Aspects of Pluralism in the Foundations of Mathematics

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

This proposal is to develop the PI's ongoing investigation into non-trivial examples of pluralism in the foundations of mathematics involving seemingly incompatible but viable approaches. Two main loci of such multiplicity are (1) the choice between classical and constructive logic (renouncing certain classical principles) in the formulation of analysis, differential geometry, etc., and (2) differing ontological frameworks as in the single fixed universe of set theory as contrasted with the multiple universe approach of category theory (CT) and related varieties of structuralism. Of particular interest under (1) is "smooth infinitesimal analysis" (SIA) which develops a non-punctiform conception of the continuum, with infinitesimals whose square is 0. Constructive logic must be used to avoid inconsistency, but the results are striking simplifications of classical analysis and a vindication of early proofs in calculus, long thought to be discredited. Problems of interpretation, however, arise due to the banning of certain classical inferences involving identity. Whether such apparent conflicts are genuine or merely apparent, and, in the latter case, how reconciliation is to be achieved, are important questions that this project will address and try to resolve. Concerning (2), although set theory with a fixed universe is traditionally taken as the background framework for developing mathematical theories, category theory's claims to provide an alternative, more general and adaptable framework deserve serious attention. This project seeks to clarify the status of CT as an autonomous alternative, as well as its relation to other structuralist approaches (e.g. the PI's modal-structuralism) that, like CT, also provide for "multiple universes" of mathematical discourse. Progress on these questions can help shape the general understanding of the nature of mathematics as well as its relation to other sciences, which exhibit analogous kinds of "pluralism." Concerning the broader impacts of this research, it should promote a conception of mathematics as a much more multi-faceted body of knowledge than is widely thought, both among scientists and the public. This can be reflected in mathematical education in a wide variety of contexts, most immediately at the college level. The PI's work illustrates the importance of tolerating, and even promoting, a multiplicity of seemingly conflicting approaches, which are not only useful but may even be necessary for comprehending a rich and complex subject matter. At the same time, as the particular mathematical theories focused on here well illustrate, there need be no sacrifice of rigor or other relevant intellectual standards in the pursuit of these multiple approaches. This project can help promote tolerance of pluralism, while maintaining appropriate standards of logic and evidence, in a wide variety of contexts, within and beyond the academic.