Problem: Naturalism wants to rule out the existence of supernatural things without ruling out the existence of mathematical things. This puts it in the awkward position of having to allow that there is more than one way in which a thing can exist or be objective, and more than one method to establish this, while also having to rule out one sort of objectivity and set of methods which are by definition different from one that they allow.
-Describing mathematics and logic as “formal systems” is tendentious. What do we call one item in the system? A formal entity or being, right? But then our metaphysics is off to the races.
-The desire to allow for math while ruling out God and soul helps to explain the arbitrary dogmatism of our time that “abstract entities are not causal”, an axiom that was established by some act of divination that everyone performed when I wasn’t looking.
-Abstract entities cause events in the same way an agents do, by being non-negations or privations having an existence prior to the event and being responsible for is existence.
-“But abstractions are static!” Maybe so, but so are conduit pipes, rail lines, and river banks while they are still as directive instruments. The shape of the bat is as integral to event-agency as the swing, and the swing takes part in the batter like the shape takes part in circularity.
-“Mathematics is a tool for science, and this establishes that it’s objective and existent”. But it isn’t and it doesn’t. Go explain to a mathematician that his study becomes pointless when it loses scientific applicability (he actually believes more or less the exact opposite). At any rate, the claim proves to much by requiring the existence of ideal gasses, frictionless surfaces, massless rest particles, black boxes…