- Mathematical entities(what the argument is about—whether they're real)
- Abstract objects that mathematicians study, like numbers, sets, functions, and geometric shapes—things that exist in mathematics but not as physical objects.
- Mode of being(Used by James to characterize the ontological status of relations)
- A way a thing is that does not differ from the thing so as to constitute another essence or thing
- ideal objects(as examples of things whose reality is being questioned)
- Abstract things that exist only in thought or as concepts, like numbers, geometric shapes, or mathematical ideas—as opposed to physical objects you can touch.
- instantiation(Lowe's trope-involving pluralism)
- The formal relation between a mode and the universal attribute the mode instantiates; it is part of a mode's essence that it instantiates the specific attribute it does
- mental representation(as another way ideal objects might exist through our minds)
- A thought, image, or idea that exists in someone's mind as a representation of something.
- nominalism(Metaphysics; opposed to realism about universals)
- The view that abstract entities such as properties or universals do not exist, and that predicative facts must be explained without appealing to such entities.
- the real and ideal spheres(as the two categories whose distinction the statement claims collapses)
- Two separate categories of existence—the 'real sphere' contains physical, concrete things; the 'ideal sphere' contains abstract things like numbers and concepts.