Status Updates From The Discovery of Dynamics: ...

The way forward for Mach was to complete the work that Kepler and Newton had begun and make dynamics completely interactive. Only such an approach could solve satisfactorily the problem of the frame of reference."

We have already been introduced to what is the most important thing about this 'suspension' - that it is arbitrary. This is the price which Newton pays for the only semi-interactive nature of his dynamics: ultimately, it leads to a loss of predictive power, as will be shown in Vol. 2

...will be spelt out in volume 2

What makes the family of inertial systems so mysterious is not so
much its specific structure (which is odd enough - rotationally absolutely
rigid but slightly undetermined translationally) but rather the manner in
which it is 'suspended' in a decidedly arbitrary manner in the midst of the observable matter. The nature of this curiously disjunct suspension of the invisible inertial systems in the observable matter...

We can conclude this discussion of the definition of mass with the
following comment of Mach: 'All uneasiness will vanish when once we
have made clear to ourselves that in the concept of mass no theory of any kind whatever is contained but simply a fact of experience.

(Mach's extreme positivistic approach. Of course, we have moved on from this.)

Mass liberated itself little by little from its predynamical geometrical origin and came to be understood as that 'something' in matter which measures its inertia. It was transformed from meaning volume into a substance, the Atlas-type prop to support the observable attribute of resistance to acceleration..

Newton gave an explicit definition of density along exactly these lines: 'Bodies are denser when their inertia is more intense, and rarer when it is more remiss.' This is clearly a totally different approach to the problem and amounts to a dynamic definition of mass: the amount of mass is measured by the resistance it offers to a given force. We
shall see that this is a clear anticipation of Mach's definition of mass

Such a mechanism was not, of course, open to the plenists like Descartes and Leibniz. They had to rely on the differing ability of matter to give way to other matter.

(Is this way of defining inertia, not quite reminiscent of how the higgs field now is said to give mass to things?)

Thus was born the concept of occult quality - an intrinsic property by which a body's motion is directly determined; occult because there was nothing in the appearance of the body to tell us that it possessed such a quality. The intellectual battle which raged in the seventeenth and early eighteenth centuries was all about such concepts

What moved Aristotle more than anything else in the direction of genuine dynamics was probably a gut awareness that three-dimensional geometry by itself could never explain why motion occurs at all. Motion is something genuinely new. Thus, where the atomists and Plato had a concept of matter determined almost exclusively by extension and shape, Aristotle introduced attributes of matter that related directly to motion

Huygens sensed the pure relativity of motion; he anticipates the
sentiments that we shall find in Vol. 2 in Berkeley's and Mach's writings.
The paper begins with the statement that 'motion between bodies is only relative'. He points out that in the case of a single body (alone in the universe) it is quite impossible to conceive any difference between motion and rest of such a body

And Newtons specific confounding by trying to find an absolute motion, when that motion was circular, and was thereby really a special case of acceleration, and not just of motion as in velocity, or displacement over time in space.

This greatly clarifies the specific limitations of the Newtonian approach. ANd the whole absolute vs relational debate. If we need a materialisation for the reference frame, but the materialisation is arbitrary. Then its materialisation goes against pure relationalism about space, and its arbitrariness and superfluity from then on, goes against the absolutist view of space.

The temptation to dispense with the particular materialization is very great. It seems rather odd that it is always necessary to have some matter defining the frame of reference. The very fact that, in principle, any matter will do helps to create the impression that the materialization of the frame of reference is superfluous.

The root cause of his (Newton's) trouble seems to have been the attempt to dispense totally with matter when defining motion. Wherever specific applications of dynamics are made, the inescapability of a definite materialized frame of reference is unavoidable. Without it all equations are void of content. With it, there is no limit to what dynamics can achieve.