A Polar Concept Argument for the Existence of Abstracta

In this paper, I present a polar concept argument for the existence of abstract objects. After recalling the fundamentals concerning the debate about the existence of abstracta, I present in a detailed way the argument for the existence of abstracta. I offer two different variations of the argument: one, deductive and the other, inductive. The argument rests primarily on the fact that our universe is well-balanced. This well-balanced property results from the fact that all instantiable polar dualities are instantiated. Hence, the abstract pole of the abstract/concrete duality must also be exemplified. Lastly, I review several objections that can be raised against the present argument. There are several famous problems about abstract entities. One of them is whether there exist any abstract objects. A second issue is concerned with the definition of which sorts of entities are genuinely abstract. A third issue relates to whether the abstract/concrete duality is exhaustive or not. The purpose of this paper is to address the first of these issues and to describe a polar concept argument that entails the existence of abstracta. Before stating the argument in detail and reviewing several objections that can be raised against it, it is worth recalling first the fundamentals of the debate about the existence of abstracta. 1. The debate about the existence of abstracta Let us recall preliminarily the main lines of the issue of whether there exist abstract objects. This latter problem rests primarily on the abstract/concrete distinction. Uncontroversially, the following objects are considered as abstracta: the natural numbers, the cosine function, the Pythagorean theorem. For this reason, I shall only be concerned in what follows with paradigm abstracta, i.e. natural numbers, setting aside other sorts of entities whose status is controversial. On the other hand, an instance of jay or of an oak-tree, the mountain in front of me, the sun, our galaxy are paradigmatically classified as concrete objects. In this context, concreta are considered as existent objects. But at this stage, agreement stops. In effect, by contrast, the mere existence of abstracta is at issue. Do abstract objects truly exist? There are two main philosophical answers to this latter question. On the one hand, some philosophers simply deny the existence of abstracta. According to the corresponding line of thought, only concrete objects exist in our universe, and abstract concepts are a mere product of our brain circuitry. Thus, natural numbers, the sine function, and so on. which are considered as paradigm cases of abstracta, exist only in our mind. The view that denies the existence of abstract objets is nominalism. On the other hand, according to a line of thought originating from Plato, abstract objects do truly exist. According to platonism, abstracta have an existence of their own, in the same way as concreta 1 E. J. Lowe (1995) distinguishes three different conceptions of abstract objects. In what follows, my concern will be with what he terms abstract1 objects, i.e. in opposition to concrete objects.