I came across this statement of Sayre’s Law: "In any dispute the intensity of feeling is inversely proportional to the value of the issues at stake." It made me wonder: What are the stakes in this math mistakes thread? The way I understand the “Math Mistakes� thread so far it seems Rick claims to have discovered some new truth about the velocity-curve whatchamacallit and Alex, Martin and Bruce claim (a) Rick doesn’t understand it and (b) Rick’s math is flawed.
Is there something really important and profound at stake here or simply egos at war? It seems to me that the parties involved are all talking past one another. Assuming there is something truly important at stake here, it seems to me that the use of a disinterested, third-party mediator might be in order. What do we need to resolve this dispute? An accomplished mathematician? A physics professor? What?
Fred Nickols
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From: Richard Marken [mailto:rsmarken@gmail.com]
Sent: Thursday, September 15, 2016 1:55 PM
To: csgnet@lists.illinois.edu
Subject: Re: Math Mistakes
[From Rick Marken (2016.09.15.1100)]
Martin Taylor (2016.09.15.13.05) –
MT: Having read Bruce’s request for Rick to restate Bruce’s explanation of the mathematical error at the heart of Rick’s problem, to show understanding, I would like to modify my request to Rick. I originally asked Rick to point out where in the material quoted below he had difficulty understanding, but since he hasn’t chosen to do that, I offer an easier alternative instead.
MT: Now I ask Rick just to provide a paraphrase of the argument.
RM: No thanks. The proper way to show that there is a problem with my analysis is to explain in in terms of my analysis. So show me how what Bruce calls my “conceptual error” and what you call my “mathematical error” affects the validity or correctness of my analysis. IN order to do this you have to know what my analysis is. As I told Bruce, you and Bruce have shown no evidence of understanding my analysis. So how about showing showing that you do understand my analysis by restating it and then explaiking exactly how my “conceptual” or “mathematical” error invalidates it.
Thanks.
Rick
It should only take four or five lines to paraphrase enough to show whether he does understand. The first of those lines might be something like this: “A basic equation for curvature is stated”. That would be enough to cover the first paragraph of the explanation.
With Rick’s paraphrases of these two explanations of the problem by Bruce and me we would have a basis for understanding why he thinks that the error either isn’t an error or is irrelevant to his claim to have demonstrated (proved?) something about the curvature power-law observations.
RM: No we wouldn’t. It’s your turn to shop that you understand my analysis.
Martin
--------material to be paraphrased, from [Martin Taylor 2016.09.13.14.55] (“they” are Gribble and Ostry)-----
Now we have to see how they came to equation (9). That’s a bit more complicated, so please bear with me.
They presumably either used someone else’s derivation or made their own, starting from one of several equivalent measures of curvature, one of which is C = 1/R where R is the radius of the osculating circle at the point of concern. Another one is developed using vector calculus, which I have no intention of introducing into this discussion. It is C = dx/dsd2y/ds2 - dy/ds * d2x/ds2, where s is distance along the curve from some arbitrary starting point.
For G+O this formula was not very convenient, because they would have had to measure these first and second derivatives of x and y with respect to distance along the curve fairly accurately. But they had a trick available, in the “chain rule” of differentiation: dx/dydy/dz = dx/dz. The "dy"s cancel out just like ordinary variables. Using the chain rule on the first derivative gives you the rule for the second derivative, and so on. For the second derivative the rule is (d2x/dy2)(dy/dz)2 = d2x/dz2.
Using the chain rule, G+O could multiply the formula for C by (ds/dz}3/(ds/dz)3 = 1, for any variable z that allowed the differentiation, to get C = ((dx/ds)(ds/dz)(d2y/ds2)(ds/dz)2)(ds/dz)3 - (dy/ds)(ds/dz)(d2x/ds2)__(ds/dz)3)/(ds/dz)3__. This formula is true (allowing for typos) for variable “z” whatever (as with the divide by zero example), but it wouldn’t have helped G+O very much, had it not been that for one particular variable they already had measures they could use. Those measures were the ds/dt velocity and the derived d2s/dt2 values they had obtained from their observations of movement. Using those measures, they could set “z” = t (time), making dx/dt = dx/ds*ds/dt. They could then take advantage of their measured velocities to substitute for ds/dt, and writeC = (dx/dtd2y/dt2)/V3 - (dy/dtd2y/dt2)/V3
Oh goody! We don’t have to measure anything new to get our curvatures. We can use the values of dx/dt and dy/dt that we got before! Very handy. … But also very confusing, because it made the published equations look as though the V3/V3 multiplier was special to the velocities they measured, whereas it was simply a convenient choice from a literally infinite variety of choices they could have made. G+O made it even more confusing in the publication by using the Newton dotty notation, which made it look as though there was something necessary about the time differentiation in the curvature equation.
When we put all this together, we come to the way this is a variant of the “divide by zero” error. That error depends on the fact that you can put any variable at all in for “x” in “x/0 = infinity”. The – shall we call it – the “curvature error” depends on the fact that you can use anything at all for V (including the measured values), provided only that V is defined as ds/dz where z is some variable for which ds/dz exists everywhere. You therefore cannot use the curvature equation in any way to determine V.
---------end material for paraphrase-------
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Richard S. Marken
“The childhood of the human race is far from over. We have a long way to go before most people will understand that what they do for others is just as important to their well-being as what they do for themselves.” – William T. Powers