How do you resolve the issue: logically reasoning about 'god' (or even physics; mathematics) when sometimes logic is not valid or is partially broken?

Question

Granted, the start of the following discussion may use some common everyday common sense logic, and therefore may not be valid in some new paradigm.

However, the question is, in the face of Russel's paradox in mathematics (therefore implying that without some creative axiomatic repairs; our lines of mathematical reasoning fail us if we care about being 100% true with respect to simple definitions), and in the face of weird quantum phenomena, how can we then use pure, conventional logic to reason about 'god'?

I could be contradicting myself with the question, because it implies the use of our standard logic. So please give some leeway.

If you understand what I am getting at, is there an algebraic way (not necessarily logically provable in the conventional sense) to resolve the conflicts of paradoxes?

If 'god' has logical paradoxes within itself, shouldn't mathematicians or philosophers be studying "paradox structure theory", or something...

Answer 1

The most straightforward way out of this conundrum is to localize logic. There are a lot of variants of this (my favorite being neo-Intuitionism) but most of them can be captured by the notion of Mathematical fictionalism.

The idea here is that reality, in particular math, as the elaboration of basic intuitive logic, (though it is really just an example) may not be a pure and consistent whole, but it has large consistent pieces that apply in a broad range of domains, as we witness when we actually do it. So only use the large consistent pieces, and retain isolation from the paradoxes. Don't allow your logic to make inferences too close to a paradox, by limiting how the effects of a paradox would 'spread' and contaminate your consistency.

This involves abandoning various aspects of classical logic, and which aspects you choose becomes a very interesting question, but one that is hard to find agreement upon. (After all, it involves violating a strong and appealing intuitive system in favor of choosing which intuitions we should apply where. The obvious answer, that all intuitions should apply everywhere, failed, and, as you note, that leaves us kind of at sea.)

The Intuitionist form looks at mathematics as the oldest form of psychology: Starting from the Kantian base that space is an aspect of human understanding, and not of reality, the subject matter of mathematics in general is not what is real, but what we as humans can readily understand, and what we find interesting about the patterns in our world.

Answer 2

shouldn't mathematicians or philosophers be studying "paradox structure theory"

And why wouldn't they also need to study meta-"paradox structure theory," meta-meta-"paradox structure theory," meta-meta-meta-"paradox structure theory," … ad infinitum?

Thus, you're left with the regress problem, which Aristotle describes in Posterior Analytics I.2:

b5. Some hold that, owing to the necessity of knowing the primary premisses, there is no scientific knowledge. Others think there is, but that all truths are demonstrable. Neither doctrine is either true or a necessary deduction from the premisses.

b8. The first school [agnostics?], assuming that there is no way of knowing other than by demonstration, maintain that an infinite regress is involved, on the ground that if behind the prior stands no primary, we could not know the posterior through the prior (wherein they are right, for one cannot traverse an infinite series): if on the other hand — they say — the series terminates and there are primary premisses, yet these are unknowable because incapable of demonstration, which according to them is the only form of knowledge. And since thus one cannot know the primary premisses, knowledge of the conclusions which follow from them is not pure scientific knowledge nor properly knowing at all, but rests on the mere supposition that the premisses are true.

b15. The other party [sophists?] agree with them as regards knowing, holding that it is only possible by demonstration, but they see no difficulty in holding that all truths are demonstrated, on the ground that demonstration may be circular and reciprocal.

b18. Our own doctrine is that not all knowledge is demonstrative: on the contrary, knowledge of the immediate premisses is independent of demonstration. (The necessity of this is obvious; for since we must know the prior premisses from which the demonstration is drawn, and since the regress must end in immediate truths, those truths must be indemonstrable.) Such, then, is our doctrine, and in addition we maintain that besides scientific knowledge there is its originative source which enables us to recognize the definitions.

^{cf. also St. Thomas Aquinas's Commentary on Aristotle's Metaphysics IV l. 6 [¶607] for a conciser resolution of the regress problem}

Aristotle's conclusion is that, analogous to Gödel, there are truths which cannot be demonstrated.

For example, there are laws of reasoning that are true but cannot be proven to be true, like the principle of non-contradiction, which is the "first indemonstrable principle" (Summa Theologica I-II q. 94 a. 2 c.):
^{quoted here}

[A] certain order is to be found in those things that are apprehended universally. For that which, before aught else, falls under apprehension, is "being," the notion of which is included in all things whatsoever a man apprehends. Wherefore the first indemonstrable principle is that "the same thing cannot be affirmed and denied at the same time," which is based on the notion of "being" and "not-being": and on this principle all others are based, as is stated in Metaph. iv, text. 9.