Stella Saes Byeol Moon

2023 April, 1 - June, 29

Affiliation: University of California, Irvine

Research for a study about:

Account of mathematical Structuralism in which mathematical Structures arise from the social practice of Mathematics

Report

During my time as an IVC fellow, I investigated an account of mathematical structuralism inspired by the works of Edmund Husserl. I argued that the Husserlian method of Besinnung can offer accounts of philosophical and methodological structuralism in mathematics - a distinction introduced by Georg Schiemer and Erich Reck (2020). In this Husserlian view, a structure is characterised as a goal of a given mathematical community, based on the community's structural methodological practices. I further considered homotopy type theory, a relatively young area of mathematical/mathematical logic, as a possible candidate to realise the Husserlian view of structuralism. I presented this work at the Logik Cafe and received helpful comments and suggestions from the audience.

I also presented a working paper on philosophy of homotopy type theory at the annual international meeting of Univalent Foundations and Homotopy Type Theory, hosted at the Vienna University of Technology. In this paper, I argued that path induction (identity elimination rule in homotopy type theory) can be given a homotopical justification based on the intended interpretation of type theory by homotopy theory. This argument is philosophically inspired by Husserlian philosophy of mathematical practice, in which the philosopher focuses on the claims made by mathematicians to clarify and answer their philosophical concerns. It contrasts with existing philosophical arguments concerning path induction, which start from philosophical perspectives, independently of the homotopical interpretation which the mathematicians have been advocating for. I received constructive feedback from mathematical experts in the field, which has been valuable in developing my project further.

The research at IVC naturally motivated my postdoctoral project at the Czech Academy of Sciences. I am currently working on philosophical interpretations of type theories, and their foundational relevance. I am truly grateful to all the members of IVC, University of Vienna, and other IVC fellows, with whom I have had such fruitful and interesting conversations on philosophy of science, mathematics, and logic.