Postgraduates work in progress

Speaker: Chihon Ley Polanco

Title: The Existence of Mathematical Objects in Plato's Ontology

Abstract: Within Plato's ontology, it is widely known that he posited a dualism of Forms and sensible particulars, where the former are the foundation of reality that gives existence and meaning to the latter through a process of participation. However, according to Aristotle's testimony, Plato also posited a third kind of entity in his ontology: the mathematical objects, called Intermediates for being between Forms and sensible particulars. These Intermediates, on the one hand, share with the Forms the fact that they are imperishable, and, on the other hand, they share with the sensible particulars the fact that there are many of them. Nevertheless, after decades of controversy, it remains open the question whether Plato in fact posited these entities or not.

In this context, my research project intends to build a case in favour of the existence of Intermediates in Plato’s ontology and philosophy of mathematics. At first glance, upon taking a look at the literature on this topic, it seems that this endeavour is hopeless. Decades of commentaries on this matter have reached little consensus among specialists. However, in recent times, instead of remaining stubbornly fixed in the frame “Did Plato posit Intermediates or not?”, new works have made progress by changing the questions with which we analyse this problem. Alternatively, the discussion has been freshened by starting to think on Plato's rationale for positing the Intermediates. My contention is that Intermediates are the best solution to the problems triggered by mathematics in Plato's ontology.

Within the flanks of this rationale, we find (1) arguments of suitability of the Intermediates (i.e. Intermediates are the /only/ objects that can account for mathematical practice in all its aspects), (2) negative arguments (i.e. how can someone do mathematics with no Intermediates and just Forms), and (3) arguments that appeal to the strong realism present in Plato, Aristotle, and Greek mathematical practice. For the sake of time, in this presentation I will focus on showing why (1) is compelling on the ground of Plato's works, and also put (3) to a test. Indeed, by means of their logical procedure, I hope to show that Greek mathematicians considered the objects of mathematics as existing in a strong sense. This last point, in addition to the fact that Plato's and Aristotle's epistemology require a strong truthmaker to make each branch of knowledge a proper science, make a case to think that mathematics, /qua/ branch of knowledge, requires an existing object to work. And if this is true, then it would be safe to claim that Plato needed Intermediates in order to complete his ontology.