Paradigm Lost...(and Regained?)

This is my attempt to explain why the discussion of the power law of movement is important, to me anyway. It is because the disagreement about how to explain the power law illustrates why PCT has had – and continues to have – so much difficulty being accepted by mainstream scientific psychologists and being understood by those who have accepted it.

The difficulty comes from taking PCT to be a new theory when, in fact, it is a new paradigm. A paradigm is a set of "past scientific achievements that some particular scientific community acknowledges for a time as supplying the foundation for its further practice (T. S. Kuhn (1996) The Structure of Scientific Revolutions, 3rd Ed., Chicago, University of Chicago Press, p. 10). “Further practice” refers to what that scientific community considers the right way to go about their business; it’s “normal science”.

So, a paradigm is the principled basis for the conduct of normal science. It is not a theory but, rather, the context in which different theories are tested. In scientific psychology the current paradigm is what I have called “the causal model” and it is the basis for the conduct of what is considered normal scientific research in psychology; research that is used to test many different theories in many different subdisciplines of psychology.

This causal model paradigm emerged in the early 1900s with the success of Behaviorism. Powers seems to have understood Behaviorism as equivalent to the causal model paradigm because he began his discussion of PCT in Behavior: The Control of Perception by contrasting it to Behaviorism. Unfortunately, the presentation of PCT as a new paradigm was somewhat conflated with his description of PCT as a new theory. But he didn’t conflate theory and paradigm in his Spadework paper (Powers, W.(1978) Quantitative analysis of purposive systems: Some spadework at the foundations of experimental psychology, Psych. Rev., 85, 417-435) when, at the end of that paper he says “Kuhn (1970) uses the term paradigm in the sense I mean when I say that control theory is a new paradigm for understanding life processes—not only individual behavior but the behavior of biochemical and social systems.”

The relevance of all this to the power law is summarized in another quote from Powers’ 1978 paper: “The difficulties faced by a new paradigm, as Kuhn explained so clearly, result not from battles over how to explain particular conceptual puzzles, but from bypassing altogether old puzzles that some people insist for a long time still need solving”. The battles over the PCT explanation of the power law reflect the difficulties faced by this new paradigm. They have been ostensibly over how to explain the power law but, in fact, they have been mainly over whether or not the power law is an “old puzzle” that no longer needed explaining.

The fact that this was the case can be seen in the nature of the reaction to the PCT explanation of the power law. If it were just a reaction to an alternative theoretical explanation of the power law then the battle would have been over which theory best accounts for the data. But the battle was mostly over whether the PCT explanation was scientifically correct, suggesting that the reaction was to a new paradigm; a paradigm that suggested that the power law was a phenomenon that no longer needed explaining.

The reaction to the PCT explanation of the power law has been consistent with what Kuhn ( and Powers) expected it to be; a reaction to a new paradigm as opposed to a new theory. A new paradigm would be seen as not just a wrong theory, but as a wrong way of doing science itself; it would be seen as not normal or not good science. I think this is one of the main reasons why PCT has had such a hard time gaining traction in scientific psychology. To the extent that PCT has gained some apparent traction it has done so by being sold as an alternative explanation of observations that have been made in the context of the current causal model paradigm; the fact that PCT is a completely new paradigm on which to base a new science of psychology has been lost.

Actually, it was over whether your interpretation of the mathematics was correct. Nothing more.

Same thing.

A very different thing, actually.

Here is the part of my Paradigm Lost post to which you were responding:

If it were just a reaction to an alternative theoretical explanation of the power law then the battle would have been over which theory best accounts for the data. But the battle was mostly over whether the PCT explanation was scientifically correct , suggesting that the reaction was to a new paradigm;

You say the battle over my PCT explanation of the power law was over whether my interpretation of the mathematics was correct (it used to be over whether the mathematics per se were correct but then the Maoz et al article came to light and the criticism had to change). Since it was not over which theory best accounts for the data, I consider that to have been a battle over whether I had done the “normal science” correctly.

Interesting how perceived history changes, isn’t it? I suggest that in itself is a topic worth studying. It has been much studied outside the realm of PCT, but I don’t remember it having been a study topic within PCT. Nor do I have much if anything to say about it in PPC. Outside of PCT, I believe the consensus is that we reassemble a historic narrative from fragments that seem to fit together. I wouldn’t be surprised if someone on this forum has studied it in a PCT context.

Martin

[I sent hours ago this reply by email. I hope it will not come twice.]

I have never yet seen a PCT explanation of the Power Law phenomenon. Have others? All I have seen has been: 1) an unjustified but very plausible assumption that the phenomenon is a side effect of control; 2) an unplausible and unscientific numerological explanation of the phenomenon; and 3) a coarse misuse of the OVB analysis meant to support the numerological explanation.

And BTW, then paradigm theory of Kuhn is no way the last or even current word about the development of science. For example, Lakatos’ theory of scientific research programs is much more robust. I don’t believe that Kuhnian based revolution romanticism is very blessing for the future fate of PCT as a scientific research program.

Eetu

I agree with Eetu. I keep asking myself why call the power law a phenomenon.

Seems to me it is extremely contrived to take something you may or may not even observe (I sure don’t know how to observe it) and come up with a supposedly mathematical, exponential formula (which varies all over the place) to define it.

Seems to me this is the creation of “scientists” in a “science” that does not have much more going for it than the epicycles defined by Ptolemy. That is all observation, with formulas created any which way to fit appearances.

Dag

I have never yet seen a PCT explanation of the Power Law phenomenon. Have others? All I have seen has been: 1) an unjustified but very plausible assumption that the phenomenon is a side effect of control; 2) an unplausible and unscientific numerological explanation of the phenomenon; and 3) a coarse misuse of the OVB analysis meant to support the numerological explanation.

And BTW, then paradigm theory of Kuhn is no way the last or even current word about the development of science. For example, Lakatos’ theory of scientific research programs is much more robust. I don’t think that Kuhn based revolution romanticism is very blessing to the future of PCT as a scientific research program.

If you’re referrring to the history of your criticism of my explanation of the power law then that history is well documented. In your published rebuttal and in many private emails to me you said that our error was a mathematical mistake that should rule out publication of our paer; the mistake was my failure to take the derivatives of the x and y position of the movement with respect to space rather than time when we computed radial curvature. The pitch now, at least according to Eetu, is that the math is fine it’s just our interpretation of the results that is mistaken.

The PCT explanation of the power law is that it is an irrelavent side effect of control. This has been demonstrated in my models of movement control described in both M&S (2017) and M&S (2018) as well as in my recent Behavioral Illusions paper (2022, See Figure 9 and the discussion thereof). The OVB analysis of M&S (2017) and Maoz et al (2006) explains why the obsered coefficient of movement is typically “in the ballpark” of its “lawful” value, -1/3.

How long since you read the published rebuttal carefully? Did you read it carefully even once? How does your statement quoted here relate to it? I confess, I cannot see any relationship.

Yes. And you said we made a mathematical mistake. Here’s the text:

What “whereas in (5) they are arbitrary parameters, valid for any velocity…” means is that the derivatives in (5) are with respect to space, not time. That’s a mathematical mistake!

The fact that you considered it a mathematical mistake can also be seen, perhaps more clearly, in a post from you to CSGNet [Martin Taylor (2017.06.05.14.19)] where you clarified to me that you wanted me to send a letter asking the editor who accepted the M&S paper “… to get an independent expert to adjudicate the mathematical question and its influence on the rest of the paper” (emphasis mine). Clearly, your hope was that such an expert would find the egregious mistake, the presses would be stopped and the offending paper would never see the light of day.

In that same post you even wrote up a verion of the letter you wanted me to write to the editor. Here it is:

This shows pretty clearly that you considered my mistake to be mathematical; a failure to note that, in the equation for curvature (1/R) the derivatives of the x and y position of the movement are with respect to space rather than time. This mathematical mistake presumably meant that the equation I derived relating velocity to curvature and affine velocity was misleading. So our OVB analysis was all wrong.

But then came the revelation that other power law researchers had derived the exact same equation for the relationship between velocity, curvature and affine velocity that we did and that one of those others, Maoz et al, did the exact same OVB analysis we did. So now I went from having made a mathematical mistake to incorrectly interpretating the results of the analysis.

This is classic Kuhnian defense against a paradigm shift.

Nothing you say here is relevant to the problem, that you took an analysis valid for any velocity as though the fact that it is valid for velocity V as proof that V is the true velocity.

I notice the same tendency in much of your work. If X is true, it is therefore the only thing that is true about the situation. I don’t remember just when, but there was a paper you wanted me to co-author because I agreed that your hypothesis worked, ignoring that I also said that a different hypothesis also worked. You wanted the paper to say that this proved your hypothesis was the correct one, contrary to the truth. I refused to be a co-author, since we both knew that your chosen one was not the only possibility, but you went ahead and published anyway,

There are very few peer reviewers who would realize that there are alternative possibilities when asked to review a paper that shows one possibility that works, on a topic on which they have not worked.

[quote=“MartinT, post:14, topic:16014, full:true”]
Nothing you say here is relevant to the problem, that you took an analysis valid for any velocity as though the fact that it is valid for velocity V as proof that V is the true velocity.

[quote]

I think I undertsand your point now. Since the derivatives in the equation for curvature are with respect to space (dx/ds) rather than time (dx/dt) the mathematical relationship between curvature and velocity that I (and Maoz et al) derived does not describe a unique relationship between these variables because movements with the same velocity profiles (measured with respect to time, dt) can have quite differnt associated curvature profiles (measured with respect to space, ds). This is because movements can traverse a distance between two points (dx) at the same speed (dt) through what is probably an inifnite number of different degrees of curvature (ds).

I think Maoz et al (2006) implicitly addressed this problem right at the start of their analysis:


Maoz et al seem to have understood that curvature (kappa) is parameterized with both t and s as per your version here:
image
where kappa = 1/R
But Maoz et al have apparently found a way, using the Frenet-Serret formulas, to paramaterize the equation for curvature with t alone, making it possible for them to derive equation (3), where curvature, kappa(t), is written as a function of velocity, v. I, of course, have no idea why the Frenet-Serret formulas allow them to do this but I looked up Frenet-Serret in Wikipedia and, in what has to be one of the most spectacular examples of irony I have seen since Kissenger won a Nobel Peace Prize, it seems to have something to do with the use of the Taylor series.

And, of course, researchers who actually test the power law hypothesis against actual data calculate the values of the variables, velocity and curvature, using the time (dx/dt) parameter version of the equations that define these variables, as in equation (3) above.

A copy ot the paper that was published can be found here. I probably asked you to be co-author because you collected the data on which it is based (and as you will see, you are acknowledged for this in the paper). And I don’t recall refusing to test only one possible hypothesis regarding your data. As you’ll see, if you read the paper, I tested three possible hypotheses, one of which – the threshold-distance control hypothesis – was yours. And (spoiler alert) it faired pretty well until the very end of the paper.

I don’t remember anything that I might label a “threshold distance control” hypothesis. Maybe you could remind me by explaining that hypothesis and what controlling a threshold distance might mean.

The description can be found in the paper which (to repeat) is found here. It’s at the top of the page that contains Figure 6.