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.