Abstract

The so-called Third Man argument in Plato’s Parmenides exposes a structural instability within the theory of Forms. If many particulars are F in virtue of a Form of F, and if the Form itself is F, then a further unifying principle appears required, generating an infinite regress. This paper offers a minimal formal reconstruction of the regress and argues that its source lies in a collapse of explanatory levels: the unifying principle is treated as belonging to the same ontological domain as the plurality it explains. To clarify this structural difficulty, two mathematical models are developed. A stratified set-theoretic framework enforces level-sensitivity by typing participation across distinct domains. A topological model interprets Forms as limit-structures determined by patterns of convergence rather than as additional members of a plurality. Both models preserve the unifying function attributed to Forms while blocking the regress-generating step. The formal analysis is then situated within a comparative interpretation of Plato, Aristotle, and al-Kind¯ı. Plato’s dialectic exposes the instability of unregulated self-application; Aristotle resolves it by denying the separate ontological status of universals; al-Kind¯ı preserves explanatory hierarchy through disciplined abstraction. The paper advances a moderate structural claim: regress arises when explanatory structure is confused with membership. Clarifying this distinction illuminates a central tension in ancient metaphysics concerning unity and multiplicity.