[From Bruce Nevin (2003.12.20 15:53 EST)]

The primary issue here is equivocation between theory and simulation, but

we have to talk about what “model” means to understand

that.

Bill Powers (2003.12.20.0812 MST)–

It may be that to a mathematician, a model is

an interpretation of a

mathematical system, but I think that is only because mathematicians

give

primacy to their own area of interest.

But of course they do. In *every* case where people talk about a

model, they give primacy to the domain that they are interested in. Their

only reason for talking about a model is because they hope it can help

them better understand the domain that interests them. Mathematicians

give primacy to the mathematical system that they are wrestling with and

use an interpretation, such as the properties of a well-understood

physical system, as an aid to understanding it. Physicists give primacy

to physical phenomena that puzzle them and use well-understood

mathematical systems as an aid to understanding them.

Talk of a model is always talk of a correspondence between a domain where

things are obvious and the domain that we want to understand better. We

can speak of that correspondence as analogy, but the term has other

connotations that can confuse the issue (e.g. the continua of analog

computing vs. the discreta of digital computing). Homology is the wrong

word because in biology it suggests common origin. A correspondence of

form, whatever we call it.

F=ma is one of the

generalizations that make up Newton’s theory.

A model incorporating this generalization is

V = 30 m/sec - 10 m/sec2 x t sec.

"[F]or a mathematician, a model is a way of interpreting a

mathematical

system.

# I think of the second statement simply as an evaluation of the equation v

v0 - a*t

Now you are taking the mathematician’s point of view, in which a model is

an interpretation of a mathematical system. Except that probably you

don’t think of an evaluation as a model.

(not f = ma).

You’re right, I misquoted Bruce Gregory. He posed a question: “If a

ball is thrown directly upward with an initial velocity of 30 m/sec, how

long will it take to reach the ground?” This question cannot be

answered by generalizations like f=ma, but only by a model such as the

equation

```
V = 30
```

m/sec - 10 m/sec2 x t sec

He said that this equation, incorporating Newton’s law of gravity, is a

model that enables us to answer that question.

By putting in specific values for the

initial

velocity and the acceleration, you generate a specific value (or series

of

values) for V. Is this what “model” is intended to mean? If so,

I think

this proposition misses the point of modeling, as I see it.

[…]

I look on mathematics as a tool for

approximating or idealizing natural phenomena, with the phenomena

taking

center stage and the mathematics acting in a supporting

role.

Of course: the phenomena are what are to be explained, not the

mathematics. You quite reasonably expect mathematics to be a

well-understood domain on which we can rely. A mathematician, however, is

wrestling with mathematical equations that are not so well understood.

The equations are the domain of interest which she hopes to explicate by

finding an interpretation or “model” for them.

For example, no one can prove Euclid’s parallel postulate (given any

straight line and a point not on it, there exists one and only one

straight line which passes through that point and never intersects the

first line, no matter how far they are extended). Let’s assume it’s not

true, maybe we can get to a contradiction and thus prove it negatively.

But hold on, pursuing that tack, Mr. Riemann has constructed a perfectly

consistent geometry in which the parallel postulate is false. Does that

make sense? How can this be called a geometry? What can it possibly mean?

Oh … great-circle lines on the surface of a sphere behave this way. Now

I understand.

As I use the

term, modeling means proposing underlying mechanisms which, if built

or

simulated, or if their behavior were deduced analytically or by any

other

rigorous means, would necessarily produce the phenomena we are trying

to

explain.

And this is the root of the equivocation between theory and simulation.

Seems to me it is the proposed underlying mechanisms that constitute the

model. These mechanisms can be described theoretically, in terms of

equations and associated generalizations, or they can be simulated in a

computer program or robot. The equations and generalizations describe the

model (the mechanisms). The simulation replicates the model (the

mechanisms).

The mechanisms underlying behavior are not directly accessible to us in

living organisms, but they are accessible to us in the mathematical

equations (and associated generalizations) of the theory and in the

constructs of a simulation. Both the theory and the simulation are

required. On the one hand, only a working simulation can generate the

behavior of the organism that is being modeled, the theory cannot itself

be a working model. On the other hand, it is necessary to show that the

simulation is principaled, that it is an instantiation of the

generalizations and mathematical equations of the theory. The

instantiation of the mechanisms and the mathematical explanation of the

mechanisms are both required.

This all seems to differ from the

classification scheme offered in your

post. I think that’s because the various authors you cite have

different

ideas about the meanings of the terms, so they see different

properties

implied by them. Is anyone “right” about these meanings?

They’re all of them right, once you understand that “model”

inherently refers to a relationship. Sure, your figure/ground point of

view depends on your needs, but it’s always a relationship of

correspondence between something well known and something murky, and the

purpose is always to use the former to explain the latter.

I know that

physicists often speak as if natural phenomena were only approximations

to

the “true” mathematical forms, but I disagree with that view.

I suspect we’d be more sympathetic if we were dealing with the same

phenomena. Physicists cannot escape noticing that they are on very

confounding ground, where their description seems to create that which it

describes. For PCT, it is much easier for us to assume without much

thinking about it that perception precedes description.

Describing

nature with mathematics is like describing a sculpture using only

cubes,

spheres, cones, cylinders, and so on – the basic idealized forms for

which

we have simple mathematical descriptions. As computer artists find out,

you

can render arbitrary forms by using enough of these idealized shapes,

but

if you look closely you will see that there are still differences.

I think you’re on wobbly ground with this analogy. The “simple

descriptions” provided by mathematics are contrasted here with what

alternative kind of description?

And if

the mathematical approximation is tweaked until the differences are

small,

you’ve lost all the elegance and simplicity of the idealized

representations.

What idealized representations? Are you comparing the mathematics of

physics with some other representation for physics which is more

idealized? (Same applies to a domain other than physics.)

For me, the central question is, “What is

the phenomenon?” The next

question is always “How does that work?” The answer to the

first question

is a detailed set of observations. And the answer to the second, as

nearly

as we can find an answer, is a proposal about an underlying

mechanism

which, if it existed, would entirely account for what we observe. We

use

mathematics where we can to assemble and test models. Sometimes we

actually

build the models so we can see them working. And other times we use

analogies – analog computers – to set up similar mechanisms in a

form

where their behavior is easy to observe, to see if they really do

reproduce

the phenomena.

Yes, this is the basic methodology of PCT. The proposed “underlying

mechanism” is a model of the organism, that is, its structure is

proposed to correspond to the relevant structure of the organism. One

sort of evidence for this is that a simulation that instantiates the

“underlying mechanism” generates the behavior that we observe

in the organism. Another sort is demonstration of physical structures in

the organism that seem to function as parts of the underlying mechanism

that the theory predicts.

This view of models is based on the idea that

they are

descriptions of underlying mechanisms, not descriptions of specific

behaviors.

So this is why you would say that

V = 30

m/sec - (10 m/sec2 * t sec)

although based on Newton’s law of universal gravitation, is not a general

model because it has specific values for initial acceleration and for the

force of gravity at the surface of the earth. But while this equation

describes specific behavior (modulo the detail of wind resistance,

humidity, etc.) it does not do so by curve fitting. It restricts the

general model provided by Newton to a very specific question. In the same

way, PCT describes a general model in theoretical terms, while any given

simulation such as the CROWD simulation or the baseball-catching

simulation restricts that general model to specific domains of behavior.

Both the theory and the simulations represent the model, the underlying

mechanisms; perhaps neither *is* the model.

```
/Bruce
```

Nevin

## ···

At 08:57 AM 12/20/2003 -0700, Bill Powers wrote: