UD+ASSA (http://fennetic.net/irc/finney.org/~hal/udassa/) is an extension/refinement of the mathematical universe idea that I find very compelling. It explains why even though "everything mathematically describe-able is real", we personally see and anticipate that some things are more likely than other things, including why physics stays the same instead of spontaneously changing, etc. (One of the original authors of the idea thinks there's issues with it, but I think it's still a very useful idea: https://www.lesswrong.com/posts/zd2DrbHApWypJD2Rz/udt2-and-a...)

Since considering it and the mathematical universe idea, the question of why anything exists at all no longer feels so mysterious. Even if there were somehow an objective universe with nothing in it, as long as the basic concepts of logic still held (which of course they do, they don't depend on the universe), then there would still be logical/mathematical structures that describe entities that see themselves as conscious.

> if the basic axioms of math are true

Isn't the problem that you can't prove axioms. We intuitively "know" them to be true but without proof.

You can't use deduction to prove them... but you use induction to show them to be statistically non-violated through experience so far.

I haven't read it, but that certainly sounds a lot like saying that the map is, in fact, the territory.