Theologian RD says: Mathematics maps onto the world but our intuition of which mathematics maps on is not infallible. The problem is with Conversational Atheist's apparent view that mathematics does not map onto the world at all. That's the view that needs defending.
Whether mathematics maps onto the world is a very interesting question. Of the infinite number of mathematical models that are able to be constructed, a few map onto the world in very useful ways. If you come up with a mathematical model, the question immediately becomes: DOES your abstract mathematical model map onto the world?
Re-read a section from the comment that prompted this entry:
Conversational Atheist says: However, Einstein, placed in a box before the first experiments in quantum mechanics were done, would never end up CONCLUDING Quantum Mechanics. He might, given infinite time, detail hundreds of thousands of possible physics on small scales [mathematical models] that includes our modern conception of QM, but he would be in no position to choose one from the others with any confidence at all. He might even pick what he thinks is the most beautiful physics at small scales, but what counts is not beauty or arguments, per se, but whether his physics [mathematical model] matches reality. -- Emphasis and [mathematical note] added
What counts is whether your mathematical model matches reality. That's the $64,000,000 question for a physicist. Long ago a person could have come up with the mathematical framework of modern day quantum mechanics by doing math. Lots of math. Now, that person hasn't done PHYSICS unless he is either consulting data others have taken, or going out and doing experiments himself.
To answer some of your questions raised in the entry itself:
RD: How do we gain knowledge of this realm? That appears very mysterious, indeed somewhat revelatory.
Are you equally perplexed by how we gain knowledge of chess? It doesn't seem very mysterious to me. We consider the relationships and interactions between entities in chess-space. Are these relationships, rules, interactions all contained in a self-consistent chess-realm similar to the relationships, rules, and interactions in the self-consistent mathematical-realm? Yes. Does this mean that there is a supernatural chess-realm, mathematical-realm, tic-tac-toe-realm, and scrabble-realm? Um... only in the most abused sense of the word "supernatural". The chess-realm does not exist in any physical way.
Is the pawn, real? Well, you can hold a wooden piece in your hand that people call "pawn", but then you should realize that you are just holding a representation of a pawn. Does this upset anyone? It shouldn't.
Is 7 real? Well, you can hold 7 apples in a bag and call it "seven", but then you should realize that you are just holding a representation of a number. Does this upset anyone? Apparently, yes.
RD: Second, what is the ontological status of this realm?
I'd say that mathematics has the same ontological status as the game of chess.
RD: And if you accept the existence of abstract numbers and their relations which are irreducible to and independent of the physical world, what about other entities like souls, spirit beings, and God? What rational reason does Conversational Atheist have to believe the supernatural realm, like the natural realm, does not far transcend his limited experiences?
If a pawn is irreducible to and independent of the physical world, what about other abstract entities like souls, spirit beings, and God? -- Sure. I'd say they have the same status as "real pawns". Could there be a supernatural pawn out in the "vast supernature of things that I don't fully comprehend"?
Sure. He might be playing Chinese Checkers with God and the Easter bunny.
But seriously, considerations in the fully abstract might have something to do with external reality -- but you have to actually check to have any confidence that your favorite abstract reasoning maps onto external reality in a useful way.