aboutlogic

Choice vs. Excluded Middle: A Constructive Paradox | aboutlogic: premises #08 Constructive mathematics is all about building things explicitly — so why does it reject the Axiom of Choice, which sounds trivial in a constructive context. In this Premises episode, Thorsten walks Deniz through Diaconescu's theorem: the surprising proof that the Axiom of Choice implies the Law of Excluded Middle, turning a seemingly innocent principle into full-blown classical logic. Using an intuitive type-theoretic explanation (starting with a very relatable glove-matching example), Thorsten builds up to Diaconescu's classic argument, touching on propositional extensionality, the difference between intensional and extensional predicates, and why the Axiom of Choice turns out to be a stronger form of "magic" than Excluded Middle itself.

Can AI prove the Riemann Hypothesis? Tudor Achim, CEO of Harmonic and creator of Aristotle — the first AI to win IMO gold and solve Erdős problems using the Lean theorem prover — joins Deniz and Thorsten to discuss how mathematical superintelligence is transforming research, education, and the very nature of proof.

Fixing Russell’s Paradox: The Birth of ZFC & Constructive Set Theory How did mathematicians fix Russell’s paradox and save set theory? In this aboutlogic: premises episode, Deniz and Thorsten explore the solutions that reshaped the foundations of mathematics. From Zermelo-Fraenkel (ZFC) axioms to constructive set theories (IZF, CZF). Discover how large cardinals, the continuum hypothesis, and the iterative conception of sets became central to modern set theory and why some mathematicians still prefer type theory for its structural and computational advantages. Your support helps us keep these conversations going! If you’d like to contribute, you can buy us a coffee here: https://buymeacoffee.com/aboutlogic Or become a channel member here on Youtube. We’d love to have your support! https://www.youtube.com/@aboutlogic Join the Discussion: Have questions or thoughts to share? Drop a comment below and engage in a discussion with fellow viewers and experts.

Homotopy Type Theory, Narya & the Future of Proof Assistants with Michael Shulman. How does homotopy type theory bridge the gap between abstract mathematics and computational proof systems? Mike Shulman (University of San Diego) joins Deniz and Thorsten to discuss his journey from topology to higher observational type theory, the development of the Narya proof assistant, and how these tools are reshaping the way we think about equality, equivalence, and computation in mathematics.

What Is a Set? A Beginner’s Guide to Set Theory | aboutlogic: premises #06 In this aboutlogic: premises episode, Deniz and Thorsten explore the foundations of set theory. From Cantor’s groundbreaking ideas to Frege’s logical foundations and Russell’s paradox. Discover how sets evolved from simple collections to a rigorous mathematical framework, and why the power set, well-ordering, and the continuum hypothesis remain some of the most fascinating (and controversial) ideas in math.

aboutlogic #18 | What really happened in the 1930s logic revolution? Jan von Plato (University of Helsinki, ERC Grantee) joins Deniz and Thorsten to uncover the hidden collaborations, misunderstandings, and lost manuscripts that shaped modern logic. From Gödel’s unpublished notes to Gentzen’s lost normalization proof and Bernays’ pivotal role in Hilbert’s school, this episode reveals how the history of logic is far richer—and more interconnected—than we often assume.

Dependent Type Theory: A Revolution in Math & Computer Science | aboutlogic: premises #05 What makes dependent type theory so powerful? In this aboutlogic: premises episode, Deniz and Thorsten explore the evolution of type theory. From simple types to Pierre Martin-Löf’s groundbreaking dependent types. Discover how this innovation transformed mathematics and computer science by allowing types to depend on values, enabling more expressive and precise reasoning.

aboutlogic #17 | Why is mathematics so effective in science? José Pérez Escobar (UNED, Madrid) joins Deniz and Thorsten to explore Wittgenstein’s philosophy of applied mathematics, the role of rules vs. structures in math, and how models shape our understanding of reality. From neuroscience to physics, José explains why mathematical models often act as rules of description rather than mere representations of reality and how this perspective resolves Wittgenstein’s "rule-following paradox." The conversation also dives into Turing’s structural view of math, the Dirac delta function controversy, and whether contradictions in mathematics are truly problematic.

Your support helps us keep these conversations going! If you’d like to contribute, you can buy us a coffee here: https://buymeacoffee.com/aboutlogic How do interactive theorem provers like Lean and Agda change the way we teach and do mathematics? In this aboutlogic: premises episode, Deniz and Thorsten discuss the role of proof assistants in education, the differences between Lean and Agda, and how AI is transforming formal verification.

aboutlogic #16 | How is mathematical language structured, and what can linguistics teach us about proofs, ambiguity, and storytelling in math? In this episode, Bernhard Fisseni and Bernhard Schröder (University of Duisburg-Essen) join Deniz and Thorsten to explore the frames, narratives, and pragmatic structures behind mathematical texts.

Inspired by our conversation with Emily Riehl on higher category theory, this aboutlogic: premises episode dives into the synthetic vs. analytic approach in mathematics. Deniz and Thorsten explore how Euclid’s geometry, category theory, and higher categories embody the synthetic approach. Focusing on abstract structures and relationships rather than concrete coordinates or definitions.

aboutlogic #15 | Emily Riehl (Johns Hopkins University) joins us to explore higher category theory, homotopy, and the role of AI in modern mathematics. From the foundations of category theory to the challenges of formalizing math with proof assistants like Lean, Emily shares her insights on synthetic vs. analytic approaches, the beauty of abstraction, and how AI is changing mathematical research.

Your support helps us keep these conversations going! If you’d like to contribute, you can buy us a coffee here: https://buymeacoffee.com/aboutlogic What is set theory—a foundation of math or a science of infinity? In this aboutlogic: premises episode, Deniz and Thorsten explore the history, paradoxes, and philosophical debates behind set theory. From Cantor’s diagonal argument to Hilbert’s Hotel and the role of ZFC, they discuss why set theory became the language of mathematics—and where its limits lie. Join the Discussion: Have questions or thoughts to share? Drop a comment below and engage in a discussion with fellow viewers and experts.

aboutlogic #14 | Turing Award winner Dana Scott joins us to discuss his groundbreaking work on lambda calculus, forcing, and Boolean-valued models and how these ideas revolutionized set theory and computability. From his collaborations with Kleene and Solovay to his thoughts on constructive mathematics, Scott shares insights into the history and future of logical foundations. Hear anecdotes about Gödel’s unpublished ideas, Einstein’s influence, and the telephone conversations that shaped modern logic.

Our weekly Premises: Behind-the-scenes thoughts, deep dives, and the ideas that didn’t fit into the main episodes. Is mathematics a discovery or a story we tell ourselves? In this first aboutlogic: premises episode, Deniz and Thorsten explore why math might be more like fiction than absolute truth and what that means for logic, proof, and the future of the field.

aboutlogic #13 | In this episode of aboutlogic, we’re joined by Joel David Hamkins, professor at the University of Notre Dame and a leading figure in set theory, mathematical logic, and the philosophy of mathematics. Joel shares his insights into the multiverse view of set theory, a perspective that challenges the traditional "universe view" by embracing a pluralistic approach to mathematical truth. We explore how this view connects to constructivism, potentialism, and the foundations of mathematics, and discuss its implications for understanding concepts like the Continuum Hypothesis (CH) and the nature of mathematical reality. Joel also reflects on the historical contingency of mathematical axioms, the role of categoricity in mathematics, and how different philosophical perspectives, such as Platonism, formalism, and fictionalism, shape the way mathematicians approach their work. Whether you're a mathematician, philosopher, or simply curious about the foundations of logic, this conversation offers a deep dive into the diverse and evolving landscape of mathematical thought.

aboutlogic #12 | In this episode of aboutlogic, we’re joined by UrsSchreiber, a senior scientist at New York University Abu Dhabi. Urs shares insights into his work at the intersection of quantum physics, topos theory, and homotopy type theory. We explore how these advanced mathematical frameworks help address fundamental questions in physics, from understanding gauge fields to the role of higher category theory in describing the universe. Urs also discusses the historical and philosophical connections between physics and logic, and how modern mathematics is shaping our understanding of reality.

aboutlogic #11 | In this special Season 1 Recap of aboutlogic, Deniz and Thorsten reflect on your comments and feedback, revisit some of the most intriguing topics, and look back at all the incredible guests from the first season. What were the highlights? What did we learn? And what’s in store for Season 2? Join us for a wrap-up filled with insights, gratitude, and a sneak peek at what’s next.

aboutlogic #10 | In this episode, we talk with Seunghyun Song and Jordi Fairhurst about the ABC conjecture, the importance of epistemic justice, and the role of linguistic justice in non-Western mathematical traditions. How do diverse perspectives reshape our understanding of mathematics? Join the Discussion: Have questions or thoughts to share? Drop a comment below and engage in a discussion with fellow viewers and experts.

aboutlogic #09 | In this episode, we talk with Andrej Bauer about the five stages of accepting intuitionistic mathematics and the challenges surrounding proofs by contradiction. Join the Discussion: Have questions or thoughts to share? Drop a comment below and engage in a discussion with fellow viewers and experts.