Foundations as Superstructure. (Reflections of a practicing mathematician)

TL;DR

This paper explores the evolving nature of mathematical foundations, emphasizing their role as flexible, historically contingent principles rooted in human intuition rather than fixed, prescriptive rules.

Contribution

It offers a reflective perspective on the history of mathematical foundations, highlighting their variability and dependence on human intuition over time.

Findings

Foundations serve as norms for knowledge transfer.

Foundations are historically variable and intuitive.

They lack prescriptive power in mathematics.

Abstract

This talk presents foundations of mathematics as a historically variable set of principles appealing to various modes of human intuition and devoid of any prescriptive/prohibitive power. At each turn of history, foundations crystallize the accepted norms of interpersonal and intergenerational transfer and justification of mathematical knowledge.