Mathematical Practices Can Be Metaphysically Laden
In this chapter I explore the reciprocal relationship between the metaphysical views mathematicians hold and their mathematical activity. I focus on the set-theoretic pluralism debate, in which set theorists disagree about the implications of their formal
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Contents 1 The Set-Theoretic Pluralism Debate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Woodin’s Ultimate-L Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Hamkins’ Multiverse View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Abstract
In this chapter I explore the reciprocal relationship between the metaphysical views mathematicians hold and their mathematical activity. I focus on the settheoretic pluralism debate, in which set theorists disagree about the implications of their formal mathematical work. As a first case study, I discuss how Woodin’s monist argument for an Ultimate-L feeds on and is fed by mathematical results and metaphysical beliefs. In a second case study, I present Hamkins’ pluralist proposal and the mathematical research projects it endows with relevance. These case studies support three claims: (1) the metaphysical views of mathematicians can shape what counts as relevant research; (2) mathematical results can shape the metaphysical beliefs of mathematicians; (3) metaphysical thought and mathematical activity develop in tandem in mathematical practices. This makes metaphysical thought an integral part of mathematical practices. Keywords
Set theory · Set-theoretic pluralism · Metaphysics · Ultimate-L · Mathematical practices · Set-theoretic multiverse C. J. Rittberg (*) Centre for Mathematical Cognition, Loughborough University, Loughborough, UK Centre for Logic and Philosophy of Science, Vrije Universiteit Brussel, Brussel, UK e-mail: [email protected] © Springer Nature Switzerland AG 2020 B. Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice, https://doi.org/10.1007/978-3-030-19071-2_22-1
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C. J. Rittberg
In this chapter, I argue that the metaphysical views mathematicians hold can shape what counts as relevant research in mathematical practices. I will furthermore show how these metaphysical views are in turn shaped by the results of formal mathematical research. This makes metaphysical thought an integral part of mathematical practices: mathematical practices are metaphysically laden. To make my argument, I present two case studies from set-theoretic practice. Set theory is arguably more heavily shaped by the metaphysical views of its practitioners than other mathematical practices. In set-theoretic practice, it is particularly visible how metaphysical beliefs and mathematical activity shape each other. This makes set theory a suitable math
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