Or are the mathematical functions merely properties of the physical? The same way that breadth, width, height are properties of a brick... the brick is not caused by those properties - they are contingent upon the existence of the brick... to be a brick means that it has properties of breadth, width, height etc.
Let me start with the qualification that (as this layman understands it), when Tegmark speaks of numbers and equations, he uses them in an abstract sense.
A simple example: 1 atom equals the abstract number 1, but 1 star also equals the abstract number 1. The actual value of these abstractions vary by the numbers (values) contained in the those single (1) objects themselves.
IMO, our Algebra illustrates the abstract nature of mathematical relationships without assigning specific values. So, our mathematics are algebraic equations, using specific numerical values. Nature does not need these abstract descriptions of its mathematical functions, it just functions that way.
A single brick is more than it's dimensions. It is also the number and weight of all the atoms contained within the brick. The atoms themselves consist of numbers of particles. It is the entirety of all the numbers (values) and their mathematical relationship (equations) within the brick that allows the brick to become explicated in reality as a brick.
I think it is the causal nature that sideshowbob has issue with (unless I'm mistaken).
I am struggling with that myself, but when we speak of causal forces we still use numbers (values) to even describe the forces themselves. The 4 fundamental *causal* forces we have identified all have different values.
Everything, everywhere, seems to be just discreet sets of discreet mathematical values with discreet mathematical relationships, which can be translated and quantified in our mathematical language with numbers and equations, a remakable ability of the human brain, but still limited by our point of observation.
But Rhesus monkeys can count just as well as humans. They just use a different way of counting, like recognizing "more or less" of something. But one can make an argument that this is a rudimentary form of algebra.
Also, a completely brainless organism such as the slime-mold can mathematically navigate a maze to a source of food. It does so by a cumbersome, but effective way of subtraction. First, it extends parts of itself into every opening it finds. When that is a closed path, it withdraws this (tendril) back to the main body, but leaves a marker, which signals a dead-end and is then ignored by the rest of the body. It follows this procedure for every opening it finds, until all dead ends are closed and only the open path to the end of the maze, where the food source is found and the entire body of the slime mold follows this open path toward the food source. Also a remarkable ability of using an abstract mathematical function, in view that the slime mold is a single celled amoeba without a brain at all. This may illustrate:
Note the fractal properties of the amoeba.
Another (in-depth) presentation:
Gravity is a set of numbers (a field) with relationship to other sets of numbers, (such as a brick) within the gravitatioal field. It is certainly causal to the brick falling. It doesn't just do that by itself. Is that not the basis for Newtionian mechanics? Einstein later proved the inadequacy of Newton's theory at very large scales.
Schrodinger proved the inadequacy of Newton's theories at very small scales.
The geometry of spacetime itself is a set of numbers, measerements. A quantum event is an exchange of numbers, but with an aspect of uncertainty to us. But that can be due to our limitation of observing the mathematical function itself. But we know that the quantum function seems to work everywhere, except perhaps in Black Holes.
Finally, i can see a clear relationship of CDT (*causal* dynamical triangulation), a hypothesis developed by Renate Loll et all, and endorsed by other physicists, such as Lee Smolin.
The elegance of this hypothesis is that it is a background independent *causality* (using the mathematical function of iteration, and tends to support Tegmark's mathematical hypothesis.
https://en.wikipedia.org/wiki/Causal_dynamical_triangulation