Potentialism: The Universe of Mathematical Objects

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The concept of potentialism in set theory has evolved over time, moving away from the notion of infinity being a core component. Potentialism now focuses on the idea that the universe of mathematical objects could be unfinished, even if there are actual infinities within it. This perspective allows for the universe to contain infinite sets at different levels of completion. Potentialism is not solely about infinity but revolves around the question of whether the mathematical universe is completed. Scholars are delving into potential set theory to explore different aspects of potentialism and its implications on the philosophical understanding of our mathematical reality.

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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