Measured versus derived variables

[From Bill Powers (931217.1940 MST)]

Bob Clark (931217.1725 EST) --

To me the relation among energy, force, distance and angle
between force and distance is simply an algebraic expression.
Given the values of any three of the four variable, the fourth
can be calculated. This is not "reduction," it is merely
rearrangement.

The point I'm trying to make is not easy to state in a sensible
way. It has to do not with the structure of the algebraic
relationships but with how they're used in describing phenomena.
What you say is, of course, algebraically true. To simplify
somewhat, given a force f and the distance d through which it
acts, one can express the energy as E = f * d. And from that, it
follows that given any two of these variables, the third can be
calculated.

Suppose I give you the weight of a bucket hung from a windlass,
and the height through which the bucket is pulled. You can
compute work done as E = w * h. But suppose I give you only w.
You can say, "Given E, I can compute h" -- but E was not given,
nor can it be observed, because we have no observation of w. E
must be determine from observations of forces acting through
distances in some other context -- it can't be found from the
data in this experiment unless both w and h are measured. So
while it is true that w = E/h, this mathematical truth is useless
because we do not know E.

Force and distance are measured variables; we can put the
algebraic variables, like w and h, into one-to-one correspondence
with the outcomes of measurements. But energy is a derived
quantity, a computed quantity, and can't be put in one-to-one
correspondence with a measurement. The only way to equate energy
directly with something else is to make a different set of
measurements, complete enough for another energy to be
calculated, and from that deduce what the energy in the first
case must be.

Suppose there is a spring on the windlass which can be wound up
with a crank, then released to raise the bucket. We can calculate
the energy stored in the spring by measuring the force exerted on
the handle of the crank and multiplying it by the distance
traveled by the crank (in circles) during the winding. Then, when
the spring is released to wind up the windlass, we can observe
the weight of the bucket and the height through which it is
lifted, and compute the work done by the spring on the bucket.
Now we can equate two derived variables: the work done on the
spring by the crank, and the work done by the spring on the
bucket. Assuming no friction, we would expect the two calculated
energies to be equal.

What is it that makes the two energies equal (assuming no
losses)? Is it "conservation of energy?" Not at all: it is the
fact that the one force, acting through one distance, can be
mechanically converted to another force, acting through another
distance. The final height of the bucket can be computed without
ever mentioning energy. The law of conservation of energy follows
from the fact that this is true. The relationship f1*d1 = f2*d2
can be proven to be true, under the given assumptions, without
ever involving any derived variables, strictly through a
mechanical analysis of the system. It can be done strictly though
operations on the values of measurable variables.

Of course once we know it is true, we can express this
relationship much more conveniently in terms of energy
relationships. A similar consideration holds in, for example,
matrix algebra. When we first learn about matrix operations, we
have to go through all the permutations of indices one by one,
painfully working out the detailed operations implied by the
conventions of matrix notation. But once we have proven all the
transformations, the laws of multiplication, taking inverses and
transposes and so on, we can then dispense with the details and
just use the matrix notations and the rules that go with them,
with about one percent of the amount of scribbling on paper (and
an even smaller percentage of the mistakes). However, when we
describe a real system using matrix notation, and try to
implement it as a hardware or software design, we find that we
must convert the matrix notation back to its expanded form,
because the real system or the computer has to perform every
detail of every operation: neither the real system nor the
computer can do matrix operations. We must go back to the level
of measurable variables and direct relationships expressed only
in terms of measurable variables.

Physics consists largely of derived variables and relationships
among them. Each derived variable is a completely-specified
function of a set of derived variables of lower order; the chain
of derivation can be followed back to direct relationships among
measured variables. Doing so would seldom be convenient or
practical, but the point is that the structure of physics is so
designed that this is always possible. Every abstract
relationship in physics could be expressed strictly in terms of
elementary relationships between measurable variables. That is to
say, every such abstract relationship SHOULD be traceable to the
level of measurements.

In fact, however, this traceability to fundamental measurements
can fail. It can fail when someone, to save time or hoping for a
lucky guess, substitutes a made-up relationship that has never
been (or can't in principle be) derived from measurements. Of
course if the mathematics has been done correctly, the
measurements should check out -- as long as no arbitrary
assumptions have been introduced, and no unobserved variables are
involved. But there is trouble, I think, when it is simply
assumed that everything the mathematics can be made to say
corresponds, somewhere at the bottom level, with observable
phenomena. There is bigger trouble when an assumption is injected
into the structure without having been defined and justified at
the level of observation.

Up to a point, physicists and others using similar frameworks can
keep in mind the relationship between abstract variables and the
measurements that comprise them. But what I see going on in
certain fields, particularly where physicists get into the
problem of explaining living systems, is a loss of the connection
between a highly abstract level and the measurement level.
Relationships among abstract variables are then imagined, not
derived from observation. Premises are introduced not because
they can be justified at the measurement level, but because of
the conclusions they allow. This destroys the whole structure,
turning it into a fantasy.

What I have been talking about here has, of course, a direct
parallel in the levels of perception. Even the highest levels of
perception are, in principle, traceable to the level of
observation, which is the level of raw sensory signals. Each
level contains perceptions that are specifiable functions of
variables of the level below.

But at each level, we have the possibility of introducing
imagination, which can provide dependencies among perceptions
that have no known counterparts at the lowest level. We can
imagine things that have never occurred, although they might
occur.

So the question is, how do we deal with imagined relationships
that look as if they have interesting implications? One way is
simply to explore the implications, under the hopeful assumption
that if we can make sense of them at higher levels, they must
have some significance in Boss Reality. The other way is to test
the imagined relationships to see if there is any control process
by which they can be realized in perception. This is the way of
science, in my opinion. If an imagined relationship can't be
produced by any action on the external world, then however
interesting it may be, elaborating on it is worthless except as
an entertainment, a game.

One such imagined relationship is fundamental to the difference
between the PCT approach and the two conventional approaches, S-R
and cognitive planning. In both of the latter fields, it is
imagined that once a command is sent to the muscles, the outcomes
are determined. All higher-level analyses are based on this
assumption. But this assumption, when tested at the level of
observation, proves to be false: there is no significant number
of cases in which consistent outcomes of motor acts (by living
systems, in real environments) can be reproduced simply by
repeating the motor acts. In almost all cases, reproducing an
outcome requires using _different_ motor acts, often very
different even to the point of reversal, a fact that reflects the
universal presence of disturbances and changes in the parameters
of the acting system.

So the entire abstract system of reasoning that depends on the
regularity of output processes should never have been developed,
if the aim was to say something true of behavior. The first step
should have been to test the premise before going on to build an
intellectual structure on it. If that test had been done, the
assumption would quickly have been proven false, and an
assumption justifiable by observation used instead. And then we
would have had a structure of thought about behavior just as
solidly based on observation as physics is, or used to be.

ยทยทยท

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Best,

Bill P.