M²SP: Metaphysics Meets Set Theoretic Practice

(Salzburg, 3 September 2020)

A contributed Symposium

to the SOPHIA 2020 (2-4 September)

(Salzburg, 3 September 2020)

A contributed Symposium

to the SOPHIA 2020 (2-4 September)

Funded with the generious support of the Evert Willem Beth Foundation

## Speaker

## Neil Barton (Konstanz): Topic TBA

Matteo de Ceglie (Salzburg): Topic TBA

Joel David Hamkins (Oxford): Topic TBA

Deborah Kant (Konstanz): Topic TBA

Collin Jakob Rittberg [TBC] (Brussel): Tobic TBA

more TBA

## Registration

No registration is required

## About

In this workshop, we want to investigate the interplay between the mathematician’s metaphysical position and their practice (especially modern foundational endeavours such as set theory and constructive foundations). In particular, we want to discuss the following questions:

Examples of issues related to these questions are the following:

Literature

Antos, Carolin; Friedman, Sy-David, Honzik, Radek & Ternullo, Claudio (2015): “Multiverse Conceptions in Set Theory”, Synthese 192.8: 2463-2488

Awodey, Steve (2014), "Structuralism, invariance, and univalence." Philosophia Mathematica 22.1: 1-11.

Friend, Michele (2019), Varieties of Pluralism and Objectivity in Mathematics. To be reprinted In: Centrone, S.; Kant, K., & Sarikaya, D. (Eds): Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts. Synthese Library, Springer.

Hellman, G. (2003), “Does Category Theory Provide a Framework for Mathematical Structuralism?”, Philosophia Mathematica, 11-2: 129–157.

Koellner, Peter (2013), "The Continuum Hypothesis", In: Edward N. Zalta (Ed.): The Stanford Encyclopedia of Philosophy,

Rittberg, C. J. (2016). Methods, goals and metaphysics in contemporary set theory. Ph.D. Thesis. https://pdfs.semanticscholar.org/0fac/03d2192e91e5b4501a5994def3e396669e7e.pdf Accessed 21.05.2019

Landry, E., Marquis, J.-P. (2005), “Categories in Context: Historical, Foundational, and Philosophical”, Philosophia Mathematica, 13-1: 1–43.

Hamkins, J. D. (2012). “The set-theoretic multiverse”. The Review of Symbolic Logic, 5.3: 416–449.

Lambek, J. (2004), “What is the world of Mathematics? Provinces of Logic Determined”, Annals of Pure and Applied Logic, 126: 1–3: 149–158.

Tsementzis, Dimitris (2017). "Univalent foundations as structuralist foundations." Synthese 194.9: 3583-3617.

Van Atten, M. (2016). Essays on Gödel's Reception of Leibniz, Husserl, and Brouwer. Springer International Publications. Chicago

Woodin, W. Hugh (1999), The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Walter de Gruyter

Woodin, W. Hugh (2001a), "The continuum hypothesis. I", Notices of the AMS, 48.6: 567–576

Woodin, W. Hugh (2001b), "The Continuum Hypothesis, Part II", Notices of the AMS, 48.7: 681–690

- Which metaphysical position justifies or motivates which foundational program? And conversely, what are the metaphysical requirements of a particular foundational program?
- Can (meta-)mathematical insights change our metaphysical beliefs or even disprove certain conceptions?
- Do philosophical ideas and mathematical practice of contemporary mathematicians match?

Examples of issues related to these questions are the following:

- How does the set theorist’s metaphysical position on set theory (e.g., universe view, multiverse view) reflect in their mathematical work? In particular, does the multiverse view forces us upon the position that the continuum hypothesis (CH) has no truth value (cf. Hamkins’s Dream Solution Argument), but we only have to understand how its truth-values change under forcing? How does the Universe view relativise the independence results we obtain by forcing? In general, how do mathematicians of different metaphysical positions interpret each other’s work? Should we settle independent questions, like CH? See Rittberg (2016) or Hamkins (2012).
- Are all models of set theory created equal? Is it problematic to restrict much attention on countable models? See, for instance, Woodin (1999, 2001a, 2001b) or Koellner (2013).
- Which foundational view explains best recent developments, like Homotopy Type Theory? See, for example, Awodey (2014) or Tsementzis (2017).
- Do new practices especially in the context of constructive mathematics motivate new philosophical positions? See Rittberg (2016).
- What consequences does all of this have for pluralism in mathematics? Is it justified to assign a unique status to part of the practice, f.i. the usage of classical logic in the meta level. See Friend (2019).
- Did early views in the philosophy of topos theory, viewing toposes as mathematical universes, forecast the multiverse view? See, for example, Lambek (2004).
- What are the connections between logical system and mathematical pluralism?
- Can classical and non-classical approaches to the foundations of mathematics be considered in the same metaphysical arena?

Literature

Antos, Carolin; Friedman, Sy-David, Honzik, Radek & Ternullo, Claudio (2015): “Multiverse Conceptions in Set Theory”, Synthese 192.8: 2463-2488

Awodey, Steve (2014), "Structuralism, invariance, and univalence." Philosophia Mathematica 22.1: 1-11.

Friend, Michele (2019), Varieties of Pluralism and Objectivity in Mathematics. To be reprinted In: Centrone, S.; Kant, K., & Sarikaya, D. (Eds): Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts. Synthese Library, Springer.

Hellman, G. (2003), “Does Category Theory Provide a Framework for Mathematical Structuralism?”, Philosophia Mathematica, 11-2: 129–157.

Koellner, Peter (2013), "The Continuum Hypothesis", In: Edward N. Zalta (Ed.): The Stanford Encyclopedia of Philosophy,

Rittberg, C. J. (2016). Methods, goals and metaphysics in contemporary set theory. Ph.D. Thesis. https://pdfs.semanticscholar.org/0fac/03d2192e91e5b4501a5994def3e396669e7e.pdf Accessed 21.05.2019

Landry, E., Marquis, J.-P. (2005), “Categories in Context: Historical, Foundational, and Philosophical”, Philosophia Mathematica, 13-1: 1–43.

Hamkins, J. D. (2012). “The set-theoretic multiverse”. The Review of Symbolic Logic, 5.3: 416–449.

Lambek, J. (2004), “What is the world of Mathematics? Provinces of Logic Determined”, Annals of Pure and Applied Logic, 126: 1–3: 149–158.

Tsementzis, Dimitris (2017). "Univalent foundations as structuralist foundations." Synthese 194.9: 3583-3617.

Van Atten, M. (2016). Essays on Gödel's Reception of Leibniz, Husserl, and Brouwer. Springer International Publications. Chicago

Woodin, W. Hugh (1999), The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Walter de Gruyter

Woodin, W. Hugh (2001a), "The continuum hypothesis. I", Notices of the AMS, 48.6: 567–576

Woodin, W. Hugh (2001b), "The Continuum Hypothesis, Part II", Notices of the AMS, 48.7: 681–690