Ben,
Thank you for the nice comments over on my essay. I responded over there. While your essay is way over my head in terms of math, the points you made that I think (?) I understood were very good! A few comments are below, but take them with a grain of salt because, as I said, it was kind of over my head. Anyways, they are:
1. I think your way of thinking as illustrated by this quote:
"What I try to do is build up fundamental physics from simple principles like cause and effect. This leads to some rather thorny mathematics, but my view is that the physical principles ought to be simple and well-motivated"
is exactly right, and I wish more physicists and thinkers in general would think this way. If we start at base principles like cause and effect, we have a better chance of building a working model of the fundamentals of the universe that can make predictions than by starting out with high level, assumption-riddled, current physics thinking and working down to more fundamental levels.
2. I totally agree that the assumption that systems evolve with respect to an independent time parameter, and that the universe has a static background structure seem unlikely. To me, if the universe is the "system", time is just the same as a sequential chain of physical events occurring within the universe, with the earlier events in the sequence corresponding to earlier times. If the events A ->B->C are sequentially followed by the events C->B->A, this doesn't mean that time is going backwards because the events C->B->A still occurred after the A->B->C sequence of events. When I hear physicists say that their equations work fine when time is negative, this doesn't mean that time in the real world (not in the equations) can actually go backwards. Also, if time exists as this separate, independent dimension somewhere, I'd like someone to point it out to me now. Where is it?! I can't see it. Also, in regard to the second assumption, I think of the universe a little more holistically where matter and energy aren't occurring against a separate space background, but rather that they're interactions between the units that make up the universe/space.
3. You mentioned on pg. 6 "This means, in particular, that spacelike sections are merely unordered sets, with no independent notion of distance or locality". If I understood this, I think I'd also agree because I think that location of something refers to its position relative to other things within a bigger set of things. That is, while a single existent state may "be" or exist as a location, it doesn't "have" a location within a bigger reference frame.
4. My own view on volume is that to physically exist, any existent state must have three dimensions, and, therefore, volume. I have trouble imagining an actual physical state in which one of the dimensions is zero. Not just infinitesimally small but actually zero. At zero, it disappears. So, three dimensions, or volume, seems to me to be a requirement of an existent state and thus a requirement for whatever existent state makes up our universe.
5. So, is a binary relation just a relationship between two elements in a set/ And, if one element causes the related element to appear, is this a causal relation? If this understanding is right, this makes a lot of sense to me because my own view of existence is that given a fundamental state of existence, whatever this is, this state will somehow cause the formation of identical states around it, these new units will cause the formation of new states around them, etc. and this expanding space of existent states is equivalent to our universe. So, in my view, I think I would say that there's a causal relation between each existent state and the existent states it causes to appear next to it. I have more on this at my website at:
https://sites.google.com/site/ralphthewebsite/filecabinet/why-things-exist-something-nothing
Sorry for the long response. Nice essay and good luck in grad. school!
Roger