Hi Efthimios,
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btw, I found your article to offer a lot of food for thought and hope you score high on the final ballot. Best of luck.
Anyway, regarding this issue, I don't think the problem here is one of infinity, it is the way we think about infinity.
First of all, I think we need to keep in mind that mathematics simply represents a model for reality. You stated, "I want to show that calculus wants its cake and eat it too when it comes to justifying motion."
I think it was Confucius who said, "Do not confuse the finger pointing to the moon for the moon itself." If we were to rely exclusively on mathematics to form our picture of reality, we would immediately find ourselves in a lot of trouble. This is why we check to make sure a mathematical solution to a problem is physically admissible. If the solution does not conform to what we observe then we do not throw out our observations, we reconsider the validity of the solution or we reevaluate our line of reasoning that led to the solution. Sometimes, a theorist cannot decide whether or not a solution to a problem is a physically admissible one(e.g. String Theory), but that is another story. The danger is relying too heavily on theory to form a complete picture of reality.
Basically then, in the context of this discussion, motion is possible and exists because we observe it to exist. Any argument or solution that infers motion is impossible must therefore either be reframed or thrown out completely. There is no way around this. Observation and experiment always has the final say in science. So, we are on shaky ground when we expect our mathematics to justify motion.
I think that the conceptual difficulties inherent in these types of arguments all lie in our ideas about casuality.
IMO, the fundamental question is not why do things happen the way they do, but why does anything happen at all? This is where Newton, motion, and the concept of inertia comes in. If you think about it critically, the first law is logically necessary in any universe where cause and effect has meaning. This is because the first law essentially reduces to a simple statement about causation--nothing happens without an impetus to action or a reason behind the impetus. If this were not the case, we would simply have random chaos and unpredictability.
The same applies to any idea of uniform motion--i.e. inertial. In the context of this discussion, there is not an infinite number of actions taking place in inertial motion because there is no action required. Uniform motion is relative and there is no preferred frame of reference. In one frame, an object may appear to undergo inertial motion. In another frame of reference, the object may appear to be at rest. If action was required to maintain inertial motion then motion would not be relative and there must be a preferred frame of reference.
Zeno's Paradox,in all it's incarnations,therefore leaves out this concept of relative motion. In one frame of reference where an object is undergoing uniform motion and must travel the segment AB, one can find another frame of reference where the object is at rest and the length of the segment is 0. Which one corresponds to reality? Zenos paradox becomes a non-sequitur when relative motion is considered.
Also, when you inquire-- ."But calculus tells us that ds/dt = at. How can that be?", the answer is because the term 'a' implies an impetus that acts to change unfiform motion or a state of rest relative to an observer. Implicit in the calculation is the concept of non-uniform motion. For each element dt, the object 'caries' with it the velocity it had at t-dt. Through impetus, an additional velocity was added to the object during the time element dt.