Evaluation of discussions in contemporary philosophy of mathematics on the axis of Plato's metaphysics and interpretation of Gödel's incompleteness theorems

In contemporary philosophy, ontology is understood as a research on what things will be accepted to "exist", and metaphysics is understood as a research on reality as it is. The ongoing debates in the contemporary philosophy of mathematics, mostly develop around the question whether mathematical objects, such as numbers and sets, exist or not whereas the metaphysical ground of these objects are ignored. "The Platonists" who accept the existence of mathematical objects, actually "presuppose" the existence of mathematical objects in a certain framework of language, but they don't go into an inquiry concerning the metaphysical nature of these objects. According to Plato, whom the Platonists took their name from, the Ideas, as beings in themselves are the ground of all mathematical and physical entities and constructions. Since the soul is acquainted with these Ideas prior to sensible experience, she can give judgments about these mathematical objects and do mathematics by carrying out mathematical constructions and entertaining an experience of them. For Plato this acquaintance of the soul with Ideas is taken as evidence to the result that the soul is similar in natüre with these Ideas and so is immortal. In contemporary philosophy of mathematics, the views presented about the foundations of mathematics suppose some mathematical objects such as extensions of concepts, formal formulas and intuitional constructions are mentioned, but the metaphysical ground of these objects are not questioned. According to Plato's metaphysics, as we have tried to show in our thesis, thanks to the acquitance of the soul with Ideas themselves, the soul can construct the sets, which are foundational mathematical objects, and therefore it can do mathematics. Kurt Gödel's "incompleteness theorems" which claim that formal systems are essentially incomplete and that they can't prove their consistency can be considered to support the tenets of Platonic metaphysics and they can be evaluated in terms of this metaphysics.