> Why should the limits of our universe have anything to do with the limits of math?

Mathematics is a human activity, and humans are part of the universe.

> Math is bigger than our universe.

What does that mean?

> If our universe were completely different, math would still be the same.

That is more than debatable.

> Eh, I can write a recipe that calls for multiple universes-worth of flour and 600 trillion years in the oven, and I can write that recipe in a minute on an index card that weighs a couple grams. I can write small programs that loop infinitely or (attempt to) use infinite memory. I can write a song that repeats ad infinitum that nobody can ever sing to completion in the entire lifetime of the universe. That's not much different than the words "the Integers" requiring 2 words but describing something that could not possibly be "fully materialized" in any finite universe. Who cares?

I think you're skipping over the point. No one is going to argue with you about "multiple universes-worth of flour" existing because to our current knowledge, the universe is finite and thus there is a finite maximum to how much flour can exist. My question revolves around the philosophical question of the existence of mathematical objects and whether they can or cannot be actually realized with the given information in the universe. This is not a question out of left field. The acceptance of the real number continuum is not even settled.

Thanks for the reference to Kolmogorov complexity.