Earlier I argued that religion is axiomatic, rational and wrong, and that science methodology pushes religion either into allegorical distinct domains through heterodoxy, or it faces direct conflict with orthodox religions.
I want to argue that any particular religion really is axiomatic and that the axioms are "arbitary" in the same sense that the axioms of geometry are arbitary, there is freedom to choose different, incompatible axiom sets that lead to different inferences.
We can consider religion as a social structure, that communicates its precepts among both its adherents and to unbelievers.
So, at an essential level, everything about a religion is reducable to a finite set of data that is communicable. To the extent that it is not communicable, it is irrelevant and can be bumped to the boredom of "non-overlapping magisteria".
Now, either the data is incompressible and therefore random, or maximally compressed, or it is compressible.
Therefore the data which constitutes the information which established the religion is redundant.
Hence it can be reduced to a set of finite irreducible elements, the axioms, some finite set of deductions made from the irreducible essentials, and redundant restatements of the same (allegorical tales and ambiguous precepts fall into these categories).
In some cases these are partially explicit (thou shalt/thou shalt not, or the Nicean creed), in many cases they are implicit, which requires some work.
Note that the set of axioms for a single religion is not unique, in general, since there are redundant sets of axioms which imply each other (some set of axioms+inferences could be substituted with different axioms+inferences, that were individually different, but communicated the same information).
Clearly not all possible inferences will have been made, yet, for any given religion. Unless some buddhists have completed some projects they're not telling us about (are there only 9.109 names of god?).
Now, any interesting religion will have strong axioms, at least as strong as the Peano axioms (which implies that arithmetic is a subset of any interesting religion...)
Any rational modern religion will not want its axiom set to be contradictory, In fact I would argue that much of theology for introspective adaptive or heterodox religions consists of exploring possible inconsistencies. (cf Christian angst over the potential contradictions of the trinity)
The reason religious axioms must not be contradictory, is because then a clever theologian can show that A=!A for any A (that reads "A equals not-A" for all propositions labeled "A" (ie any and all of them), which means any statement and its opposite can be show to be true at the same time).
ie in a contradictory religion everything is permitted (and, annoyingly also forbidden).
That is not good. (cf the problem of "thou shalt not kill" - tricky that one, open to lots of interpretations).
So... we therefore conclude that any reasonable religion is incomplete! It contains unprovable statements.
Such can of course be added as additional axioms (does any mainstream religion have the equivalent of the axiom of choice? It seems they ought to).
The trouble with adding axioms is that they may introduce contradictions, or new unprovable statments, in fact in general they will.
We assume that apparent contradictions in actual religions is simply due to an incomplete process, ie that the respective theologians or religious authorities have not completed their process of establishing an orthodoxy (which in general can never be completed anyway) or simply made mistakes that will eventually be rectified through religious methodology.
It is in this formal sense that any religion is axiomatic and rational.
It is simply asserting that the religion is strong enough to be interesting, finite in its actual presentation, and can not harbour contradictions or it becomes meaningless.
The problem, such as it is, is then that any religion as actually presented, is amenable to scientific methodology, and therefore must either face tests that can falsify the axioms (assuming the inferences were made correctly), or it must retreat into untestable allegories or infinitely rare or deferred instantiations (ie miracles occur, but only as one-offs, or in such a way that they are never testable as supernatural occurrences).
Hm, the world is natural except for a subset of measure zero. I can see the appeal...
That still leaves a lot of axiomatic theories of religion that are essentially untestable, it is sufficient to make statements about distinct domains that can never be tested by physical measurement.
But that is silly. Isn't it?
I would be interested in someone asserting that some particular religion is parsimonious enough in its axiomatic basis that it is weaker than the Peano or ZFC axioms, I do not believe that is the case for any interesting religion.
It is far easier to show that most religions are internally contradictory, but then no one believes in most religions.
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Ah, you're talking about axioms, not axions.
Postulate a god that is all-knowing, without limits. Then the god knows the past all way back to the beginning of time, and knows the future all the way to the end of time. As a consequence, the god is necessarily powerless to change anything -- past, present, or future -- since any act would change the facts and thus prove his knowledge faulty.
Yet people who postulate gods with many such infinite powers have no problem with necessary contradictions, as they decide dead people live invisibly in the sky, watching us with invisible eyes (the optical problems here are mind-boggling), listening to our whining with intangible ears (think of the acoustics and harmonics of ethereal organs and how sympathetic vibrations would arise), and somehow obeying the laws of physics while violating the same laws of physics.
Religious people call this 'transcendence'. Loosely translated it means 'insane reasoning'. Self-contradiction is a big part of the fun they have with their entertainment. But be wary of them: they take their entertainment very seriously. Usually.
Well, you can of course always do religion badly.
In fact, most religion is an incoherent contradictory mess.
But, what I am trying to explore in a roundabout way is why religion is successful, in spite of the fact that it is wrong and a mess.
Religion seems to increase fitness for societies, and adaptive religions do well over interestingly long time scales.
Now, in so far as religion tries to address issues amenable to scientific methodology, the process of religion is amenable to scientific analysis. So, if a religion, for example, claims to provide a coherent account of how the universe functions, or some explanatory power, we can analyse it.
More generally, the underlying assumptions of any given religion can be analysed, summarised and the implications studied. Both within the framework of religion (theology) and from the outside.
Since religion does stuff, it affects human behaviour and it affects the process of doing science, this can be interesting to do.
I find it interesting that in so far as religion tries to rationalise and be coherent it necessarily must be either inconsistent or incomplete. The former is catastrophics for a rational religion, the latter is problematic.