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Justifying Alternative Foundations for Mathematics

Title: Justifying Alternative Foundations for Mathematics.
Name(s): Ifland, Jared M., author
Type of Resource: text
Genre: Text
Bachelor Thesis
Date Issued: 2020-07-31
Physical Form: computer
online resource
Extent: 1 online resource
Language(s): English
Abstract/Description: It is inarguable that mathematics serves a quintessential role in the natural sciences and that ZFC — extended by large cardinal axioms — provides a foundation for vast swaths of contemporary mathematics. As such, it is understandable that the naturalistic philosopher may inquire into the ontological status of mathematical entities and sets. Many have argued that the indispensability of mathematics from scientific enterprise warrants belief in mathematical platonism, but it is unclear how knowledge of entities that exist independently of the natural world is possible. Furthermore, indispensability arguments are notoriously antithetical to mathematical practice: mathematicians typically do not refer to scientific applications to justify the truth of mathematical propositions. In Defending the Axioms: On the Philosophical Foundations of Set Theory, Maddy focuses her attention on these issues specifically with regard to set theory. Any metaphysical position which states that the truth value of any set-theoretic proposition is dependent on some objective reality (including but not limited to mathematical platonism, ante rem structuralism, in re structuralism, and neo-logicism) is termed Robust Realism, and evidently inherits the epistemological and methodological problems associated with platonism. She introduces two methodologically equivalent but ontologically distinct positions that presumably respect set-theoretic practice termed Thin Realism and Arealism, each having a realist and anti-realist bent, respectively. I argue that Thin Realism does not enjoy the same success when applied to broader mathematical practice. To do so, it is argued that alternative foundations to set theory are useful and utilized by mathematicians, and ergo Thin Realism must be revised in a pluralistic fashion; in doing so, this presents new challenges regarding how the putative unity of mathematics may be accounted for.
Identifier: FSU_libsubv1_scholarship_submission_1596226699_20ef7537 (IID)
Keywords: Philosophy of Mathematics, Philosophy of Science, Foundations of Mathematics
Persistent Link to This Record:
Host Institution: FSU

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Ifland, J. M. (2020). Justifying Alternative Foundations for Mathematics. Retrieved from