From Landsburg's "Accounting for Numbers," point 9:

numbers are indeed just “out there” and that they are directly accessible to our intuitions.

[Emphasis mine.]

I'm conflicted on whether to think the world is actually physical in a special way, or simply mathematical in such a way that all mathematical structures exist and are equally real. My greatest sticking point to accepting the latter position is the emphasized part of the quote.

Okay, if I grant that numbers are "out there," we do seem to interact with them via a cognitive algorithm we call "intuition." But how the heck does that work!? Nothing I know about cognitive science suggests that intuitions are somehow specially equipped to interact with numbers, or are fundamentally different than the brain's other cognitive algorithms.

The crux of the argument for a mathematical universe - so far as I understand it - is parsimony. But in order for parsimony to be the relevant criteria, both hypotheses need have the same experimental predictions. In other words, if the mathematical universe hypothesis is correct, we would expect it to be consistent with our non-mathematical universe understanding of cognitive science. But with respect to intuitions, this doesn't seem to be the case.

Without having a satisfying account of how intuitions work in a mathematical universe, the MUH isn't consistent with my current understanding of reality. Is there such an explanation of intuitions? If not, maybe a more fruitful question to ask might be:

What kind of cognitive algorithms generate the feeling that numbers are "out there"?