The explanatory function of abstract objects: their nature and cognoscibility
We will consider the argument that claims that mathematical entities are indispensable on behalf of their explanatory function, an argument that was formulated by Baker (2001, 2003, 2005), and Colyvan (2001, 2007), to propose that it also applies (via extrapolation) to other kinds of abstract objects that are clear examples of cultural products. We understand that hand in hand with the explanatory indispensability proposal, it is possible to formulate the concrete/abstract distinction in a new way. Such a new way would go in combination with some version of the abstraction principles (schema), and would add to the already established ways identified by Lewis (1986) (also Lowe 1999; 2002 chapter 20). As a result of this project, we would have the following ways of establishing the concrete/abstract contrast: a) the way of negation (what an abstract object is not); b) the way of paradigmatic cases; c) the way of coincidence with some other distinction (i.e., universal/particular); d) the way of abstraction; e) the way of the explanatory function.
In this Project we will vindicate that this last way, the way of the explanatory function, countenances a generalized argument to postulate abstract entities of diverse ilk. We believe this has to be so because, in general, explanations of many different phenomena demand positing abstract objects; this is so, regardless of the fact that the abstract objects posited can vary together with the phenomena-or with the aspects of the phenomena-to be explained. With this generalized argument as a guiding principle, we will approach the topic of abstract objects by taking into account two levels of analysis: i) analysis of how some of the various abstract objects contribute to the explanation of different problems; ii) study of abstract objects in themselves starting from some study cases in order to clarify their ontological status, their nature and how is it possible for us to get to know them. We will pursue these issues in three philosophical areas in which the appeal to abstract entities is ubiquitous: the philosophy of mind and language, the philosophy of mathematics, and the philosophy of empirical sciences. We will concentrate, essentially, in the analysis of the explanatory function of propositions, concepts, numbers, theoretical entities and theories.
The inquiry about all the issues mentioned will be done by paying attention to: 1) the unavoidability of abstract objects in human discourses; 2) the ontological indispensability of abstract objects; 3) the heterogeneous character of the category of abstract objects; and 4) the dependencies of certain essential properties of abstract objects of a certain type on their theoretical or explanatory function.
Besides, in the light of those analyses, we will try to consider the issues related to: the case for or against the existence of abstract objects, and the clarification of its nature (when it is assumed its existence as different of that of concrete entities); and, the study about the way in which we would come to grasp and know them (if they existed). There is an important literature about nature and cognoscibility of abstract objects; the novelty in our research on these aspects relies on the fact that it will be done from the perspective of the theoretical and explanatory roles of abstract entities with respect to different issues.