This post combines some of my recent statements about metaphysics with the Principle of Verifiability.
Mathematical systems are based on axioms, propositions that are assumed to be true without proof. From these axioms we can derive theorems, or truths which follow from those axioms. Pick different axioms, and you either get a different mathematical system, or you get an inconsistent system.
A physical theory is a mathematical system that is augmented with axioms of empirical fact and with a correspondence function from mathematical propositions to empirical ones. The empirical axioms are assumed to be true because they have been observed. Physical theories make predictions by stating which additional empirical axioms would be compatible with the system of propositions. If we observe some empirical axiom that is not consistent with the theory, then the theory is "falsified." Physical theories are never intended to apply over all possible empirical observations. So, a falsifying experiment does not necessarily falsify the theory completely, it might just limit its domain of applicability. Newtonian mechanics has been falsified for speeds close to the speed of light, but this has only limited its domain of applicability. We still use Newtonian mechanics to build cars and bridges.
A fictional world is a physical theory of a world in which the empirical axioms are not fixed by our experience, but fixed by our design. Such a system could make predictions about what other fictional empirical axioms could be added. For example, we know from fictional empirical facts that some life forms in the Star Wars universe are immune to Jedi mind tricks.
Products of Systems
I can take two independent mathematical systems and compose them into a new space of propositions. For example, I take two algebraic systems, R and F, and create a new system R×F ("the product of R and F"). The propositions of R×F are like (Ri, Fj).
Propositions in R do not contradict those in F, even if they are written using the same strings of symbols. For example, there might be a proposition (x = 7, x = 5) in R×F. This is perfectly consistent because the symbol x has a different context. Propositions in R are meaningless in F because they have no logical consequences in F. This is analogous to getting two consecutive algebra problems in a workbook, one with x = 5 and one with x = 7. No problem.
Suppose that systems R and F share some axioms. I can combine R and F into a single system by creating a new system, R+F ("R union F"), which is founded on the axioms of both systems and contains every proposition of R and F. There is no guarantee that R+F will be consistent (e.g., if R contains x = 5 and F contains x = 7).
Suppose that R+F happens to be consistent. I can factor R+F into R×F by creating the two proper subsets of the axioms of R+F, in this case the set of axioms of R and the set of axioms of F, and rebuilding them independently as R and F.
Now suppose that R+F is a physical theory. If I factor R+F into R×F, and R contains no empirical axioms, then R is purely mathematical, and F is a physical theory. R contains mathematics which is extraneous to F. Any empirical axiom can be added to R without contradiction, but not every empirical axiom can be added to F without contradiction. That is, I can factor a physical theory with extraneous mathematics into pure mathematics plus a refined physical theory. This process of refinement aims to isolate just those axioms of a theory that have bearing on empirical axioms that might later be added to the system.
Suppose I have a system of Newtonian mechanics augmented with the mathematics of noncommutive algebra (NCA), I can factor this (N+NCA) into one system of Newtonian mechanics and another system of NCA (N×NCA). The NCA system doesn't have any correspondence with empirical facts, so it is extraneous to Newtonian mechanics. In other words, NCA doesn't do anything for my theory of mechanics. It just adds irrelevant propositions to it. I could have started from a system containing Newtonian mechanics and any random mathematical system that was without a correspondence function. The lack of a correspondence function linked to the axioms of this mathematical system allows me to factor out the extraneous mathematics.
Meaning of Strings of Symbols
Suppose you provide me with a string of symbols and claim it to be a meaningful proposition, P (e.g., P: E = mc2). I must ask in which system does P have meaning? Certainly, I could locate some mathematical system M and some choice of symbols for M where this proposition has some meaning. But, by choosing an alternate symbolic representation of M, P could have the opposite meaning. So, by itself, P has no meaning. It only has meaning in the context of a logical system where P can be related to other propositions in a symbol-independent way. P only has distinct meaning in a system when we designate what strings of symbols are consistent or inconsistent with it. That is, we must designate what propositions, if true, would be consistent with P and which would falsify it. If we cannot do this, then we cannot claim to know the meaning of P. In other words, the meaning of a proposition is its method of verification or falsification.
When I give you a string of symbols representing a proposition, your brain is working to find a context (a logical system and a symbolic representation of that system) in which the proposition has meaning. The goal of logical positivism is to make these meanings clear. Intuitively, we can almost always find a meaning for a string of symbols. Unfortunately, intuition is sometimes subjective and misleading.
If I provide you with a set of propositions like 'God is good' and 'God created the universe outside of time and out of nothing', you can only make sense of these propositions by confusing different systems. You create one logical system, G, in which God is a symbol, and in which propositions like 'God is greater than everything' are true. Then you use symbols like 'universe', 'create', 'nothing', 'time', and 'good' from the system of empirical science, E. So you try to make sense of the propositions in the context of G+E. Unfortunately, because propositions about G are compatible with every possible empirical axiom, G+E is factorable into G×E. In other words, propositions about God and reality are factorable into mathematical statements and scientific reality. The metaphysical propositions drop out from experience just as any extraneous mathematics would drop out of a physical theory. Propositions about God and reality are no more about the world than propositions about noncommutative algebra and reality.
Okay, it's a first draft.
I need to refine my definition of "product". I think I can say that
P+(R×F) = (P+R)×(P+F)
(R-F)+(F-R)+(R×F) = (R+F)×(R+F) = R+F
Where R-F is the set of axioms of R not in F.