Functions, Relations, and Types

The concept of a function, stemming from the work of Leibniz and brought into its modern form by 19th-century mathematicians and logicians, is of indisputable significance in mathematics, physics, economics, and other sciences. By contrast, at least since Bradley and Russell, the attention of philosophers has tended to be directed at relations rather than at functions. This divergence gives rise to several questions. To begin with, it raises the question of whether theorizing about functions, as we find it in mathematics and theoretical computer science, might have lessons also for the philosopher who tries to gain a grasp of the ontology of relations. It further raises the question of whether philosophical theorizing about the identity conditions of properties and relations might carry over in some fruitful way to the theory of functions, and thereby inform those branches of knowledge in which the notion of a function plays a more prominent role. In addition, it raises the question of to what extent considerations about our cognitive grasp of functions and mathematical operations are applicable to the question of what it takes to have a cognitive grasp of a property or relation. Our aim in this project will be to attack this complex of questions from three sides, roughly corresponding to (a) ontology, (b) philosophical logic, and (c) philosophy of mind.