Mathematical Interpretation of Plato's Third Man Argument

Abstract: The main aim of this article is to defend the thesis that Plato apprehended the structure of incommensurable magnitudes in a way that these magnitudes correspond in a unique and well defined manner to the modern concept of the "Dedekind cut". Thus, the notion of convergence is consistent with Plato's apprehension of mathematical concepts, and in particular these of "density" of magnitudes and the complete continuum in the sense that they include incommensurable cuts. For this purpose I discuss and interpret, in a new perspective, the mathematical framework and the logic of the Third Man Argument (TMA) that appears in Plato's "Parmenides" as well as mathematical concepts from other Platonic dialogues. I claim that in this perspective the apparent infinite sequence of F-Forms, that it is generated by repetitive applications of the TMA, converges (in a mathematical sense) to a unique F-Form for the particular predicate. I also claim and prove that within this framework the logic of the TMA is consistent with that of the Third Bed Argument (TBA) as presented in Plato's "Republic". This supports Plato's intention for assuming a unique Form per Predicate; that is, the Uniqueness thesis.