In this lecture, I want to take a break from
all the chaos and talk about different ways

of thinking about mathematical models.

And my main point in this lecture is simply
that there are different ways of thinking and

using mathematical models across the sciences
and social sciences.

I'm not going to argue that there's one way
that's right or that some ways of using models

are better than others, but rather that there
are lots of different ways that models get used

that I think is a good thing, but it's worth
being aware of.

So let me start with something that maybe
isn't even a model per se, and that's Newton's

Law of Gravitation. so F=GM_1M_2/r squared.

This describes how two objects separated by
a distance r, their masses are M_1 and M_2,

it describes how strong the forces of attraction
is between them.

So this is described as the law of nature or
the law of physics, is often thought of as a

true law, a true and accurate description.

It's pretty fundamental, it's almost mechanistic,
it describes sort of exactly what is going on.

Perhaps one could explain things in terms of
quantum gravity or general relativity, but this

is sort of a fundamental explanation, this is
really what's happening.

So another equation that one could use, that
maybe also functions a little bit as a model

might be the Law for Metabolic Scaling.
So Y=Y_0*M^(3/4).

This would describe how the metabolic rate
Y depends on the body mass M for essentially

all living organisms.

So this is also an equation, is less mechanistic
in the sense that it merely describes, quite

accurately, but it's a description of a relationship
between two variables.

It doesn't necessary tell you how or why there's
this relationship between body mass and metabolic

rate, but it does say that there is this
relationship.

But even not thinking so much about models
but about equations, equations are used in

different ways or maybe thought of in different
ways as well.

Some might be mechanistic or fundamental
and fundamental doesn't necessary mean better

but maybe sort of at the bottom of a layer
of explanations,

and others might describe a relationship 
between variables, but not give a sense of

why that relationship is the way it is.

So when one takes mathematical modelling
into fields outside of physics, sometimes there's

a thought, maybe it's just physicists who might 
think this, I come from a physics background,

that models and equations might function
similarly in other fields or contexts, that

we're making some statement of truth, a
statement that one can make quantitative

predictions with.

But models very often aren't used that way,
they're used for other purposes, so models

and this is how the logistic equation is
used, that we've been studying, so models

like the logistic equation I think of
as more like a caricature.

We're not trying to capture the truth, or
the complete truth or the entire story about

something, but rather just a quick sketch
or a minimal impression of an object or

phenomena that captures something of importance.

So here on the screen is a caricature. I'm
not sure of the original source of this, several

places on the web, but you probably recognize
this guy. It's a sketch or a caricature of Barack Obama.

So the purpose of this caricature is not to
have as accurate or complete a portrayal of

President Obama as possible.

One probably wouldn't critique this and say
it is not accurate enough. The sign of a good

caricature is if you've somehow highlighted
or augmented some feature of a person or

an object or whatever it is you're studying that it 
sort of becomes immediately becomes recognizable.

And if you can do that, then you've probably
captured some true or useful or interesting

feature of the thing you're making a 
caricature of.

So I think models can often function in the
same way as a caricature.

For the logistic equation, we're not trying to
model everything possible about an ecosystem,

even a simple and sort of made-up ecosystem
where we have rabbits living on an island,

but instead we're just trying to do sort
of a caricature.

Some sort of a very simple model with a few
strokes, that can somehow capture some feature

about the ups and downs of populations.

Here's some other ways to think about different
ways that models might function.

So, for one example, consider the idea of
a field guide.

So field guides, these are books maybe of
birds, or trees, or who knows what and the

goal of it is that if you're outside and you
see a bird, and say "Gee I wonder what type

of bird that was," you can look in the field
guide and then you can look at pictures of

birds and figure out what bird it is you're
seeing over there and say

"It's cool, I know what that is now."

So you would think that a good field guide
should have photographs in it, because

photographs are true or complete
representations of the image of the bird.

Of course, they are not entirely complete
representations and photographs can be

altered, all that sort of thing, but you
might nevertheless think, oh, a photograph

that's the way to go.

But actually photographs are really poor
things to use when trying to identify birds

or plants or whatever.

So as a quick example, here's a field guide
for marine mammals that wash up on shore

that I borrowed from a colleague down the
hall and it's filled not with photographs,

but with pictures.

Pictures of whales, that's a fin whale.

So that if you find a big whale washed up on shore,
and you wondered, "Gee what type of whale is this?"

You could look and these drawings are more
useful than photographs.

So that's a little bit weird because a drawing
you would think, well a person's doing it and

leaving out lots of detail so it's less accurate.

And maybe that's true, that's sort of the point,
that a good drawing, like a good caricature

highlights certain things, sort of recognizable
or important features so you can tell fin whales

apart from minke whales or blue whales, or
whatever it is you're trying to do.

So you wouldn't critique a drawing like this
and say, "Oh you left out all these details."

But the point of a drawing like that is to
leave out all those details so you can highlight

something else.

So some mathematical models function in
that way.

Maybe the logistic equation is another such
example.

The point isn't to include all of the details,
the point is to leave out some of the details,

that don't matter, and along the way, figuring
out what those details are that don't matter,

to get a recognizable sketch of a phenomena,
physical, biological, social that you're trying

to learn something about, what can you leave
out and still have it be recognizable.

Surely that's a useful and important way of
learning about that process.

On the other hand, there are settings in
which photographs are useful.

So suppose a student comes to me and says,
"Dave, I saw a purple unicorn in my dorm room."

And that would be awesome, who doesn't love
purple unicorns, but I'm not sure I would believe

the student. And so I might say, "That's nice,
what evidence do you have that there really

was a purple unicorn in your room?" And if
the student said, "Oh here's a picture I drew

of it," even if it was an awesome picture,
that wouldn't necessarily help me believe

that there was a unicorn in the room.

But if they showed me a photograph, even
knowing that people can do great things

with photoshop, that would be evidence I
would take more seriously.

So in some contexts the photo is more useful,
in other contexts, a painting or a good drawing

is more useful.

So the point is the type of representation
one might seek of an object or phenomena

depends on the goals and on the contexts.

Here's another example to help us think about
models.

These are two different models of the human
body.

So the first model, suppose a medical student
needs to learn about surgery and what organs

look like, so what might she practice on
instead of cutting up a real human?

Well one thing that's often done is medical
students work on fetal pigs.

And so a fetal pig is, I guess, a good model
of a human body.

If you cut it open, the organs kind of look
like our organs and they're kind of in the

same places that our organs are.

So a fetal pig is a model of the human body,
and a very good one for the medical student.

On the other hand, consider a fashion designer.
And he's designing clothes, and he needs to

know what the clothes look like on a person
so he might have a model of a person and

he would use typically a mannequin. So it's
just a big plastic thing that's shaped like a human.

And that works very well for his purposes.
A mannequin is a good model of the human

body for the fashion designer. So which is
a better model of a human?

Well hopefully that question doesn't make
sense. It depends on the context, what you're

trying to model.

It certainly would be a bad idea if the fashion
designer designs clothes using a fetal pig

as a model. And it would probably be an even
worse idea if medical students practice surgery

and physiology on mannequins that are usually
empty inside.

So anyway, the point is, again, that models
serve different purposes and it doesn't always

make sense to say which model is better or
worse, without specifying the contexts and goals

that you have and why you're using or making
a model in the first place.

Finally, I should mention that this example
is not original with me, but I don't know who

thought it up, I read it a long time ago, and
I can't remember where so I don't know how

to attribute it, so I'm sorry but anyway if
you've thought it up, let me know and I'll

attribute you, or if you've read this before,
let me know cause I can't for the life of me

remember where this idea is from.

Anyway, different types of models are used
in different ways in different circumstances.

That's the point.

So to summarize.

Sometimes, particularly in physics, we use
models as a direct accounting of some aspect

of reality. A direct and complete accounting.

And these models are often, and these equations
are often not merely descriptive, but explanatory

and perhaps mechanistic and usually predictive.

So Newton's Law of Gravity, perhaps it explains
gravity at some level and it certainly is predictive.

Newton's Law of Gravity predicts all sorts
of things about celestial orbits, along with

Newton's Second Law of Motion and can be
used to send satellites out to very precise

locations in the solar system.

But outside of some basic physics, models
and equations are used in a lot of different

ways, and often they're more used in this
caricature or sketch sort of way.

And so for these sorts of models, I don't
think asking if a model is true or false is

a useful or even meaningful question.

Is the fetal pig a true model of a human being?
Well at some level all models are false.

At another level, all models are true because
presumably all models capture some aspect

of the object under study.

So I think a more useful set of questions
to ask a model is is it helpful or not helpful?

The fetal pig is probably really helpful to
the medical student, and surely a mannequin

is helpful to someone who's designing clothes.

Does a model tend to enlighten, or does it
obscure the thing under study?

Not that it's always easy to tell, but those
are the questions that one would wanna ask.

Models are often useful for suggesting certain
minimal mechanisms or key essential features.

Figuring out what you can subtract from a
representation but still keep the thing there

that you're trying to study.

So, in any event, models are used in a lot
of different ways.

There are some really good articles and essays
on this topic, and it's a huge topic and philosophy

of science, so there's lots more to read in
the segment following this video.

I have a page with links to articles I particularly
recommend. Maybe you know some good articles,

let me know and I can post links to those as
well.

So a lot of the models that we'll work with in this
course are more, sort of sketch type model.

The logistic equation, and the Lorenz equations
which we'll talk about more in subsequent units.

I wouldn't take them too seriously as direct
representation, say, of rabbit population

dynamics on an island, but they're more like
sketches or caricatures that we can use to

ask questions about models in general and look for,
say, general features of a physical phenomena

and they are very useful simple examples to
dig deeper into dynamical systems and learn

about chaos and sensitive dependence on initial
conditions and topics like that.