Discussion of the speed-curvature Power Law

[rsmarken]

@bnhpct

@rsmarken

I believe he is saying that our mistake was to consider the term in parentheses in equation 1 below to be the same as the term in parentheses in equation 2. The difference that we supposedly missed is that the derivatives of x and y in equation 1 are with respect to time while those in equation 2 are with respect to space.

A quick fact check. Again, I claim no mathematical expertise, but in [Martin Taylor 2016.09.15.13.41], it appears to me that Martin’s objection is that (1) is with respect to a particular trajectory while (2) is an abstraction over an infinite set of possible trajectories…

Yes, and his justification for saying that is at the end of that post [[Martin Taylor 2016.09.15.13.41]] where he says:

MT: Actually one does not have to use mathematics to see the falsity of Rick’s point 4*. As has been pointed out many times, curvature has no relation to time, whereas velocity does. At the risk of inducing boredom by repetition, curvature has the dimension (1/length), while velocity has a dimension (length/time). Although they may be related in experimental observations, they are not, and could not be, mathematically related.

• My point 4. “So power law researchers are using linear regression to determine whether there is a linear relationship between variables that are mathematically linearly related, per equation 2.”

Martin made this point (that curvature is related to space while velocity is related to time) in his comment on Marken & Shaffer (2017) in Exp. Brain Res. That paper was the basis for my description of what I believed Martin thought was our mistake.

@bnhpct

That was in 2016, eight years ago. Is there someplace else that Martin abandons this objection and replaces it with an objection that the derivatives of one equation are in respect to time and the derivatives of the other are in respect to space?

Actually it was six, not eight, years ago and the objections to my analysis of the power law have been stated in many different ways during that time and I’ve tried to answer them substantively but, apparently, my answers have never satisfied my opponents. And vice versa.

So after six years of debate about the power law, some of which has taken place on CSGNet, some as published papers in journals (Exp. Brain Res.), and many now in IAPCT Discourse, we are clearly no closer to agreement than we were when this all started.

However, I think the fact that this discussion has been so persistent and so heated suggests that there is something very basic at stack here. I was getting at what it is in my earlier post where I quoted Bill on PCT as a paradigm shift. I think the power law is an excellent example of what Bill was talking about – a problem that no longer needs to be solved due to the paradigm shift that is PCT. I think the power law debate shows that some people think of PCT as a paradigm shift, and some don’t.