Sketching how what we might call ‘real platonism’ would differ from received ‘platonism’ in philosophy of math. (Rough draft, in no particular order, with no attempt to reduce the theses to a minimal basis.)

1. Forms, always plural.
2. Forms are not abstractions.
3. Forms have an object-like side, but they also have concept-like, point-like, relation-like, number-like, and, especially, structure-like sides. Taking forms as property-like is primarily good for kicking off investigation, since one can always frame a preliminary problem on that basis. Only some forms have a vantage-point from which they can helpfully be seen as property-like further.
4. The finite simple groups, the natural numbers, and the continuum are forms.
5. Knowledge of forms depends on a degree of participation in them. Philosophy that would engage with forms must allow mathematics to reinterpret the received concepts with which we would at first like to grasp mathematics. Mathematical philosophy can not be assimilated to logic per se, nor to language, nor science, nor epistemology, nor mathematical practice. A philosophy of mathematics is also a mathematics of philosophy.
6. The point at which participation becomes necessary for cognition is the point at which the multiplicity of the form sets a limit to the compressibility of the concept. An ur-example: being-two quickly shows that it can not be treated as a monadic predicate. (Hippias Major) Beyond the limit of compressibility, no general schema of unity suffices to unify the multiple. Every form is one in its own way. Every form defines a thick identity.
7. Thus there are forms of knowledge of forms. These necessarily lead to metamathematical problems, including the relations of proof and truth, consistency and completeness, and the other metalogical dualities. These are the modern shapes of the duality of the principles of the One and the Indefinite Dyad, or Multiplicity.
8. Participation (methexis) and separation (khorismos) go together. To succeed in asking a question that’s compelled to be about something other than its own inscription is to be uncertain, in the first instance, what one is asking about. But this is a departure from the internality of the ‘exterior’ (naive reference) toward the exteriority of the ‘interior’ (the externalism of reflection itself). It compels real work on indeterminacy-of-reference problems, rather than taking them as justification for the end of thinking.
9. The form of the Good is diagonalization, that second pass at the name of the One, which shows the One’s difference from the One-All. Equivalently, the form of the Good is that there is not a being that is exactly what it is of and is of exactly what it is. Equivalently, the form of the Good is that there is not a Form of forms. A Form of forms could exist only as the phantasm of a ruling metacognitive omniscience, aka Evil. That there is no Form of forms (which theorem has the form of a knowable truth, not the form of a being) is the transition from metamathematics to something apparently quite different—metaethics.
10. Real platonism prefers the possibility of questions to the mystical vanishing of questions.
11. Real platonism prefers the potentiality of questions to the guarantee of answers.

January 24, 2022. Last modified September 24, 2022.