Infinity and Intensionality: Towards a New Synthesis

The project aims to systematically and comprehensively explore two sets of questions that are normally only studied separately. The first cluster concerns the theory of infinite sets, which form the basis of today's mathematics. This foundation is threatened by various internal tensions and paradoxes, which have received a lot of attention. The second set of questions concerns “intentional” concepts such as assertion, property, and relation. Assertions, for example, represent the content of our beliefs and desires, and thus play a central role in linguistics, psychology, and philosophy. In line with our overall hypothesis, we have developed a new synthesis where the two clusters of questions are explored in a holistic way, with concepts and theories from each cluster being applied to the second cluster. To answer the first set of questions, we have supplemented set theory with intentional concepts of collection and generality. We have developed theories in which a collection is defined via a membership criterion rather than directly via its members. In the opposite direction, we have developed theories of intentional concepts that draw on ideas developed by some prominent critics of modern set theory (notably Poincaré and Weyl). We have therefore largely confirmed our hypothesis that substantive progress requires a holistic exploration of these two sets of questions. Two remaining goals are to apply our work to (1) develop theories of properties that can be used in the foundations of semantics; and (2) to build bridges between our existing theories, which make use of modal logic, and more user-friendly theories where modal logic is not included.

We aim to undertake an integrated investigation of two clusters of questions—concerned with infinity and intensionality—which have so far only been studied in isolation. Our overarching hypothesis is that real progress can only be made by means of a new synthesis, where the two clusters of questions are tackled in a unified way, thus bringing concepts and theories from each cluster to bear on the other one. The first cluster of questions concerns the theory of infinite sets, which is the standard foundation for nearly all of today’s mathematics. This foundation is threatened by various internal tensions and paradoxes, which have attracted much philosophical attention. The second cluster concerns intensional notion such as propositions, properties, and relations. Propositions, representing the contents of beliefs and desires, are a central concern in linguistics, psychology, and philosophy. To address the first cluster, we need to supplement standard set theory with intensional notions of collection, number, and generality. But to develop satisfactory theories of intensional notions, we need to draw on concepts and theories developed by critics of infinitary set theory. Thus, real progress can only be made by an integrated investigation of the two clusters of questions. Our project will therefore develop a new synthesis in our theorizing about infinity and intensionality. Our objectives are to develop - a novel and distinctive foundation and philosophy of mathematics, which retains the theory of infinite sets but supplements it with intensional notions of collection, number, and generality - new theories of the intensional notions of proposition, property, and relation, of great significance for the foundations and philosophy of semantics and psychology - a novel and surprising connection between areas so far only investigated separately In research areas known for their male dominance, our interdisciplinary team has, very unusually, a perfect gender balance.