1/ A few random and very disjointed thoughts: 1) There is logical ontology and there is physical ontology. An object of logical ontology is anything that can be designated by the subject of a subject/predicate assertion.

2/ The objects of physical ontology count among the objects of logical ontology. In addition, they are structured by the form/matter schema. Physical ontology is what Husserl would call a regional ontology. Logical ontology corresponds roughly to #Husserl's formal ontology.

3/ If we were to construct an ontological hierarchy, the logical bunch would be at the top or very near the top. It is, you might say, at a high generic level.

4/ Taking a step further, the way to distinguish objects in the real or physical world from non-physical objects, such as mathematical, especially formal logical, truth theoretical and model theoretical objects is that we apply a similar schema to the former but not to the latter

5/ That is to say, real objects are structured according to a form/substrate schema, whereas unreal objects are not. My use of substrate here is mostly Aristotelian.

6/ The difference is that by ὑποκείμενον #Aristotle also meant anything about which we can make a predicative assertion. It is what underlies or bears a property. In Aristotle's sense anything about which we can say something can be (or have) a substrate.

7/ In my usage mathematical objects do not have a substrate. Incidentally the form/substrate schema is compatible with atomism, since there is no reason that a substrate should be continuous, that it can't be made up of particles.

8/ We can sense and feel what substrate is, but providing a rigorous account of substrate is a challenge.

9/ This account shows that we need not take seriously the claims of mathematical realism: Mathematical objects may be real in one sense but they are not structured according to the form/substrate schema.

10/ It sets us on the road to understanding why the ontological conclusions drawn by the likes of #Leibniz, #Kripke and #Plantinga (and perhaps the Wittgenstein of the Tractatus as well as #Gödel) are absurd.

11/ For anyone who might have been wondering, this last paragraph is the money shot.