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A Mathematical Model for Plato's Theory of Forms

机译：柏拉图形式理论的数学模型

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Aims/ Objectives: In this article we construct a mathematical/topological framework for comprehending fundamental concepts in Plato's theory of Forms; specically the dual processes of: 1. The participation/partaking-methexis of the many particulars predicated as F to the Form-essence F, according to their degree of participation to it. 2. The presence-parousia of the Form-essence F to the particulars predicated as F, in analogy to their degree of participation to F as in 1. The theoretical foundation of our model is primarily based on a combination of both the Approximationist and Predicationalist approaches for Plato's theory of Forms, taking into account the degree of participation of the particulars to the Form, that are predicated to. In constructing our model we assume that there exists exactly one Form corresponding to every predicate that has a Form (Plato's `uniqueness thesis'), and to support our main theses we analyze textual evidence from various Platonic works. The mathematical model is founded on the dual notions of projective and inductive topologies, and their projective and inductive limits respectively.
机译：目的/目标：在本文中，我们构建了一个数学/拓扑框架，用于理解柏拉图形式理论中的基本概念。特别是以下双重过程：1.根据形式参与F的形式，将许多要件F预测为F的参与/参与方式。 2.形式本质F对以F表示的细节的存在比喻，类似于它们对F的参与程度，如在1中所述。我们模型的理论基础主要是基于近似论和谓词论的结合柏拉图形式理论的方法，要考虑到具体细节对形式的参与程度。在构建模型时，我们假设存在一个与每个谓词相对应的形式（柏拉图的“唯一性命题”），而该形式恰好存在，为了支持我们的主要观点，我们分析了各种柏拉图著作的文本证据。数学模型建立在射影和归纳拓扑的双重概念以及它们的射影和归纳极限的基础上。

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