrapolate.
It takes a blueprint.
Intrinsic in there was
an absolute clear opinion
as to what variability is,
which is to say it's noise.
It's junk in the system.
It's a pain in the rear.
And it's stuff you
want to get rid of.
What is intrinsic in this
whole reductive view is noise
represents instrument
error; instrument
error, instrument in the
largest metaphorical sense.
Instrument, somebody's
observation to machinery.
Variability represents
noise, represents the system
you use to measure
stuff, to observe stuff
not working perfectly.
It represents something
you want to avoid.
And what was also intrinsic
in that inductive view
is the surest way to avoid that
is to become more reductive.
The notion that the closer you
look to a phenomenon, the more
detail you see it
with, the more you
are looking at a
more reduced level
closer to the component
parts, the closer
you will be to seeing
what's actually going on.
And as you look
closer and closer,
variability should disappear.
Because the variability is
just noise in the system.
And if you're trying to measure
people's body temperature
by being up in a
Zeppelin and looking
at people with your
sort of binoculars
and trying to see if
they're sweating or not
and come up with
an estimate then,
that's going to be a lot
more variable than if you now
do something more
reductive, like put
your hand on their forehead.
Oh, do they feel hot or not?
And thus, there will
be less variability.
And even less if you now
invent yourself a thermometer.
This whole notion that as
you get fancier techniques
for examining a phenomenon,
as you get techniques that
allow you to be more
and more reductive
and look closer and closer,
there will be less variability.
Because sitting way down
at the bottom of all
this reductive processes there
is an iconic, an absolute,
an idealized norm as
to what the answer is.
If you see anybody
not having 98.6
it's because there's noise
in your measurement systems.
Variability is noise.
Variability is
something to get rid of.
And the way to get
rid of variability
is to become more reductive.
Variability is discrepancy
from seeing what
the actual true measure is.
And that has been
a driving force
essentially in all
of science in terms
of the notion of inventing
new ways to look at things.
More powerful microscopes,
more powerful ways
of measuring the levels
of something or other
in the bloodstream, all of
them built around the notion
that the closer we look
to the component parts,
the closer we will be to
seeing how the system really
truly works and be able
to finally see what truly
is going on without the noise.
Because all noise
is is discrepancy
from what is truly going on.
So in that view, what
you, of course, wind up
having is an extension of that
in beginning to think about
how, like, bodies work
in biology and all that,
as you begin to look at
that as an example of a very
complicated system.
And, of course,
what you then have
in a reductionist view is
if you want to understand
how the body works, you need to
understand how the organs work.
And if you want to
understand how organs work,
you need to understand
how cells work,
and cells all the way
down to molecules,
all the way down there.
And the notion that the
closer you get to all the way
down there, and once you
understand things down
at that level, the
purer, the more accurate
your answers will be.
And all you do then is add
the pieces together and out
comes your whole body.
So where does that
begin to cause problems?
The fact that the
body simply can't
work that way, a whole bunch
of realms in which reductionism
has to fail when you're
looking at biological systems.
One example of this,
the first one, and this
is immediately jumping
into neurobiology,
this was classic work, work done
by these two neurobiologists.
Anyone who was in
BioCore sort of
went through me haranguing
about how great these guys were,
a pair of neuroscientists,
Hubel and Wiesel,
absolute giants in the field.
In the 1950s, up to
the 1960s, everybody
thought that they had discovered
exactly how the cortex worked.
And what they found was a
phenomenally clean reductive
world of how you
extracted information
from the visual
world around you.
I will spare you the details
because it's not important.
But what they
basically showed was
you could find individual cells
in the retina that corresponded
to individual neurons
in the simplest
part of the visual cortex.
And between them, you had
simple point for point reductive
relationships.
If you stimulated
this one retinal cell,
it's associated neuron in
this part of the cortex
would get excited and
have an action potential.
If you shift your
electrode over a smidgen
and stimulate the
one right next to it,
the neuron right next to this
one is going to get stimulated.
In other words, if you know
the starting state, which
receptors in the eye
have been stimulated,
you have 100% predictability
of which neurons appear,
are going to fire.
And the converse, know
which ones fired here.
And you have
complete information
about the starting state.
And what they did was
begin to build on that.
They showed that insofar as
that first layer of the cortex
had this one to one
correspondence with one cell
here to one cell there, what
did individual neurons know
about in this simple part
of the visual cortex?
These neurons knew
how to recognize dots.
Each neuron could
recognize a dot
and one dot only and was the
only neuron that recognized it.
This was a point for
point reductive system.
And take all those individual
little neuron component parts,
each of which then knew
something about one dot,
and put them together
and you could
begin to get some information
about what just hit the eye.
What they then showed was
the next layer of the cortex.
And again to simplify
things as much as possible,
what they began to
see was you would now
stimulate one of
those retinal cells
and one neuron in the
first layer of cortex
would get excited.
Nothing would happen
in the second layer.
Shift over, stimulate
the next one over,
the next one over, nothing in
the next part of the cortex,
over and over and over,
over and over and over,
and then suddenly one of the
neurons in the second cortex
gets excited.
If and only if you first
stimulate this photoreceptor,
followed by this, followed
by this, followed by this,
followed by this.
What does that neuron
know about, light moving
in a straight line
in this direction.
That part of the cortex
could extract the information
from that first layer and
put them together and get
different sorts of information.
And, thus, you would
have another neuron there
that would code for an angle
that was slightly different,
and another one there.
And then ones for
different parts
of the cortex, different
parts of the visual system,
and very long lines or
very slow moving lights
or things like that.
What do neurons in that
second layer know about?
They know how to
recognize straight lines.
And you could see again,
this is a reductive system.
Because you know the wiring
that goes from one layer,
from the eye, to this
layer, to this layer.
And thus, if you know
what's going on here,
you can work backwards and
know what's happening there
and what's happening
in the eye and the same
with the other direction.
A point for point
system, where now you're
beginning to extract a higher
level, a hierarchy of analysis,
but the same exact reductionism.
And just to then begin to show
what they then went on to,
again this is very
simplified, now
you begin to get neurons here.
One of them will
respond to this line.
Another will respond
to this line.
Another will respond
to this line.
If and only if
these three neurons
are firing
simultaneously, one neuron
in the next layer of the
visual cortex would fire.
What do neurons
there know about?
Each one knows about a
curve and one curve only.
The same exact
thing again, which
is point for point reductionism.
If you want to
understand the system,
you need to understand how
every single neuron is wired
to every next one in line.
And once you got that,
all you need to know
is what information,
what activity,
is happening at any level.
And you've got 100% knowledge
of what will be going on here,
and here, and here, a
purely reductive system.
Everybody loved this.
This was the greatest stuff
that happened in neurobiology.
This was arguably the most
important work in neurobiology
between, like, 1950, 1975.
The two of them got
their Nobel Prize.
People would have given them
a dozen if they could of.
Because what had
they just solved,
they just showed how the brain
processes sensory information,
how it extracts information
from the world around
and turns it into complex
bits of sensory information.
Because it was completely
obvious at this point what
was going to happen,
which was above this,
there would be a
layer that had neurons
that could respond to a
certain number of curves
simultaneously.
It can start seeing
three dimensions.
And then above that is one
where the three dimensions
are changing over time.
It can detect movement of
a three-dimensional object.
And the notion was you
would be able to just go
up a layer after layer of
reductive pointilist wiring
and way up on top, you would
have this super-duper layer
of visual cortical processing.
And all the way up on
top, somewhere up there,
you would now have a neuron that
knew one thing and one thing
only, knew how to recognize
your grandmother's face
at this angle.
And the notion would be
that right next to it
was another neuron that
recognized your grandmother's
face at this angle,
and then one like this.
And right behind, with
the rows of neurons
recognizing your grandfather.
And everybody
decided this is it.
Just take this world
of Hubel and Wiesel's
stepwise extraction of
information and keep going.
And that's how the brain
winds up recognizing faces.
And meanwhile,
people subsequently
showed in the auditory cortex
that the correspondence
between one cochlear
cell, one hair cell there,
recognizing a single note,
up to chords, up to OK.
So now you go up
enough layers there
and you will find a layer
that your single neurons then
know your grandmother's
favorite symphony.
And that's it.
All the way up, you
would eventually
find neurons that were
specialized in really
complex sensory information.
And all you had to
do was just keep
going like this in this
purely reductive way
and you've got it
up there on top.
And people at the
time actually did
refer to these as
grandmother neurons.
The notion that
enough layers up here,
you would get neurons
that responded to a really
complex thing and only to that.
And it was the only neuron
that responded to that point
for point, one thing
and one thing only.
And that all you needed to
do was just keep doing this
and you would eventually
get neurons that
recognized your grandmother.
So right around the time
that Hubel and Wiesel got
their third layer here, and
this took them about 15 years,
they decided to go
study something else
in the visual system.
And that turned out to be
at least as interesting
as this stuff.
But everybody else
leapt in at that point,
to try to find the next
layer, and the layer,
and the next layer.
And Hubel and Wiesel had
shown a remarkable bit
of wisdom or
intuition by bailing
on the field at that point.
Because to this
day, hardly anybody
has ever shown the existence
of a grandmother neuron
all the way down there.
They simply don't exist.
OK, that's a lie.
They do exist.
But there's very few of them.
There's sparse coding.
Occasionally, you find
neurons that show grandmother
neuron-like processes,
neurons, a single neuron that
will respond to a face
and only one type of face,
way up there in many layers
of visual cortical processing.
There are some of those.
And there was a paper in
Nature some years ago,
which was one of the weirdest
papers I have ever seen.
Really interesting in
terms of what it showed,
but weird from the
standpoint of what
were these folks thinking
to actually test this?
And they were recording
from the very upper layers
of visual cortex
in monkey brains.
And they found some
neurons that responded.
One neuron would respond
only to a certain human face,
encoding a
grandmother-type neuron.
Here's the bizarre
thing in that paper.
What they discovered
were neurons
in the brains of
these rhesus monkeys,
where there would be a single
grandmotheresque neuron.
And what they found
was a neuron that
would respond to a picture
of Jennifer Aniston.
You think I'm being sarcastic.
They found a Jennifer
Aniston neuron,
which would respond
to a photograph of her
at all sorts of different
angles, a caricature, all
of that.
They went and showed the
grandmother specificity of this
by showing that-- and this
is in the paper-- it did not
respond to Julia Roberts.
It did not respond to Brad Pitt.
Brad's very
meaningfully with that.
It did not respond to
Jennifer Aniston and Brad Pitt
in the same picture.
And God knows what was going on
with Angelina Jolie with that.
OK, that shows
how bizarre it is.
The one other thing
they discovered
this neuron would respond to was
a picture of the Sydney Opera
House.
What's up with that?
So this is, to knowledge, almost
a perfect reductive grandmother
neuron.
The bizarre thing being, what
made these guys figure, I know,
let's go get a picture
of Jennifer Aniston
and show it to the rhesus
monkey and see what happens?
Where did that come from?
I recall, there
was not a whole lot
of illuminating information
in the methods section
as to where those sort
of pictures came from.
But, so there are some these.
Some of them do exist, cases
of what people in the field
call sparse coding, where
you only need a few neurons
to recognize some really fancy
things, like Jennifer Aniston.
Nonetheless, the vast, vast
majority of the attempts
to find grandmother
neurons failed dismally,
for a very simple reason.
OK.
How many neurons do you need
where each one knows one dot
and one dot only?
You need the exact
same number of neurons
as the number of
photoreceptors in your retina.
How many neurons do
you need in this layer
that turns these into lines?
Well, you need one
that's going to respond
to a line of this
length, and then one that
will respond to this length,
and one of this length,
and one of that length,
and the one that's
a slightly different angle.
You need, like, 10 times more
neurons in this layer than this
to be able to pull
off that processing.
How many neurons do
you need in this layer?
Like, tenfold, 100-fold
more than here.
And how come you don't
even have the next layer,
let alone the grandmother
neurons in these three numbers
because you run out of neurons.
There's not enough neurons
in the brain, let alone
the visual cortex, to
process stuff that way.
You can't have a
layer above that
because you've run
out of neurons.
In other words, there's not
enough neurons in the brain
to do face recognition in
a point for point reductive
manner.
The system breaks
down because of lack
of enough numbers of things.
And what people
have been doing ever
since then, what's become
the dominant sort of approach
in that field, is an explicitly
non-reductive approach, which
is now looking at something
called neural networks.
Information, the really
fancy complex information,
like what everybody
else from Friends
looked like, except
for the Aniston neuron.
Really complex
information is not
coded for in a single
protein, a single synapse,
a single neuron.
It's coded for in patterns,
in patterns of activation
across hundreds,
thousands of neurons,
networks that are interacting.
So you have a complete crashing
and burning of what up to here
seemed like the greatest
demonstration of point
for point reductive processing
of sensory information, which
just led you all the way
up to grandmother neurons.
And they basically don't
exist because you run out
of neurons at this point.
You can't solve the problem
of recognizing faces
by using reductive
component partner biology.
The next domain where
it falls apart as well.
OK, what do we have here?
Either we've got
a canal on Mars,
or we've got a frost patten on
a window, or we've got a tree,
or we've got a pattern of, like,
long branching of some sort.
What do we have?
We have a bifurcating system.
And the characteristic
of a bifurcating system
is it is scale-free.
On a certain level, if this is
what the drainage line looks--
you know, the Nile emptying out
into sort of the Mediterranean
as seen by a satellite.
And if you're looking
at the dendrites
on one single neuron with
an appropriate microscope,
if you look at it formally in
terms of the branching pattern,
you can't tell which
one you are looking at.
The complexity of the
branching is scale-free.
So it turns out some of
the most important things
we have in our bodies
are bifurcating systems.
All the branch points on
neurons are bifurcating.
And maybe what we'll post
is some amazing pictures
of bifurcating dendritic
trees of neurons.
And they're called
dendritic trees
because they look
just like trees.
And when they get more
complex for some reason,
you are said to have increased
their arborization, using terms
straight out of treeology, so
neuronal complexity in terms
of its branch points.
Look at the circulatory
system and it's the same thing
as a bifurcating system.
You've got your descending
aorta, which is ascending here.
And then it splits into
two, and splits into two,
and splits in-- and
it a tree, bifurcating
into a whole bunch of little
capillaries at the end.
You look at the
pulmonary system and it
is the exact same
bifurcation, coming down
your trachea, which
splits into two bronchos,
and then splits into
branchioles or whatever.
And it's the same exact thing.
So you've got this theme
of bifurcating systems
throughout living
systems in the body.
And notice the
difference in scale.
If this is the blood vessels,
we are talking about millions
of cells making up the
blood vessel walls.
But here, we are
seeing potentially
the exact same
complexity of branching
in a neuron that's a single
cell, independent of scale.
And you can have just as
complex branching coming off
of a single neuron as the
branching you're getting
in a gazillion
different capillaries
in the tree of
projections it comes from,
the same exact
degree of complexity.
So now we come to
the problem here,
which is so how does
the body code for that?
How does the body give
out the instructions
as to how to make a
bifurcating system?
And this is where you
immediately run into trouble.
What's a world we're
sort of oriented to?
In a purely reductive
world, there
is some sort of set of rules
telling an aorta, that's
growing like this, that there
is some gene or genes which
specify the point
where this bifurcates.
And this bifurcates
and it's two inches
in diameter or something.
Meanwhile, at a
later point, where
it's about an inch in
diameter, it bifurcates.
It's a different
gene or set of genes
that specifies this bifurcation
and another type of gene that
specifies the next one.
And those are obviously going
to be totally different sorts
of genes than specifying
the same branching
patterns of the neuron.
This is one cell.
Here, you're having to
specify thousands of cells
and at what point do they
stop adhering in a certain way
so you can make a split there.
And you look at this
and suddenly you've
got the same problem,
which is there's not
enough genes in the
genome that exist
to be able to code this way.
You can't code for
bifurcating systems
in a living organism that covers
completely different scales.
From one cell up to
zillions of cells,
you can't code for it in a point
for point reductive way, where
the points down at the
bottom, the component parts,
are individual genes.
You can't code for
recognizing your grandmother
with simple reductive
component parts of neurons.
You can't code for the
bifurcating systems in the body
because these will bifurcate
out to millions of capillaries
and there's only 20,000 genes.
The reductive approach
breaks down here as well.
Reductive point for
point approaches
break down when you
get to the cortex,
trying to do something fancy,
like recognizing faces, instead
of dots.
And it breaks down when you
look at bifurcating systems that
have to have a plan to split,
and then split, and then split
a million times to get all
the capillaries out there.
You can't code for it
with a reductive approach.
If there's not enough neurons,
there's not enough genes.
The next way in which
reductionism fails,
and the notion that if you
know the starting state,
you'll know the complex version
and the other way around,
all of that, and we've
already gotten this.
We got this back when, in the
molecular genetics lectures,
which is the role of
chance in these systems.
All of that stuff we heard
about, about sort of molecules
vibrating, Brownian motion.
And what that winds up
doing is when cells split,
it's going to be unequal
distributions of mitochondria.
It's going to be things of that.
Sheer chance is
going to throw off
your ability to deal with
a reductive point for point
system.
You take identical
twins and they're
each at the, like,
fertilized egg stage.
And what you know in
a reductive world is
when it splits in
two in this twin,
and splits in two in
this twin, this cell
is going to be identical
to this one, this one
identical with this, all the
way down to single molecules
because this is a
reductive world in terms
of how they split.
And what we know is by
the time a cell splits
for the first
time, this split is
going to distribute
the mitochondria
between these two
differently than distributed
between these two.
Even at the first
cell division, chance
is throwing off this ability
to know the starting state
and know what the complex
system is going to be.
So reductionism breaks down
there as well, the fact
that chance plays a role
in any of these systems.
The mitochondria wind
up dividing unequally,
the transcription factors.
You remember all that
stuff from there.
The same exact thing
with transposons,
with genes jumping around.
You throw in that
randomizing chance element
into there as well.
You can't take the
starting states
and wind up building on it.
An example in
behavior, a guy named
Ivan Chase did this really
interesting research
with dominance behaviors,
the emergence of dominance
in different species.
OK.
So you are going to have a
colony of, like, 10 fish.
And what you do initially
is each one of them
is in a tank of their own.
And you set up a
round-robin tournament.
You get every possible
pairing of fish.
You put them against each other.
And you see which one is
dominant of that pair.
So you've done all of that.
And you were able to derive
a dominance hierarchy, where
the number one fish is
the one that dominated all
the other nine in those
dyadic interactions.
Number two dominated
eight of them, so on.
It is simply a process,
a syllogistic expansion,
to be able to then generate
a dominance hierarchy;
pure, I know the starting
state, every single dyad
and what the outcome was.
I can now predict what
the dominance hierarchy
is going to be when you
put all the fish together.
And what he sees, of
course, is once you actually
get the fish together
in a social group,
there is no
resemblance whatsoever.
The dyadic pairing
dominance outcomes
has zero predictability over
what the actual dominance
hierarchy is going to be like.
Why should that be?
Because chance plays
a role as well.
You are fish and you've learned
this transitivity stuff,
as fish are able to do, at least
in Professor Fernald's lab.
And they're able to
do, if he defeats him
and he defeats me, I
better give that guy
a subordinating gesture.
We've now just fit together
two of those pieces,
establishing the
dyad beforehand.
But what if the guy happens
to be facing the other way
and doesn't see
him dominating him?
And you've just lost the chance.
Chance interactions wind
up driving the system.
Random movement of the animals
and such winds up meaning
knowing the starting
states of the dominance
relations of every
single dyad gives you
zero predictability of
what the complex system is
going to look like.
So what we have
over and over here
is, amid this wonderful
Westernized focus
on reductionism--
and this is going
to tell us exactly how
complex systems work,
and the starting
state, and the form,
we're seeing here over and
over in biological systems,
ranging from behavior of entire
organisms, down to genes,
reductive systems break down
because there's simply not
enough pieces in
there to explain
complex function in a point for
point reductive component part,
broken down, add them up
together afterward way.
And there's no way
to deal with the fact
that chance plays a role
in biological systems.
So what have we
just gotten to here?
500 years or so into
this reductive program,
what we're seeing is if you kind
of are interested in behavior,
or the brain, or any of
that stuff, what you've just
discovered is the most
interesting domains
of brain function, of
genetic regulation,
the most interesting
stuff can't be regulated
in a classical reductive way.
It breaks down there.
It can't be that way.
It's got to be something else.
So what this will do
now is transition us
into this whole issue
of chaotic systems.
What happens when you have a
system that is not reductive,
where there is
non-linear nonadditivity,
where you suddenly have
a very different picture?
If a clock is broken,
you take the pieces apart
and you find the one tooth or
one gear there that's missing
and you fix that.
And you now are able to put
the pieces back together
in an additive way and you
will have fixed the clock.
A clock can be fixed
using reductive point
for point knowledge.
Now, you have a problem
with something else.
You have a cloud that doesn't
rain enough during a drought.
How are you going to
figure out what's wrong?
I know.
Let's divide the
cloud in half and then
get better tools so we can
divide each half into half,
and each half into half.
And eventually, we'll get like
one molecule worth of cloud
and a gazillion of them.
And we understand how
each one of them works.
And put them together
and then we'll
understand why
there's a drought.
It doesn't work that way.
Reductive approaches can
be used to fix clocks.
Reductive approaches can't
be used to understand
why clouds don't rain.
And the whole point of all
chaos in these lectures
here is when you look at the
interesting complex biological
systems, they're clouds.
They're not clocks.
You need a whole different
explanatory system.
So let's take a
five-minute break.
And we will transition
to beginning
to look at what chaoticism
is about, about this science.
Showing that
Westernized reductionism
is really good for fairly
uncomplicated systems
breaking down in
component parts.
The whole world of this
stuff we find interesting,
it can't work because there's
not enough component parts.
There's not a blueprint
that has enough elements
in it and because of
the role of chance.
And what this
transitions us into
are non-linear systems,
nonadditive systems,
where you break something
down to its component parts,
and you study all these
pieces, and you put it
back together again, and
it's going to be different.
They've added up differently.
You understand the starting
point in the system
and you are going to have
no predictability about what
the complex form is about
because the pieces don't add up
in a straight linear manner.
OK.
What do I mean by this?
As we begin to approach
this, what is chaotism about?
Here, we have a
distinctive thing.
We have a difference between two
different ways in which things
can be deterministic.
Here's a-- no.
You're just coming up
with a number series.
And there's a rule.
There's a rule which
determines what
the next number is going to
be in the sequence, which
is just add 1.
This is a determinist system.
It is a periodic one, in
that knowing what the rule is
and knowing which
point you're at,
someone could say
what's it going
to be in 15 steps
down the line there?
And you don't need to say, well,
number one is going to be 5,
number two is going
to be 6, number-- you
don't need to do that.
You've recognized
a periodicity that
allows you to predict
pieces way down
the line simply by applying
the same determinist
rule over and over.
This would be a system that's
both determinist and periodic.
Now, in contrast,
you can get a system
which is determinist,
but aperiodic,
which is where we're
heading very quickly.
You have some system where
there's a sequence of numbers.
There's a sequence of places
on a three-dimensional matrix.
There's a sequence.
And there are rules for
how you go from each step
to the next one.
But the thing is, you can't
just apply the same rule
over and over.
You cannot sit here and say,
if we start at number five,
and given what the rule is, I
know what it's going to look
like 10 of these down.
The only way to
know is to see what
five produces, as the
first step, what that
produces as the second step.
You cannot see periodicity.
You cannot see
patterns that repeat.
The only way to know
the complex form
is to go stepwise and apply
the rule over and over again.
Because the relationship
between any one step
here is going to be
different from any other one.
Here, it's always the same.
Each one is one higher,
straight-forward, additive.
You just keep doing it
over and over again.
In an aperiodic
system you have rules.
It's determinist.
But the rules are
such that the spaces,
the difference with each
step, is not constant.
The only way you can know
what the number is going
to be like x number
of rounds down
is you got to do
this, and then this,
and then this, and then this.
This is an aperiodic system.
At any given one
of these points,
the rules exist for
you to know what
the next one is going to be.
But the rules
don't exist for you
to know what the one
two down is going to be,
unless you figure
out what this one is.
You've got to march through
in that sort of way.
It's aperiodic.
There are no patterns that can
be used over and over again.
The third version
is one that people
mistake for what I'm
talking about here, which
is a system which is
nondeterminist because there
is randomness in there.
And in that one, going from
this one to the next one,
there is no rule.
It's totally random what the
next number is going to be.
And the one after that
is totally random.
In that one, you have
no predictability.
You're going to have to go
every single step down the line.
But it's not a
determinist system.
There's no set of
rules that are being
applied over and over again.
The nature of this one does
not specify the nature of that.
It's not determinist
in that way.
That's not what we're going
to be interested in here,
where randomness comes in.
What all of these non-linear
systems are about,
these chaotic systems are ones
where they are determinist.
There's rules for how you
go every step along the way.
But the relationship, given
any given step, is non-linear.
They're not identical.
And thus, the only
way to know what's
happening two down
from here is to know
what's happening one down,
the only way to do that.
And thus by definition, this
cannot be a system where
knowing the starting state
allows you to know the mature
system.
Without having to go through
every single calculation
and knowing the mature
state, doesn't tell you
what the starting
state was unless you
are willing to do all the back
calculations because it's not
reductive in that sense.
Where would this begin to
sort of manifest itself?
OK.
Into the Chaos book, and I
think it was page 27 or so,
that you get the water wheel
coming up for the first time.
And go and look at this
picture, obsess over it,
understand what
that page is about.
Because it begins to
show how these properties
of non-linear aperiodicity wind
up producing chaotic systems.
OK.
So you've got this water wheel.
And it's got these buckets here.
And they've got
holes in the bottom.
And you can have a very
simple steady state.
You just put in a little
bit of water, such
that the water is basically,
as soon as it gets here,
it's running out.
It's coming out
in the same state.
This never fills.
The water wheel doesn't turn.
Now, you begin to fill
up at a higher rate.
And what that does is it's
a little bit asymmetrical.
This is heavy enough that it now
begins to push the wheel down.
And as it's going
down, this next one
is getting filled,
and this next one.
And all the while,
it's emptying out.
So a constant input of water,
a rate of things emptying,
the wheel turns.
It's possible to get a rate
at which you're pouring water
into the system, where
it will do precisely
that for the rest of time.
It will turn at a set speed.
It is a steady state.
It is in the equilibrium state.
It is stable such that
if you sit here right now
and somebody tells you, in
a circumstance like this,
the wheel is turning this
fast, in this direction,
with this force,
you can sit there
and you can tell them
thus exactly what it
will be doing 4,070 years
from now on Tuesday afternoon.
It is a periodic system.
You don't have to
sit there and go
through every second between
now and 4,000 years from now.
It is steady state.
And you can apply a periodic--
there's periodicity.
There's a reductive
quality to it.
Now, what you do is
you put in the water
with a little more force.
And what begins to
happen is the wheel
turns faster because
the wheel is filling up
with water faster.
So it's moving this way faster.
And that's great.
That's totally logical.
But at some point,
if you're doing that,
there's going to be too
little time for these buckets
to empty out.
They're going to start
having more water
when they're coming
up on this side
because it's moving fast enough,
but they are not emptying.
And at some point, there's going
to be enough water left here
that it will suddenly
change direction.
OK.
It's possible, if you get the
water pressure just right,
that you can get a steady
state pattern there.
It will go around
three times when you're
putting water with a speed.
It will go around three times at
this speed and with this force.
And when it has gone
around 3.73 times,
it will change
direction for 1.7 turns.
And then it will go
3.7 times around.
And once again,
it's a periodicity.
It's simply a periodicity
with two components to it,
two changes of directions.
The first time,
you're going this way.
And, oops, this fills up.
So you have your first
change of direction.
And then, at some
point, the balance
is such that you get your
second change of direction
and you start the
process all over again.
There is a pattern, a
periodic pattern to it,
that just happens to
have two components.
You've doubled the number
of components in it
by putting more
force in the system.
But once you
understand that rule,
OK, this speed at this rate,
with this force of water,
it's going to change
direction at this point
for this length of time.
Then it's going to
change direction again.
Knowing that, you can
now sit here and be
told exactly how full,
how fast, with what force.
And you could now
tell exactly what
this water wheel is going to
be doing 4,700 years from now.
It is still a reductive
periodic system.
Now, you put in the water
with even more force.
And what you begin to
see is, as the wheel
is moving fast
enough, it will have
sort of this first
spin in this direction
and then it will
change direction.
And because the buckets are now
emptying out that much slower
compared to the rate at
which the water is coming in,
it will change direction
once more, and once more
back this way.
And you're back to
your starting point.
In other words now, we have a
completely periodic reductive
system that has four
components in it,
four changes of
directions before you
get back to exactly the starting
point and do it all over again.
You have simply gotten a more
complex version of a totally
predictable periodic system.
And what you see is, as you put
in more and more water force
in there, you keep getting
doublings of your periods.
You will now get
spinning, where it
goes through eight
changes of directions
before it starts the exact thing
over again, and doing that.
And you can still
predict 4,000 years
from now, 1632, all of that.
And throughout,
you can be graphing
on a sort of way
of representing it.
This is the simple system here.
That's the single rotation.
Here's when you get
a first doubling.
Then it does
something like that.
Here's when you're--
you get the point.
You can represent it that way.
And you see it is still, you
let it keep running like this
and there will be the same
periodicity, the same pattern,
that will go over and
over for the rest of time.
It's still a reductive
periodic system.
It's just gotten
more complicated.
And then, somewhere in
the doubling process,
something happens.
And the something
that happens is
it becomes a non-linear
chaotic system.
As you increase the force
on the system, the force
here being the force of
water coming in there,
at some point with the
force of water increasing,
it's going to stop this
perfect periodic doubling
of the components and
it's going to shift over
to a chaotic pattern now.
How do you define that
as a chaotic pattern?
It will shift over to
a pattern of spinning
this way and then back for
a while, and going then.
It will generate a pattern
which never repeats.
There's no periodicity anymore.
It generates a pattern that
is going to be infinitely
different along the way.
Because you're putting that
much force in the system,
it has become chaotic.
And what is obvious here, as
an implication, is knowing here
gives you no predictive
value's ability
of what's happening
4,000 years from now.
The only way to know what's
happening 4,000 years from now
is to study what the wheel
does for 4,000 years.
You can't sit there and
recognize a periodicity
and just do it over
again and again.
And what the
discovery of chaos was
about was that in structured,
reductive, linear systems,
when you increase the
amount of force on it,
there is a doubling and
quadrupling and all of that.
It just gets more
complicated and reductive.
That there is a transition
point, where it suddenly
becomes chaotic and the
pattern never, ever repeats.
And there is no predictability.
And sort of the founding
generation of chaosists,
this is what they were showing,
with things like water wheels,
where you can see
the exact same thing.
You have a cylinder.
And what you're
applying in here--
and it's filled with water.
And you're applying
heat to the system.
And what you begin to get
after awhile is convections
or whatever.
Stuff moves.
And as you heat it even
more, changes direction.
It's the same thing.
And at some point, when
the heat gets high enough,
it breaks into boiling.
It breaks into turbulence.
It breaks into a chaotic system,
where there's no periodicity.
There is no repeating
of these patterns.
And an amazing insight by
one of the first people
in the field, this guy
Yorke, was that any time you
see periodicity
of an odd number,
you've just guaranteed that
you've entered chaotic terrain.
That, as he called
it, period three,
as soon as you're going
instead of a single period,
a double, four, eight,
whatever, as soon
as you see the first evidence
assumes any system like this
begins to have three components
before the pattern repeats,
it's about to disappear
into a chaoticism.
So this is what a chaotic
system is about, which
is you have a starting state.
And as you increase the
force on the system,
the periodicity, the
predictability, breaks down.
And eventually, you get
as the critical thing,
a pattern which never repeats.
And thus, the only way to
know what that pattern will
be doing x amount of
time down the line
is to run the system
from now until time x.
There is no
predictability from here
as to what's happening at x.
You gotta sit there
and march through it
because it's an aperiodic
system, rather than one
like this.
So what the entire sort of
starting point for chaosists
was, that you get these
nonlinear systems.
And people had been noting them,
mathematicians, physicists,
whatever, in systems like
that for a long time.
And the longer they do,
right around the point
that things would
become chaotic,
they would say, well, this is
just getting perturbed by now.
It's not functioning
properly anymore
because it's not working in
a linear, periodic manner.
Something's wrong with
the system, something
along the lines of
noise and variability.
We will stop studying
it up to that point.
And if you say, we're
going to stop studying it
until it gets to that point,
the last bit of periodicity,
what do you come away with, a
very distorted view that all
the interesting
things in the world
work in reductive
periodic systems.
Because what you've
just done is say,
I'm getting totally
disturbed by these nonlinear
chaotic things that
happen at an extreme.
So I'm going to decide
they are just anomalies.
And we're only going
to study in this domain
and reach the conclusion
that the entire world works
in this domain.
Kind of like behavior
geneticists, who say, oh,
I want to understand the
heritability of a trait.
And I want to understand
it very cleanly.
So we're going to study it
in only one environment.
Because if you study in
a bunch of environments,
it gets noisy, and
variable, and messy data.
No, it doesn't.
It is showing you that
the heritability is zilch.
It's showing you that you have
just artificially excluded
your ability to see
what's actually happening.
And the founding
generation of the chaosists
took the stance
that what you've got
is all the interesting
stuff about complex systems
out there, are all functioning
out in the chaotic realm.
And what science has been
spending forever doing
is looking the other way and
pretending it's not there.
And restricting the
studying of complex systems
to just these first
baby step domains
of the periodic doubling.
Most of the world is doing this.
And most of science has worked
very, very hard to ignore this.
So once you get this, you begin
to get some really interesting
implications.
So now you find a way.
You get one of these
chaotic systems
and you first study
it when it is still
in the nice straightforward
periodic way.
A little bit of
water is coming in.
And it's turning like
this, at a set speed.
And come back 4,000
years from now
and it will still be
doing it the exact thing.
It's a great periodic system.
And you can come up with a
graph of which direction it's
turning, would be here or here.
And how fast, with what
work force, whatever.
And you will come up
with a single dot, which
represents this rotating in this
direction, at this set speed.
And this is this point
of stability, this point
of complete predictability.
A feature of a periodic
system like this,
when it's in this boring
linear reductive way,
is you can mess with it.
And after a little
bit of time, it
will settle back
into the same system.
You briefly hold the water wheel
and that throws things off.
And then you let it go and it
goes back to what it was doing.
And it will take a
little bit of time
for it to go back
to where it started.
And a way of viewing
this graphically
is it's doing this forever.
And now, you go mess with it.
You hold it.
You turn off the water
for a second, whatever.
And for a while, it does
this, and it does this,
and does this, and does this.
And eventually, it gets
pulled back to this spot.
It goes back to this point of
stability, of predictability.
It is attracted to this point.
And, thus, the linear systems
like this have attractors.
Something where, when
you mess with it,
the system will equilibrate
and go back to where it was,
attracted to the "real"
solution to the problem.
And if you are looking
at it at any point here,
and it's not here, because
it is here instead,
and it is here instead,
and here instead,
all that is noise in the system.
And you're in the process
of getting rid of the noise,
back to the pure, perfect state,
the pure, perfect description
of how the system works.
So now, you look at what's
going on when instead you've
got it to the point of
chaos, a chaotic system.
And what you see is--
OK, let's assume that was
where the attractor was.
And what you see is
when you're mapping
the speed, the direction
that it's turning,
the force, all of
that, it's doing this.
And it will do this for a
while, and will do this.
And it will reach that critical
point where suddenly it
changes direction.
And will do this for
a while, and then it
will change direction again,
and then this, and this.
And what you have is this
butterfly wing pattern,
that became one of the
iconic images in early chaos.
What do you have here?
You've got a description of
how the system is working now.
Once it's hit this
chaotism, it's
not settling down into
a repeating pattern.
The fact that it is never
here, and staying there,
that's the business that
you could never predict.
It is constantly oscillating.
It is constantly chaotic.
So now you ask,
and you say, well,
that's pretty strange
because it's not actually
touching the spot.
But it just keeps
going around it.
It's clearly pulled to it, but
in a very different sort of way
than when you get a perturbation
and quickly it does this.
This is being attracted back
to this pure starting point.
And here, it never
actually quite gets there.
But it's sort of
being pulled by it.
What do we have here?
We have a strange attractor.
And that was the terminology
that came into the field.
A regular old attractor
is one that will pull down
to a single stable point.
This is the predictable,
utterly predictable, state
of the system right now.
A stranger attractor is one
that has to do with the fact
that it's going to
oscillate like this.
But it's never going to settle
down into a single point.
And suddenly, there's a very
different implication there.
Because here, when
you're not yet
at this spot, what's this spot?
It's noise.
Its variability.
And hang on and it's going to go
away because it will eventually
reach the real answer.
In systems with
strange attractors,
what do you make
of the variability?
It's not noise.
It is the phenomenon.
There is no absolute
pure answer in there.
There is not some idealized,
the real correct answer.
And you're just
fluxing around here.
And if only you had better
control of the system,
you would eventually get
it to look like this.
This is a myth.
This is imaginary.
In complex systems,
there is no answer
as to what you are
supposed to be observing
and everything else
is variable noise.
This is the system itself.
A critical expansion on that,
so you're looking at this
and you're saying,
OK, what is this?
This is measuring
in whatever units
of time, where the wheel is,
what direction, what speed, all
of that.
It's a whole bunch
of data points.
And the data points would
just keep doing this forever
and ever, unpredictably.
Wait a second.
And you say, at some point
it's got to cross here,
this spot here.
And at some point
it's circling around
and will circle and come
through exactly that point.
And right now at that point, if
you apply the same equations,
the same determinist
rules, right at that point,
the next point should be this.
And the next point
should be this.
And what have you just done?
You're beginning
to repeat yourself.
You've just had periodicity.
Wait, this really isn't chaotic.
As soon as it hits the
same point that was there
previously, it suddenly
should repeat the pattern
all over again.
It's periodic.
It stopped being
a chaotic system.
How can this be?
Because they've got to
touch the same points.
Look at all that.
And this was the
next critical concept
in the field, which is you can
look at this point and maybe
its coordinates, 6, 3, in a
standard graph or whatever.
And that's what
the coordinate was
the first time it was there.
And now, spinning around, it's
just come there a second time.
It's back in the same spot.
Oh, no, it's repeating.
It isn't chaotic
and unpredictable
and going on for an infinity.
And it just fell apart.
It doesn't really work
this way, until you
look a little bit closer.
And you look a
little bit closer.
And it turns out
this is not 6 and 3.
This coordinate
was actually 3.7.
And you look closer here.
And this one was 3.8.
In other words, it's not
really in the exact same spot.
It never gets to the
same exact spot again.
Where is it then?
Well, we really measured it now.
And, in fact, both
of them are 3.7.
It's the exact same.
And look a little bit closer,
an order of magnitude closer,
and there be it.
And take it out a
million decimal places
and they still look like
they're in the exact same point,
a million small places
out in terms of accuracy.
And a million and one is where
they will differ a little bit.
And thus, they're
not in the same spot.
Critical next concept with this,
which is if that's the case
and there are a million
decimal places out there,
they are differing by one
decimal place way out there,
what that means is the
fact that one of them,
4,000 digits out there,
is 8.2-- and notice,
an 8.3 and a gazillion
zeroes before that-- that
means at some point this is
going to function differently
than this.
This will produce a
different spot than that.
They won't go to
the same next place
because they're actually
different numbers.
And if a million digits
out, that tiny difference,
will change the functioning of
that, a million minus 1 digits
out there, that will
then potentially
change the functioning of
a million minus 2 digits
out there, all the way up.
In other words, the consequences
of this tiny little difference
gets amplified.
And this is what's called
the butterfly effect.
In the standard sort
of jargon in the field,
the butterfly effect is the fact
that the way a butterfly flaps
its wings in Korea will change
the weather system in Indiana.
And this is absolutely the case.
Because of these
butterfly effects,
the very local consequences
of something like a butterfly
flapped its wings versus if
it hadn't flapped its wings,
would have you just done to
air movement on the planet?
You are a million
digits out there.
And you've just changed
that this very last digit
went from 3 to 4 because the
butterfly flapped its wings.
And that's going to cause
a difference one digit
before that, and one digit.
And these are already
beginning to differ.
And by the time it gets
up to any level higher,
it's differing enough that the
next spot will differ as well.
What are you doing?
This is why the pattern
can never repeat
and why it's good
to do this instead.
Because on some level
of magnification
in a chaotic system
like this, you never
have the exact same location,
the exact same coordinates,
occur a second time.
Somewhere, however number
of digits out there you
need to go, the two
of them will differ.
And that difference can
potentially amplify upward
in a butterfly effect.
If you do it that
way, you suddenly
have a very different view.
Which is, all the way out, a
gazillion decimal places out
there, and they
differ like this,
this isn't noise in the system.
This isn't variability.
This is intrinsic to the system.
And the fact that this
will now expand, amplify
the consequences, is
why the whole system
is unpredictable because
of these butterfly effects.
OK.
Let me just think about-- oh,
so the critical point here
is in this strange attractor
realm, not only is there
no predictability.
That's an important point.
And the closer you
look, you will still
see the same degree
of noise variability.
Noise is not something
that goes away.
But almost philosophically,
this critical point
in a system, a linear boring
system with an attractor,
this is the answer.
In chaotic systems,
there's no real answer.
And if you're out
here, it doesn't
mean that you're not correct,
you're not quite there.
There is no there there.
The notion that there is
a solution at the center
is a fabrication of the data
oscillating around there.
There is no correct
idealized answer in there.
This is the idealized
answer, which
is a completely
unpredictable system.
OK.
What's intrinsic in
this now is the fact
that even though you now look
an order of magnitude closer,
there's still this
variability stuff, which
can amplify, as a
butterfly effect up,
and make a huge difference.
And you look an order
of magnitude closer
and there's still a
butterfly effect potentially.
And it doesn't matter how
many digits you go down,
how closely you look, how
good your reductive tools are,
the variability is still
going to exist down there.
And the way to
describe this here
is, thus, this is a
scale-free system.
The nature of variability
is exactly the same
if you're looking
at a whole number
or if you're looking
at a number taken out
to three decimal places or
three googleplex places,
and that sort of thing.
That it's independent of
how many steps down there
you're examining
the system, the fact
that there will
still be a difference
and it can still
butterfly its way up
to make a difference there,
means that all of this stuff
is scale-free.
It doesn't matter how closely
you are looking at it.
In other words, the whole
reductive philosophy
of the closer you look,
the better your measurement
tools, the more variability
is going to go away,
in a scale-free chaotic
system, regardless
of what degree of reduction,
of what degree of detail
you are looking at the system
with, the amount of noise
is going to proportionally
remain the same
because it's not noise.
It is the system.
So you don't have this reductive
notion that all we need to do
is get better tools and
look closer and closer
and noise will go away.
Because it isn't noise.
This is the phenomenon
at any scale you look at.
And this introduces the notion,
thus, of what a fractal is.
Because a fractal is a complex
pattern, a visual pattern,
an equation that produces a
pattern, things of that sort,
where it is scale-free.
The appearance of it
is the same no matter
what scale you look at.
The complexity of it
is the same no matter
how close you look at it.
The degree of
variability is the same
because it's not variability.
It's intrinsic to the system.
OK, ways to define a
fractal, because this has
becomes sort of a very
trendy sort of subject.
And there's a number
of different ways
to think about it.
Most formally, what a
fractal is is information
that codes for a pattern.
Where, for example, it can
code for a pattern that
is a line-- a line-- and thus
is a one-dimensional object.
But where the line is moving
around with such complexity,
with such an infinite amount of
complexity, because even if you
look closer, it's going to be
just as complex, proportionally
closer and closer.
In other words, this is going
to be an infinitely long line
in a finite space.
Which begins to make it sound
kind of more eventually,
this is more
beginning to resemble
a two-dimensional object.
What a fractal is is some
object, some property, that's
a fraction of a dimension.
If this goes on
infinitely, it's a line.
But it's really much
more than a line.
But it's not quite
two-dimensional.
It's 1.3 dimensions.
It's a fractal.
A fractal is a system that has
fractional dimensions to it,
where just the infinite
amount of complexity
the closer you look, it's
still going to be like this,
means that this is a
line like no other line.
It's one that's infinitely
long, packed in a finite space.
That's more than a line.
But it's not quite
a two dimensional.
It's a fraction.
It's a fraction of a dimension.
That's the formal mathematical
definition of a fractal.
For our purposes, it's instead,
as well, no matter how close
or how far away you look at
it, the amount of variability
is the same.
And thus, we have absolutely
a classic fractal system
if these are canals on Mars
or if these are the dendrites
of a single neuron.
No matter what dimension,
with what resolution
you are looking at
it with, the degree
of complexity, the
degree of variability,
remains proportionally constant.
It's a fractal.
Bifurcating systems
are classic fractals.
And the variability
within the system
is constant regardless of
what scale you are looking at.
And, thus, is telling
you the variability
isn't noise variability.
It's instead, what the
system is actually about.
So I will look here.
OK.
So you see these
fractal properties.
The circulatory
system is a fractal
with roughly the same
degree of complexity
as is the pulmonary system
in its branch points,
as in the dendrites
of a single neuron,
as are the branches of a tree.
They're all fractals.
Their complexity,
their variability,
is independent of scale.
This is what you
begin to see in all
these physiological
systems, these fractals that
have equivalent
scales of complexity
at infinitely different,
vastly different,
scales of magnification.
And as a hint of where we're
heading on Friday, what that
begins to tell you
is you can solve
this problem of coding for these
vastly bifurcating systems.
Where this is made
of a billion cells,
and this is a single
cell, you can use
some are these similar rules.
And all the rule has
to be is scale-free.
We will see exactly what
I mean by that on Friday.
But this begins to
solve the problem of,
there's not enough
genes in there.
And what this
introduces is the notion
of there being fractal genes,
genes that give instructions
that are independent of scale.
And we'll see lots more
about that on Friday.
So you've got these fractal
systems all over the place,
where the point over,
and over, and over again
is the variability
isn't variability noise.
It is what the
system is all about.
There is no absolute state,
where the closer you get,
the more it suddenly is going
to seem clean and nonvariable.
An example of this,
an example of this
in the biological literature.
And this was actually a study
I did about 15 years ago,
with probably the most obsessive
undergraduate I've met in all
my years here at Stanford.
Which was great, because
the study was only doable
because he was
out-of-his-mind obsessive.
Here's what the study was about.
I was thinking about, well, all
of this chaos, fractal stuff,
amid all of us functioning
with a standard model
that, God, if only we
could measure this down
to the single cell
level, then we'd really
know what's happening, because
that's going to be so much
better than blood values,
because that's noisy working.
Working with that
model of the mirror
reductive you get, the cleaner
the data are going to be.
And then, here's this
whole other world
of these non-linear
fractal systems saying,
it shouldn't work that way.
So it struck me to do the study.
And what I wanted
to do was study
the data generated in
the scientific literature
at different scales
of reductionism
and see what happens
to the variability.
And the point here was to make
it as, like, well controlled
as possible.
I felt for some subject, that
was sort of Bio 150 related.
And came up with the
notion that won't it
be interesting to look at what
are the effects of testosterone
on behavior?
Just taking any such sort of
question out of this course,
what are the effects?
And you can answer that
on the level of societies.
OK.
People who are
agriculturalists tend
to have different
testosterone levels
than hunters, all of that.
How does that affect
things like behavior?
You can ask that on the
level of a single individual.
What does a person's
testosterone levels
tell you about behavior?
You can ask on an
organ system level.
What's happening
to blood pressure
throughout the body
and cerebral oxygen
delivery as a
functioning system,
down to a single organ,
what's happening to the brain,
down to a single
cell or molecule?
All the way down, you
see the logic of it.
So what we did-- and
I use "we" in the,
like, most parasitic way
possible-- what we did
was to go to the literature,
the scientific literature.
And for a reason, we
picked the literature, not
a contemporary one at the time,
but one that was 10 years old.
And we looked at every
journal out there
that we could come
up with, that ever
had papers in the
realm of-- that
could be interpreted as
the effects of testosterone
on behavior.
Up from anthropology
journals, comparisons
between different
groups; down to people
doing X-ray crystallography
on testosterone receptors.
And what this mad
man I had hanging out
with me at that point did was
go through every single one
of those papers.
And first classify it, is
this an organismal one?
Is this a multiorganismal one?
Is this a cellular, is this
a subcellular, all of that.
And then measure how variable
were the data in that study.
This was a drag.
Because what he
had to do was-- OK.
So there would be some figure
in one of these papers,
looking like this.
And what this tells you is, for
this group, here's the average.
And this is a measure of how
much variability there was.
And this tells you
there was a lot more
variability in this measure
than this one, all of that.
You could come up
with something--
and do not worry about
the details here--
something called a
coefficient of variation.
Which is, you ask how
much variability is there
relative to the
total size of this?
And thus, what you will get
is, say, a circumstance,
if this is 100 units
high; and in this case,
this is 10 units high;
and in this case,
this is 50 units high.
In this case, your coefficient
of variation would be 50%.
Your variance is half the size
of the thing you're studying.
And in this case, the
coefficient of variation
would only be 10%.
It's much less noisy data.
So what he did with
his little ruler
there, in the next 3 and 1/2
years, with nothing else to do,
he went through these
hundreds of papers.
And for every single figure,
he measured what was the mean
and what was the error bar
in every single figure.
And, thus, what
was the coefficient
of variation for that
piece of inflow of data,
for that figure, for
that entire paper,
that would have eleventy
different bars of data
in there?
And he's measuring away
and going mad from this.
And eventually, what
he could then do
was stick in the average
coefficient of variation
in all of the papers in
the organismal category,
and all the papers in
the cellular category,
and all the way down.
Reductive science,
what's the prediction?
As we go from the big
organismal papers,
all the way down to the
subcellular, submolecular ones,
the noise, the variability,
the coefficient of variation
should be decreasing as
you get more reductive.
That would be the
traditional interpretation.
The chaotic fractal
interpretation would be,
it's not noise.
It's not noise that you want to
get rid of with better tools.
It is intrinsic--
I'm going to leave it
there-- oscillatory stuff,
which is the system, rather
than the discrepancy
from the system.
It would predict that the
relative amount of noise,
the variability, the
coefficient of variation,
shouldn't be trending
towards decreasing.
There shouldn't really
be a relationship
between what level you
are examining phenomenon
and the amount of variability.
So an insane amount
of work later,
this was his, like, what he
spent the early 1990s on,
producing this one figure here.
[LAUGHTER]
This was it.
This was it.
He wept with pride and happiness
when we finally saw this,
going from, this
is the coefficient
of variation on all the data
in all the papers that year
that were at the
organismal level,
about an 18% coefficient
of variation.
Organ system, single organ,
multicellular, single cell,
subcellular, is there a trend
towards variability decreasing?
Absolutely not.
It's not going anywhere.
It's remaining fairly constant.
Looking over scales of
magnifications ranging
from societies, down
to crystallography
on single molecules, the
data don't get cleaner.
They don't get less variable.
Because it's a fractal system.
One additional
possibility there was
that part of what was
going in was you're
looking at the entire
literature in that year.
And as we all know,
some of those papers
are going to be kind of garbage
and not very good science.
Maybe that's the
noise in the system.
And there's enough noise that
it swamps every single one
of these levels.
Here was the advantage that
we had been looking at papers
published 10 years before.
You could now see how many
times that paper was cited
in the subsequent 10 years.
In other words, you
could find the papers
that were considered
by people in that field
to be the really good ones,
versus the ones that were junk.
Now if we do the whole analysis
only on the papers that
were in the top 10
percentile of influence,
the best papers in the
field, and it winds up
looking exactly like this.
It's a fractal system.
As you get closer and
closer to measuring
what's really
happening, wow, down
to the level of
single molecules,
you don't get any cleaner data
because it's a fractal system.
It's a chaotic fractal system.
So that was real interesting.
What was even more
interesting was
after that, when we tried
to publish the paper.
So got it together and
we wrote up this paper.
And we sent it off to, like,
one of my favorite neuroscience
journals.
And after a couple
of weeks, the editor
wrote back, saying this
is totally cool stuff.
This is really interesting.
This has really made
me rethink some stuff.
This is very stimulating stuff.
I don't see what it has to
do with our journal though.
So we really can't publish it.
So then I sent it off to my
favorite endocrinology journal.
And two weeks later,
back comes a letter
from the editor, saying,
whoa, totally cool stuff.
Come over for dinner and bring
that piece of paper with you.
But I don't quite see what
it has to do with our field.
And marching our
way up and down,
all the different
relevant journals
in those different
disciplines, and each time
they come back saying, whoa,
isn't that interesting.
I can't wait to
tell my friend, who
is the social anthropologist
or the x-ray crystallographer.
But I don't see what it has to
do with our particular field
here.
And we couldn't get it
published in a journal that
specialized in any single
one of these levels.
So ultimately, we
published in this journal,
this sort of philosophical
proceedings of medicine
and biology.
As far as I can tell, I was
the first person under age 80
to ever publish a
paper in that journal.
[LAUGHTER]
It's the journal of
somewhat demented, senile,
elderly emeritus
professors, who were now
writing their philosophical
pieces because they're not
generating data any more.
I broke the age barrier
on that journal.
And they published in there.
And in the years
since, it has had,
like, zero impact on the
literature in terms of anybody
quoting it, in terms
of anybody citing it.
Actually that's not true.
There's this
mathematician in Moscow.
This guy started writing
to me about two months
after the paper came out.
And he basically said, this
was the most wonderful thing
he had ever read.
And I had transformed his
life, and he loved me.
And he's been writing about
once every three weeks since.
His English is not very good.
Either he wants to adopt me
or he wants me to adopt him.
I'm not quite sure in that.
[LAUGHTER]
But as far as saying,
that's like the only person
who noticed that
it there, showing
this point that this
is, but what does it
have to do with our discipline?
So what we see here is a prelude
to Friday, what fractals do
is blow apart the notion
that get even more
reductive than they
were back in the 1500s
and we'll get better data.
It doesn't get
better because it's
the same degree of variability.
Because the variability is
the-- whoa, where did that go?
The variability is the
system, rather than
discrepancy from the system.
Fractals show that as well.
What we'll begin
to see on Friday
is how you can now
use fractal systems
to solve some of those problems
of not enough neurons, not
enough genes.
OK.
So on first past, what did
today seem to be about?
This endless
trashing of reductive
science and, you know, that's
how you fix a broken clock.
But the world of really
interesting things
are like broken clouds and are
non-linear aperiodic systems,
that are just as interesting
and just as complex,
no matter what scale
you look at them at.
Hooray, most of science
makes no sense whatsoever.
Hooray for us, we
are the vanguard.
OK.
That's a drag.
Because that really doesn't
accomplish a whole lot
because what there has
to be is a substitution.
So where is the
actual predictability?
Where is the actual
insight coming from?
And that's what
Friday is about, which
is this whole field of
complexity, emergence,
what we'll look at there.
But the last point here winds
up being, OK, OK, terrific.
You've convinced me.
You have trashed linear
additive periodic systems.
And 500 years' worth
of books of science
need to be burned at the stake.
Is classical reductionism
good for anything?
Yes.
We already know it is.
It's good for when
clocks are broken.
No, no, no.
I mean is it good for
anything in the realm of stuff
that we're interested in,
in how, like, biology works,
in how behavior
works, any of that?
And what you get is, it's very
useful and very effective,
if you're not very picky,
if you are not very precise.
An example.
There's a miserable
disease out there,
that's wiping out
people left and right.
And it's this viral disease.
And you're trying to
figure out how to come up
with a vaccine for it.
And you finally come
up with a vaccine.
And you start
distributing it to people.
And what you see is exactly
what Jonas Salk saw,
was that it did wonders
for preventing polio.
But one in 560
kids, instead would
get a worst case of polio.
So you could now ask a
very reductive question,
which is, is this vaccine,
what happened in that one kid?
And what we'll see
at that level is
if you're trying to ask
that, that's not going to be.
If we understand what
happened in that one kid,
we're going have to understand
their individual cells, all
the way down, to get
better and better numbers.
Because this OK, we've
just entered the world
where it's actually a
nonlinear chaotic system.
Where's the reductionism useful?
On the average, it's
a whole lot better
that kids got this vaccine than
they didn't get the vaccine.
You want reductive classical
dead-right male reproductive
predictable science.
It's if you have a
community of kids,
who get injected with this,
versus a community that
doesn't, they're
going to be healthier.
Don't ask me about one
particular individual, let
alone one particular
individual's immune system.
The reductionism
breaks down there.
But if fancy, satisfying
science for you counts as,
do they tend to be healthier
than them, reduction is great.
That is sufficient.
Suppose your question
is, well, is there
a certain time of
the year, what's
the weather likely going to
be in January versus in June?
You're never going to
have reductive tools that
can tell you on any particular
day what the temperature is
going to be three
years from now.
But if all you want to know is,
in general, it's warmer in June
than in January, reductive tools
that exist now are sufficient.
And when you look at what
people do in their research,
in the labs and places
like this, in my lab,
all of that, when you sit
there and you say, whoa,
we just learned and
are trying to, like,
figure out the cure
of some disease
by finding the
mutation, whoa, that's
really a reductive approach.
And we just learned
that's gibberish.
Whoa, everybody toss
out what they do.
It's actually quite
useful because you're not
very picky about what
you want out of it.
My lab, for example,
studies what
happens in brains after
there's a stroke and trying
to figure out gene therapy
stuff that you can do.
And we can't tell you
why one rat out of eight
is not going to be
helped by this procedure.
Whereas, three rats are
wonderful, and so on.
Oh, I know.
Let's look at their
are single molecules.
That's not going to do it.
Reductionism breaks apart.
But it's perfectly fine at
the level that's useful here,
which is, on the average,
is this something
plausibly that you might want
to do to humans somewhere
down the line?
Most of what the science
research is about--
and when you look at
the labs around here
and you look at
the labs you work
in, if those of you, who are
lots of you, do research,
it's reductionism is
a perfectly good thing
to use because you're not being
too picky what-- you do not
want to come up with
an explanation for how
every neuron in every
developing grasshopper on Earth
bifurcates.
At this point, you want to
know, in general, somewhere
around hours three
to four after the egg
starts or the grasshopper
parents meet and fall in love,
or whatever it is,
that, in general, that's
when you begin to get
differentiation here.
Your science is great.
So as long as most
of what we're asking
is this sort of science,
where you just kind
of a need to have a
general predictive sense,
reductionism is just fine.
But nonetheless, underneath
all of it, when you really
want to understand the systems,
it's anything but reductionism.
And probably the most
important philosophical point
is when you look at these
interesting complex systems,
there is not a "the answer."
There is not a "the solution" to
what is this water wheel going
to be doing?
And thus, all of us, who
are out there, not quite
matching the perfect
one, it is not
that we are being deviating
from what the real answer, what
the real norm is supposed to be.
The variation is what
it's supposed to be.
All of the things
that are interesting,
when you measure them,
and they look like this,
it's not because they
are failing to be what
they're supposed to be and
match the norm, they're--
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