Abolishing Platonism in Multiverse Theories

Axiomathes 32 (2):321-343 (2020)
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Abstract

A debated issue in the mathematical foundations in at least the last two decades is whether one can plausibly argue for the merits of treating undecidable questions of mathematics, e.g., the Continuum Hypothesis, by relying on the existence of a plurality of set-theoretical universes except for a single one, i.e., the well-known set-theoretical universe V associated with the cumulative hierarchy of sets. The multiverse approach has some varying versions of the general concept of multiverse yet my intention is to primarily address ontological multiversism as advocated, for instance, by Hamkins or Väätänen, precisely for the reason that they proclaim, to the one or the other extent, ontological preoccupations for the introduction of respective multiverse theories. Taking also into account Woodin’s and Steel’s multiverse versions, I take up an argumentation against multiversism, and in a certain sense against platonism in mathematical foundations, mainly on subjectively founded grounds, while keeping an eye on Clarke-Doane’s concern with Benacerraf’s challenge. I note that even though the paper is rather technically constructed in arguing against multiversism, the non-negligible philosophical part is influenced to a certain extent by a phenomenologically motivated view of the matter.

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Stathis Livadas
University of Patras (Alumnus)

Citations of this work

Against the countable transitive model approach to forcing.Matteo de Ceglie - 2021 - In Martin Blicha & Igor Sedlár (eds.), The Logica Yearbook 2020. College Publications.

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References found in this work

Non-standard Analysis.Gert Heinz Müller - 2016 - Princeton University Press.
Mathematical truth.Paul Benacerraf - 1973 - Journal of Philosophy 70 (19):661-679.
Naturalism in mathematics.Penelope Maddy - 1997 - New York: Oxford University Press.
The set-theoretic multiverse.Joel David Hamkins - 2012 - Review of Symbolic Logic 5 (3):416-449.

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