Cristi,
I am sorry I can't address all the issues and tend to focus on those which concern me most, but that is the nature of the game. Since we don't see the particular issue of the properties of points from the same point of view, If I may, I would like to put the problem in a broader, historical context and use that to illuminate the actual issue which led me to the issue of dimensionless points.
As I see it, math is the study of pattern and order, then using this to project out to make connections with other patterns and predict how processes with evolve. Given this reality is highly complex and often chaotic, the math is also enormously complex.
Now one of the oldest sciences is cosmology, the study of the celestial objects and their actions. This has been going on for tens of thousands of years and while it only broadly occurred to astronomers that the earth was not the center of the cosmos about 500 years ago, the cosmology of epicycles, placing terra firma at the center of the cosmic coordinate system, was extremely accurate and effective. There was nothing wrong with the math, because any point can be the center of a coordinate system. As individual beings, each of us is the center of our view of the entire cosmos, so logically we could construct a model with everything moving relative to our position. It would be an effective representation of our experience of reality.
The problem is trying to deduce a physical explanation for this perceived order and with cosmology, a significant factor was that we are not the center point of the larger universe. The sun is the general centerpoint of the solar system, while the Milky Way revolves around the black hole at its center, the name of which currently eludes me.
So can you see how math informs physics, but is a necessary and useful mapping of the observed order, not an underlaying basis of that order? That when we assume the perceived order is foundational, without remaining skeptical, it creates problems. Remember we still see the sun as rising in the east and setting in the west.
So the issue which originally led me to this view is time. Since our minds function by forming a sequence of perceptions, we think of time as the point of the present "flowing" past to future. Physics codifies this as measures of duration, between events and then assumes duration is evidence of some underlaying dimension, which is then correlated to measures of distance. Then assuming that dimension of duration functions as a fourth coordinate.
Now mathematically it is all very functional, but there are physics issues, such as assuming time is symmetric, meaning there is no difference which direction it goes, it would be the same duration. As well as possibly all events still extant on this underlaying dimension, but since the math works, one is just supposed to "shut up and calculate."
As I see it though, it is not so much the present moving past to future, but change turning future to past, as in tomorrow becomes yesterday, because the earth turns. Duration is simply the state of the present, as events coalesce and dissolve, meaning past and future do not physically exist.
Therefore time is an effect of action, similar to temperature. Time being measures of individual frequency, while temperature is an effect of masses of amplitudes and frequencies.
We could correlate temperature and volume, using ideal gas laws, but assume ourselves to be more objective about temperature, since it is only the basis of our emotions, bodily functions and environment, not the narrative flow of our thought process.
So it is a similar problem to epicycles, in that the cause is going the other direction from the position we perceive. Earth turning west to east, as with future becoming past.
Time is asymmetric because it is a measure of action and action is inertial. The earth turns one direction, not both.
Different clocks can run at different rates and remain in the same present, because they are separate actions. All things being equal, a faster clock uses more energy, as with metabolism.
Simultaneity was dismissed by observing different events with be perceived in different order from different points of view, arguing all these events must still exist along that fourth dimension. Yet this is no more consequential than seeing the moon as it was a moment ago, simultaneous with seeing stars as they were years ago. It is the energy that is conserved, not the information it carries. It is that this information changes is what creates the effect of time in the first place. It is the very fact that the energy manifesting an event is radiated away, that we can observe it having happened, as well as why it no longer exists.
So, not to be too persistent, can you see why I have issues with the "shut up and calculate" crowd?
Regards,
John