Bogdan Dembiński

The subject of my address is a presentation of the crucial threads of Plato’s theory of Ideas. Accordingly, this proposal conjoins topics of epistemology and ontology, but foremost allows us to consider the issue of interconnection between mathematics and ontology. I will present Plato’s conception of measure-Idea, which has crucial significance for understanding the status of being of Ideas. The face of Plato’s teachings about Ideas that will also be analyzed is linked with his “unwritten doctrine”. This theory refers mainly to teaching about numbers and ideal figures in addition to tutelage about principles of being, the One and the Infinite Dyad. It will let us depict in a precise way the very essence of Plato’s understanding of mathematics and explain the status of being which he attributed to mathematical objects. It will relate to critique of the interpretative stand declaring that in Plato’s philosophy we deal with so-called “two worlds” theory. Epistemological topics will also be given much attention. I consider my address as an invitation to discussion about the contemporary understanding of Plato’s philosophy, including the part of it that is accepted in many current versions of mathematical Platonism.

In this article I analyze the issue of many levels of reality that are studied by natural sciences. Particularly interesting is the level of mathematics and the question of the relationship between mathematics and the structure of the real world. The mathematical nature of the world has been considered since ancient times and is the subject of ongoing research for philosophers of science to this day. One of the viewpoints in this field is mathematical Platonism. In contemporary philosophy it is widely accepted that according to Plato mathematics is the domain of ideal beings that are eternal and unalterable and exist independently from the subject’s beliefs and decisions. Two issues seem to be important here. The first issue concerns the question: was Plato really a proponent of present-day mathematical Platonism? The second one is of greater importance: how mathematics influences our understanding of the nature of the world on its many ontological levels? In the article I consider three issues: the Platonic theory of “two worlds”, the method of building a mathematical structure, and the ontology of mathematics.

In this article I analyze a traditional interpretation of Platonic philosophy, which assumes a "theory of two worlds." I try to prove that it is difficult to accept such an interpretation. If one can say that the on tic status of ideas differs from the ontic status of undefined matter (the phenomenon is always for Plato a relation, a compositum, of idea and undefined matter), one nevertheless cannot say that accepting this necessarily results in accepting two independently existing worlds. For then what would phenomena be, if they lacked the ideas that are the source of their determination; and what would ideas themselves be, if they referred to something other than the subject of their determination? Plato always opted for the existence of one world, although this world was for him ontically complex.

The analysis inherent in this text is basic for the construction of Proclus’s system, namely the problem of the relation: the One/Multiplicity. This question is central and constitutes the most meaningful systems of Greek philosophy. The relation: the One/Multiplicity is the starting point in constructing metaphysical systems, it makes that all that is its consequence must have and has a relational nature. This eliminates thinking about the world in terms of isolated, autonomic, and unrelated objects. The considerations in question have shown the most important stages in the problems: the One/Multiplicity, starting from Plato, through Plotinus Porphyry, Iamblichus, down to Proclus

In this article I analyze a traditional interpretation of Platonic philosophy, which assumes a "theory of two worlds." I try to prove that it is difficult to accept such an interpretation. If one can say that the on tic status of ideas differs from the ontic status of undefined matter, one nevertheless cannot say that accepting this necessarily results in accepting two independently existing worlds. For then what would phenomena be, if they lacked the ideas that are the source of their determination; and what would ideas themselves be, if they referred to something other than the subject of their determination? Plato always opted for the existence of one world, although this world was for him ontically complex.

In this article I analyze a traditional interpretation of Platonic philosophy, which assumes a "theory of two worlds." I try to prove that it is difficult to accept such an interpretation. If one can say that the on tic status of ideas differs from the ontic status of undefined matter (the phenomenon is always for Plato a relation, a compositum, of idea and undefined matter), one nevertheless cannot say that accepting this necessarily results in accepting two independently existing worlds. For then what would phenomena be, if they lacked the ideas that are the source of their determination; and what would ideas themselves be, if they referred to something other than the subject of their determination? Plato always opted for the existence of one world, although this world was for him ontically complex.