Arithmetic and Geometric Relations in Plato’s Metaphysics

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When we hear about Forms in the Phaedo, (74a-79b) Socrates makes it very clear that even though sensible objects have their characters by sharing or participating in Forms, Forms and sensible objects are two ontologically different and distinct objects. Forms are eternal, unchanging, moreover, they are not extended in space nor do they have any sensible characteristics. They are simple, unitary objects. In contrast to Forms, sensible objects change, they are located in space and are perceptible. They are complex, composite objects. The inevitable result of this distinction presented in the Phaedo is that a common character cannot be shared in the same way between Forms and sensible particulars. The Form of Beauty, for example, is beautiful in an incommensurable way to the beauty of Helen. To understand why Helen is beautiful, we must think of her in a particular context, historical or literary. We can consider her beauty from a particular point of view, from Hecuba, Menelaus or Paris. We can focus on the youth of her skin or the sheen of her hair or the color of her eyes, if these characteristics are available for discovery by us. The beauty of Helen is contextualized within time and space. None of the instances of her beauty will be identical to the Form. Sensible objects have their characters in geometric relations. Forms have their characters in simple arithmetic relations. So how are Forms and sensible objects related? Forms are the boundaries, the organization or limit of the characters we find in sensible objects but they are not identical to those characters. I suggest here that we think of the Form-sensible particular relation as analogous to the relations we find in early Greek mathematics and use the problems that concerned the great philosophers and mathematicians during Plato’s time to reexamine Plato’s metaphysical claims about the relation of Forms and sensible objects in the Phaedo, Republic, the Parmenides and the Sophist. Of course to give an extensive account will require several papers. Here, I give my first sketches of an analysis, using Greek accounts of arithmetic and geometric relations and how they might inform our understanding of Plato’s metaphysics.


International Meeting of Philosophy and Mathematics in Plato


Paris, France


College of Arts and Sciences


Dr. Sophia Stone, assistant professor in the College of Arts and Sciences, presented her paper, Arithmetic and Geometric Relations in Plato's Metaphysics at the International Meeting of Philosophy and Mathematics in Plato at the École Normale Supérieure in Paris, France. Her paper explores Plato and themes in Greek Mathematics.

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