Information

[From Bruce Abbott (970419.1450 EST)]

Rick Marken (970418.2340) --

Me:

If I am not able to design an experiment so that it produces clear
and consistent results, I give up on the experiment until a
cleverer person comes along and does it right.

Bruce Abbott (970418.1100 EST)

Well, that explains why PCT has not advanced beyond simple
tracking experiments in over 25 years. You leave all the
really hard work to others -- and there ain't no others.

How true. But I reassure myself with the knowledge that
reinforcement theory has not advanced beyond simple feeding experiments
in over 60 years.

No, those "feeding experiments" have become quite sophisticated, and what
has been learned from them could fill many large volumes. Why has there not
been a similar explosion in the tracking field? Most studies can be
implemented with nothing more than a PC and a single willing participant,
and an enormous amount of data can be produced in short order (e.g., 3600
data points in 2 minutes). (For comparison, those "simple feeding
experiments" usually take months to complete, sometimes over a year.) Even
ol' unfunded Rick Marken should be able to crank out several such studies a
year in his spare time. So why the lack of activity?

Bruce Abbott (970418.2005 EST)

there is no use arguing about it; it is an empirical question. We
could answer it with appropriate data from a simple tracking study.

I doubt it. We did such studies a couple years ago; Martin is convinced
that they showed that there is information about the disturbance in the
perceptual signal; I (and I think Bill and Tom)
am convinced that they showed that there is no such information in
the perceptual signal.

How can this be a matter of opinion if you have the data? Did anyone
compute the quantity of information? What was it?

We can design such studies again if you like. But I think it
might be nice to first discuss a higher level question: Why should
anyone care whether or not there is information about the disturbance in
perception? If there is such information it is certainly not
clear what it would be doing there.

Martin Taylor already answered that question. You are reifying
"information" into a "something" that has something to "do." Information is
just a metric for system performance. If I said that the output of a sensor
conveys information about the termperature, would you ask me what the
information is "doing there" in the wire?

The question is whether this alternative metric provides insights into
system performance that are not as easy to obtain using other metrics (like
the correlation between d and p). I'd like to see Martin give it a shot.
What can it hurt?

It may be, however, that the information about the disturbance is
something that control systems don't want. All control systems might
have a reference level of 0 for the amount of information about the
disturbance that they want to receive.

You seem to be on a reification _binge_, Rick. Control systems don't have
wants with respect to information, nor reference levels. In fact, control
systems don't "want" anything. They just do what they do. If this action
happens to bring p near to r, they don't become any happier, and they don't
feel sad or upset when these actions fail. They are not little people.

But a perfect control system _would_ allow _no_ information about a
disturbance to p to show up in p. That's what control systems _do_:
ideally, they isolate their controlled inputs from the effects of
disturbance, and isolation means that one could learn nothing about the
disturbance merely by monitoring p. On the other hand, if r were known to
be constant, then any wiggles in p would indicate a disturbance to p.

I can tell you why I _don't_ like the idea that there is information
about the disturbance in perception. I don't like it because it
sounds to me like an attempt to conceptualize control in S-R terms.
The information-based models of control that I have seen assume that
information about the disturbance is an aspect of the sensory input
to the system. This information _guides_ the outputs that compensate
for that disturbance.

That's just metaphorical speech. Information is just a metric. It implies
nothing whatever about S-R or causality. You should be objecting to the
concept of gain: it sounds like an attempt to conceptualize control in
capitalist economic terms. (No doubt the term was invented by a Republican
economist.)

Of course, there might actually be information about the disturbance in
perception. If so, then I wonder what is it there for?

Arrgh. Let's rephrase this. "Of course, there might actually be some
shared variance between the disturbance and the perception. If so, then I
wonder what it is there for?" The steering wheel on my daughter's care
shakes at under 20 mph. I wonder what the shaking is there for. Do these
questions make any sense? No.

I note that you had _no_ comment about my claim that a system
provided a sensor to detect the effect of its own actions on
its perception of the CV would be capable of providing almost
full information about the disturbance. Don't want to admit I
have it right?

Both Bill and I noted that if a system (other than the one doing the
controlling) has access to both the perceptual signal, p, and output
variable, o, (assuming all other functions are constants) then it would
be trivially easy for that system to deduce the disturbance, since p = o
+ d and d = o - p. The system doing the controlling
could compute the estimated value of d too, of course, and everything
would be fine as long as the system "provided" this information to some
other system and didn't use it itself as its perceptual or
output signal.

Well, you know Rick, the lack of comment does not indicate agreement; in
fact it _can_ be interpreted as the opposite, if one is disposed to do so.
I for one would find it most helpful if you explicitly indicate when you
_agree_ with some observation or conclusion of mine as well as when you
disagree. I'm not a mind reader.

Of course, in a real situation, you would have to know a lot more
than just the output signal to deduce the disturbance variable(s);
you would have to know the feedback function, the number of
disturbing variables influencing the controlled variable and the
function relating EACH disturbing variable to the controlled
variable.

Yes, of course, with respect to the first. As for the rest, that depends on
whether you need to know each disturbance individually or only their
composite effect.

Regards,

Bruce

[From Rick Marken (970420.0030 PDT)]

Me:

But I reassure myself with the knowledge that reinforcement theory
has not advanced beyond simple feeding experiments in over 60 years.

Bruce Abbott (970419.1450 EST) --

No, those "feeding experiments" have become quite sophisticated,
and what has been learned from them could fill many large volumes.

I wasn't putting down feeding experiments. I was pointing out that
reinforcement theorists, like control theorists, study the phenomenon of
interest to them using a technique that has proven useful;
feeding experiments for reinforcement theorists; tracking
experiments for control theorists. Of course, since reinforcement
theorists are unaware of the fact that they are studying a
side-effect of the phenomenon of control, they have filled many
large volumes with the results of sophisticated experiments that
tell us almost nothing about what the subjects of these experiments are
doing.

Why has there not been a similar explosion in the tracking
field?...Even ol' unfunded Rick Marken should be able to crank
out several such studies a year in his spare time. So why the
lack of activity?

Gee. No need to mock my obvious failings. I know I haven't done much to
advance the study of control. That's why I get excited when real smart
people like you get interested in PCT. I figure that's when things will
really take off.

I've been involved with PCT for about 16 years and the meager
results of my efforts are published in _Mind Readings_. There are about
11 research studies reported in that book so my PCT research output is
about 11 studies in 16 years or .68 studies/year. Not much, I agree. But
you can't get blood out of a turnip. You, on the other hand, have been
involved with PCT for nearly 3 years and the results of your efforts are
nowhere to be seen. So your research output is exactly 0 studies in 3
years or 0.0 studies/year. If I were you
I'd feel pretty embarassed to be bested by the likes of me;-)

The question is whether this alternative metric [information]
provides insights into system performance that are not as easy
to obtain using other metrics (like the correlation between d
and p). I'd like to see Martin give it a shot. What can it hurt?

We would all like to see the insights information theory provides.
But so far (after 4 years or so) no insights have been forthcoming. I'm
happy to drop the information discussion until we get these insights.

if r [reference signal value] were known to be constant, then
any wiggles in p would indicate a disturbance to p.

Oops. Can't drop it. Gotta point out that this is false. p is at all
times the _combined_ result of output and disturbance (in the simplest
case, p = o + d). So any variation in p less a constant is variation in
o+d less a constant. So wiggles in r-p do not indicate a disturbance to
p; they indicate the combined effects o and d on p.
The wiggles could be entirely the result of variations in output,
with d = 0.

Information is just a metric. It implies nothing whatever about
S-R or causality.

Maybe not. But the psycholgical models I know that are based on
information theory are are all open-loop, causal models. According to
these models, sensory information is used by a central processor as the
basis for making decisions and calculating actions. This is an
open-loop model of behavior.

Information theory was designed to measure the quality of a
communication channel, which is an open-loop system; messages
are encoded at the input to the channel and the encoded messages
are carried across the channel to a decoder which reconstructs the
messages. Information theory allows us to measure how well the
decoded messages match the messages that were actually encoded. I don't
see how this analytic technique is relevant to the behavior
of a closed loop system, where the output of the system (like the
decoded messages) has an effect on the inputs (the encoded messages)
that are causing those outputs.

Best

Rick

[From Bruce Abbott (970420.1045 EST)]

Rick Marken (970420.0030 PDT) --

Bruce Abbott (970419.1450 EST)

No, those "feeding experiments" have become quite sophisticated,
and what has been learned from them could fill many large volumes.

I wasn't putting down feeding experiments. I was pointing out that
reinforcement theorists, like control theorists, study the phenomenon of
interest to them using a technique that has proven useful;
feeding experiments for reinforcement theorists; tracking
experiments for control theorists. Of course, since reinforcement
theorists are unaware of the fact that they are studying a
side-effect of the phenomenon of control, they have filled many
large volumes with the results of sophisticated experiments that
tell us almost nothing about what the subjects of these experiments are
doing.

There is nothing inherently wrong with doing tracking experiments, but those
experiments should be providing new basic knowledge about the relevant
systems, providing a broadening foundation for further progress. So far
there hasn't been much done beyond some simple demonstrations showing that
people are good at controlling cursor position, and that their performance
is nicely modeled by a simple proportional controller with leaky
integration. These are important demonstrations of principle, but they
haven't told us much about how the human system is actually organized; very
little has been done to investigate the nature of the hierarchy, for example.

As for the results of reinforcement experiments, as you know, I see things a
bit differently than you. Experiments conducted to identify the
"reinforcer" in a given situation tell us quite a bit about what the subject
was attempting to control, in my view.

Why has there not been a similar explosion in the tracking
field?...Even ol' unfunded Rick Marken should be able to crank
out several such studies a year in his spare time. So why the
lack of activity?

Gee. No need to mock my obvious failings. I know I haven't done much to
advance the study of control. That's why I get excited when real smart
people like you get interested in PCT. I figure that's when things will
really take off.

I'm not mocking your "failings," nor am I casting aspersions on your
intelligence. I am asking why an intelligent scholar well-versed in control
theory has not produced more in the way of empirical research into control.
It clearly isn't for lack of motivation, and such research doesn't appear to
require a lot of subjects, time, or sophisticated and expensive equipment.
Might it have something to do with the nature of the subject matter?

I've been involved with PCT for about 16 years and the meager
results of my efforts are published in _Mind Readings_. There are about
11 research studies reported in that book so my PCT research output is
about 11 studies in 16 years or .68 studies/year. Not much, I agree. But
you can't get blood out of a turnip. You, on the other hand, have been
involved with PCT for nearly 3 years and the results of your efforts are
nowhere to be seen. So your research output is exactly 0 studies in 3
years or 0.0 studies/year. If I were you
I'd feel pretty embarassed to be bested by the likes of me;-)

How about summarizing what you learned about the human control system from
those 11 studies?

I've now been involved in PCT since November 1994, or a little under 2.5
years. I spent the first year and a half developing my understanding of
control theory (with a lot of help from Bill P., you, and many other
CSGnetters), learning how do program control simulations, programming,
conducting, and analyzing several tracking experiments, and constructing
control models capable of generating certain results known from the operant
conditioning literature. I spent the next year conducting experiments to
understand what is going on in ratio schedules of reinforcement and to
investigate the control of nutrient level (as reflected in body weight) in
lab rats. In these studies, data were collected _daily_ for nearly a year.
Given the massive amount of data collected, it is going to take me a while
to get it all sorted out. That brings us up to now.

For some reason, I don't feel embarrassed at all.

if r [reference signal value] were known to be constant, then
any wiggles in p would indicate a disturbance to p.

Oops. Can't drop it. Gotta point out that this is false. p is at all
times the _combined_ result of output and disturbance (in the simplest
case, p = o + d). So any variation in p less a constant is variation in
o+d less a constant. So wiggles in r-p do not indicate a disturbance to
p; they indicate the combined effects o and d on p.
The wiggles could be entirely the result of variations in output,
with d = 0.

If r is fixed, d = 0, and output varies, then o is itself a disturbance to
p. Output varies because the residual effects of earlier variations in d
are still reverberating in the loop. So what I said is _not_ false: if p
wiggles, then p is being disturbed. By definition.

Regards,

Bruce

[From Rick Marken (970420.1100)]

Bruce Abbott (970420.1045 EST)

very little has been done to investigate the nature of the
hierarchy, for example.

Well, if you want more investigations, why don't YOU do them. I
think the tiny number of PCT based tracking studies that have been done
(and published) tell us more about human nature that all the books found
in the BF section of the library put together.

As for the results of reinforcement experiments, as you know, I
see things a bit differently than you. Experiments conducted
to identify the "reinforcer" in a given situation tell us quite
a bit about what the subject was attempting to control, in my
view.

They may _suggest_ possible controlled variables but I can't see
how they could possibly tell us "quite a bit" about what the subject
is attempting to control. I think that all we can say from the results
of 60 years worth of reinforcement experiments is that animals
probably control some aspect of their food and water intake.

I'm not mocking your "failings," nor am I casting aspersions on
your intelligence. I am asking why an intelligent scholar
well-versed in control theory has not produced more in the way
of empirical research into control.

I can only produce the amount of research that I want to (and can)
produce. In fact, I have produced more research than is reported in
_Mind Readings_ but it's very difficult to get this stuff published,
largely because it is virtually impossible to get it past reviewers who
(like you) see this research as contributing very little to our
understanding of behavior.

Might it have something to do with the nature of the subject
matter?

Yes, of course. Conventional psychologists seem to think that
PCT experiments don't tell them much; certainly nothing they didn't
already know. So it's hard to publish PCT experiments or get
conventional psychologists to pay attention to them. To me, science is,
in large part, a social enterprise; a collective effort to
understand the world around us (including ourselves and other people in
that world). If the members of this scientific society show no interest
in (and, in many cases, active hostility towards) one's research
offerings, the inclination is to do stop offering.

How about summarizing what you learned about the human control
system from those 11 studies?

Gee, I've been doing that on the net here for years. I've learned
that people control perceptual input variables by varying their actions,
as necessary, to produce intended perceptions. I've learned that you
can't tell what variable an organism is controlling by just looking for
patterns in its actions or in the results of those actions. I've learned
that multiple control systems can automatically coordinate their outputs
to produce intended results. I have learned
that the only way to determine what a person is controlling is by
testing for controlled variables. I have learned that people select the
consequences their actions and that the appearance that actions are
selected by their consequences is an illusion. I have leaned that the
controlling done by one control system can be changed by the controlling
done by other control systems in the same person.

I've now been involved in PCT since November 1994, or a little
under 2.5 years. I spent the first year and a half developing
my understanding of control theory

It seems to me that you've spent most of your time trying to set me
straight about conventional psychology.

I spent the next year conducting experiments to understand what
is going on in ratio schedules of reinforcement and to
investigate the control of nutrient level (as reflected in body
weight) in lab rats...Given the massive amount of data collected,
it is going to take me a while to get it all sorted out.

I think it's great that you are doing this research. I look forward
to your report of the results. Based on Bill Powers' report on the
status of the research (at the CSG meeting nearly 1 year ago!) it
seems like there are plently of interersting findings that should
be brought to the attention of the animal behavior community. I hope
you're planning to write this up and submit it for publication soon.

If r is fixed, d = 0, and output varies, then o is itself a
disturbance to p. Output varies because the residual effects
of earlier variations in d are still reverberating in the loop.
So what I said is _not_ false: if p wiggles, then p is being
disturbed. By definition.

Actually, this is NOT the definition of disturbance. See Bill Powers
(970419.1830 MST).

Best

Rick

[From Bruce Abbott (970420.2230 EST)]

Rick Marken (970420.1100)]

Bruce Abbott (970420.1045 EST)

very little has been done to investigate the nature of the
hierarchy, for example.

Well, if you want more investigations, why don't YOU do them.

O.K. What do you suggest, boss?

As for the results of reinforcement experiments, as you know, I
see things a bit differently than you. Experiments conducted
to identify the "reinforcer" in a given situation tell us quite
a bit about what the subject was attempting to control, in my
view.

They may _suggest_ possible controlled variables but I can't see
how they could possibly tell us "quite a bit" about what the subject
is attempting to control. I think that all we can say from the results
of 60 years worth of reinforcement experiments is that animals
probably control some aspect of their food and water intake.

You really don't know much about it, or you wouldn't say that. However,
definitely _lacking_ is any physical model of the system that would
_explain_ the many interesting behavioral phenomena this research has
disclosed. That's the next step, and what PCT promises to do.

I am asking why an intelligent scholar

well-versed in control theory has not produced more in the way
of empirical research into control.

I can only produce the amount of research that I want to (and can)
produce. In fact, I have produced more research than is reported in
_Mind Readings_ but it's very difficult to get this stuff published,
largely because it is virtually impossible to get it past reviewers who
(like you) see this research as contributing very little to our
understanding of behavior.

I didn't say that. I asked why there seems to have been so little progress
beyond the basic tracking studies.

Might it have something to do with the nature of the subject
matter?

Yes, of course. Conventional psychologists seem to think that
PCT experiments don't tell them much; certainly nothing they didn't
already know. So it's hard to publish PCT experiments or get
conventional psychologists to pay attention to them.

No, no, you've changed the context of my questions. I wasn't asking why
more studies have been _published_, but why more have not been _conducted_.
Reviewers and journal editors may make it difficult to get a study
published, but they certainly can't stop you from doing the research.

How about summarizing what you learned about the human control
system from those 11 studies?

Gee, I've been doing that on the net here for years. I've learned
that people control perceptual input variables by varying their actions,
as necessary, to produce intended perceptions. I've learned that you
can't tell what variable an organism is controlling by just looking for
patterns in its actions or in the results of those actions. I've learned
that multiple control systems can automatically coordinate their outputs
to produce intended results. I have learned
that the only way to determine what a person is controlling is by
testing for controlled variables. I have learned that people select the
consequences their actions and that the appearance that actions are
selected by their consequences is an illusion. I have leaned that the
controlling done by one control system can be changed by the controlling
done by other control systems in the same person.

Most of those things are implications of PCT, not empirical results. I
didn't ask you to summarize what you've learned about control, I asked you
to summarize your _research findings_. Didn't one study, for example, show
that suddenly reversing the sign of the environmental function produced a
runaway action for about 0.5 seconds, after which the participant regains
control by reversing the sign of the output function? That's the sort of
empirical stuff I'm asking about.

I've now been involved in PCT since November 1994, or a little
under 2.5 years. I spent the first year and a half developing
my understanding of control theory

It seems to me that you've spent most of your time trying to set me
straight about conventional psychology.

Yes, an obvious example of maladaptive behavior that supports neither
reinforcement theory nor the reorganization principle. It's probably a
manifestation of brain damage.

I think it's great that you are doing this research. I look forward
to your report of the results. Based on Bill Powers' report on the
status of the research (at the CSG meeting nearly 1 year ago!) it
seems like there are plently of interersting findings that should
be brought to the attention of the animal behavior community. I hope
you're planning to write this up and submit it for publication soon.

I certainly could use a publication or two at the moment, so I'm hoping so, too.

If r is fixed, d = 0, and output varies, then o is itself a
disturbance to p. Output varies because the residual effects
of earlier variations in d are still reverberating in the loop.
So what I said is _not_ false: if p wiggles, then p is being
disturbed. By definition.

Actually, this is NOT the definition of disturbance. See Bill Powers
(970419.1830 MST).

Disturbance it is; THE disturbance it isn't. When _I_ use a word, it means
what I want it to mean; that's why I pay it so well. (;->

Let's not quibble over words; the reverberations of previous disturbance
appear in the form of output wiggles, which shake p from its tranquility.
If an action itself results in the production of further error, it is a
disturbance in my book. It makes more sense to me to call this a form of
disturbance when, for example, it leads to oscillations, damped or undamped.
After the first impulse, the system keeps disturbing _itself_.

Regards,

Bruce

[From Rick Marken (970425.1730)]

Bill Powers (970422.1622 MST)

I'll agree that Rick's point is different from mine, but it is not wrong...
Rick is wrong in saying that d CANNOT (by any means) be deduced from
knowledge of the variables, parameters and functions in the control loop.

I appreciate your efforts to provide Martin with some spiritual fulfillment
but Rick (me;-)) never said that "d CANNOT (by any means) be deduced from
knowledge of the variables, parameters and functions in the control loop".
In fact, I (like you) have always pointed out that it is rather easy to
compute d given p and knowledge of all the other relevant functions and
variables' in a control loop. Indeed, earlier on the very same day that you
said the kind words above to Martin, I [Rick Marken (970422.0920)] told
Hans (another fellow whose soul seems to sustain itself on the hope that
"Rick is wrong"):

If p = f(u) + d and you are given p and f(u) then I have no doubt that even
you could solve for d.

Of course, I am going with Hans' assumption that g(d) = 1*d. I know, in
other words, that if p = f(u) + g(d) then you would have to have f(u)
_and_ g(d) (not just d) to solve for p.

Love

Rick

[Hans Blom, 970428g]

(Rick Marken (970425.1730))

Hans (another fellow whose soul seems to sustain itself on the hope
that "Rick is wrong"):

On the contrary, Rick! My only hope is that you will see the light,
some time soon now ;-).

Hans

[Martin Taylor 930617 17:35]
(Rick Marken 930617.0900)

Martin Taylor (930617 09:50) correctly answered my three questions.
But he confuses me once again. In answer to:

3) Can you do it if I also give you the output function that was in effect at
the time? Yes or no?

Martin says:

Yes, if you assert that the output function includes the environmental
path between the ECS and the CEV to which d is applied, and that the
function is defined precisely and is monotonic.

This is true. But a control system that worked under these restrictions
would be working in a very contrived environment. In fact, it would
not even have to be a control system; a calculated output system
would do since the outputs that it generates are guaranteed to counter
the effects of the disturbing variable.

A misunderstanding here. All I asked was to know whether the output
function was, say 10 * integral(error), or 100 * error^3 or whatever.
The requirement was that this function be fixed. The assertion that
the perceptual signal contains the information about the disturbance
comes from the fact of control and the stability (relative to the
disturbance) of the output function. And that stability has to extend
as far as the CEV. It has to be a control system, not a calculated
output system, or none of the argument works (it won't, either).

I agree that it is a most unnatural situation to have a fixed output
function. I tried to persuade you of that in the earlier discussion
on information, but you insisted that it was not so--that the output
function would normally be fixed. My argument, which I think you now
accept, was that control can be attained with an output function that
is not only nonlinear, but which is also time-variant and uncertain.
The information-theory view on this is that the time-variance and the
uncertainty of the output function (i.e. the transform between error
and the effect on the CEV) has to be considered as information that
the control system must deal with, reducing the ability to control
against the effects of the disturbance. Numerically, it is subtracted
from the controllable information in the disturbance.

So you are saying that the perceptual signal carries information about the
relationship between output and CEV. Yet, in your answer to question 3)
you said that you could not reconstruct the disturbance unless you were
given information about the relationship between the output and CEV (as
part of the specification of the output function).

This is why I want to stop talking about information and start talking
about how control systems really work.

That "This is why" is the non-sequitur that bugs me. I know I have a
problem getting you interested. I have a problem getting Bill P interested,
though he has asked a lot of penetrating questions that have helped me
refine my concepts. But I hope I will succeed, and not go off into
isolation mode, as you are tempted to do in respect of "mainstream"
psychologists.

The point is, and always has been, that the "knowledge" of the output
function is implicit in the fact of control. It is explicit only in
the view of a modeller--an outsider view. It is the outsider who has
to know the form of the function in order to do the calculation. The
control system does it by correcting for error. If the output function is
fixed, the amount of information involved in its specification very
soon is swamped by the stream of information from the CEV (i.e. from
the disturbance, if the reference is fixed). It vanishes from the
calculation, and does not affect the ability of the system to control
except insofar as a non-linear output function has a gain that differs
for different levels of output. But if the output function is time-varying,
information about it has to be factored into the calculation of the
ability of the control system to control. That matters. And in a
control hierarchy in which reference levels for lower ECSs come from
several independent higher ECSs, it matters a lot.

I know exactly
how control systems operate; I can build control systems that operate
in environments where the connection between output and CEV is
constantly changing and I can do it without understanding what is
meant be information.

Well, you are one (or more) up on me. I don't know "exactly" how
control systems operate, especially if there are lots of them connected
in nonlinear ways. I have some notions, however. Can you specify
beforehand how such a net will respond, given limits on the perceptual
resolution and bandwidth of its various perceptual input functions, and
given a variety of behaviours of the multiple disturbances to which
the various CEVs will be subjected? Maybe you can. I'm not claiming
that information theory will tell you how such a system will operate,
but I am claiming that it can provide you with limits on how well
the different disturbances can be controlled.

I, too, want to talk about how control systems "really" work. I'm trying
to lay the groundwork for doing so from a viewpoint complementary to
the one you take, but supportive of it. Didn't thermodynamics help the
understanding of many-body systems that were hard to understand using
Newtonian mechanics? Newtonian mechanics might be correctly applied,
and would give good prediction if the calculations could be carried through
(with infinite precision, as we now know). But the thermodynamic approach
provided different insights into the behaviour of such systems, insights
that would not have been obtained even with an exact Newtonian calculation.
In addition, it produced global results that could actually be used,
approximate though they might be. I think we are in the same situation.

Martin