[Martin Taylor 2019.03.27.23.03]
···
This is getting almost as weird as the
curvature discussion. I don’t know who is on which side of Alice’s
Looking Glass any more. I asked why it is that whenever I say I
agree with you, you insist that I don’t, and you tun that around
into my getting upset when you say you don’t agree with me. Anyone
who read the thread so far will understand the difference between
those two concepts.
As for this "disturbance causes output" and Bill's message, Bill
mistakenly thought that by “disturbance” I meant "causes of
disturbance, which are permanently unknowable and irrelevant.
Can you seriously contend that when a disturbance changes the
variable you call the controlled environmental variable, the
output does not change to compensate? That seems to be what you
are insisting today. What you will insist tomorrow is anyone’s
guess, but I really wonder what kind of control would happen if
the output did not change to compensate for changes in the
disturbance.
Finally, about "information from the disturbance", you have never
understood that information analysis is a generalization of
variance analysis, which you have no qualms about using. For my
part, I have never understood what so horrifies you about that
simple mathematical fact. The point is and always was very simple.
The disturbance and the output are highly correlated negatively,
the disturbance and the perception have a very small correlation.
The reference and the perception are highly correlated. Those are
fine statements when the variables have a Gaussian probability
distribution, not when their distributions are far from Gaussian.
They don’t work very well, for example, when the disturbance or
the reference makes irregular step changes among a small number of
possibilities.
All of those statements have precise equivalent in
informational/uncertainty terms: The disturbance and the output
have low mutual uncertainty (high mutual information relative to
their individual uncertainties), the disturbance and the
perception have low mutual information, and the reference and
perception have low mutual uncertainty. When all the variables
have a Gaussian (Normal) distribution, the variance and
informational ways of reporting the relationships are exact
synonyms. Numerically they differ by a single fixed constant whose
value I am not bothering to look up right now. The
information.uncertainty statements remain valid no matter whatever
the distributions of the variables.
What's so hard about that? It seems to me preferable to use
statements and mathematics that remain valid over statements that
are their direct equivalents but that are restricted to one kind
of probability distribution.
But I don't suppose you have any better answer than you had at the
time, just to repeat over and over that “there is no information
from the disturbance in the output”, in the same way as in the
curvature discussion you insisted that because the velocity
formula was the same in two equations, the velocity value is also
identical in the two equations, when ANY velocity could be plugged
into one of the equations leaving the equation still true. You
never once made any apparent effort to show how your many critic
were wrong in their demonstrations and analyses. All you did was
repeat “I’m right and you are wrong, and moreover you don’t
understand PCT.”
Maths was never your strong point, which is no problem, until you
persist in asserting the truth of some idea that got into your
head based on a faulty mathematical analysis, long after the error
has been demonstrated to you in a variety of ways.
As I said up front, I have lost interest in this thread because of
the way meanings have been weirdly twisted and distorted into
unrecognizable forms recently. If you wish to comment seriously on
the relationship or lack thereof between PCT and Free
Energy/Predictive Coding, and I find your comment sensible, I may
contribute further to the thread, but I am not prepared to
continue trying to follow the tortuous and tortured ways you find
to prove that I disagree with you when I say that I agree with
what you write.
Martin
[Rick Marken 2019-03-27_15:19:12]
[Martin Taylor 2019 03 26.22 55]
MT: Second: What makes you think I control a perception ofmy level of agreement with you?
RM: Because when I didn't agree with you you got veryupset; my lack of agreement was clearly a disturbance to a
perception you were controlling that could be called “Rick’s
level of agreement with me”.Â
MT: I don't. I do control forsupporting what I consider to be correctly stated,
RM: If that were true then you wouldn't care whether ornot I agreed with you.
MT: I have no idea why or when you got into your head theidea that i think or ever thought that “the disturbance
causes the output”…
RM: From all the "information in the disturbance is thebasis of output" discussions on CSGNet. And just recently
you said that a change in the disturbance leads to a change
in output.
MT: So far as I can remember, thefirst time I heard this idea “disturbance causes the
output” applied to me (and to Bruce Abbott) was when we
showed you the glaring error in your mathematical argument
in your curvature analysis.
RM: You'e been consistently arguing that the disturbancecauses output from the time you got on the net. I have
copied to the end of this message a post from Bill Powers to
you from 1996. It’s worth reading the whole thing since it
not only shows what Bill thought of your ideas about the
role of the disturbance relative to that of the output of a
control system but it also shows what Bill thought of your
style of argument.
Â
MT: Why did I call you "an enemy ofPCT"? By not withdrawing that curvature paper, you laid
open to a wider audience the suspicion that PCT research
is done by people not long out of mathematical
kindergarten, a situation not likely to induce people to
learn more about its power and beauty.
RM: Well, of course, I should have withdrawn mypublished, peer reviewed paper when you told me too and when
I didn’t the only appropriate thing to do was to call me the
“enemy of PCT”.Â
Â
MT: By allowing the paper to standand neither withdrawing it before publication nor
retracting it afterwards, and even continuing later to
claim that the paper presented a PCT explanation of the
curvature power-law effect, you did act like an enemy of
PCT.
RM: Funny, I thought it was your mathematical“explanation” (or justification) of the power law was
completely irrelevant to a PCT explanation of the power
law. Â
RM: But enough of this. Here's Bill's post. Knowing you,I’m sure you will be able to spin it as the highest of
praise for your genius. God I wish he were still here.
Best
Rick
 =========
Date:Â Â Â Â Â Â Â Â Tue, 18 Jun1996 01:01:43 -0600
Reply-To: "ControlSystems Group Network (CSGnet)"
             <CSGNET@POSTOFFICE.CSO.UIUC.EDU> Sender: "ControlSystems Group Network (CSGnet)"
             <CSGNET@POSTOFFICE.CSO.UIUC.EDU> From: "William T.Powers" POWERS_W@FORTLEWIS.EDU
Subject:Â Â Â Â Â Re:information blah blah perception blah blah disturbance
blah…To: Multiplerecipients of list CSGNET
             <CSGNET@POSTOFFICE.CSO.UIUC.EDU>[From Bill Powers (960617.1500 MDT)]
Martin Taylor 960617 15:45 –
What was said in 1992 seems to be mostly a matter ofwhat you now think
were your most important points then, and thus what youselect from the
stream of communications to show that you were rightall along. One
great difficulty is that your arguments then did notimpress me any more
than they do now, yet you cite them as if they settledthe matter once
and for all -- and as if I had agreed that they did.Let’s just look at
a few of them.
    In fact, [the discussion] arose out of my 921218analysis of the
    informational basis of PCT:
    "The central theme of PCT is that a perception inan ECS should be
    maintained as close as possible to a referencevalue. In other
    words, the information provided by the perception,given knowledge
    of the reference, should be as low as possible."
This is not and never was the central theme of PCT as Isee it. The
central theme of PCT is that organisms control theirperceptions by
acting on the environment. How well they control themdepends on the
parameters of the control system. There is no "should"involved.
Organisms control as well as they control, neitherbetter nor worse.
I have no idea what you mean by saying "the informationprovided by the
perception, given knowledge of the reference".Information and knowledge
are not the same thing, and anyway what is there in acontrol system
that can evaluate the information in a perception, withor without
"knowledge" of the reference? You're talkinggobbledygook.
    Later, dogmatic assertions were made that there isno information
    about the disturbance in the perceptual signal,assertions that we
    proved false, using experimental simulationsagreed to be effective
    for the purpose.
They were not adequate for the purpose except in yourown mind. You and
Randall agreed they were effective; Rick and I did not.In fact there is
no way to tell what the disturbing variable is fromknowledge of the
variables in the control loop (perception, reference,error, output) or
from the forms of the functions in the loop(perceptual, comparison,
output, feedback). The reason is very simple: exactlythe same
perturbation of the loop can arise from an infinity ofdifferent
disturbing variables acting (singly or together)through an infinity of
different disturbance functions.
In your first demonstration, you employed astep-disturbance acting
through a unity disturbance function. This led to astep-change in the
perceptual signal, which you then assumed representedthe true
disturbance. But it did not. The same step-change inthe perceptual
signal could have been created by an infinity ofdifferent disturbances
acting through different disturbance functions. Thereis no possibility
that one could work backward from knowledge of theperceptual signal to
deduce the nature of an unknown disturbing variable orvariables acting
via unknown functions. There is simply no information(in any sense of
the word) about the disturbing variables in theperceptual signal.
Every now and then you seem to wake up and say "Oh, OFCOURSE there is
no information about the CAUSE of a perturbation in theperceptual
signal. How could you ever have thought I would suggestsuch a silly
thing? Please read what I say and you will notattribute such foolish
ideas to me." And then you turn right back to the sametheme and claim,
as above, that there really is information in theperceptual signal
about the disturbance, and that you proved it.
I can't account for this except by guessing that youare shifting
meanings of "disturbance" between one set of statementsand the other.
One sense refers to the proximal perturbation of theinput to the
perceptual function that results from whatever distaldisturbing
variables happen to be acting. The other sense (which Ialways mean by
"the disturbance") refers to the changes in the distaldisturbing
variables themselves. Your statement about informationin the perceptual
signal about "the disturbance" cannot apply to thedistal disturbing
variable. It applies trivially to the proximalvariable, because the
proximal variable is exactly what we mean by a CEV. Tosay that the
perceptual signal contains information about the stateof the CEV is a
tautology, because that relationship defines the natureof the
perceptual input function. As I tried to point outfour years ago, if y
is the sum of a, b, c ... d, then there is no way towork backward from
knowledge of y to the state of a, b, c, and so on. Youcould have
exactly the same value of y arising from an infinity ofcombinations of
a, b .. d. A control system can control y if it canvary one of the
variables on which y depends. To do so, it does notneed to know
anything about the states of the other variables onwhich y depends.
Nothing. NADA.
    At least, they were agreed to be effective untilthe results showed
    the dogma to be false. Then, and only then, wereirrelevant
    objections raised.
This somewhat scurrilous allegation rests on ourinitial difference in
conceiving the conditions of the "challenge." Rick andI were assuming
that you would be given only the state of theperceptual signal. You
then proceeded to use your own assumptions about theforms of all the
functions, including the disturbing function, and thevalues of all the
variables and signals, including the reference signal,to deduce the
only remaining unknown, the disturbing variable.
Rick sent you some lists of numbers on severaloccasions, representing
the state of the perceptual signal in a working controlmodel, and
challenged you to deduce the behavior of the disturbingvariable from
knowledge of the behavior of the perceptual signal (Isee that he is
offering to do this again). If the perceptual signalhad contained
information about the disturbance, you should have beenable to use that
information to deduce the behavior of the disturbance.Obviously, you
could not do this. Rick's challenge should have beencompletely
sufficient to show you how we conceived of thechallenge in general.
What you did was to permit yourself to use all kinds ofknowledge that
Rick and I were ruling out. Our objections were quiterelevant to our
understanding of the phrase "information in theperceptual signal about
the disturbance."
You citing you:
    The fact that the fixed functions were the outputfunction and the
    feedback function of the control loop is neitherhere nor there.
    The fact that they don't vary as a function of thewaveform of the
    disturbance is what matters. The only varying itemused was the
    perceptual signal.
You citing me:
>>You forgot to mention the form of the inputfunction, the function
>>relating the disturbing variable to thecontrolled variable, and the
>>setting of the reference signal, all of whichyou must also know.
You now:
    And could you now, after three years ofconsideration, tell me
    which of these varies in a manner coordinated withvariations in
    the disturbing influence on the CEV? If you cancorrectly assert
    that any one of these contains information aboutthe fluctuations
    of the disturbance, then and only then can youcriticize the
    demonstration experiment and the derivedconclusion.
Wait a minute. You're saying that I can't criticizeyour experiment and
its conclusion if I can't correctly assert that anyvariable or function
in the control loop but the perceptual signal "variesin a manner
coordinated with variations in the disturbing influenceon the CEV." If
I've untangled this set of nested negatives correctly,you’re saying
that the perceptual signal _does_ vary in a“coordinated” way (whatever
that means) with the disturbing influence.
But this is exactly what I am trying to tell you isyour primary
mistake. The perceptual signal does NOT vary in a waythat correlates
with any particular disturbing variable. At one momentthere might be a
single disturbing variable acting through a simplelinear function; at
the next there might be twelve disturbing variablesacting through a set
of functions ranging from square to square root toexponential. The
control system will behave no differently in any case.It simply senses
the controlled variable and acts according todeviations of its
perception from the momentary setting of the referencesignal.
Furthermore, given complete knowledge of everything inthe control loop,
but not of the environment beyond the input quantitiesthemselves, you
could certainly deduce the state of a hypotheticaldisturbing variable
based on assuming a hypothetical disturbance function.But this would be
a complete fiction; it would not be a "reconstruction"of the true
disturbing variable. Your chances of guessing correctlywhat the actual
number of disturbances is, and what their individualwaveforms are, and
how each one is linked to have an effect on thecontrolled variable, are
essentially zero. And the control system can't do this,either.
    But (as I said those long years ago as well), isit not absurd to
    ask the control system, which has but a singlescalar value for its
    perceptual signal, to _know_ (perceive,understand,…) anything
    other than the value of the CEV. Is it not a redherring to suggest
    that anything in the discussion hinges on thisabsurdity?
I use "know" in a loose way, to be sure. I say that asystem “knows”
about something outside it if there is a variableinside the system that
covaries with the external something. A photocell“knows” about light
intensity, but not about color. In a simple controlsystem, the only
"knowledge" that exists is the perceptual signal. Andit is “knowledge”
only in the sense that it represents the value of afunction of some set
of input quantities.
Since this is the only knowledge that the system itselfhas, it is
absurd to say (as you have said) that the system "usesinformation" that
is "contained in" the perceptual signal. All thecontrol system needs is
the perceptual signal itself. It does not have toperform any operations
to detect or manipulate measures of information. So whois being absurd
here?
    You should be stating that "as the precision ofopposition to the
    disturbance increases, so the information aboutthe disturbance
    remaining in the perceptual signal decreases" andthen you would
    see it as a perfectly straightforward,self-evident proposition, in
    place of a paradox contrary to reason.
But that is contrary to the idea that the controlsystem uses the
information in the perceptual signal to construct anoutput that
precisely opposes the effects of the disturbance on theinput quantity.
The paradox lies in claiming that control -- theprecise opposition to
the effects of an unknown disturbing variable orvariables – relies on
information in the perceptual signal, and also to saythat the better
the control, the less information there is in theperceptual signal. In
the limit, according to this way of looking at thesystem, control would
be perfect if there were NO information in theperceptual signal. But in
that case, what would be the basis for constructing theoutput?
Well, let’s move on.
    Firstly, consider a predictable world. PCT is notnecessary,
    because the desired effects can be achieved byexecuting a
    prespecified series of actions.
I thought this was silly in 1992 and I still do. If theworld is
predictable, this does not mean that any organism iscapable of
predicting it. Furthermore, as I pointed out back then,even if the
world is predictable, a control system is still thefastest and least
complex way to control it. Suppose the muscles werecalibrated perfectly
and the organism somehow could carry out thecalculations necessary to
generate the muscle tensions required to produce anyposition of the
limbs. Yes, in principle one could do an open-loopcalculation involving
all the inverse kinematics and dynamics, but at whatcost? Probably a
large portion of the brain would have to be devoted toperforming this
calculation over and over in real time. But the sameresult can be
achieved, for all practical purposes, using a few verysimple negative
feedback control systems which do only a few elementarycalculations. So
even in a perfectly predictable world, the controlsystem is still the
system of choice. To say that the world is predictableis not to say
that it is simple or that a given organism is capableof predicting it.
Your assumption is not tenable. Unfortunately, youinsist that it is
correct, and go on from there.
    At the other extreme, consider a random world, inwhich the state
    at t+delta is unpredictable from the state at t.ÂPCT is not
    possible. There is no set of actions in the worldthat will change
    the information at the sensors.
There is no information at the sensors. Information, asyou have said a
number of times, depends on the nature of the receiver.It does not
exist independently in the environment. If the receiveris monitoring
the mean noise level of the sensor signals, acting atrandom can raise
or lower that noise level, since random acts imposed ona random world
will add in quadrature to the net effect. Control wouldstill be
possible, if not very useful.
    Now consider a realistic (i.e. chaotic) world.
Fine. But you are assuming at this point that PCT wouldnot be necessary
in a predictable world, which is false. That vitiatesthe strength of
this orderly argument. You are equating "predictable"with “simple” or
"understandable." In fact, you are attributingpredictableness to the
environment, as if it were a property of theenvironment and not a
function of the organism’s capacities to predict.
    At time t one looks at the state of the world, andthe
    probabilities of the various possible states att+delta are thereby
    made different from what they would have been hadyou not looked at
    time t. If one makes an action A at time t, theprobability
    distributions of states at time t+delta aredifferent from what
    they would have been if action A had not occurred,and moreover,
    that difference is reflected in the probabilitiesof states of the
    sensor systems observing the state of the world.ÂAction A can
    inform the sensors. PCT is possible.
When you start talking like a quantum physicist youlose me. This whole
way of dealing with phenomena strikes me as awkward andugly. And
anyway, I don't have to follow your arguments anyfurther, since you
have made a basic mistake in saying that in apredictable world, PCT
would not be necessary.
    Things become more interesting when we go up alevel in the
    hierarchy. Now we have to consider the source ofinformation as
    being the error signals of the lower ECSs, giventhat the higher
    level has no direct sensory access to the world
Not the error signals: the perceptual signals. Theseare not the same
thing, even though you try to make them the same:
    Even though the higher ECSs may well take assensory input the
    perceptual signals of the lower ECSs, neverthelessthe information
    content (unpredictability) of those perceptualsignals is that of
    the error, since the higher ECSs have informationabout their
    Actions (the references supplied to the lowerECSs) just as the
    lower ones have information about their Actions inthe world.
This is patching up your argument as you go. The erroris the difference
between the reference signal and the perceptual signal.If the higher
system is in the imagination mode, it is not receivingthe perceptual
signal. If it is in the action mode, it is notreceiving a copy of its
own output. When you try to design a system can canoperate in both
modes at once, you run into all sorts of problems. ButI don’t expect
that such niggling details will deflect you.
    (Unexpected events provide moments of highinformation content, but
    they can't happen often, or we are back in theuncontrollable
    world.)
So you are still assuming that disturbances have to bepredictable for
control to work?
    What does this mean? Firstly, the higher ECSs donot need one or
    both of high speed or high precision. The lowerECSs can take care
    of things at high information rates, leaving tothe higher ECSs
    precisely those things that are not predicted bythem–complexities
    of the world, and specifically things of a KINDthat they do not
    incorporate in their predictions. In other words,the information
    argument does not specify what Bill's elevenlevels are, but it
    does make it clear why there should BE level ofthe hierarchy that
    have quite different characteristics in theirperceptual input
    functions.
If information theory could really, out of its ownpremises, come up
with these predictions, that would be impressive. Butit can’t because
it didn't. You're solving a problem to which youalready know the
answer, and throwing in all the assumptions needed tomake your
"prediction" come out right. Those assumptions are notcontained in
information theory. What does information theory haveto say about
“kinds” of perceptions? Nothing.
Another item
    In your comment, you take it to refer to how afunctioning ECS is
    to be designed, and that the perceptual bandwidthshould be low.
    If the perceptual bandwidth is low, then the ECSwill have
    difficulty matching the perceptual signal to thereference signal,
    and thus the error signal will have highinformation content.
First I have never said that the perceptual bandwidthshould be low.
They are what they are. And second, if the perceptualbandwidth is low,
the ECS will have an easier time in matching theperceptual signal to
the reference signal, and the error signal, in yourparlance, will have
a low information content. Your deduction here isexactly the opposite
of what would happen. Of course if the reference signalvaried rapidly,
the error signal would also vary rapidly and containmore information –
but why would a reference signal from a higher, slowersystem vary more
rapidly than the perceptual signal of a lower, fastersystem?
    Now it is true that if the perceptual signal haslower bandwidth
    than the reference signal and the same resolution,then the error
    signal will in part be predictable, thus havinglower information
    content than would appear on the surface. But Ihad the
    presumption that we are always dealing with anorganism with high
    bandwidth perceptual pathways, so I forgot toinsert that caveat.
By your argument, a completely random error signalwould have the lowest
predictability of all, and thus contain the mostinformation. But so
what? The control system would not work with a randomerror signal.
    Well, given last year's experience, I didn'texpect my information-
    theory posting to be understood, and I wasn'tdisappointed in my
    expectation. Is it worth trying some more?
No, it is not. You don't have a clear and rigorousargument that can be
built up from basic principles without any outsideassumptions to carry
you across the rough spots. If you knew what you weretalking about, you
would be able to explain it clearly.
Lastly:
    The situation is different if we take afull-blooded outside view
    of the action of a CEV. It is from this kind ofview that we argue
    that the disturbance provides information thatpasses through the
    perceptual signal to the output signal. From theoutside we can
    see the disturbing variable do whatever it does toaffect the CEV,
    and we can see the ECS modifying its output tobring the perceptual
    signal back to its controlled value. From outsidewe can see the
    reference signal of the ECS changing, and theouput changing to
    move the CEV so that the perceptual signal comesto its new
    controlled value. From outside, the argumentsabout there being no
    information from the disturbance in the perceptualsignal lose
    their force.
So from the outside view, it is the information fromthe disturbance
that passes through the perceptual signal to the outputsignal, with the
result of modifying the output to bring the perceptualsignal back to
its controlled value? This takes us back to theoriginal information-in-
perception argument. If the information in theperception decreases as
the output comes to oppose the effects of thedisturbance more
precisely, how can it be the information passingthrough the perception
to the output that is responsible for the increase inprecision? Does
precision improve as the amount of information on whichit is based
decreases? What you are saying may make perfect senseto you, but to me
is is nonsense.
One more peanut:
    [Allan Randall 930325 12:40] to Rick Marken
    > >Are we also agreed that this disturbance,while defined in this
    > >external point of view, is nonethelessdefined in terms of the
    > >CEV, which is defined according to theinternal point of view?
    >
    > Say what? Why not just say CEV(t) = d(t) +o(t). If that’s what
    > the above sentence means then I agree withit.
    The point is that the disturbance d(t), ifseparated out from o(t),
    is not a meaningful quantity to the ECS. It ismeaningful only to
    the external observer. By drawing an arrow markedd(t) you are
    talking about something the ECS has no directaccess to. From the
    perspective of the ECS, only the variation in theCEV matters. It
    cannot separate out its own output from thedisturbance. On the
    other hand, this disturbance is defined in termsof the CEV, since
    only things in the world that affect the CEV canbe said to be
    disturbance.
It is not the disturbance that is defined in terms ofthe CEV, but the
effect of the disturbance. As you say, all that mattersis the value of
the CEV itself. Words like "meaningful" are justnoises. Talking about
the ECS "having access to" something is just a noise.My whole point is
that the ECS does NOT have "access" to the disturbanced(t). Nor does it
have "access" to the form of the function relating d(t)to its effect on
the CEV. Nor is the linking function or the nature andnumber of d(t)
variables necessarily the same from one moment to thenext.
The basic problem in the "information aboutdisturbance" argument is
that you keep forgetting that a given fluctuation ofthe CEV can be
produced by many different independent variables in theenvironment,
acting through many different paths, even from onemoment to the next.
All your arguments are based on the (often apparentlyunconscious)
assumption that there is a _single_ disturbing variableacting through a
_known and invariant_ disturbance function on the CEV.When that
assumption is true, your conclusions follow trivially,but you are
dealing only with a special case set up to MAKE yourarguments true. In
general, a control system _however intelligent andcomplex_ cannot know
what is causing a CEV to vary at any given time. All itcan know – that
is, all that can be represented by its perceptualsignal – is the
current state of the CEV. And that is all that it needsto know.
    If Signal X matches the disturbance, theperceptual signal must be
    the route from which the mystery function M(r, p)gets the
    information about the disturbance. Right?
    Now let the function M be indentical to O(R-P).ÂSignal X will then
    be the negative of the output signal, which is thedisturbance.
    The only question here is whether O(error) is afunction or a
    magical mystery tourgoodie. I prefer to think weare dealing with
    physical systems, and that O is a function.Therefore, information
    about the disturbance is in the perceptual signal,and moreover, it
    is there in extractable form.
    QED.
See what I mean? This sloppy analysis omits two things:the form of the
function through which even a single disturbance actson the CEV, and
the number of such functions with disturbing quantitiesoperating
simultaneously. What you have shown is that if youassume a single
disturbance acting through a unity transfer function,you can deduce its
value from knowledge of all other signals and functionsin the system.
Big surprise! But you have not shown that there is onlyone disturbance,
or that the form of the disturbance function is asimple multiplier of
1. You're in such a hurry to get to your triumphant“QED” that you
overlook an elementary omission in setting up yourimaginary experiment.
Enough. I'm just not up to following through all thesearguments which
are made up on the spur of the moment to meet aparticular case and then
forgotten about when the same principle comes up in adifferent context.
What I am hearing are arguments for the sake ofarguing, for the sake of
appearing to win an argument. I've been picking holesin your arguments
for a good four years now, with no discernible effect.I know when I am
trying to alter a controlled variable that is beingmaintained by a
strong and active system, although I may be somewhatslow to admit that
I can’t budge it.
This time I am going to stick to my oft-brokenresolution: no more
participation in this line of discussion.
Best,
Bill P.
–
Richard S. MarkenÂ
"Perfectionis achieved not when you have
nothing more to add, but when you
have
nothing left to take away.�
   Â
            --Antoine de
Saint-Exupery


