Post by Ben on Mar 28, 2023 6:58:55 GMT
Miles has made (at least) the following arguments:
1. It is mathematically valid to calculate the circumference of a circle using a non-Euclidean metric. No one disputes this as non-Euclidean metrics have been known for some time. But that is all his "proof" is - replacing the Euclidean distance with taxicab distance in the path-integral formula and performing a simple calculation. Obviously, the problem with the taxicab metric is it's not how the real world works: it is faster to move along the diagonal of a square than along its sides. You have to show how to reconcile your calculation with the fact that objects in the real world don't all move like taxicabs.
2. Calculations of discrete quantities don't necessarily apply "at the limit." Likewise, "Real bodies traveling curves do not travel the limit of the hypotenuses, they travel the limit of the orthogonal legs." Maybe so, but that leaves us in the same boat as before.
Consider the diagonal of a unit square (lowest blue).
An object traveling straight on the diagonal travels the Euclidean distance, which is sqrt(2). I hope there is no dispute about that. If its path curves slightly (cyan), the distance will increase slightly as well. If we used the taxicab metric the distance would bounce up to 2. So you can't just jump to another metric for a curve or your measurement will be highly discontinuous and way off.
Likewise, if you move along two sides of the square (outermost red), you will travel the length of the two sides, which is 2. If your path bows in slightly (uppermost blue), common sense and experience would dictate that your distance diminish as well, getting closer to the diagonal. The taxicab metric says it would remain constant at 2, and that is all but inconceivable.
I had considered the possibility of a metric that combines the Euclidean and taxicab distances, with weights based on curvature. That is mathematically possible but again, very counterintuitive.
The bigger problem however is that the taxicab metric is sensitive to rotation.
Suppose you have a square tile and you move along its diagonal.
The path is exactly the same, but the distance you've travelled is now shorter! Because your basis vectors have changed, it is now sqrt(2)*s.
Obviously, the choice of basis vectors has to be determined by something. If it is controlled by the way you're facing, as in the second case here, then you're back in the Euclidean world. Otherwise it is just arbitrary.
To put it another way: if a body travels the limit of the orthogonal legs, then WHICH orthogonal legs does it travel the limit of? Different legs will give you different answers. It doesn't matter on a circle, because of rotational symmetry, but to measure arbitrary curves, the only way to get a consistent (basis-independent) answer is going to be to decompose the curve into a component with rotational symmetry and a component without. The taxicab distance of the first component would be incorporated somehow with the Euclidean distance of the second.
That might be doable, but I couldn't begin to formalize it. Maybe Miles has a beautiful picture in his head of how it would work, but it doesn't amount to much if he can't provide any formulas or algorithms. A few lines of actual math here would be much more valuable than pages of invective. He just needs to show how to calculate distance travelled and then experiments can work out whether it's accurate.
3. Abstruse statements along the lines of, "Time adds a degree of freedom"; "You can't express a velocity as a curve over time" and so on. I have read and re-read his arguments here, and can only conclude that he is confused.
Let's start with the basics. In classical mechanics, position, velocity, and acceleration are generally expressed as functions of time. That means that if you know the time, you have a rule that will tell you where something is and how it's moving. Velocity and acceleration are derivatives, or linear transformations approximating a function at a point. For example, velocity gives a linear approximation to the position function near a point. Being linear is not the same thing as being constant. If a derivative is zero then the derivand is constant. In circular motion, velocity is not constant because acceleration is nonzero. Its acceleration also changes with time. In fact, its Taylor expansion is infinite, meaning that you can take derivatives indefinitely and never obtain a constant.
Miles argues that because circular motion accelerates (i.e., its velocity changes), the position function cannot be approximated by a linear velocity vector. He seems to think that change over time conflicts with linear approximation, but doesn't explain why. I really cannot follow the reasoning here. He also doesn't explain how his taxicab calculation solves this. As I see it, it is just going to introduce error of more or less magnitude depending on your choice of coordinates.
In relativity, the position and motion of objects are expressed differently because the timing of events can differ among reference frames. You and I might have a different idea of when an event has happened. In that context, the classical model of position and movement as functions of time is incomplete. That's what is meant by time adding a degree of freedom. But that doesn't really matter in everyday circumstances, like riding a bicycle, where you are far below the speed of light.
Miles quotes a calculation from Wikipedia that contains three variables (x, y, and t) and claims that it has three degrees of freedom and that is why it evaluates to 4, rather than pi. This is simply not correct. x and y are defined there as functions of t, which means they are not degrees of freedom. A path parameterised over time has only one degree of freedom: time. The reason the integral evaluates to 4 is that the definitions of x and y - which he omits - specify motion in a cycloid. If they defined motion in a circle, the form of the integral would be the same, but the result would be pi.
I am sorry to say, but glaring errors such as these aren't acceptable coming from someone who claims to have debunked Euler and corrected Einstein. He will probably call me a spook for saying that, so I will add the following: I like him; I agree with him on a lot of things; I don't think he is a crank; and I think he is human and imperfect like all of us. His mathematical training was limited, overshadowed by his time in philosophy, and that is nothing to be ashamed of. But his ego gets in the way of seeing it.
And I am not claiming to be a great mathematician either. It's not my profession and I could learn a lot more if I wanted to.
4. Math used by NASA and others "blowing up" and needing to be fudged. Frankly, there is not much to say here, owing to a lack of specifics. But if the error is as fundamental as he says then I think it is unlikely that throwing in fudge factors would fix much.
5. Experiments. There is not much to say here either. The tubes are not very exact and the tracks gave an answer that no one expected (3.2something, if I recall). Miles went to great length to argue that the track was underestimating, but we can't really be sure. As far as I'm concerned the experimental evidence for his hypothesis is still not there.
With all of that said, I will allow that he might have had a real insight that no one else has. He is a strong physical/intuitive thinker and he may see things that are difficult to get across in writing. But the burden is still on him to make his point clearly and concisely, and he has definitely not done that.
1. It is mathematically valid to calculate the circumference of a circle using a non-Euclidean metric. No one disputes this as non-Euclidean metrics have been known for some time. But that is all his "proof" is - replacing the Euclidean distance with taxicab distance in the path-integral formula and performing a simple calculation. Obviously, the problem with the taxicab metric is it's not how the real world works: it is faster to move along the diagonal of a square than along its sides. You have to show how to reconcile your calculation with the fact that objects in the real world don't all move like taxicabs.
2. Calculations of discrete quantities don't necessarily apply "at the limit." Likewise, "Real bodies traveling curves do not travel the limit of the hypotenuses, they travel the limit of the orthogonal legs." Maybe so, but that leaves us in the same boat as before.
Consider the diagonal of a unit square (lowest blue).
An object traveling straight on the diagonal travels the Euclidean distance, which is sqrt(2). I hope there is no dispute about that. If its path curves slightly (cyan), the distance will increase slightly as well. If we used the taxicab metric the distance would bounce up to 2. So you can't just jump to another metric for a curve or your measurement will be highly discontinuous and way off.
Likewise, if you move along two sides of the square (outermost red), you will travel the length of the two sides, which is 2. If your path bows in slightly (uppermost blue), common sense and experience would dictate that your distance diminish as well, getting closer to the diagonal. The taxicab metric says it would remain constant at 2, and that is all but inconceivable.
I had considered the possibility of a metric that combines the Euclidean and taxicab distances, with weights based on curvature. That is mathematically possible but again, very counterintuitive.
The bigger problem however is that the taxicab metric is sensitive to rotation.
Suppose you have a square tile and you move along its diagonal.
Suppose your coordinate basis vectors run parallel to the sides of the square. Going from one corner to the other, according to the taxicab metric, you will travel twice the length of a single side, or 2s.
Now turn the square (and the basis vectors) 45 degrees and traverse the same path as before.
Now turn the square (and the basis vectors) 45 degrees and traverse the same path as before.
The path is exactly the same, but the distance you've travelled is now shorter! Because your basis vectors have changed, it is now sqrt(2)*s.
Obviously, the choice of basis vectors has to be determined by something. If it is controlled by the way you're facing, as in the second case here, then you're back in the Euclidean world. Otherwise it is just arbitrary.
To put it another way: if a body travels the limit of the orthogonal legs, then WHICH orthogonal legs does it travel the limit of? Different legs will give you different answers. It doesn't matter on a circle, because of rotational symmetry, but to measure arbitrary curves, the only way to get a consistent (basis-independent) answer is going to be to decompose the curve into a component with rotational symmetry and a component without. The taxicab distance of the first component would be incorporated somehow with the Euclidean distance of the second.
That might be doable, but I couldn't begin to formalize it. Maybe Miles has a beautiful picture in his head of how it would work, but it doesn't amount to much if he can't provide any formulas or algorithms. A few lines of actual math here would be much more valuable than pages of invective. He just needs to show how to calculate distance travelled and then experiments can work out whether it's accurate.
3. Abstruse statements along the lines of, "Time adds a degree of freedom"; "You can't express a velocity as a curve over time" and so on. I have read and re-read his arguments here, and can only conclude that he is confused.
Let's start with the basics. In classical mechanics, position, velocity, and acceleration are generally expressed as functions of time. That means that if you know the time, you have a rule that will tell you where something is and how it's moving. Velocity and acceleration are derivatives, or linear transformations approximating a function at a point. For example, velocity gives a linear approximation to the position function near a point. Being linear is not the same thing as being constant. If a derivative is zero then the derivand is constant. In circular motion, velocity is not constant because acceleration is nonzero. Its acceleration also changes with time. In fact, its Taylor expansion is infinite, meaning that you can take derivatives indefinitely and never obtain a constant.
Miles argues that because circular motion accelerates (i.e., its velocity changes), the position function cannot be approximated by a linear velocity vector. He seems to think that change over time conflicts with linear approximation, but doesn't explain why. I really cannot follow the reasoning here. He also doesn't explain how his taxicab calculation solves this. As I see it, it is just going to introduce error of more or less magnitude depending on your choice of coordinates.
In relativity, the position and motion of objects are expressed differently because the timing of events can differ among reference frames. You and I might have a different idea of when an event has happened. In that context, the classical model of position and movement as functions of time is incomplete. That's what is meant by time adding a degree of freedom. But that doesn't really matter in everyday circumstances, like riding a bicycle, where you are far below the speed of light.
Miles quotes a calculation from Wikipedia that contains three variables (x, y, and t) and claims that it has three degrees of freedom and that is why it evaluates to 4, rather than pi. This is simply not correct. x and y are defined there as functions of t, which means they are not degrees of freedom. A path parameterised over time has only one degree of freedom: time. The reason the integral evaluates to 4 is that the definitions of x and y - which he omits - specify motion in a cycloid. If they defined motion in a circle, the form of the integral would be the same, but the result would be pi.
I am sorry to say, but glaring errors such as these aren't acceptable coming from someone who claims to have debunked Euler and corrected Einstein. He will probably call me a spook for saying that, so I will add the following: I like him; I agree with him on a lot of things; I don't think he is a crank; and I think he is human and imperfect like all of us. His mathematical training was limited, overshadowed by his time in philosophy, and that is nothing to be ashamed of. But his ego gets in the way of seeing it.
And I am not claiming to be a great mathematician either. It's not my profession and I could learn a lot more if I wanted to.
4. Math used by NASA and others "blowing up" and needing to be fudged. Frankly, there is not much to say here, owing to a lack of specifics. But if the error is as fundamental as he says then I think it is unlikely that throwing in fudge factors would fix much.
5. Experiments. There is not much to say here either. The tubes are not very exact and the tracks gave an answer that no one expected (3.2something, if I recall). Miles went to great length to argue that the track was underestimating, but we can't really be sure. As far as I'm concerned the experimental evidence for his hypothesis is still not there.
With all of that said, I will allow that he might have had a real insight that no one else has. He is a strong physical/intuitive thinker and he may see things that are difficult to get across in writing. But the burden is still on him to make his point clearly and concisely, and he has definitely not done that.
