Mathematical platonism and set-theoretic indeterminacy

In this work, I will be looking at the issues raised by set-theoretic indeterminacy for a Gődelian platonist, who holds that there is a universe of independently existing math- ematical objects and that there are objective unique truth-values for any set-theoretic statement. After careful consideration of the philosophical and mathematical issues involved, I claim that Gődelian platonism is untenable. In Chapter 1, I examine dif- ferent forms of mathematical platonism and I elucidate their features. In particular, I distinguish between a substantive form (Gődel's platonism) and an operational form (anti-constructivism). I also make it clear that I will be concerned with set-theoretic Gődelian platonism. In Chapter 2, I examine the indeterminacy phenomenon in set theory through a detailed analysis of the most famous open conjecture, the Continuum Hypothesis (CH). In Chapter 3, I move on to describe the main philosophical orien- tations with regard to the indeterminacy phenomenon and I show how model-theoretic relativity is the main source of trouble for platonism. In Chapter 4, I examine the the- oretical ancestry of Gődel's conceptions (which may date back to Cantor's philosophy of the infinite) and Gődel's philosophy of indeterminacy. In Chapter 5 and Chapter 6, I deal with, respectively, Maddy's set-theoretic naturalism and plenitudinous platonism (in the form presented by Balaguer, FEP), and I raise some objections against these conceptual frameworks. In Chapter 7, I propose abandoning ontological platonism and I defend a mild form of conceptual realism resting upon the notion of non-arbitrary expansions. Finally, in Chapter 8, I tackle the problem of insolubility in contempo- rary set theory and I advise that operational platonism, qua anti-constructivism, as described in Chapter 1, is the only bit of platonism which could be upheld.EThOS - Electronic Theses Online ServiceGBUnited Kingdo