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Justifying Alternative Foundations for Mathematics
Title:  Justifying Alternative Foundations for Mathematics. 
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Name(s):  Ifland, Jared M., author  
Type of Resource:  text  
Genre: 
Text Bachelor Thesis 

Date Issued:  20200731  
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 settheoretic proposition is dependent on some objective reality (including but not limited to mathematical platonism, ante rem structuralism, in re structuralism, and neologicism) 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 settheoretic practice termed Thin Realism and Arealism, each having a realist and antirealist 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:  http://purl.flvc.org/fsu/fd/FSU_libsubv1_scholarship_submission_1596226699_20ef7537  
Host Institution:  FSU 
Ifland, J. M. (2020). Justifying Alternative Foundations for Mathematics. Retrieved from http://purl.flvc.org/fsu/fd/FSU_libsubv1_scholarship_submission_1596226699_20ef7537