Hello Israel! Nice to see you here again. I´ll do my best to comment your thoughts, one by one. I´d like to point out that maybe I´m not the most indicated person to give arguments for the relational conception of motion. You can look at the references or email Barbour, Smolin and others for a deeper exposition. Anyway, above all I can say the relational picture is a viable alternative, and just that. We assume it and derive the consequences.
1. " He was aware that for practical purposes, what we measure with our instruments are relative motions relative spaces and relatives times, but behind these relative quantities there was also true motion, true space and true time. So, contrary to what most people erroneously believe, Newton was also a relativist. "
Yes, in this sense, he was relativist. You can see more on Newton´s discussion on absolute x relative in the "Scholium" of his Principia. You may use the word "relativistic" in this sense, though I think this could lead to some confusion. But let´s go on.
2. " Newtonian mechanics is relativistic, why? Because all spatial quantities are defined relative to a system of reference"
This is necessary by definition. Position have to be measured against something, be it an invisible background or a visible object.
3."There are an infinite set of frames of references, among these we can find space itself as the absolute frame (space as substance not as an empty container)."
The space as substance, or ether, is not directly accessible to observation. That is why Leibniz attacked this conceptions with the relational picture of motion.
4."So, if you have understood this, then you should understand that the quantities q_i(t) are not measured relative to absolute space but relative to a system of reference."
Of course, but for Newton, this system of reference could possibly not be observed. We can approximate it by the distant stars, or for the earth, depending on the problem we´re working on. This invisible background, or ether, is exactly what he calls absoulte space. For Newton, the system of reference could be defined anywhere in this "etherial container". Think of the bucket experiment.
5." Now, the fact that absolute space cannot be observed doesn't mean that it doesn't exist."
Perfectly. The relational conception of motion is not "obvious" or "necessary". It is simply an alternative for dynamical theories.
6. "So relationalism and relativism are not the same thing. Newton was absolutist and at the same time relativist but not relationalist."
Yes, agreed.
7."I agree that it can be superfluous but when you denied the absolute space we arrive at logical contradictions (such as the twin paradox in SR, please see Daryl Janzen essay for a deep discussion of this topic, perhaps you will realized how paradoxical SR is)"
I must admit I cannot understand why the twin "paradox" is a problem at all, and also, why it is a consequence of denying absolute space (?). The resolutions I have found seem reasonable. Maybe we should discuss this point more after I read Daryl´s essay. Lorentz invariance has given us not only direct empirical confirmation, but also QFT, the prediction of anti-particles, and so on. I find very hard to see how can SR be paradoxical.
8."When you say that we have a configuration of particles I have to ask you: in what physical thing do you think those particles are embedded? What is the physical thing that mediates between those particles"
Why should they be embedded somewhere? When we open our eyes and observe the universe directly, all we see are particle separations Rij. So why don´t we do physics with this information only? This would result in a simpler, more predictive theory. Why our mathematical descriptions should depart from the accesible empirical experience? Please notice I´m arguing as a relationist, but I´m aware this kind of discussion has been happening (verbally) for centuries. Relational and absolute conceptions of motion are not "obvious" or "necessary". Maybe they´re not even the only alterntives!(remember my essay last year?) I´m just considering the relational picture as a hypothesis and speculating on the consequences.
9."In relation to your formulation, you introduce a parameter of change called gamma. My question is: what is the physical meaning of gamma? In practice, how is this parameter measured? As far as I can see, this formulation is an attempt to get rid of the notion of time, but in this attempt you have to introduce a new parameter. In this sense, I only see an exchange of one parameter (t) for another (gamma)"
This gamma could be a watch´s pointer, or the rotation of the earth, for instance. It could be any visible parameter. Then we should have reparametrization invariance, somthing which does not happen in Newtonian theory. There´s one special choice of gamma that reduces to what Newton called absolute time, but here it is defined by the delta x of the bodies in the system (the equations are in the essay). This is just an "interpretation" of Jacobi´s principle. Now by applying best-matching, we arrive at a fully relational conception of motion. All "invisible" information is eliminated from the theory.
10."Finally, I'd like to invite you to read my essay and leave some comments."
Thanks, I´ve been very busy, but next week I´ll have time to read essays and comment carefully.
Best regards
Daniel