I found your question when searching for the same answers. I am somewhat perplexed that the hypothesis of the mathematical universe, and Tegmark's book and papers, have not resulted in a flurry of philosophical debate on the subject. It is common scientific thought that biological and mental processes are chemical in nature and that chemical processes boil down to physics. And physics certainly distills down to math. So ought not everything be as its basis math, or emerge from mathematical constructs such as logic and pattern? It is only our famously unreliable and subjective experiences and culture that persuade us there is anything else that is fundamental. Of all things in time and space it is only math (and logic) that might escape time and space. It is only mathematics that can be eternal. We should ask, not why math is fundamental to our universe and everything in it, but why anything else should exist at all except as illusion. And we should ask how the physical world and the mental world emerge from the abstract and pure realm of Plato's ideal forms, how the world we see emerges from the perfection and purity that is logic itself. Everything we see, think, feel, and love, everything that is breathtakingly beautiful and everything sad and unfair, is at it's heart just math being math. The concept is not, or should not be a source of despair; rather it is a concept that is liberating. And to me it rings true, and seems natural and reassuring.
So to really answer your question, I have found no meaningful exploration of the subject of the mathematical universe, except derisive criticism. Nothing that delves into it while supporting the hypothesis, only negative criticisms that to me, so far, don't fly.