Tag Archives: pluralism

The ontology of mathematics, Japan Association for the Philosophy of Science, June 2022

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I shall give the Invited Lecture for the Annual Meeting (online) of the Japanese Association for the Philosophy of Science, 18-19 June 2022.

Abstract. What is the nature of mathematical ontology—what does it mean to make existence assertions in mathematics? Is there an ideal mathematical realm, a mathematical universe, that those assertions are about? Perhaps there is more than one. Does every mathematical assertion ultimately have a definitive truth value? I shall lay out some of the back-and-forth in what is currently a vigorous debate taking place in the philosophy of set theory concerning pluralism in the set-theoretic foundations, concerning whether there is just one set-theoretic universe underlying our mathematical claims or whether there is a diversity of possible set-theoretic conceptions.

Slides – The ontology of mathematics – Hamkins – Japan Association Phil Science – June 2022

Pluralism in the ontology of mathematics, MaMuPhi, Paris, February 2022

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This will be a talk for the conference L’indépendance mathématique et ses limites logiques, an instance of the MAMUPHI seminar (mathématiques – musique – philosophie), organized by Mirna Džamonja, 12 February 2022. Most talks will be in-person in Paris, but my talk will be on Zoom via https://u-pec-fr.zoom.us/j/86448599486 at 4:30 pm CET (10:30 am EST).


Abstract: What is the nature of mathematical ontology—what does it mean to make existence assertions in mathematics? Is there an ideal mathematical realm, a mathematical universe, that those assertions are about? Perhaps there is more than one. Does every mathematical assertion ultimately have a definitive truth value? I shall lay out some of the back-and-forth in what is currently a vigorous debate taking place in the philosophy of set theory concerning pluralism in the set-theoretic foundations, concerning whether there is just one set-theoretic universe underlying our mathematical claims or whether there is a diversity of possible set-theoretic conceptions.

Is there just one mathematical universe? DRIFT, Amsterdam, May 2019

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This will be a talk for the Wijsgerig Festival DRIFT 2019, held in Amsterdam May 11, 2019. The theme of the conference is: Ontology.

Kamanasish Debnath [CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0)]

Abstract. What does it mean to make existence assertions in mathematics?
Is there a mathematical universe, perhaps an ideal mathematical reality, that the assertions are about? Is there possibly more than one such universe? Does every mathematical assertion ultimately have a definitive truth value? I shall lay out some of the back-and-forth in what is currently a vigorous debate taking place in the philosophy of set theory concerning pluralism in the set-theoretic foundations, concerning whether there is just one set-theoretic universe underlying our mathematical claims or whether there is a diversity of possible set-theoretic worlds.

Slides: Is there more than one mathematical universeDownload
At the conference venue in the Vondelpark, Amsterdam

Pluralism-inspired mathematics, including a recent breakthrough in set-theoretic geology, Set-theoretic Pluralism Symposium, Aberdeen, July 2016

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Set-theoretic Pluralism, Symposium I, July 12-17, 2016, at the University of Aberdeen.  My talk will be the final talk of the conference.

University of AberdeenAbstract. I shall discuss several bits of pluralism-inspired mathematics, including especially an account of Toshimichi Usuba’s recent proof of the strong downward-directed grounds DDG hypothesis, which asserts that the collection of ground models of the set-theoretic universe is downward directed. This breakthrough settles several of what were the main open questions of set-theoretic geology. It implies, for example, that the mantle is a model of ZFC and is identical to the generic mantle and that it is therefore the largest forcing-invariant class. Usuba’s analysis also happens to show that the existence of certain very large cardinals outright implies that there is a smallest ground model of the universe, an unexpected connection between large cardinals and forcing. In addition to these results, I shall present several other instances of pluralism-inspired mathematics, including a few elementary but surprising results that I hope will be entertaining.

Slides | Set-theoretic Pluralism Network | Conference program