One major breach is that belief systems cannot justify the 'truth' of their principal axioms (Godel's Incompleteness Theorem) so 'rational' argument which ensues is irrelevent. The only aspect that matters about such systems is whether they are pragmatically 'functional'.
Gödel's incompleteness theorems
From Wikipedia, the free encyclopedia
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of the natural numbers. For any such consistent formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.
I completely agree with that theorem because, "No consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of the natural numbers. For any such consistent formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system" is a true statement for this reason:
Looking at a system or algorithm (like the universe) from inside the system (which is the point of view we are physically required to have because our bodies are a system "with intelligence" embedded in and whose existence depends on an interaction with the universe (for instance, "the higgs mechanism of the higgs field" gives matter mass.) Being able to completely understand the system from that view forces one into a philosophical view (naive realism) that by itself can never be complete because of the physical nature of the view which is a system of information embedded and an integral part of the larger and more encompassing system.
And since, we are embedded in the system of the universe with an immeasurable number of other embedded systems (many of which have so much undetectable complexity embedded in their own system some of which reach the complexity reached the level of endowing intelligence on that system) they can alter the larger system (the universe) in unpredictable ways.
So saying that,"The only aspect that matters about such systems is whether they are pragmatically 'functional'." is correct when determining the purpose of the system from that limited view. That is the view the Roman Catholic philosopher Thomas Aquinas took when he used "Natural Law" in developing the understandings that are now professed by the Roman Catholic Church.
But, he did not use religion to develop Natural Law, instead he used Natural Law to "validate" religion. That validation now gives an intelligent person a reason to look for an all encompassing purpose for the system and the hope that the purpose is understandable and attainable. What that does is give one an incentive to put natural law or naive realism (which is the philosophy of natural science) in an all encompassing context of objective idealism, which is the context of religion.
So, I think Rorty is wrong when he points out:
As Rorty points out, the functionality of religious belief systems is limited to emotionality or psychological comfort. They have no longer any place in what we call science.
Because, an incentive for an all "encompassing philosophy" to put an "embedded philosophy" into context does not have to be emotional or religious in nature (though for some people it can and should be).
As for me it is an intellectual pursuit to more fully understand the "universal" system my bodily "system" is embedded in. That pursuit has given me an incentive to look for an "objective intelligence" behind the universal system we view as the algorithm called the universe.
The wonderful thing is the search has lead me to a person that has claimed to be the objective creator of the algorithm we call the universe as He created it from a position outside of the system. But, He then subjected Himself to the system as he entered the system (as the man Jesus Christ), so as to deliver the "Complete Philosophy of Objective Idealism" (and deliver it in a way that only the creator of the algorithm can) necessary to put science and natural law in its logical context.
So, can we discuss naive realism put naturalism in context and how objective idealism puts naive realism in context. and, then in a purely scientific way use those philosophies to describe the natural world?
PS I through naturalism in in the end because I like to lead the discussion