Sean Carroll's Mindscape: Science, Society, Philosophy, Culture, Arts, and Ideas 282 | Joel David Hamkins on Puzzles of Reality and Infinity
23 snips
Jul 15, 2024 Joel David Hamkins, the John Cardinal O'Hara Professor of Logic at Notre Dame, navigates the perplexing world of infinity and mathematical truths. He discusses the evolution of mathematical thought, from Cantor’s groundbreaking work to Gödel's Incompleteness Theorems. The conversation dives into the concept of the mathematical multiverse and how different axioms shape our understanding of reality. Hamkins also addresses the Continuum Hypothesis and its implications, highlighting the philosophical challenges posed by infinity in mathematics.
AI Snips
Chapters
Books
Transcript
Episode notes
Mathematical Platonism and Pluralism
- Platonism in mathematics no longer implies a singular universe.
- Realists can believe in multiple, incompatible mathematical universes, like different set theories.
If-Thenism and Foundations
- "If-thenism" views mathematical truth as relative to chosen axioms.
- Strong foundations seek fundamental truths, while weak foundations prioritize wider applicability.
Models, Axioms, and Infinity
- Different models can satisfy the same axioms, like non-standard integers modeling arithmetic.
- The continuum hypothesis asks if there's an infinity between countable and uncountable sets.
