Event Detail

Fri Apr 19, 2024
60 Evans
4:10–5 PM
Logic Colloquium
Giovanni Sambin (Università di Padova)
Positive Topology. A New Practice in Constructive Mathematics

Since my first serious encounter with axiomatic set theory ZFC (at Berkeley as a PhD student in 1971-72), I have never been able to make its strong idealizations my own. This has led me, after more than fifty years of small steps, to a conception of mathematics that is radically different from the common one.

The main idea is to really accept that mathematics is a human construction—the outcome of a dynamic process—along with all its consequences. Then it can be seen as a conquered and local truth, which means certain and reliable information, and not as a given and universal truth. It becomes crucial to avoid principles such as the Law of the Excluded Middle, the Power Set Axiom, and the Axiom of Choice, that destroy the difference in information, respectively, between $\exists$ and $\neg \forall \neg$, inductively generated domains and not, operations and functions. The resulting dynamic-minimalist foundation provides a solution to foundational problems. In particular, the corresponding formal system MF is provably consistent.

This talk (with the same title as a book to appear next summer with Oxford UP) gives an overview of what mathematics, in particular topology, has been developed using this foundation. Most importantly, it illustrates the vast new regions of mathematical thought that emerge, in particular the co-presence and connection between real-effective and ideal-infinitary aspects, in particular between discrete and continuum, made possible via pointfree topology.

The challenge now is to show that we could arrive at a new Kuhnian paradigm in mathematics. This will inevitably require a considerable collective effort.