If we accept materialism, then we say that every part of existence can be described by mathematics. This is so because physical laws consist of mathematical equations, and according to materialism existence is described by physical laws. We therefore say that under materialism, existence is mathematical.
If we reject materialism, we must nonetheless acknowledge that some of existence is material and is described by science. We may reject materialism and nonetheless maintain that all of existence is mathematical. If we reject this as well, then we are forced to conclude that some of existence is mathematical, and some of it is not. It cannot be that none of existence is mathematical, because some of existence is described by science.
This leaves us with two theories of existence. Either it is wholly mathematical, or it is part mathematical and part unmathematical. In the latter case we need not posit that there is a sharp distinction between the two; there could be room for a fuzzy border.
Let us call these views the Mathematical View, and the Divided View.
If we accept materialism, then we accept the Mathematical View. We only accept the Divided View in a case in which we reject materialism. Here we will take “rejecting materialism” to mean “including nonphysical categories in our ontology.” This is the only act which we may seriously call a rejection of materialism. To the category “physical entities” we add, for instance, the category “mental entities,” or the category “spiritual entities.”
It is odd to notice that these three categories seem to be exhaustive for almost any ontology, and even more strongly seem to be exhaustive for the entities which we directly experience. We experience physical entities such as tables and chairs. We experience mental entities such as thoughts and feelings. We experience spiritual entities such as moments of grace/enlightenment. There does not seem to be a fourth category. Put differently, there do not seem to be any entities which do not readily fall into at least one of these three categories.
Some ontologies, of course, use only one of these categories. We have materialism, in which we posit that all entities are really physical entities. We have idealism, in which we posit that all entities are really mental entities. We may also imagine some kind of spiritual monism in which we posit that all entities are really spiritual entities; Advaitism may be such a philosophy. Furthermore it is easy to imagine ontologies using only two of these categories, or consolidating two categories into one (e.g. merging “mental” and “spiritual”). But the odd fact remains that there does not seem to be a fourth category.
I draw the following tentative conclusions. Every ontology will have to posit some combination of these three categories (either including all three, reducing the number by exclusion and/or merging, or increasing the number by splitting). Furthermore, each category will be either wholly mathematical, wholly unmathematical, or somewhere in between.
No monistic ontology can have its one category be wholly unmathematical, because the parts of the world described by science are mathematical, and those parts of the world will be contained in the one category of any monistic ontology.
Furthermore, any ontology which contains the category “physical” will probably have that category be wholly mathematical. (This conclusion could only be altered upon very detailed philosophical inquiry.)
It is difficult to imagine that the category “spiritual” could be wholly mathematical, and so any ontology which contains the category “spiritual” will probably have that category not wholly mathematical. Most likely it will be wholly unmathematical.
What are likely or obvious candidates for ontologies?
1. The materialist ontology, in which there is one category, “physical,” and it is wholly mathematical.
2. The idealist ontology, in which there is one category, “mental,” and it is partially (perhaps wholly) mathematical.
3. An ontology in which all three categories are present, “physical” is wholly mathematical, “mental” is partially mathematical, and “spiritual” is wholly unmathematical. Let us call this a “physical-mental-spiritual ontology.”
One thing that these inquiries show is the great philosophical difficulty that arises in rejecting materialism. If we take the claims of religions seriously then we almost certainly do reject materialism, but in doing this we cast ourselves out into very confusing and uncertain territory.