Dear Ben
I will now try to explain the point I was trying to make. Let´s place ourselves in the 17th century and try to build physics from the scratch, that is, for the sake of the argument, let´s ignore any complications due to modern physics. I must firts say that Mach´s thoughts are philosophical. For someone coming from a math background this may seem extremely vague. But Mach´s philosophy has tight relations with GR, and this makes it very important.
Due to our stable enviroment, we were led to think that there was an invisible space and time background. The rotation of the earth provided a parameter to which all motion should be labeled and the distant stars provided an infinite grid to which all distances should be measured. It was then very natural to take the mathematical gadgets such as R³ to model the physical world. Then you can build equations between 'stuff' defined on R³ such as ma=-grad(V) but this theory cannot be tested because this R³ spatial grid cannot be seen... a closer look reveals that the distant stars also seem to move! There is no epistemological way to identify the grid. The best that can be done is to find a visible object, measure distances upon it and check if ma=-grad(V) would hold (such an object would provide an inertial frame of reference). But even so we could not identify a grid defined this way with absolute space, because the inertial frame object can be moving with constant velocity in relation to absolute space.
So, was absolute space a historical mistake? Could we formulate a physical theory without unobservable structure, in a purely relational (because all we see and measure are relations) way? This is what Mach intended, though he never wrote down a complete physical theory. But then, Barbour has shown that, in a sense, it suffices to impose a relational framework to a 3-metric field to RECOVER general relativity from first principles.
If one recovers GR from relational first principles, it becomes compelling to study the origin of these first principles... what´s the origin of Mach´s thoughts? What I have been thinking is that maybe it is possible to see Mach´s procedure as a part of something bigger.
Mach´s unease with classical mechanics can be seen as coming from the following: what do we mean when we say an object´s position at a time t is (x,y,z)? Mach´s criterion of meaning is OBSERVATION. To say that the position is (x,y,z) at a time t can only MEAN that the relative distance between the object and a reference is (x,y,z) and that a clock (which is a physical object) has marked t. For Mach, all statements of classical mechanics should MEAN only what can be observed upon then. Unobservable statements should be cut off from the start, because they don´t MEAN anything. What Mach was ultimately searching was MEANING, using OBSERVATION as criterion. And remarkably, this leads to GR via Barbour´s argument!
But the relational philosophy is not complete: it gives time and position a meaning upon the concept of physical object. But the concept of physical object is still MEANINGLESS! This is why I proposed to try to build a dynamics where time, space, motion and objects all gain their meaning upon each other, to see what would be the correct mathematical structure amd what result we would achieve. I call a completely meaningful description of the universe ''semantically complete''. Now one of the mysteries of QM is the nature of observation: why is observation so different from other physical phenomena? What causes the wave function to collapse? How can we classify a physical process as an ''observation''? Category theory now comes into the game: imagine a category where objects are semantically complete descriptions of the universe and functors that send such descriptions to descrptions that MEAN the same (let´s call them semantic functors). Now it seems that the concept of ''observation'' could be given a precise mathematical meaning: it is ''that thing that is used to build the semantic functor''! And relational physics can be simply stated as ''the square commutes'' as I put in my essay! This is the outline of what I was thinking.
I think the first step should be to cast barbour´s relational physics in a category theoretic framework. Barbour´s procedure of eliminating absolute structures is his method of best-matching, as I explained in the essay. I was thinking of trying to find the origin of best matching via category theoretical considerations. I don´t know if this is possible, but my intuition tell me it is. Once we do that, then I think everything would be easier to understand. Maybe you will find that interesting. I´ve seen that Derek Wise was trying to find relations between Barbour´s theory and Cartan geometry, which has some relations to category theory, we could investigate that. Anyway, I will be waiting for your feedback.
Best Regards,
Daniel