Lev,
I read a newspaper story here in Detroit, concerning a giant T-shirt that public school teachers had made to adorn the "Spirit of Detroit" sculpture downtown.
Accompanying the story was a picture of the shirt all spread out, with a caption that began "Up to 300 yards of material were used ..."
"Up to"? I imagine that the caption writer was working from an earlier press release that estimated the amount of material needed. But here is the actual shirt, all constructed. If all the information we have is in the picture and caption, the shirt may have anywhere from a fraction of a yard to 300. If it has 300 plus a fraction, however small, the narrative that was true, before the shirt was made, is false after the shirt is made. If it is important to know (obviously, in this trivial example, it isn't, but the principle still applies) we measure the material to the best of our technical ability. Perhaps there are weather conditions that cause it to shrink or expand, so we may decide on either an average or a median measure under varying conditions. After all said and done, we declare the account of the event true or false, but not both. Wait, though - this account was written _after_ the shirt was constructed, and one would think that it has a definite value, a "real" outcome. Yet, even though the outcome has already been determined (the shirt is made) and its representation (photo & caption) exist simultaneously, our knowledge of the state of the shirt is in limbo until after the event and its representation are reconciled by measurement. There is only one right answer.
My point: As you make clear with the independence of semantics & syntax, the gulf between representation and truth is not bridgeable. A representation is neither true nor false; ergo, there is no numerical representation that can tell us the truth, so long as the truth is probabilistic.
Trouble is, we know that measured truth _is_ probabilistic. One of the contributors to quantum theory (I forget whom) characterized quantum mechanics as "something somewhere is doing we don't know what." The measured is never certain enough, only good enough. We don't need to know true from false. Except ...
For cosmological models.
I think Brendan made a good call in naming this thread. It makes a difference, when considering the origin, whether we start with something or nothing. If "something," our problem is rather like botany or zoology -- naming particles and properties, identifying and studying the characteristics that demarcate one particle from another. If "nothing," our problem is one of structure itself--and that is structure without matter, because when we measure changes in the state of a system, it is always from mass points, never space points. We know how spacetime changes matter dynamics; we do not know (in fact, general relativity forbids it in principle) that matter changes spacetime.
To a theorist, and particularly a mathematical analyst, matter is a contaminant in an otherwise perfectly symmetrical world. We want all our space to behave smoothly, on an infinitely flat plane, where we manipulate points and lines at will -- we are descendants of Euclid. If mass doesn't want to cooperate, we coopt mass for energy and spread it symmetrically over the plane. If we encounter enough curvature caused by physical effects, to be troublesome, no problem -- we add dimensions. The n-dimension distribution (generalized Poincare Conjecture) is massless, with average energy content zero.
My own strategy toward this problem (my "time barrier" preprint linked earlier) is to give the vacuum an n-dimensional sphere kissing order in which time (information) is dissipative over n Euclidean dimensions. The math works out (by a purely algebraic method) to where the ordinary matter content of the universe (4.59%) becomes a probability 0.0459 that our universe has any matter content at all. (As a consequence, my model anchors probability theory to a length 1 continuous field in n-dimension space, with asymptotic approach to 1).
As with the giant T-shirt, the result we get after observation (WMAP data) is consistent with the range of observation; i.e., 4 dimensions. The 4-dimension horizon, I have found, is identical to a 10-dimension limit of an n-dimension field.
So--a numerical analysis _does_ get us "somewhere." My interest in your research, Lev, is that the numbers don't tell us what "something" is doing. The dynamics of structure must always add a " 1" dimension to account for our knowledge of change in the system. The classical dynamic system is 3 1. The added dimension is of course, time.
We agree between us that time is identical to information. However, do we mean the same thing by "information?" I mean, quantum information bytes, physical information, i.e., the same as described by Jacobson-Verlinde. I think, by your definition of structs if I understand it, that we do mean the same thing in different terms--"pairs of primitives" correspond, do they not, to quantum information? It is understood, though, that the state of your structs is positive definite and not in superposition, and quantum state measure is replaced by struct dynamics. Which leads back to the main question:
Something or Nothing?
(1) If something, structs are the product of broken symmetry, evolution is top down, and cosmology is botany.
(2) If nothing, structs are a primitive, self organized, dynamic relation between time and space, and evolution is bottom up.
My program is compatible with structs IFF (2). In that context, 0 1 spacetime relates the n-dimension set to the measure function -- with your representation of the natural numbers in fig 12, pg 30, the connected dots representing evolutionary events and therefore dynamic results. I can see that these events can both structure and decompose structs (according to the initial states of the structs) just as in biological evolution. Your definitions that follow imply novel structures from primitive representations, with a non-perturbative mapping.
How am I doing so far?
Tom
