Sean Carroll's Mindscape: Science, Society, Philosophy, Culture, Arts, and Ideas

282 | Joel David Hamkins on Puzzles of Reality and Infinity

Jul 15, 2024

01:18:23

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Mathematician and philosopher Joel David Hamkins discusses the puzzles of reality and infinity, exploring different perspectives on mathematics, set theory multiverse, Gödel's Incompleteness Theorems, and the debate between realism and anti-realism. The conversation delves into the continuum hypothesis, sizes of infinity, models in arithmetic and set theory, Turing machines, halting problems, and the intricate relationship between mathematics and philosophy.

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Quick takeaways

Infinity in mathematics leads to counterintuitive results and challenges clarity.

Gödel's incompleteness theorem raises deep philosophical questions about provability and axioms.

Set-theoretic multiverse suggests pluralistic mathematical realities beyond uniqueness.

Deep dives

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Multiple Mathematical Universes

00:34

Consistent Theory Complexity

00:24

Potentialism: The Universe of Mathematical Objects

01:22

Limitations of Complete Systems

00:30

Intro

2min

Exploration of Mathematics, Axiomatization, and Incompleteness

4min

Philosophy of Mathematics and Set Theory Multiverse

9min

Exploring the Foundations of Mathematics Multiverse

3min

Debate on Weak and Strong Foundations in Mathematics

11min

Exploring the Continuum Hypothesis and Set Theory

7min

Exploring Axioms, Models, and Theorems in Mathematics

18min

Exploring Turing machines, halting problems, and computational models

4min

Exploring the Intricacies of Mathematical Systems and Philosophical Implications

24min

The philosophy of mathematics would be so much easier if it weren't for infinity. The concept seems natural, but taking it seriously opens the door to counterintuitive results. As mathematician and philosopher Joel David Hamkins says in this conversation, when we say that the natural numbers are "0, 1, 2, 3, and so on," that "and so on" is hopelessly vague. We talk about different ways to think about the puzzles of infinity, how they might be resolved, and implications for mathematical realism.

Blog post with transcript: https://www.preposterousuniverse.com/podcast/2024/07/15/282-joel-david-hamkins-on-puzzles-of-reality-and-infinity/

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Joel David Hamkins received his Ph.D. in mathematics from the University of California, Berkeley. He is currently the John Cardinal O'Hara Professor of Logic at the University of Notre Dame. He is a pioneer of the idea of the set theory multiverse. He is the top-rated user by reputation score on MathOverflow. He is currently working on The Book of Infinity, to be published by MIT Press.

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