Abstract

There is a traditional epistemological problem concerning abstract entities, entities with which, by their very nature, we do not have causal relations. This problem takes the form of a question: how can we refer to or know anything about abstract entities? A paradigm source of such an epistemological conundrum is provided by mathematics. An analysis of Aristotle's philosophy of mathematics throws some light on the nature of mathematical abstraction. In contrast to Plato, Aristotle held that mathematical objects do not have a separate existence from sensible bodies. Although not separate from sensible bodies, mathematical objects are nonetheless, according to Aristotle, separable in thought from them. Moreover, both the physicist and the mathematician study, Aristotle argues, the same objects. What distinguishes physics from mathematics is the fact that, in studying his objects, the mathematician takes a different stand from the physicist: The former "strips off" from physical objects all of their sensible qualities leaving only the quantitative and continuous in one, two or three dimensions. Thus according to Aristotle the mathematician studies physical objects not qua objects having certain sensible qualities, but qua quantitative and continuous. For Aristotle, then, mathematical objects are not something over and above sensible bodies. In this way, Aristotle's account of mathematics attempts to eschew the above-mentioned epistemological riddle. A further analysis of the qua locution in Aristotle's work shows some similarities between it and the mathematical theory of categories. It is argued that several basic concepts of category theory are abstract objects in Aristotle's sense. It is hoped that the present work would constitute the basis for further research that would show category theory itself as a mathematical object obtained by a process of abstraction. In this manner, for those philosophers and mathematicians alike who, in one form or another, propose category theory as a foundation for mathematics, Aristotle's efforts in trying to overcome the epistemological problems so deeply-rooted in the realm of the abstract, is shown to be of relevance for contemporary philosophy of mathematics, instead of only remaining of scholarly interest