Thank you for delving into the realism vs anti-realism questions through the lens of set theory! I’m glad to better understand the extent to set theory bears on these issues. In your Realism and Anti-realism article you referenced at the end, you mentioned Michael Dummett. I read (as best I could) his Logical Basis of Metaphysics. He concludes the book with a “parable” in which someone in effect asks God “what is truth?” and in response, God in effect shrugs his shoulders. (More precisely, I think Dummett had God denying the law of the excluded middle—another maddening discussion.)
I have to admit I’m on the anti-realist side. Or at least I agree with Wittgenstein and Dummett arguing against realism because we humans can’t overcome the “linguistic” obstacles to realism—as I think you have too, in your piece on Wittgenstein and his Remarks on the Foundations of Mathematics, in which he asks “why should we abandon mathematics just because we find a fault in it?” Studying the linguistic obstacle arguably reveals obstacles in semantics, meaning, and conceptualizability.
Philosophers generally agree science has proven a powerful tool. So as a pragmatic step, I’m tempted to see the human mind as a product of evolution. As such, there seems no reason to believe humanity’s conceptualizability can be stretched to encompass the infinite, be it the universe, the value of pi, or the quantum world. We fall short because we are a mutational incremental step forward in conceptual ability. Given other creatures sense worlds of smell or sense electric fields and we can’t, why should we suppose our minds have stumbled into some kind of complete conceptualizability of logic, reality and knowledge? Does Nature owe us more than enough tools to survive?
I’m all in favor of pushing every frontier as far as we can, but if we encounter the problems you describe in this piece, maybe we can surmount them, maybe not, and when we hit that point, maybe we can view and address the situation pragmatically, like Wittgenstein did. Maybe our conceptual limitations are the cause of the paradoxes we discover. That’s why I can see God (and Wittgenstein) shrugging when we ask for the resolution of paradoxes. No doubt God (speaking figuratively of course) sees paradoxes as unparadoxical?
