Sciences? Of what, exactly?

This question is nuanced with an objective factual bent, which doesn't always exist in philosophical discussion, which I personally appreciate.

Scientifically speaking, metaphysics today can be considered everything not physics. For if it were physical, physicists are on it. Unfortunately, this leaves metaphysics with the left overs. It used to include all the goodies too, but the scientists took them to the moon leaving the rest of the metaphysicists without anything really real. This explains why older metaphysics is broader and often more concrete. Today physics takes all the concrete and physical problems under its belt, and flat out objectively objects to anything claimed concrete that has nothing real to show for it.

So then the issue is if metaphysics is obsolete. Not only is the not-physical category important, but it's also what sustains the physical category. When a physicist encounters an idea or a hypothesis, the first test of falsifiability can be considered logistically a test for metaphysics or physics. If physics, head to lab. If metaphysics head to books (until there is something physical to test in the lab).

Next, regarding the word ontology. Ontology, by definition, is the analysis of existence. Again, as we go further back, ontology and metaphysics were closer and broader. This was before we had all the better ideas that have advanced science and civilization.

Ontology cannot be dismissed, because encompasses the debate of existence at the highest level of abstraction. In fact, it's a race to the top. In contrast, if we were to discuss existence physically, it would be the race to the bottom: to the lowest level of abstraction and highest level of concreteness. To the atoms and quarks and strings we go.

Scientifically speaking, ontology today is about language and abstraction itself. Ludwig Wittgenstein in a sense took philosophy to a higher abstract plane by discerning philosophical discussion from concrete logic. Meaning, even here, as we "game" these words with all of our input, the ideas set forth are not necessarily good or bad, but are permitted to exist even for the sake of duking it out, existence being ontological.

In simple terms, abstraction is merely a naming of a pattern. So languages are systems of abstraction and all communication that relies on language relies on abstraction. This gets us to information theory and computation, both 20th century newborns.

Ontologically speaking, existence can only be the product of abstraction. Abstraction is the act of circling a common pattern and naming it, and is a prerequisite for "apples" or anything for that matter. And computationally speaking, abstractions are the smallest unit of logical value. Logic requires language, and language is made up of words, which are all abstractions.

This answer can be referred to as "this answer". "This answer" exists, ontologically speaking. And scientifically speaking, "this answer" refers to something real, physical, and here, that is worth referring to.

So again, science offers a valuable constraint. Without it, ontology would cover any idea about existence, even anything metaphysical. But science points ontology to abstraction.

Sorry for any rough edges, but hopefully this adds to the discussion.

I don't think we should assume the world is mathematical. A more accurate statement would be that the world can be expressed mathematically. Out of all the ways nature can be represented, modeled, and expressed, mathematics is one of them. Language and art are others. Biologically the world is presented and expressed to us as experience and emotion. If all of this were to also have a mathematical representation, great. It would be because they are all part of the natural world. But they also pre-exist their mathematically representations because for much of it we still need to figure it out.

And not only can most of us do without mathematical representations, often they're not even relevant.

Since nature can be expressed mathematically, those who assume the world is mathematical do have a mathematical viewpoint of nature at hand they can point to and say "see!". Good. But to then not be able to point to anything else would be their loss.