In his papers, Peter Lynds assumes that if we make short the interval in time in which we see a moving object, and we reduce this time interval to a single instant in time, the object should remain "static", and hence, it could not change its position at a later stage. It would remain frozen.
He tries to say that there would be no difference between a moving and stationary object if we take a snapshot in that very precise instant. But obviously there is a difference, so it is clear that such a thing as a single instant in time cannot exist . That is his line of reasoning.
He states, as we have told, that there is nothing in that "mental" snapshot that could indicate that the object is moving.
However, a body, in Classical Mechanics, has some assignable properties (as a "label" on them), such as electrical charge, mass, and energy.
The difference between the two bodies, that which we imagine is moving, and the other which is static, when we take a mental snapshot, and see them "frozen" in time, is the kynetic energy which they have associated. Not matter that, in order to make a measurement of this energetic content, we should try to determine that energy taking measurements during a time interval. We can imagine that, at this precise instant in time, the moving body has a kinetic energy associated with it, and there is nothing of "logical neccesity" that contradicts this. The direction of movement and speed is associated to linear momentum, another physical magnitude assignable to the body at any instant.
Lynds tries to justify its postulate with this logical reasoning, but of course, and because of the reasons above, we believe he doesn't succeed. So simply there is the formulation of a postulate, that there cannot be a "static" instant in time, which, apart of not being a quite clearly exposed concept, leads to nowhere: how are the equations and methods of physics changed in order to accomodate to this new principle?
He later tries to reinforce his assumptions saying that:
"With some thought it should become
clear that no matter how small the time interval, or how slowly an
object moves during that interval, it is still in motion and it's
position is constantly changing, so it can't have a determined
relative position at any time, whether during a interval, however
small, or at an instant. Indeed, if it did, it couldn't be in motion."
But this can be clearly rebutted. Now we replace space and time for x and y (horizontal and vertical axes) in the text above:
"With some thought it should become
clear that no matter how small the Y interval, or how small the slope of the point during that interval, the slope still exists and it's
x-position is constantly changing, so it can't have a determined
relative x-position at any y-position, whether during a interval, however
small, or at a y-point. Indeed, if it did, the slope couldn't exist"
And, as every math student knows, this reasoning is wrong.
Note: We change the concept of being in motion for the equivalent concept of some given continuous function having a slope. The object having motion is now a point in a 2-dimensional space, having coordinates x and y.
It should be noted that Lynds reasonings can be applied both to the Physics of movement and to mathematical slopes, there is nothing that is intrinsically attached to movement in his reasonings. Thus, the objection arisen by some referees that he did not apply well Calculus seem to be clearly demonstrated with this substitution of terms and concepts.
So there are 2 points here: the second is obviously a mistake on appying infinitesimal calculus. The first one, and much more subtle, is some kind of methapysical reasoning, in which he tries to demonstrate that if we accept the concepts of "instansts" in time, we cannot have an object moving. But we believe we have demonstrated so.
In fact, a dynamical system, it is completely determined, at least in classical mechanics, if we give both spatial coordinates and momentum values in a given instant of time. Thus linear momentum is an "attached" characteristic of a body, as can be its mass or charge or spin.
There is another point that, at first glance, may be even more convincing. See here. However, note that this reasoning may be tricky. A moving reference system, acording to Lynds, may have "problems" on defining precise locations in space.
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Cesar Sirvent