282 | Joel David Hamkins on Puzzles of Reality and Infinity

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

Sean Carroll | Wondery

Society & Culture, Physics, Philosophy, Science, Ideas, Society

4.8 • 4.4K Ratings

🗓️ 15 July 2024

⏱️ 79 minutes

🔗️ Recording | iTunes | RSS

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Summary

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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Transcript

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0:38.0	Hello everyone welcome to the Mindscape Podcast. I'm your host Sean Carroll.
0:42.0	Mathematics has always been the intellectual subject that has a reputation of being the most precise and rigorous and well formulated and completely grounded, right? You have axioms and then you prove things with 100% rigor, ideally anyway.
0:59.0	But you know, starting in the 1800s, there were these events that kind of might shake your face. In the There was the discovery of non-Uclitian geometry, right?
1:14.0	Euclid thought he had figured out geometry,
1:17.0	and people generally had the idea that he was putting his finger on something true.
1:21.0	This is what we call geometry and we realized that there were
1:24.8	different axioms you could choose that would give you different kinds of geometry
1:29.5	spherical geometry, hyperbolic geometry, etc.
1:32.5	But okay, we can still handle that.
1:35.3	Then Cantor comes along and shows that not only is infinity
	...

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