Testing simultaneous hypotheses about CV

[From Bill Powers (970608.0300 MDT)]

Hans Blom, 970528b--

I think I understood something last night which explains my attitude
toward MCT, as well as toward many aspects of modern physics. I hope
I can explain it. It's the difference between a physical system and
a mathematical model of a physical system. I think that the two have
been confused with each other.

It's the difference between a physical MODEL OF A system and a
mathematical model of a system, Bill. We have nothing but models.
It's _all_ perception: we do not have access to the "real world"; all
we ultimately have of "reality" are the "nerve currents" that
originate in our sensors. We combine those sensory experiences one
way and we have what we call a _physical_ model, in which we have
"things" (higher level perceptions) like "bodies" with a "length",
"width", "mass", "velocity", and so on, all higher level perceptions
as well. We can also combine those sensory experiences in another way
and have different higher level perception such as a _mathematical_
model with "things" like "functions", "constants", "variables",
"differences" or "differentials", etc. Different models of the same
part of "reality", maybe. The difference between models arises
because we focus on different properties of that reality -- e.g. how
a "material body" "moves" through "space" and/or "time" versus the
"mathematical function" that uses a "differential equation" to
describe that movement.

I agree with all this, yet there is more to the story. What you're
describing are two kinds of symbolic representations of experience, yet
there is a difference in that the two sets of symbols refer to different
_levels_ of experience. By "experience" I mean what you refer to in the
first sentence above by the phrase "of a system." We have, on the one
hand, a MODEL, which is constructed of words and rules for manipulating
them. And we have, on the other hand, THAT TO WHICH THE MODEL REFERS,
which you designate by "the system" and which I would identify as the
world of lower-level (non-symbolic) perceptions.

The entire MCT model exists (as does any model described in
mathematical, logical, or verbal terms) at the level of symbols and
symbol manipulations. But there are inputs to the model which are not
themselves symbols: they are what you term _observations_. These
observations constitute a world that exists independently of how it is
symbolized -- or to say the same thing differently, that can be
symbolized in any number of different ways, some formal and some
informal.

Suppose you're looking at some nearby familiar object like the
computer's mouse, and you reach out one finger to touch it. You see
something you name as "a mouse", and you say you see "a finger" "moving"
"toward" "the mouse." At some point you feel "a touch" that rises very
quickly from zero to some definite magnitude. All these terms in quotes
are symbolic representations of experiences. We form sentences out of
these terms: "my finger moves toward the mouse until I feel it touching
the mouse." That sentence is itself another experience, but it is at a
different level from the experiences being symbolized. The symbolic
experience is "about" the non-symbolic ones; it is a function of them,
given the rules for symbolizing that are in effect.

However, we can create the very same experiences without naming them,
without forming sentences about them. In that case we experience
perceptions undergoing changes and appearing in certain states, but
without the accompanying commentary. Then we are looking at the
observations, the perceptions that underly the symbols in terms of which
we normally (as human beings) think about them. A dog, I presume, has
similar lower-level experiences, but it does not create the same kind of
symbolic commentary about them (if any at all).

An apparent digression:

In a discrete model, the observations that are symbolized are symbolized
in terms of mutually-exclusive categories: the value of an observation
is either 10 units or some other number that is not 10. If the next
observation of the same variable yields 12 units, then there are two
values of the variable, but there are no values between 10 and 12. It is
possible to build up a mathematical system in which only discrete values
of observations are considered; in which the variables are inherently
categorical in both identity and magnitude. This is the kind of
mathematical system behind digital computers. The measures in this
system correspond to the ordinal numbers or the integers.

In an analog model, the variables are categorical only in terms of their
identities; their magnitudes are represented on a continuous scale. The
mathematics involved is that of algebra or differential equations, in
which continuity of changes is assumed. That is, if a variable changes
from 10 units of magnitude to 12 units, it is assumed that it went
through an infinity of values in getting from 10 to 12. Measures in this
system correspond to the real numbers.

Zeno's Paradox shows the difference between these two mathematical
systems.

If the turtle must always halve the remaining distance between itself
and the wall, there is an infinite number of halvings that must occur,
so the turtle can never get all the way to the wall. This paradox is
caused by thinking of each halving of the distance as a discrete event.
In the world of discrete events, nothing can happen _between_ events;
there is no way to define physical time. Events simply occur one after
another, so if there is an infinite number of events to be got through
the sequence will never end. A computer set up to simulate the turtle's
motion in terms of successive halvings of the distance to the wall will
never complete its computation.

The solution of Zeno's Paradox requires the introduction of continuity,
both in space and in time. Out of this comes the concept of _velocity_,
the ratio of distance moved to elapsed time on the real number scale.
Now we can see that the turtle, moving at constant velocity, requires
half as much time to move half the remaining distance, so the rate at
which successive halvings of the distance occurs rises to infinity as
the turtle approaches the wall. The paradox disappears: the time
required to move through a distance d at a velocity v is simply d/v.

In order for the paradox to disappear, we must go _all the way_ from the
discrete representation to the continuous one. This is what Liebnitz and
Newton realized in their concept of the infinitesimal. As long as
physical processes are represented in terms of discrete states, no
matter how small the steps between states Zeno's Paradox persists. It is
only in the limit, where dt actually goes to zero, that physical time
and velocity have meaning.

Back to the main point:

The discrete representation involves categorizing magnitudes of
observations; the analog representation deals in continuously-variable
magnitudes. To translate between models based on these two kinds of
observations (and their corresponding mathematical rules), we introduce
a continuous scale of time, so that discrete observations can be placed
on that time scale and located in the continuous representation. So in
effect, the continuous representation is senior to the discrete one:
continuous time does not exist in the discrete representation. The
dimension of time is missing from the discrete representation; it
belongs to the other, the analog, side. Discrete events can occur at
_any_ time on the continuous time scale, yet that time cannot be
represented in the discrete system (without implicitly introducing
continuity).

We can treat the discrete representation as an approximation to the
continuous one by thinking in terms of "samples" which occur at instants
on the real number scale, but there is no unique way to translate in the
opposite direction: given a continuous variation, there is no way to
translate to any particular or unique sequence of discrete values. The
same history of continuous variations can be translated into any number
of different discrete variations simply by locating the samples at
arbitrary instants on the continuous time scale (remembering that
"uniform intervals" has meaning only in the continuous world).

Why is the continuous representation senior to the discrete one? Because
the world of lower-level perceptions operates in the continuum, not in
discrete steps. Nerve-impulses can occur at _any_ time; they are not
clocked. There is no defineable least unit of time. So observations of
the world are basically continuous below the level where the nervous
system creates categorical variables -- continuous, although somewhat
noisy.

When we speak of observations of the physical world, therefore, we mean
the world as it appears in our lower levels of perception, where there
are no discrete categories. We refer to a continuously-variable
perception of the distance between finger and mouse, not to a series of
discrete positions. And we refer to the experience itself, not to a
symbolic representation of it. We have a choice of symbolic systems in
which to represent these experiences, one that could be called a
"physical" model (meaning a continuous representation) and the other a
"conceptual" model (meaning a model cast in terms of discrete
categories). But _neither_ kind of model is the same thing as the world
of continuously-variable experience that it is supposed to represent.

Best,

Bill P.