I never seem to get my point across when I try to say what I’m about to say (and it doesn’t help that my own views on this have changed somewhat during the years that I’ve been a member here), so if it seems confused and hard to understand rest assured that it’s probably my fault, not yours.
I think you are making an unstated assumption that needs to be brought to light. Specifically, I think you are assuming that mathematics (and in particular logic) is an accurate model of reality. But this is still an assumption. It’s a very attractive assumption, as pretty much all of us have great difficulty imagining a world where logic fails to function. It’s also an assumption supported by a great deal of evidence, in the scientific sense. But nevertheless, it’s an assumption, and like Newton’s Laws of Motion it could be wrong, or at least not the whole story.
Now, imagining “what would it be like if logic failed to function” is about as useful, practically, as imagining “what if the sun were to instantly turn into cheese tomorrow”. It’s just not something worth worrying about, except in certain metaphysical discussions.
Like this one. I think that your thesis is making the mistake of assuming that logic must hold, and that if logic dictates something then it must be true. CalMeacham has made this point already, and you’ve responded to it, but I don’t think your response really addressed this problem.
Logic and mathematics, in their modern formulations, are nothing but poetry in a fancy alphabet. They’re just games, built on air. The fact that those games correctly model everything we’ve seen reality do so far doesn’t change that fact. Now, 999 times out of a thousand “correctly modelling what we’ve seen of reality” is sufficient. But you’re trying to go a little bit further than that: you’re trying to derive a real-world conclusion out of nothing but mathematics and logic. Assuming that logic and mathematics are universal papers over this fact, but it’s still there: what you’re trying to do, in the end, is make God vanish in a puff of logic, and that’s never going to work.
And as usual it appears I’ve used a lot of words to say not a hell of a lot, but let me end with this: saying mathematics transcends reality doesn’t make it more true. In a very real sense, it makes it less true instead.