aboutlogic

Category Theory: Algebra on Steroids? | aboutlogic: premises #11
Is category theory just algebra on steroids? In this premises, Thorsten and Deniz give a pedestrian-speed introduction to category theory: structural mathematics, universal properties, duality and why products always come with coproducts, co-natural numbers, diagram chasing, and why homotopy type theory leads straight to higher categories. They also ask how category theory and type theory fit together and why Thorsten sees category theory as a way to structure ideas rather than the source of them.

Why Didn't Homotopy Type Theory Take Over the World? | aboutlogic #25
From ordinal analysis to homotopy type theory: Ulrik Buchholtz (University of Nottingham) talks with Deniz and Thorsten about his journey from algebraic topology to Sol Feferman's unfolding program and from there into univalent foundations. Why do ordinals measure the strength of mathematical theories? What makes a definition predicative? And what is the one open problem that keeps homotopy type theory from taking over mathematics?

Is Unique Choice Really a Choice? (Follow-up to Giovanni Sambin) | aboutlogic premises #10
If there is only one possible choice, are you making a choice at all? In this Premises, Thorsten and Deniz pick up the debate from our conversation with Giovanni Sambin: why does his minimalist foundation reject the Axiom of Unique Choice, while homotopy type theory proves it, and why does Lean make it so hard to use constructively?

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What does set theory have to do with reality? Giovanni Sambin, Professor at the University of Padua and a leading figure in constructive mathematics, couldn't accept the standard axiomatic foundations of mathematics. So he spent nearly 50 years building an alternative from the ground up. He joins Deniz and Thorsten to discuss his new book Positive Topology, the culmination of decades of work connecting Per Martin-Löf's type theory, point-free topology, and his own "minimalist foundation" for mathematics.

Season 2 Recap: From Gödel to AI-Proved Math | aboutlogic #22
Nine guests, four emerging themes, and one big question running through all of it: what happens to mathematics when symbolic reasoning and statistical AI finally meet? In this Season 2 recap, Thorsten Altenkirch and Deniz Sarikaya look back at conversations with Urs Schreiber, Joel David Hamkins, Dana Scott, Emily Riehl, Bernhard Fisseni, Bernhard Schröder, José Pérez Escobar, Jan von Plato, Mike Shulman, Tudor Achim, and Neil Ghani — and preview where Season 3 is headed.